THEORETICAL STUDIES ON THE SPECTROSCOPY OF SOME INTRAGROUP IVA HETERONUCLEAR DIATOMIC MOLECULES AND THEIR IONS

THEORETICAL STUDIES ON THE SPECTROSCOPY

OF SOME INTRAGROUP IVA HETERONUCLEAR

DIATOMIC MOLECULES AND THEIR IONS

A

THESIS

SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY (SCIENCE)

OF

JADAVPUR UNIVERSITY

BY

ANUP PRAMANIK, M.Sc.

DEPARTMENT OF CHEMISTRY

PHYSICAL CHEMISTRY SECTION

JADAVPUR UNIVERSITY

KOLKATA – 700032

INDIA

CERTIFICATE FROM THE SUPERVISOR(S) This is to certify that the thesis entitled “THEORETICAL STUDIES ON THE SPECTROSCOPY OF SOME INTRAGROUP IVA HETERONUCLEAR DIATOMIC MOLECULES AND THEIR IONS” submitted by Sri / Smt. ANUP PRAMANIK, who got his/her name registered on 25.06.2007 for the award of Ph.D. (Science) degree of Jadavpur University, is based upon his own work under the supervision of PROF. DR. KALYAN KUMAR DAS and that neither this thesis nor any part of it has been submitted for either any degree / diploma or any other academic award anywhere before. (Signature of the Supervisor(s), date with official seal Prof. Dr. Kalyan Kumar Das, Department of Chemistry, Physical Chemistry Section, Jadavpur University, Kolkata – 700 032, India.

TO

MY FRIEND PHILOSOPHER AND GUIDE

“KOUSIK-UNCLE”

Acknowledgements The research work presented in the thesis has been performed in the Department of Chemistry,

Physical Chemistry Section, Jadavpur University since January, 2006. I would like to take the

opportunity to convey my thanks to the people whose constant help and encouragement have

finally laid me to complete the thesis.

I express my warmest gratitude to my supervisor, Prof. Dr. Kalyan Kumar Das for his kind

cooperation and thoughtful advices. What he has done for me is really beyond my expectation. In

each step I have learnt from him how to utilize the valuable times of our life properly. Great

scientific attitude as well as nice behavior of him is truly rememberable.

The financial support provided by CSIR, Govt. of India is gratefully acknowledged. Without

this it was impossible to carry out my research work, whatever I have done.

I am indebted to Prof. Dilip Kumar Bhattacharyya and Dr. Biplab Bhattacharjee for their

moral supports and valuable discussions. I am also thankful to the Head, other teaching and non-

teaching staffs of the department of Chemistry, Jadavpur University. Library and laboratory

facilities of this university are also gratefully acknowledged.

A lot of thanks to my lab-mates, Mr. Amartya Banerjee, Ms. Susmita Chakrabarti for their

ever helping hands and cooperation. The former guy deserves a speciality for his philosophical

sense and critical analysis, which helped me a lot during my research period.

My heartiest love and respect to my parents and other family members. Specially, my sincere

thanks to my mother, and my wife, Mitali. They have provided me continuous supports and all

kinds of facilities. I can’t make him dishonored by expressing only my thanks to Kousik-uncle

who induced the philosophy of science in my mind. It brings a great pleasure to me to dedicate

the thesis to him.

Date:

Department of chemistry, ANUP PRAMANIK

Physical Chemistry Section,

Jadavpur University,

Kolkata – 700032, India

Contents

Introduction 1

Plan of the thesis 4

1. A brief review of the electronic structure theory of atoms and

molecules

1.1 Introduction 7

1.2 The Schrdinger equation 8

1.3 The variational principle 9

1.4 The Hartree-Fock model 10

1.5 Basis sets 12

1.6 Relativistic effects 13

1.7 Electron correlation energy and post Hartree-Fock treatments 15

1.8 References 20

2. Brief review of the computational methodology: details of the

Configuration Interaction method

2.1 Introduction 22

2.2 Relativistic corrections

2.2.1 The Dirac equation 24

2.2.2 Effective core potential 25

2.2.3 Spin-orbit coupling 29

2.3 Computational methodology

2.3.1 Configuration selection technique 31

2.3.2 Role of unselected configurations 32

2.3.3 Spin-orbit interaction 34

2.3.4 Calculation of spectroscopic constants 36

2.3.5 Estimation of radiative lifetime 37

2.4 References 38

i

3. Electronic structure and spectroscopic properties of the SiC radical

3.1 Introduction 42

3.2 Computational details

3.2.1 RECPs and basis sets 44

3.2.2 SCF MOs and CI 44


3.3 Results and discussion

3.3.1 Spectroscopic constants and potential energy curves of Λ-S states 46

3.3.2 Spectroscopic constants and potential energy curves of Ω states 56

3.3.3 Dipole moments and transition properties 59

3.4 Summary 62

3.5 References 64

4. Electronic structure and spectroscopic properties of SiC+ and SiC−

4.1 Introduction 66






4.3.1 Spectroscopic constants and potential energy curves of Λ-S states

A. SiC+ 69

B. SiC− 75

4.3.2 Spectroscopic constants and potential energy curves of Ω states 81

4.3.3 Transition properties

A. SiC+ 83

B. SiC− 85

4.3.4 Dipole moments, ionization potentials, and electron affinities 86

4.4 Summary 89

4.5 References 91

5. Electronic structure and spectroscopic properties of SnC and SnC+

5.1 Introduction 93

ii







A. SnC 97

B. SnC+ 105

5.3.2 Spectroscopic constants and potential energy curves of Ω states

A. SnC 110

B. SnC+ 113


A. SnC 117

B. SnC+ 120

5.3.4 Dipole moments and ionization energies 122

5.4 Summary 125

5.5 References 127

6. Electronic structure and spectroscopic properties of PbC and PbC+

6.1 Introduction 129







A. PbC 132

B. PbC+ 138


A. PbC 142

B. PbC+ 147


A. PbC 151

B. PbC+ 154

iii

6.3.4 Dipole moments and ionization energies 157

6.3.5 Comparison of some spectroscopic properties of MC and MC+

(M = Si, Sn, Pb) 159

6.4 Summary 163

6.5 References 164

Conclusion 166

List of publications 169

iv

INTRODUCTION

The interpretation and understanding of every experimental finding requires the knowl-

edge of theoretical background. A large number of experimental results can be brought into

together by theoretical interpretation and suitable formulation. So, necessity of theoretical

research is urged by its own demand. A chemical problem can be solved theoretically by

proper use of physical laws and mathematical methods, often by the use of computer memory.

Large number of computational methods have been developed over the years for the com-

plete solution of chemical problems. Quantum mechanics is one such tool, which has been

developed enormously with the advancement of computer hardware and softwares. Modern

electronic structure theory, which is based on quantum mechanics, is capable of providing

reliable predictions of quantities of chemical interest. It is not surprising that, the variational

methods could be applied to systems as large as XeF6, azulene, and guanine-cytosine base

pair. Now a days, with the help of computation, a large number of organic molecules, such

as protein, DNA, RNA etc. are designed theoretically. Such attempts are very much helpful

to the experimentalists to reach the goal of real synthesis with prior experiences of chemical

hazards. Moreover, where we are bound to our experimental limit, theoretical investigation

is the only tool to interpret the natural observations. Thus structural chemistry, which is

based on spectroscopic measurement, is equally balanced by experimental results as well as

theoretical predictions.

The space trajectory and other dynamical properties of macroscopic objects can be well

studied by classical mechanics.1 However, classical mechanics fails in the domain of submi-

croscopic world of atom and its constituents. In the beginning of the nineteenth century,

Planck’s idea of quantization was brought into a new field of mechanics mainly by Heisenberg

and Schrodinger. The new mechanics, revealed as quantum mechanics2−10, is the the suc-

cessful treatment to describe the structural and dynamic properties of subatomic particles.

Now, if the velocity of the object is comparable to that of light, one must use the relativistic

mechanics of Einstein which takes into consideration of variation of mass with velocity. So,

subatomic particles of low mass and having very high velocity, comparable to that of light,

need to use of relativistic quantum mechanics, derived by Dirac. This uses the modified

Hamiltonian, containing various relativistic correction terms including mass-velocity, spin-

orbit, Darwin correction and Breit interaction. Thus, depending upon the mass and velocity,

the dynamics of a particle is governed by suitable mechanics and consequently it requires a

proper mathematical treatment.

The behavior of electrons in atoms or molecules are described by quantum mechanics.

1

Their space trajectories are described on the basis of probabilistic interpretation, accord-

ing to which the stationary sates of them are fitted with time independent Schrodinger

equations. On solving these quantum mechanical equations, which are second order differ-

ential in nature, one may get the electronic structure of atoms and molecules. By using

Born-Oppenheimer approximation, in which electronic motions are treated separately from

nuclear motion, the electronic Hamiltonian can be resolved and consequently the solution of

it gives the structural aspects and spectroscopic information of a molecule11−16. The task is

no longer a simple one, specially for molecules having heavier atoms, there involves a large

scale relativistic effects in hamiltonian and hence proper treatment is essential for that.17−19

A number of algorithms have been developed for this purpose over the past few decades

with the improvement of enormous computing facilities. Many of them give the results with

reasonably good accuracy. Large efforts are required for a bit of improvement of computed

result. Moreover, an enormous volume of computation may be necessary for this purpose.

So, there are always limitations in the accuracy. Parallel efforts are also being given for the

development of computing facilities. Thus, the real challenge is to exploit these developments

and carry out theoretical research to reach the stage more close to reality.

2

References

1 H. Goldstein, Classical Mechanics, Addition-Wesley, Reading, Mass., 1950.

2 L. Pauling, E.B. Wilson, Introduction to Quantum Mechanics, McGraw-Hill, 1935.

3 H. Eyring, J. Walter, G.E. Kimball, Quantum Chemistry, Wiley, New York, 1944.

4 L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1968.

5 J.P. Lowe, Quantum Chemistry, Academic Press, New York, 1978.

6 D.A. McQuarrie, Quantum Chemistry, University Science, Mill Valley, Calif. 1983.

7 P.W. Atkins, Molecular Quantum Mechanics, Oxford University Press, New York, 1983.

8 A. Hinchliffe, Computational Quantum Chemistry, Wiley, New York, 1988.

9 F.L. Pillar, Elementary Quantum Chemistry, McGraw-Hill, New York, 1990.

10 I.N. Levine, Quantum Chemistry, Printice-Hall, N.J., 1991.

11 R.G. Parr, Quantum theory of Molecular Electronic Structure, Benjamin, New York, 1963.

12 J.A. Pople, D.L. Beveridge, Approximate Molecular Orbital Theory, McGraw-Hill,

New York, 1970.

13 J.N. Murrell, A.J. Harget, Semiempirical Self-Consistent-Field Molecular Orbital

Theories of Molecules, Wiley-Interscience, New York, 1971.

14 R.S. Mulliken, W.C. Ermler, Diatomic Molecules, Academic Press, New York, 1977.

15 R.S. Mulliken, W.C. Ermler, Polyatomic Molecules, Academic Press, New York, 1981.

16 W.J. Hehre, L. Radom, P.V.R. Schleyer, and J.A. Pople, Ab Initio Molecular Orbital

Theory, Wiley-Interscience, New York, 1986.

17 P. Pyykko, Relativistic Theory of Atoms and Molecules, Springer-Verlag, Berlin and

New York, 1986.

18 K. Balasubramanian, Relativistic Effects in Chemistry Part A. Theory and Techniques,

Wiley-Interscience, New York, 1997.

19 K. Balasubramanian, Relativistic Effects in Chemistry Part B. Applications to molecules

and Clusters, Wiley-Interscience, New York, 1997.

3

PLAN OF THESIS

The aim of Quantum Chemistry is to have some information about chemical bonds. En-

ergetic information of chemical bonds involving permutation of all elements in the entire

periodic table have been collected over the years by many experimental scientists. Besides

their applications, the simple diatomic molecules draw special interest in contributing the in-

formation about their bond length, bond energy etc. Intragroup IVA heteronuclear diatomic

molecules have generated a special interest in recent years because of their possible applica-

tions in catalysis, sensor films and mostly, they are the building blocks of cluster materials

which have interesting solid state properties. Inspite of the facilities of many modern sophis-

ticated instruments like high resolution spectrophotometer, laser vaporization, supersonic

jet expansion, matrix isolation etc. this type of molecules are rarely studied in experi-

ment because of the difficulty of their isolation in gas phase. Even for the simplest of them

(SiC), ab initio calculations were performed before the experimental detection. Very recently

seven of the intragroup IVA diatomics have been energetically characterized by Knudesen

effusion mass spectroscopic (KEMS) technique. To verify the available experimental data

and to predict the spectroscopic characteristics it is common practice to use quantum me-

chanical techniques like configuration interaction (CI), complete active space self consistent

field (CASSCF), couple-cluster, many-body perturbation theories, density functional theo-

ries (DFT) etc. The present thesis aims to study the electronic structure and spectroscopic

properties of intragroup IVA heteronuclear diatomic molecules, specially the carbides of Si,

Sn, Pb, and some of their ions.

The multireference singles and doubles configuration interaction (MRDCI) calculations

have been performed in the present thesis using relativistic effective core potentials and

suitable Gaussian basis functions for the participating atoms. Potential energy curves of

some low-lying Λ-S as well as Ω states of the molecules and ions are constructed from the

estimated full-CI energies. Many avoided crossing interactions have been properly studied

by analyzing the CI state functions. Spectroscopic constants like re, ωe, and Te values are

calculated by fitting the potential energy curves. The variation of dipole moment functions

of some low-lying states and transition moment functions involving ground and some of the

excited states are followed against bond distances and subsequently radiative lifetimes of

few low-lying states are computed for neutral as well as the ionic species. Vertical ionization

energies (VIE) of the neutral species are reported. Electron affinity of SiC have also been

verified from the MRDCI studies of SiC and its anion.

Chapter 1 gives an overview of the basic quantum mechanics like time dependent

4

Schrodinger equation, variational principle, Hartree-Fock model, basis sets, relativistic effect,

electron correlation and post Hartree-Fock methods like MCSCF, CI etc.

Chapter 2 describes the computational methodologies which are used in the calculations

throughout the thesis. For many electron atoms, it is not possible to carry out all electron CI

calculations. So the effective core potential method is used. The details of MRDCI method

are discussed in this chapter. The configuration selection technique is also discussed in this

chapter along with the corrections due to the unselected configurations. The method of

including the spin-orbit coupling at the CI level is mentioned. The methods of estimation

of the spectroscopic constants and radiative lifetimes are also described here.

Chapter 3 deals with the results obtained from the calculations of SiC. Potential energy

curves and spectroscopic constants of a large number of Λ-S states of singlet, triplet, and

quintet spin multiplicities are reported and compared with the existing data. The ground-

state dissociation energy of the species is computed and verified with the experimental

results. E3Π is found to be an important one which have not been studied before. The

important transitions like A–X, B–X, C–X, D–X, E–X etc. are studied, at the same time

radiative lifetimes of some excited states are also reported. Dipole moments of some low-

lying states are computed. Finally, the effects of spin-orbit coupling on the spectroscopic

properties of SiC are discussed in the chapter.

Chapter 4 describes the effects of removal and addition of an electron to the neutral

silicon carbide using MRDCI methodologies. Ionization energies and electron affinities of

SiC are reported in this chapter. Spectroscopic aspects of the SiC+ and SiC− ions are

studied in detail. The ground states of SiC+ and SiC− are 4Σ− and 2Σ+, respectively. Thus,

the quartet-quartet transitions for SiC+ and the doublet-doublet transitions for SiC− are

of special interest. No experimental data are known, but a very few theoretical results are

available for comparison. The spectroscopic constants of low-lying states of both the species

up to an energy level of 6 eV are reported.

Chapter 5 contains the results of the electronic structure and spectroscopic properties

of SnC and SnC+. Spectroscopic constants and some other properties of these species, at

both Λ–S and Ω levels, are reported. Because of the heavier mass of Sn, the Ω states have

become more important as compared to those of SiC. Hence, the spin-forbidden transitions

are given a special attention. Radiative lifetimes of some low-lying states of these species

have been reported in this chapter.

5

In Chapter 6, we have discussed electronic structure and spectroscopic properties of

PbC and PbC+. Lead is the heaviest element of group IVA and consequently the spin-orbit

coupling has been found to be the most prominent. Hence, spectroscopic properties of the

Ω states are thoroughly studied. This chapter includes dipole and transition dipole moment

functions of some low-lying states. Many spin-forbidden transitions are computed, and a

comparison of the spectroscopic properties of all three carbides and their monopositive ions

has also been made in the last part of this chapter.

6

CHAPTER – 1

A BRIEF REVIEW OF THE ELECTRONIC

STRUCTURE THEORY OF ATOMS AND

MOLECULES

1.1. Introduction

As we have mentioned earlier, behaviors of electrons in an atom or molecule are described

by stationary state wave functions as given in time independent Schrodinger equations. To

have solutions of such equations is always a difficult task. If we neglect all the relativistic

effects and consider the electrons to be moving in a fixed nuclear framework as in the Born-

Oppenheimer approximation, the problem becomes much easier to deal with. But still it

is a formidable task to solve these equations because of the involvement of a large number

of inter-electronic interaction terms in the Hamoltonian. Approximate methods have been

developed over the years to solve the non-relativistic Schrodinger equation for determining

the electronic structure of the molecular systems as accurately as possible. As an approxima-

tion, firstly these two body terms are converted into separate one-electron potentials. This

transforms the many body problem into the effective one-body problems which is popularly

known as independent particle model. The model gives the best possible solution in which

the wave functions are represented by the antisymmetrized product of one-electron functions,

commonly called orbitals. Next, for the construction of the single configuration state, it has

to satisfy two criteria; the wave function would have minimum energy in its neighborhood,

and the orbitals must have maximum overlap. The minimum energy criterion is fulfilled by

Hartree-Fock model, based on the variational principle. On the other hand, the maximum

overlap criterion with the exact wave function leads to the Brueckner approximation.1 The

last one is not practicable to implement as it requires the knowledge of exact wave func-

tions, while it is easier to implement the Hartree-Fock theory in practice. The minimum

energy wave function of a given class can be obtained from the variational principle which

can be applied in quantum chemistry. There are several post-Hartree-Fock methodologies

which may then be applied for estimating the electron correlation missing in the Hartree-

Fock approximation. However, it requires rigorous mathematical calculations and numerical

methods. The basic principles and techniques, which are used in the electronic structure

theory of atoms and molecules, are briefly discussed in the following sections.

7

1.2. The Schrodinger equation

In quantum mechanics, dynamics of a system is described by the time dependent Schrodinger

equation

Hψ= ih(∂ψ/∂t), (1.1)

where H is the Hamiltonian operator consisting of the kinetic and potential energy operators

of the system and ψ is called state function which is the function of space coordinate (r) and

time (t).

Now, the state of a many-body system (say, consisting n number of electrons) is given by

the wave function

ψ=ψ(r1, r2, r3.....rn; t)

and the probability density is written as

P (r1, r2, r3.....rn; t)= |ψ(r1, r2, r3, .....rn; t) |2 .

The equation (1.1) shows how the wave function evolves in time. Now, the time-independent

Hamiltonian operator of the n-electron system in the absence of any external field but only

with the Coulomb interactions among the electrons can be written as follows

H=− h2

2meΣi(∂

2/∂x2i+∂

2/∂y2i +∂2/∂z2

i ) +ΣijQiQj

|ri − rj |. (1.2)

Since H does not contain time explicitly, one can apply the method of separation of variables.

Time dependent and time independent part of the wave function can be separated for the

stationary state problem,

ψ(r1, r2....., rn; t)=Ψ(r1, r2....., rn; t)e−iEt/h. (1.3)

This gives rise to time-independent Schrodinger equation,

HΨ=EΨ. (1.4)

There are 3n coordinates in the wave function Ψ for n electron system. In addition to

the spatial coordinates, if we consider spin coordinates into account the total number of

coordinates become 4n, where the spin is restricted to the value ±12. In the relativistic

8

treatment, the spin of the electrons appears naturally, and it is sometimes considered as an

intrinsic angular momentum of the particle.

The symmetry restriction is to be imposed on to the wave functions. The only acceptable

solutions of the equation (1.4) are those with appropriate symmetry on the application of

two particle permutation operator. For electronic systems, the wave functions must be

antisymmetric with respect to interchange of the coordinates of any pair of electrons. Time

independent Schrodinger equation will provide many solutions for stationary states. The

lowest energy state is obviously the ground state.

Schrodinger equations are simplified for stationary state problems by using Born-Oppenhei-

mer approximation,2 in which the nuclear coordinates are kept frozen and the electronic part

is solved at a fixed nuclear geometry.

Hel(r;R)Ψel(r;R)=Eel(R)Ψel(r;R) (1.5)

1.3. The variational principle

The Schrodinger equation cannot be solved exactly for many electron systems because

the variables are not separable. Many approximate methods are employed for getting the

solutions. The variational principle3 provides one such approximate technique to solve the

time-independent Schrodinger equation HΨk=EΨk, say for the k-th stationary state. In the

linear variational principle, a trial wave function (Ψk) is expanded as a linear combination

of basis functions χi

Ψk =Σiciχi, (1.6)

where ci denotes the expansion coefficients. If the functions χi form a complete set, one

obtains the true wave function of the system. However, truncation is needed for practical

purpose.

The energy functional is given as

Ek=〈Ψk | H | Ψk〉/ 〈Ψk|Ψk〉

=ΣijcicjHij/Σijcicj〈χi|χj〉. (1.7)

The linear variational principle ensures that the trial energy calculated above is always higher

9

than the true energy of the system. In other words, there is an upper bound to the electronic

energy. The energy functional, Ek is minimized with respect to all ci parameters. One gets

the exact energy if the trial function is the exact solution to the Schrodinger equation i.e.

Ψk. The above mentioned first-order variational conditions give the secular equations in the

matrix form as

Hc=Ec. (1.8)

From the computational view point, one must construct the Hamiltonian matrix element for

a given basis set used to expand the wave function in the form of the equation (1.6). The

diagonalization of the the H-matrix has to be done next to get eigenvalues and eigenfunctions.

The matrix elements may be computed by different semiemperical or abinitio methods.

1.4. The Hartree-Fock model

Electronic motions in atoms or molecules are correlated mainly because of the Coulombic

interactions among the electrons. Equation (1.2) suggests that the molecular Hamiltonian is

independent of three or higher body interaction terms. However, the two-electron interaction

terms are the most important and difficult to compute in the electronic structure theory.

Three or higher body interactions are approximated to zero.

In the independent particle model, one may write the wave function in terms of product

of orbitals, which are functions of both space and spin-coordinates of electrons.

Ψ(r1, r2, r3....) =Φ1(r1)Φ2(r2)Φ3(r3)..... (1.9)

Here the electron-electron repulsion term is taken into consideration in an indirect manner.

Each electron has been considered to be moving in the mean-field of the remaining (n-1)

electrons. This gives the following set of one-electron equations

hiΦi(ri)=εiΦi(ri), i=1,2,3...n (1.10)

where hi is an effective one-electron operator for the i-th electron which includes the mean-

field interaction with other electrons. The sum of all orbital energies (εi) differs from the

total energy of the system. A self-consistent field (SCF) method is employed to solve these

one-electron equations (1.10) iteratively.5,6

10

One must employ the antisymmetry requirement for the many electron wave function in

the next step to ensure the incorporation of Pauli exclusion principle. The antisymmetry

requirement is fulfilled if the product of one-electron functions is written in the form of a

Slater determinant.

Ψ(r1, r2, r3....)=|Φ 1(r1)Φ2(r2)Φ3(r3)....Φn(rn)| (1.11)

The use of complete set of orbitals gives rise to a complete set of determinants those span the

full space of the antisymmetric many-electron wave functions. Such an independent particle

model is known as the Hartree-Fock (HF) model.

Two rows of the determinant will be identical if two electrons possess the same coordinates

and hence the probability of such event is zero. So, in the Hartree-Fock model, electrons

must have different coordinates. The determinantal form of the wave function leads to a

certain correlation between their positions and movements (Fermi Correlation) for electrons

with the same spin. Moreover, the orbitals in the determinant must be linearly independent

and orthonormal. Instead of solving one 3n-dimensional equation, one has to solve n 3-

dimensional differential equations7 in the Hartree-Fock approximation. The one-electron

Fock operator (hi) has the following form:

hiΦi=Ti + VN + VC + VX ,

where

Ti =−12h2( ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2)Φi,

VN =−Σα( Zα

|r−rα|)Φi,

VC =Σj

∫dr2(

Φ∗j (r2)Φj(r2)Φi(r1)

r12),

VX =−Σj

∫dr2(

φ∗j (r2)φj(r1)Φi(r2)

r12). (1.12)

Ti is the kinetic energy, VN corresponds to all electron-nuclei attraction, VC is the Coulomb

interaction and VX is the exchange interaction. This last term VX is the outcome of the

antisymmetry requirement and has no classical analogue.

The integration of the spin-components must be considered in addition to the spatial

coordinates as each electron is associated with spin. The spin-orthogonality has a major role

in deciding the zero or nonzero value of the integrals. The Coulombic interaction between

11

electrons will always occur, but the exchange interaction has non-zero values only between

electrons of the same spin.

Although the Hartree-Fock equations are one electron equations, the Fock operator (hi)

itself is a function of all other orbitals in the system. Iterative methods are employed to solve

the equations. At first, one has to guess the trial orbitals which are used to compute the Fock

operator, and one-electron equations are then solved. A new set of orbitals is constructed

from the resulting orbitals and the iteration is continued. A convergence problem may be

encountered if the initial guess of trial orbitals is not good enough. Different numerical

techniques such as damping, scaling etc. are employed to achieve the convergence in those

situations. The SCF method has become a very important technique for modeling a variety

of many electron systems as it generates a number of symmetry adapted molecular orbitals

which are used as basis functions. At the lowest level, the closed-shell Hartree-Fock theory

gives good result for the ground state of molecules in the close vicinity at the equilibrium

configuration.

1.5. Basis sets

In the molecular orbital (MO) theory, the probability density for the electron in a molecule

is described by a set of MOs φi which are constructed from the set of atomic orbitals (AO)

of the constituent atoms in the molecule. The individual molecular orbital (φi) can be

expressed as linear combinations of a set of one-electron basis functions χj centered on

each atom

φi =∑nj=1 Cjiχj,

where Cji terms denote the expansion coefficients. To represent the MOs exactly, the basis

functions χj should form a complete set, hence an infinite number of basis functions is

required which is not possible in practice. So a finite number of basis functions is chosen

and their choice is important for the satisfactory representation of the molecular orbitals.

One may use Slater type orbitals (STO)8

χSTO =Yl,m(θ,Φ)e−αr (1.13)

as basis functions. However, these STOs are not often suitable for the numerical work.

12

Boys9 proposed another type of functions, namely, the Gaussian-type functions

glmn =N(x−xo)l(y−yo)m(z−zo)ne−α(r−ro)2 , (1.14)

where N is the normalization constant, l, m, n are positive integers and α is orbital exponent

which is also positive. The function glmn denotes s, p, d-type of Gaussian depending upon

the value of l+m+n=0, 1, 2, respectively.

The evaluation of various two-electron integrals in the Hamiltonian matrix elements is

the most difficult part in the MO calculation. Furthermore, the number of these integrals

increases rapidly with the number of basis functions. Wherever possible the symmetry of the

molecule may be used to reduce the number of integrals to a large extent. Instead of using a

single Gaussian function one can use a linear combination of a small number of Gaussians,

χj =∑i dijgi,

where gi s are Gaussians centered on the same atom and having the same l, m, n values as

one another with different α values. χj is called a contracted Gaussian function and gi s are

called primitive Gaussians, and dij terms are the suitable coefficients. The use of contracted

Gaussians instead of primitive Gaussians reduces the number of variational coefficients to

be determined. However, at the Hartree-Fock level, the number of MOs generated does not

pose much problem. But in the large scale post Hartree-Fock calculations such as MCSCF

and CI, the number of MOs can not be kept too large as it generates enormous number of

configurations for a given electronic state.

1.6. Relativistic effects

The velocity of light in classical mechanics is considered as infinite compared to that of

the object and the light does not interact with the object of measurement. Assuming the

velocity of light to be infinite, if the light is allowed to interact with the matter, one gets

the non-relativistic quantum mechanics through Heisenberg’s uncertainty principle . If both

the assumptions are relaxed, i.e., the velocity of light is finite relative to that of the object

and there exists an interaction with the object, one should include relativistic corrections.

In other words, if a particle moves at a velocity which is comparable with that of light, the

non-relativistic quantum mechanics is no longer accurate. Actually, the mass of the system

13

determines the extent of the relativistic correction necessary in the calculations. The energy

of the one-electron atom in the ground state with atomic number Z is -Z2/2 in atomic units.

The average velocity of the electron is of the order of Z which can be easily shown from virial

theorem. The velocity of light is about 137 in atomic unit which indicates that relativistic

effects can not be neglected for atoms of heavier masses.

Non-relativistic quantum mechanical methods are quite satisfactory for most of the molecu-

les having lighter atoms in the first and second row of the periodic table. But the situation

is not similar for molecules containing atoms of higher rows in the periodic table where the

relativistic effect comes into play. The inner electrons of heavy atoms attain faster speed

due to large nuclear charges, and the speed is comparable with that of light. As for exam-

ple the 1s electron of the Au atom acquires about 60% of the speed of light. As the core

electrons are subjected to larger nuclear charges, the relativistic effects10−14 are significantly

large for them. These core electrons in turn affect the valence space which is significant

for the chemical bonding. Therefore, the chemical bonding and spectroscopic properties of

these molecules are expected to change to a large extent due to the heavy nuclear masses.

Different types of relativistic corrections are made for heavy atoms and molecules. These

are mass-velocity correction, Darwin correction, spin-orbit correction, spin-spin interaction,

Breit interaction etc. The dominant part of the relativistic correction is the mass-velocity

correction which arises due to the variation in mass of the electron with its speed as it

compares the speed of light. The relativistic mass is written as

m= m0√1− v2

c2

.

The basic equation in the relativistic quantum mechanics is the Dirac equation

(−ih∂/∂t−V+cα · π+βmc2)Ψ=0, (1.15)

where π =−ih(∂/∂x, ∂/∂y, ∂/∂z),

α=

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

,

0 0 0 −i0 0 −i 0

0 −i 0 0

−i 0 0 0

,

0 0 1 0

0 0 0 −1

1 0 0 0

0 −1 0 0

,

14

β=

1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −1

,

and Ψ=

Ψ1

Ψ2

Ψ3

Ψ4

.

For stationary states, time-independent one-electron Dirac equation takes the following form

(−V+cα · π+βmc2)Ψ=EΨ. (1.16)

Results of the solution of the Dirac equation for one-electron systems are in excellent agree-

ment with the experimental data but for many-electron systems, the applications of the

Dirac equation are not so simple. Many approximate schemes based on variation as well as

perturbation have been developed. In the next chapter of the thesis, we have reviewed the

method of computation for many electron atoms and molecules with the consideration of

relativistic effects.

1.7. Electron correlation energy and post Hartree-Focktreatments

Hartree-Fock wave functions are written as Slater determinants of one electron functions

under the independent particle model. The best possible determinant is chosen by using

variational methods. Therefore, it means that each electron experiences an effective mean

electrostatic field of all other electrons, but the motion and instantaneous positions of these

electrons are not explicitly correlated. The approximation is a crude one, though it works

well in some cases, especially for the ground state of the molecule. To obtain more accurate

results for studying the electronic structure and properties in the low-lying excited states

of the molecule, one must go beyond the Hartree-Fock approximation . Therefore, the

approximation by the mean effective field is not sufficient15,16 as the electron-correlation in

the post Hartree-Fock calculations becomes very important.

The Hartree-Fock model does not produce accurate results because of the inadequacy

15

of including correlations between the motions of electrons. The wave function written in

the single determinant form does not take into account of the electron correlation between

electrons of opposite spin. The correlation of the motions of electrons having the same spin

is partially, but not completely accounted for by virtue of the determinantal form of the

wave function. There are many qualitative deficiencies in the description of the electronic

structure of many electron system due to the omission of the correlation between electrons of

opposite spin. The closed-shell Hartree-Fock calculations do not describe the dissociation of

molecules correctly. The difference between the true non relativistic energy and the Hartree-

Fock energy is the measure of the correlation energy.

Eexact-EHF=Ecorrelation

Therefore, one has to achieve this amount of the correlation energy by some other means

for getting better results. The post Hartree-Fock methods are thus employed in quantum

chemistry so that the electron correlation17−19 which has been left out in the HF treatment

can be obtained. The methods like configuration interaction (CI), Many body perturbation

theory (MBPT), Coupled cluster (CC), density functional theory (DFT) are among the

post-Hartree Fock methods employed in quantum chemistry.

The CI method is one of the most useful methods which extend beyond the Hartree-Fock

model. In this method, the main concern is the choice of important configurations and

elimination of others at the optimum level so that the volume of the computations does not

increase very rapidly with the molecular size. Another requirement is the size consistency i.e.

the method must provide additive results when applied to an assembly of isolated molecules.

It is advantageous if the variational method can be applied as it ensures the upper boundness

of the total energy.

In order to incorporate the electron correlation within the variational principle, the wave

function is expressed as a linear combination of several Slater determinants, each of which

represents an individual electronic configuration. The variational method determines the

best possible combination. Multiconfiguration-SCF (MCSCF) method exploits this tech-

nique in which both expansion coefficients and orbitals forming the determinants are opti-

mized variationally while in the CI methodology, only CI coefficients are optimized. Variants

of MCSCF and CI methods are available and used to solve the actual problem depending

upon the computational capability and the desired accuracy. In recent years, many efficient

computational techniques are available in literature to tackle the problem in carrying out

16

MCSCF and CI calculations.

The lower energy molecular orbitals are generally occupied while the higher vacant ones

are virtual orbitals. One can generate antisymmetric many electron functions which have

different orbital occupancies. Each such many electron antisymmetrized function is a Slater

determinant or a linear combination of such determinants. As a result spin-adapted config-

uration state functions (CSF) can be formed.

The ground-state configuration is that distribution of electrons among the MOs which

possess the lowest energy. For even number of electrons, say, 2n, the single configuration

wave function is written as,

1Φ(0)G =A |φ1α(1)φ1β(2)........φnα(2n− 1)φnβ(2n)|; S=0, Ms=0.

For odd number of electrons (2n+1), the single configuration functions are,

2Φ(0)G =A |φ 1α(1)φ1β(2)........φnα(2n− 1)φnβ(2n)φpα(2n+ 1)|; S=1/2, Ms=1/2

and

2Φ(0)G =A |φ1α(1)φ1β(2)........φnα(2n− 1)φnβ(2n)φpβ(2n+ 1)|; S=1/2, Ms=-1/2.

The singly excited configurations are those distributions in which an electron has been pro-

moted from an occupied MO say, φk to a vacant MO φs. For 2n number of electrons

corresponding to singlet single-configuration wave function may be written as:

1Φ(1)k→s=

1√2[A |φ1α(1)φ1β(2).......φkα(2k − 1)φsβ(2k)........φnα(2n− 1)φnβ(2n)|

– A |φ1α(1)φ1β(2).......φkβ(2k − 1)φsα(2k)........φnα(2n− 1)φnβ(2n)|]; S=0, Ms=0.

Triplet state wave functions are,

3Φ(1)k→s=

1√2[A |φ1α(1)φ1β(2).......φkα(2k − 1)φsβ(2k)........φnα(2n− 1)φnβ(2n)|

+ A |φ1α(1)φ1β(2).......φkβ(2k − 1)φsα(2k)........φnα(2n− 1)φnβ(2n)|]; S=1, Ms=0

and

3Φ(1)k→s=A |φ1α(1).......φkα(2k − 1)φsα(2k)........φnβ(2n)|; S=1, Ms=1

3Φ(1)k→s=A |φ1α(1).......φkβ(2k − 1)φsβ(2k)........φnβ(2n)|; S=1, Ms=-1.

The doubly excited configurations are those distributions which are obtained by promoting

17

2 electrons from an occupied MO of 1Ψ(0)G to one vacant MO. The single-configuration wave

function is,

1Φ(2)k→s=A |φ1α(1)φ1β(2).......φsα(2k − 1)φsβ(2k)........φnα(2n− 1)φnβ(2n)|; S=0, Ms=0.

A linear combination of these CSF gives the CI wave function.

Ψ=∑iCiΦi (1.17)

The variation of coefficients Ci to minimize the energy functional leads to the determinantal

equation,

det(Hij-ESij)=0. (1.18)

It is important that only those CSF will contribute in the linear combination which have

the same angular momentum eigenvalues as that of the state Ψ or the CSF will have the

same symmetry properties (symmetry eigenvalue) as that of the state Ψ. The number of

configurations increases with the number of electrons and number of basis functions. For n

electrons and p number of basis functions, the number of CSF is roughly proportional to pn.

A CI calculation that includes all possible CSF with proper symmetry is a full CI calcula-

tion. Due to large number of CSF, full CI calculations are not possible to carry out except for

very small molecules with small basis set. There exist variants of CI calculations to choose

the proper configurations which will contribute largely to Ψ. It is, thus possible to perform

limited or truncated CI calculations.20 The simplest way of limiting the CI expansion is to

truncate the series in a given level of excitation. The truncated wave function can be written

as

Ψ=Φo+Σs>0CsΦs (1.19)

where Φ0 denotes the single determinant Hartree-Fock wave function while other determi-

nants are denoted by Φs. Ψ becomes Φ0, if no excitation is allowed, and one gets the HF

energy. The inclusion of all single excitations gives the CI wave function, ΨCIS as,

ΨCIS=C0Φ0 + Σocci Σvirt

a Cai Φa

i , (1.20)

where the excitation is indicated by i→a. If only single excitations are included it does not

improve the wave function or energy much. In the next step, the CI is limited with double

18

excitation only and the wave function may be written as,

ΨCID=C0Φ0 + ΣΣocci<j ΣΣvirt

a<b Cabij Φab

ij . (1.21)

If both single and double excitations are included in the next higher level of theory, it gives

the variational trial function as,

ΨCISD =C0Φ0+ Σocci Σvirt

a Cai Φa

i + ΣΣocci<j ΣΣvirt

a<b Cabij Φab

ij . (1.22)

These CI coefficients are optimized variationally. Multireference singles and doubles configu-

ration interaction (MRDCI) method, which includes the relativistic effects with and without

spin-orbit coupling, has been employed in the present work.

Size consistency is the modest requirement for a moderate system and all forms of the

truncated CI do not have this requirement. However, the full CI is size consistent, so also

for pair and coupled-pair theories. But the major disadvantage is that these pair theories

do not use variational principle. These are based on perturbative schemes, hence the total

energy obtained from these theories may be lower than the true energies. Sinanoglu and

Nesbet have introduced the independent electron-pair approximation (IEPA).15 These two

authors have used different terminology and formulations though the final results are same.

Sinanoglu termed his theory as Many-Electron Theory (MET) and Nesbet’s theory is called

Bethe-Goldstone Theory. Many body perturbation theory (MBPT) is also used for solving

infinite systems. For large systems, one needs a theory which is size consistent.

Molecular properties often may be expressed as derivatives of the total energy with respect

to parameters that correspond to perturbations of the system. The dipole moment of a

molecule is defined as the first derivative of the energy with respect to an external electric

field. The force constant for the molecular vibration is expressed as a second derivative

with respect to the displacement of nuclei in the molecule. Other electronic properties are

also defined in a similar way. It is, therefore, important to compute the potential energy

curves or surfaces of the ground and low-lying excited states which show the total energy as a

function of the coordinates of the nuclei. It is the most challenging task for both theoreticians

and experimentalists to construct such potential energy curves or surfaces. In the past few

decades, numerous working algorithms and computer codes13,14 have been developed for this

purpose.

19

1.8. References

1 R. Carbo, M. Klobukowski, Self-Consistent Field: Theory and Applications, Elsevier,

1990.

2 M. Born, J.R. Oppenheimer, Ann. Physik. 84, 457 (1927).

3 S.T. Epstein, The Variation Method in Quantum Chemistry, Academic Press, New York,

1974.

4 J.A. Pople, D.L. Beveridge, Approximate Molecular Orbital Theories, McGraw-Hill, New

York, 1970.

5 G.A. Segal, Ed. Semiempirical Methods of Electronic Structure Calculation, Part A and

B (vols. 7 and 8 of Modern Theoretical Chemistry, W. Miller et al. eds.), Plenum, New

York, 1977.

6 C.C.J. Roothaan, Rev. Mod. Phys. 23, 69 (1951).

7 B.A. Heβ, C.M. Marian, S.D. Peyerimhoff, Ab initio Calculation of spin-orbit Effects in

Molecules Including Electron Correlation.

8 J.C. Slater, Phys. Rev. 36, 57 (1930).

9 S.F. Boys, Proc. Roy. Soc. (London) A200, 542 (1950).

10 M. Krauss, W.J. Stevens, Annu. Rev. Phys. Chem. 35, 357 (1984).

11 P.A. Christiansen, W.C. Ermler, K.S. Pitzer, Annu. Rev. Phys. Chem. 36, 407 (1985).

12 K. Balasubramanian, K.S. Pitzer, Adv. Chem. Phys. 67, 287 (1987).

13 K. Balasubramanian, Relativistic Effects in Chemistry Part A. Theory and Techniques,

Wiley-Interscience, New York, p301, 1997.

14 K. Balasubramanian, Relativistic Effects in Chemistry Part B. Applications to Molecules

and Clusters, Wiley-Interscience, New York, p527, 1997.

15 A. Szabo, N.S. Ostland, Modern Quantum Chemistry, McGraw-Hill, New York, 1989.

16 S. Wilson, Electron Correlation in Molecules, Oxford University Press, New York, 1984.

17 H.F. Schaefer, Ed. Method of Electronic Structure Theory (vol.3 of Modern Theoretical

Chemistry, W. Miller et. al. eds.), Plenum, New York, 1977.

18 H.F. Schaefer, Ed. Applications of Electronic Structure Theory (vol.3 of Modern

20

Theoretical Chemistry, W. Miller et. al. eds.), Plenum, New York, 1977.

19 S.P. McGlynn, L.G. Vanquickenborne, M. Kinoshita, D.G. Carroll, Introduction to

Applied Quantum Chemistry, Holt, Rinehart and Winston Inc., New York, 1972.

20 W. Hehre, L. Radom, P.V.R. Schleyer, J.A. Pople, Ab initio Molecular Orbital Theory,

John Wiley and Sons, New York, 1986.

21

CHAPTER – 2

BRIEF REVIEW OF THE COMPUTATIONAL

METHODOLOGY: DETAILS OF THE

CONFIGURATION INTERACTION METHOD

2.1. Introduction

In the previous chapter we have mentioned that, if the velocity of light is assumed to be

infinite compared to that of a particle, one can use non-relativistic quantum mechanics to

describe its motion. The dynamics of the electrons in molecules consisting lighter atoms can

be treated accurately in non-relativistic quantum mechanics. But the treatment is somewhat

different for the molecules consisting of heavy atoms. The inner electrons in the heavy atoms

of the molecule attain a very high speed which cannot be neglected compared to that of light.

The non-relativistic quantum mechanics is no longer very accurate in this situation and

one should use the relativistic quantum mechanics.1−20 In general, non-relativistic quantum

mechanical methods based on ab initio techniques provide satisfactory results for most of the

molecules containing light elements in the first two rows of the periodic table. On the other

hand, relativistic effects become more important for molecules containing atoms of higher

rows in the periodic table. The difference is due to the fact that the inner electrons of very

heavy elements are subjected to large nuclear charges which increase their speed to such an

extent that is comparable with the speed of light.

Relativistic quantum mechanical treatment1,2 considers both finite velocity of light com-

pared to that of electrons and the interactions between them. The relativistic effect is

classified as mass-velocity correction, Darwin correction, spin-orbit interaction, spin-spin in-

teraction, Breit interaction etc.3 The mass-velocity correction on the kinetic energy of the

electron due to the variation of its mass with the velocity is the major part of the total rel-

ativistic correction. The inner s-orbitals, which are closest to the nucleus, experience higher

nuclear charge of the heavy atoms. Hence they contract because of the mass-velocity cor-

rection. This in turn, shrinks the outer s-orbitals due to orthogonality. The p-orbitals also

shrink due to the same reason, but to a lesser extent since the angular momentum allows

the electrons to keep away from the nucleus. If the coupling between the spin and orbital

angular momentum of the electron is strong, the spin-orbit correction is to be considered.

The correction becomes large for electronic states of molecules containing heavy atoms with

open-shell configurations. The electronic structure and spectroscopic properties may change

significantly because of the strong spin-orbit coupling. The two-electron counter part of the

spin-orbit interaction is known as Breit interaction. The Darwin correction is a characteristic

outcome of the Dirac’s relativistic equation, and as such it does not have any simple physical

significance. The spin-orbit interaction in the ground state of the gold atom is small but that

22

of lead is large because of open-shell configurations. The spin-orbit coupling not only splits

the electronic states into sub-states but also allows them to mix with electronic states, which

otherwise do not mix in the absence of the spin-orbit coupling. For example, the ground

state of the Pb atom in the absence of spin-orbit interaction is 3P, while its excited states

arising from the same electronic configuration are of 1D and 1S symmetries. The spin-orbit

coupling splits the 3P state into 3P0, 3P1, 3P2 components. The magnitude of this spin-orbit

splitting for Pb is as large as 1000 cm−1. Furthermore, 3P0 mixes with 1S0, similarly 3P2 and1D2 components mix together. The spin-orbit contamination is very large for heavy atoms

such as Pb, Pt, Au, etc. As a result, the spin-orbit interaction may change spectroscopic

properties of molecules containing heavy elements to a large extent. Sometimes a number of

important changes may take place in the potential energy surfaces (curves) e.g., appearance

of shoulders, barriers, double minima etc., due to some avoided curve-crossings. The higher

order interactions in the relativistic effect are generally ignored for the electronic and spec-

troscopic properties in the valence region. However, in case of fine structure calculations,

these smaller interactions become important and contribute significantly.

The relativistic effects can alter the nature of the chemical bonding in molecules containing

heavy atoms to a large extent. Some bonds may be weakened and some may be strengthened

depending upon the particular situation. The dissociation energies of heavy molecules are

found to change due to relativistic corrections. For instance, the dissociation energy of Au2 is

larger than that of Ag2 in contrary to the usual trend. This anomaly is due to the relativistic

contraction and stabilization of the 6s orbital of the gold atom. The well known lanthanide

contraction (i.e. the decrease of radii from La to Lu) is attributed to incomplete shielding of

the 4f shell. This effect is partly due to relativistic effects. Comparing the non-relativistic

and relativistic corrections, it has been found that a contribution4 of about 27% comes from

the relativistic effects in the form of lanthanide contraction.

All-electron molecular Dirac-Fock (DF) calculations are not easy to carry out as they in-

volve a large number of electrons. Moreover, additional integrals are generated due to each

molecular spinor having both large and small components. An enormous volume of compu-

tation is involved at the CI level because of the configurations generated from excitation of

valence electrons from shells with different angular momentum. An additional configuration

mixing takes place because of the relativistic interactions. The basis sets for the atoms in the

molecule must be sufficiently large so that the result of the CI calculation would be accurate

and acceptable for explaining the observed data. On the other hand, all-electron calcula-

23

tions become increasingly difficult with the increase in the size of the basis sets. Another

problem with all-electron DF calculations is the nature of the one-electron four-component

spinors. In general, these components are complex quantities. The behavior of these spinors

near the nucleus is difficult to describe by using the conventional basis functions. It has

been found5 that the large component of a molecular spinor having s or p1/2 population be-

haves like 1/rξ near a point nucleus, while the small component behaves like Z/r1+ξ, where

ξ=Z2α2/2. Moreover, the energy expectation values obtained from all-electron relativistic

calculations do not have the property of being upper bounds to the total energy. Hence,

there can be a variational collapse in attempting to use the variational principle. Ab initio

based all-electron calculations have been carried out by using the Dirac-Fock formalism for

heavy elements by Desclaus6 in 1973. An extensive configuration interaction is required for

this purpose. The relativistic effects on the orbital energies, and thus an excitation energies,

ionization potentials, and electron affinities have a direct influence on the chemically relevant

data. The relativistic effect may also change the bonding properties of the molecule as well.

2.2. Relativistic corrections

2.2.1 The Dirac equation

The relativistic quantum mechanical methods are based on Dirac equation which is an

analogue of Schrodinger equation. The Dirac Hamiltonian for a many-electron system can

be written as

HD=ΣihD(i) + Σi<j1rij

, (2.1)

where hD(i) is the one-electron Dirac Hamiltonian in the following form

hD(i)=αi· pi + βic2 − Z

ri. (2.2)

The wave function in the Dirac equation is a multi-component quantity, two components

describing the spin degrees of freedom of the electron, and other two components describing

the spin degrees of freedom for a charge-conjugated particle, loosely speaking, a positron.

In general, the Dirac equation has states of positive energy and infinite states of negative

energies, which are interpreted by Dirac as filled by an infinite number of electrons in the

ground state. Therefore, there exists four coupled first order differential equations for the

24

four components of the wave function.

The one-particle Dirac Hamiltonian involves 4 × 4 matrices instead of scalar functions

and differential operators. The solution is, therefore, a vector of four components, which are

called spinors.

Ψnkm = 1r

(Pnk(r) χkm(θ, φ)

iQnk(r) χ−km(θ, φ)

), (2.3)

where

χkm(θ, φ) = Σσ=± 12C(l

12 j; m− σ, σ)Y m−σ

λ (θ, φ)Φσ12

,

Y m−σλ is a spherical harmonics, Φ

1212

= α =

(1

0

)and Φ

− 12

12

= β =

(0

1

)are Pauli spinors,

and C(l12 j; m−σ, σ) are Clebsch-Gordan coefficients, k is the relativistic quantum number

k = (j + 12) for j = (l − 1

2)

−(j + 12) for j = (l + 1

2),

and λ is defined as

λ = k for j = (l − 12)

−(k + 1) for j = (l + 12).

The Pnk and Qnk are the large and small components, respectively. Thus for the central

force field V(r), the coupled differential equation of Dirac can be represented as

dPnkdr

+ kPnkr−(

2α

+α[V (r)−εnk])Qnk=0 (2.4)

dQnkdr−kQnk

r+α[V (r)−εnk]Pnk=0. (2.5)

In the non-relativistic limit (c →∞), the above coupled equation becomes the Schrodinger

equation if the small components Qnk are neglected. The small components are responsible

for the relativistic effects and make a significant contribution in the core region, while the

effect of these components in the valence region can be ignored.11−20

2.2.2 Effective core potential

A reliable pseudo potential method, known as relativistic effective potential method

25

(ECP) is used to perform relativistic quantum mechanical calculation. The basic assumption

used in the ECP method is the frozen core approximation which is nothing but the core-

valence separability. In the ECP method, the interaction of the valence electrons with the

core electrons are represented by effective potentials or pseudo potentials thereby reducing

the number of electrons in the calculation. These effective potentials may be relativistic or

non-relativistic depending on the nature of the wave function from which they are generated.

The calculations may be carried out semiempirically or by the ab initio methodologies.

In general, effective potentials replace the valence orbitals with pseudo-orbitals which

are smooth and nodeless in the core region but approximately resemble the true valence

orbitals at large radii. Therefore, one freezes not only the core orbitals but also that fraction

of valence electron density responsible for the inner oscillatory behavior. Once a nodeless

orbital has been generated, the one-electron atomic Fock equation is inverted to produce a

local operator which represents the core-valence interaction. Based on Phillips and Kleinman

transformation21, many studies have been made to construct effective potentials.

In the ECP method, the explicit core-valence orthogonality constraints are replaced by a

modified valence Hamiltonian. If the potential generated by core electrons is written as Vc,

the one-electron valence wave equation takes the following form.

(h+Vc)φv = Evφv (2.6)

Phillips and Kleinman21 suggested that φv can be written as

φv = χv−Σc〈χv|φc〉φc (2.7)

so that φv is orthogonal to φc.

Substituting φv into the one-electron eigenfunction,

(h+Vc+VEP )χv = Evχv, (2.8)

where VEP = Σc(E−Ec)|φc〉〈φc| is referred to as the Phillips-Kleinman pseudo-potential, χv

is known as the pseudo-orbital, and |φc〉〈φc| is the projection operator of the core orbitals.

When VEP is obtained from the non-relativistic atomic wave function, it will become non-

relativistic model potential. The details of these aspects can be found in the review of Krauss

and Stevens.13

Ab initio based relativistic effective core potentials (RECP) have been derived from the

26

numerical DF calculations of the atoms.22 As already mentioned, the solution of the DF

equation is a set of four component spinors. After partitioning the spinors as core and

valence, the overall many-electron relativistic wave function for a single configuration can be

written as

Ψ = A[(ψc1 ψc2 · · · ψcm)(ψv1 ψ

v2 · · · ψvn)], (2.9)

where A is the antisymmetrizer, ψc1 · · · ψcm and ψv1 · · · ψvn are core and valence orbitals,

respectively with m and n being the number of core and valence electrons. The total energy

is partitioned into core, valence, and core-valence interaction energies.

ET = Ec+Ev+Ecv (2.10)

One can show that

Ev + Ecv = 〈ψRv |Hrelv |ψRv 〉

where,

Hrelv = ΣihD(i) + Σc(Jc(i)−Kc(i)) + Σi<j

1rij

,

i and j indices refer to valence electrons. The core and valence orbital sets are assumed to

be orthogonal. The DF equation for a single electron is then given by

[hD+Σc(Jc−Kc)]ψv = εvψv+Σcψcεcv, (2.11)

where εcvs are the off-diagonal Lagrange multipliers

εcv = 〈ψv|hD + Σc(Jc −Kc)|ψc〉.

Defining the core-projection operator and the pseudo-orbital in the same way as done in

the Phillips-Kleinman method for relativistic spinor wave functions, one obtains relativistic

pseudo-orbitals and RECP.

χRv = ψRv + ΣcacψRc ,

ψRv = (1− P )χRv

P = Σc|ψc〉〈ψc|

27

V PK = −PHrelv P+PHrel

v P+εvP , (2.12)

where,

(Hrelv + V PK)χRv = εvχ

Rv

(hD + U core)χRv = εvχv

U core = Σc(Jc−Kc)+VPK , (2.13)

which is the relativistic effective potential of a 4 × 4 matrix that operates on the nodeless

four-component spinor χRv . The small components in the valence region are neglected. One

can use the non-relativistic kinetic energy operator along with relativistic large components

in an equation from which valence-level relativistic core potentials are generated.

Now, one-electron radial equation becomes

(−12∇2 − Z

r+ UEP )χ′v = εvχ

′v,

where χ′v is a two-component pseudo-function having a large radial components. For many

electron system, the equation becomes

(−12∇2−Z

r+UEP+Wvv′)χ

′v = εvχ

′v, (2.14)

where Wvv′ represents Coulomb and exchange potential involving pseudo-spinor χ′v and all

other pseudo-spinors. The RECP can be expressed by introducing the lj-dependent radial

potential URECPlj as

URECP = Σ∞l=0 Σ|l+ 1

2|

j=|l− 12| Σ

jm=−j U

RECPlj (r)|ljm〉〈ljm|. (2.15)

The projection operator |ljm〉〈ljm| is comprised of Pauli two-component spinors. The sum-

mation over l to ∞ is impractical to carry out, it requires potentials of all excited states

of the atom. A good approximation is to stop the summation at the maximum l (=L) and

maximum j (=J) values. After modification, the equation can be written in the following

form

URECP = URECPLJ (r)+ΣL−1

l=0 Σ|l+ 1

2|

j=|l− 12| Σ

jm=−j(U

RECPlj −URECP

LJ )|ljm〉〈ljm|. (2.16)

Relativistic calculations (single configuration SCF) on several diatomic molecules like

Au2, PbS, PbSe+ etc. have been carried out by Pitzer and his group.22−26 However, it has

28

now become possible to perform calculations at the multiconfiguration SCF (MCSCF) and

configuration interaction (CI) levels using the RECP of the constituent atoms.

The potentials generated in the above method are numerical potentials. However, Gaus-

sian analytic fit of these potentials is more desirable and useful. The Gaussian expansion of

these numerical potentials are proposed by Kahn, Baybutt, and Trular27 as

URECPLJ (r)− URECP

lj (r) = r−2ΣNi=0Cir

niexp(−αir2),

where Ci, ni, and αi are chosen such that best fitted results are obtained.

The RECP can be averaged with respect to spin. The averaged RECP (ARECP) takes

the following form.

UARECP (r) = UARECPL (r) + ΣL

l=0 Σlm=−l[U

ARECPl (r)− UARECP

L (r)]|lm〉〈lm|,

(2.17)

where

UARECPL = 1

(2l+1)[lURECP

l,l− 12

(r) + (l + 1)URECPl,l+ 1

2

(r)].

The advantages with ARECP are as follows:

a) These potentials can be used in standard molecular calculations which are based on

the Λ–S coupling.

b) These ARECP potentials may be interpreted as containing relativistic effects present

in the Dirac Hamiltonian except the spin-orbit coupling.

c) These potentials resemble non-relativistic effective potentials, and can be introduced

into the CI calculations.

2.2.3 Spin-orbit coupling

The spin-orbit operator has been defined by Hafner and Schwarz5 as the difference of

(l + 12) and (l − 1

2) relativistic effective potentials.

HSO = ΣL−1l=1 ∆URECP

l (r)[

l(2l+1)

Σl+ 1

2

m=−l− 12

l, l + 12,m〉〈l, l + 1

2,m

− l+12l+1

Σl− 1

2

m=−l− 12

l, l−12,m〉〈l, l−1

2,m

](2.18)

29

where, ∆URECPl (r) = URECP

l,l+ 12

(r) − URECPl,l− 1

2

(r). These spin-orbit operators and ARECP are

used in molecular calculations.

Hay and co-workers28−34 have published Gaussian fits of RECP without spin-orbit cou-

pling for all elements in the periodic table. Gaussian analytic fits of ARECP and spin-orbit

operators for Li to Ar have been derived by Pacios and Christiansen.35 Similarly, for other

elements these numbers are computed by other authors.36−38 MCSCF and CI calculations of

heavy molecules are now easy to carry out because of the availability of these potentials in

literature.

A large number of post Hartree-Fock calculations (like CAS-MCSCF, variants of CI)

on homonuclear and heteronuclear heavy diatomic molecules have been made by several

authors.17,18,39 These calculations demonstrate the requirement to consider the relativistic

effects including the spin-orbit coupling. From the results of SCF, MCSCF, and CI calcu-

lations, it has been found that the relativistic bond contractions for heavy molecules are

substantial. For molecules like AuH and AgH, the bond lengths contract by about 0.25 A

and 0.08 A, respectively.40−42 Christiansen and Pitzer43 have performed MCSCF calculations

on T l2 and T l+2 in the Ω–Ω coupling scheme.

2.3. Computational methodology

In the present thesis, a series of CI calculations on SiC, SnC, PbC, and some of their

ions have been performed. For such species, RECPs are required to be included in the

calculations. However, we have used either semi-core or full-core RECPs depending upon

the nature of the problem. Optimized Gaussian atomic orbital basis sets compatible with the

RECPs are employed for this purpose. These RECPs and basis sets are available in literature.

In some cases additional polarization and diffuse functions are included in the basis set.

The present CI calculations are based on multireference singles and doubles configuration

interaction (MRDCI).

In the first step, we perform a series of self-consistent-field (SCF) calculations for a partic-

ular molecular symmetry at each internuclear bond distance from 2-3 a0 upto the dissociation

limit, say ∼15 a0. The choice of the molecular state for this purpose would be such that, SCF

calculations converge at each bond length, and it would generate reasonably good optimized

SCF-MOs which can be used as one electron basis functions for the subsequent MRDCI

30

calculations. As it is not possible to work with the actual symmetry C∞v of the heteronu-

clear diatomic molecules, calculations are performed in the C2v subgroup. At first, the CI

calculations are done without considering spin-orbit coupling. The spin-orbit coupling is

introduced in the second step. A brief description of the MRDCI method is given in the

following section.

2.3.1 Configuration selection technique

It is well known that the dimension of a full-CI calculation increases enormously with the

number of the basis functions. For many electron problem it is impossible to carry out full CI

calculations. So, a simple straightforward approach to the attainment of multiconfiguration

wave functions is not practicable. An alternate approach to the CI treatment lies in the

proper selection of important configurations which are to be considered explicitly in a given

secular equation. It is possible to successfully implement this technique without losing much

accuracy.

The objective of the configuration-selection technique is to select those configurations

which contribute significantly to the total electronic wave function. The relatively less im-

portant configurations are also identified. Another important aspect of this method is to

predict accurately the amount of error occurred in omitting the configurations which are

not selected in the process. The contribution in energy for the unselected configurations

can be estimated. A set of dominant or main configurations (referred to as reference con-

figurations), φn must be chosen, and it forms the basis for the selection of the remaining

configurations. We must generate all configurations from the reference set by single and

double excitations. The energy lowering because of the inclusion of a test configuration, say

φt in the reference set φm, is obtained from the solution of the secular equation which is

larger than the previous by one.

Alternatively, one can estimate this energy lowering perturbatively using the expression44

∆E = Em+H2mt

(Em−Et) . (2.19)

The equation (2.19) has been used for the first time by Whitten and Hackmeyer45 as a

configuration selection method in non-relativistic CI calculations. Many selection techniques

based on the above-mentioned equation have been developed and used successfully by Bender

and Davidson,46 and extensively by Buenker and Peyerimhoff.47−54 The perturbative method

31

has the advantage of involving less computation, while the variational method can check the

energy-lowering simultaneously for all roots of that symmetry. We have used variational

methods of Buenker and co-workers for our computational purpose. It is also necessary to

obtain an accurate prediction of the total energy contribution of those configurations which

are not chosen for the final CI calculations.

Therefore, each test configuration φt is tested against every member of the main reference

set φm. The configuration φt, for which ∆E exceeds some cut-off or threshold value (T),

is selected for the final CI calculations. The choice of main configurations and the estimation

of the effect of adding more configurations to the initial set are to be made quite accurately.

It is important to make sure that the set of main configurations is sufficiently large to allow

for a realistic representation of all states. The desired number of roots is obtained from each

of the secular equations involving φm and the test configuration. The energy-lowerings are

computed separately for each of the roots. The maximal energy-lowerings for all the roots

are compared with some threshold value for the purpose of determining whether the test

species should be included in the final CI. The choice of the magnitude of the threshold is

also important in this regard. A smaller threshold will be required to solve larger number of

configurations in the final CI. Generally, threshold is kept in the order of microhartree.

2.3.2 Role of unselected configurations

Once the selection criterion is fixed, one must be able to accurately compute the effect

of those configurations which are not included in the final CI calculations. It is, therefore,

required to estimate the threshold as a parameter in the final results by means of some

reliable method. Davidson’s55,56 suggestion is to compare the sum of the lowering of the

neglected test species with the total energy change effected by the CI in order to get at least

a semiquantitative estimate of the consequence of employing a nonzero threshold at different

nuclear geometries.

The method for extrapolating the CI energies to zero-threshold can be formulated as

follows. Let the total configuration-set in the CI space at zero-threshold is partitioned into

three subsets φm, φs, and φr which are main, selected, and rejected configurations,

respectively. The zero order wave function ψ and that obtained in the final truncated CI

for a given threshold can be defined as

ψ = ΣmCmφm, (2.20)

32

ψ(T ) = ΣmCm(T )φm+ΣsCs(T )φs. (2.21)

The wave function ψ ≡ ψ(0) for T=0 is obtained as

ψ(T=0) = ΣmCm(0)φm+ΣsCs(0)φs+ΣrCr(0)φr. (2.22)

The φr configurations are weakly interacting when T is quite small, and φm is sufficiently

representative. With some approximations, the energy-lowering ∆Er by adding φr to the

original set of main configurations φm can be expressed as

∆Er = 〈ψrHψr〉 -〈ψHψ〉

≈ 2ReCr(0)〈ψHφr〉+Cr(0)2〈φrHφr〉−〈ψHψ〉 (2.23)

with ψr ≈ ψ + Cr(0)ψr.

Estimating Cr(0) from the first order perturbation theory, the equation simplifies as

∆Er ≈ Cr(0)〈ψHφr〉.

Total energy E ≡ E(0) can be obtained as

E = 〈ψHψ〉 ≈ E(T )+ΣrCr(0)〈ψ(T )Hφr〉 (2.24)

Therefore, employing a small value of T and a sufficiently representative set of main config-

urations φm, it is possible to estimate the CI energy at T=0.

As we are using variational principle in determining the energy roots and wave functions,

it is assured that the CI energy (E) decreases as T lowers. But E(T → 0) does not remain

constant as T changes. A general expression of E (T → 0) can be written with a scaling

factor λ

Eλ(T → 0) = E(T )+λΣr∆Er(T ), (2.25)

where λ is an arbitrary constant which is chosen such that Eλ(T → 0) is the most slowly

varying function of T. The λ factor takes care of the internal coupling and relaxation pro-

cesses in the CI performed with T=0. The secular equations are then solved for a series of

T. The [E(r) +λΣr∆Er(T )] curves are constructed for a given root at various values of λ as

a function of T. The optimum λ value for each root has been estimated.

33

The extrapolated energy is expected to be same, provided an appropriate set of reference

configurations is chosen. It is well known that in the full-CI treatment, the nature of the

molecular orbital basis set does not have much importance. Once the CI energy extrapolated

to zero threshold is obtained, one can use Davidson’s correction to estimate the contribution

due to higher order excitations. The estimated full-CI energy may be expressed as

E(full-CI) = E(T=0)+(1−ΣmC2m)(E(m)−E(T = 0)), (2.26)

provided the contribution of the reference species in the total CI expansion is above 90% i.e,

ΣC2m is more than 0.9. This has been extended later for a multireference case by Peyerimhoff

and Buenker.57 The correction has been shown to be good for the lack of size retentivity

of the energy with single and double excitations.58,59 The Table Direct-CI version of the

MRDCI codes of the Kerbs and Buenker60 are used in some cases. All properties other

than energy may be evaluated from the wave functions corresponding to the MRDCI space.

Results are found to be quite satisfactory for one-electron properties like dipole moments,

quadrupole moments, transition moments between various electronic states.

2.3.3 Spin-orbit interaction

Once the CI energies and wave functions of different Λ–S states are obtained, the next

step is to include the spin-orbit effects at the CI level of treatment. MRDCI wave functions

are employed as basis for the representation of the relativistic Hamiltonian including the

spin-orbit coupling. The spin-orbit interaction may be introduced in different ways.

In one way, the selected spatial configurations of different types for a given spatial sym-

metry are multiplied with spin functions which transform according to irreducible repre-

sentations of the C2v double group. In the actual computations, heteronuclear diatomic

molecules are always considered to be in the C2v subgroup. The correlation with the ac-

tual symmetry (C∞v) group is done afterward. The C2v double group for even number of

electrons consists of four irreducible representations namely, A1, A2, B1, and B2 of which

B1 and B2 are degenerate. For odd number of electrons, the irreducible representations in

C2v double group are either E1 or E2 which are degenerate. The resulting products of the

spatial configurations and spin functions, are grouped according to the symmetry. The spin-

orbit matrix elements between different configuration state functions (CSFs) are calculated.

These results are combined with the previous results for the spin-independent operator to

34

form a Hamiltonian matrix representation for each of the irreducible representations in the

C2v double group. The resulting secular equations are four to six times larger than the origi-

nal Λ–S CI treatments. However, the diagonalization can be done in relatively few iterations

by using Davidson’s algorithm61 because of the availability of very good starting vectors

from the spin-independent calculations. The resulting energies are used directly without

adding perturbation correction analogous to those employed for the Λ–S CI treatment. For

small number of active electrons, this procedure is found to be accurate. However, for large

number of electrons one can add the spin-orbit interaction in a different way.

In the alternative way, the appropriately estimated full-CI energies from the Λ–S CI treat-

ment are placed in the diagonals of the CI matrix. The off-diagonal matrix elements are

obtained by employing pairs of selected CI wave functions with Ms=S and applying spin-

projection techniques and the Wigner-Eckart theorem.62−65 It is now required to diagonalize

small CI matrices, called super-CI corresponding to A1, A2, and B1 (or B2), or E1 (or E2)

representations of the C2v and C′2v double group depending upon the number of electrons.

The dimensions of the secular equations in the super-CI treatment depend on the number

of roots of the Λ–S states involved in the spin-orbit interaction. Such a two-step method

is mostly used in the calculations carried out in the present thesis. This two-step method

for the inclusion of the spin-orbit interaction has become useful, especially, to analyze the

Ω-components in terms of different Λ–S eigenfunctions. However, the disadvantage of this

simpler two-step method of introducing the spin-orbit coupling is that the spin-orbit correc-

tion is not treated at the same level as the non-relativistic terms.

The spin wave functions corresponding to odd number of electrons belong to E represen-

tation of C′2v double point group. The character table for C′2v double point group66 is given

below.

C′2v E C2(z) σxz σyz R Operator Spin function

A1 1 1 1 1 1 z αβ - βα

A2 1 1 -1 -1 1 lz, sz αβ+βα

B1 1 -1 1 -1 1 x, ly, sy αα+ββ

B2 1 -1 -1 1 1 y, lx, sx αα - ββ

E 2 0 0 0 -2 α, β

The symmetry adapted spin wave functions66 for even number of electrons with fixed S and

35

MS values and those transforming according to irreducible representations of C2v point group

may be tabulated as follows.

S MS | S, MS〉 | S, |MS|, R〉0 0 αβ A1(0)=|0〉1 1 αα B1(1)= 1√

2(|+1〉+|-1〉)

-1 ββ B2(1)= 1√2(|+1〉 - |-1〉)

0 1√2P [αβ] A2(0)=|0〉

2 2 αααα A1(2)= 1√2(|+2〉+|-2〉)

-2 ββββ A2(2)= 1√2(|+2〉 - |-2〉)

1 1√4P [αααβ] B1(1)= 1√

2(|+1〉 - |-1〉)

-1 1√4P [βββα] B2(1)= 1√

2(|+1〉+|-1〉)

0 1√6P [ααββ] A1(0)=|0〉

3 3 αααααα B1(3)= 1√2(|+3〉+|-3〉)

-3 ββββββ B2(3)= 1√2(|+3〉 - |-3〉)

2 1√6P [αααααβ] A1(2)= 1√

2(|+2〉 - |-2〉)

-2 1√6P [βββββα] A2(2)= 1√

2(|+2〉+|-2〉)

1 1√15P [ααααββ] B1(1)= 1√

2(|+1〉+|-1〉)

-1 1√15P [ββββαα] B2(1)= 1√

2(|+1〉 - |-1〉)

0 1√20P [αααβββ] A2(0)=|0〉

Where P [α...β] denotes a sum of all possible permutations of α and β spins. In the fourth

column a short | MS〉 notation is used for the | S, MS〉 spin-functions from column 3.

2.3.4 Calculation of spectroscopic constants

CI energies of various Λ–S electronic states of the molecule are estimated from the short

range of internuclear distance, 2.5-3.0 a0 to the long range, say 15.0 or 20.0 a0 depending

upon the system. Around the equilibrium bond length, small grids (typically 0.05 to 0.1 a0)

are chosen, while in the dissociation limit of the molecule, the larger grids are used for

calculations. Spin-orbit corrections are carried out at each of these bond distances. MRDCI

calculations are capable of giving a reliable description of the entire potential energy curve

all the way from a short bond distance to the dissociation limit. The computed bound

36

state potential energy curves are fitted into polynomials for substitution in Schrodinger

equation for nuclear motion. One dimensional nuclear Schrodinger equations are then solved

numerically by using Neumerov-Cooley method67,68 to obtain the desired vibrational energies

and wave functions.

2.3.5 Estimation of radiative lifetime

The electric dipole transition moments between two states involved in the transition, are

calculated as a function of internuclear separation of the diatomic species. Knowing these

transition moments, the Einstein’s spontaneous emission coefficients Av′v′′ (sec−1) between

different vibrational levels (designated as v′) of the upper electronic states and those (desig-

nated as v′′) of the lower electronic state are obtained from the following standard formula

Av′v′′ = ge′e′′ 2.1419× 1010(∆E)3Sv′v′′ ,

where ∆E (transition energy) is in a.u., and

Sv′v′′ = 〈χv′′(r)Be′e′′(r)χv′(r)〉 2

is obtained by using a polynomial fit to the discrete data for the electronic transition moment

Be′e′′ and the vibrational wave functions χv(r) generated for the respective pairs of electronic

states. The factor ge′e′′ is derived from the fact that the transition moment used in the

calculation is only for one component of a given degenerate system. The radiative lifetime

(τv′) of the vibrational level (v′) of the upper electronic state is obtained as

τv′ =(Σv′′Av′v′′

)−1, (2.27)

where the sum runs over all vibrational levels which can be reached among the lower-lying

electronic states. The oscillator strength (f) for a given absorption process may also be

obtained from the following expression

fv′v′′ = 23g′e′e′′∆ESv′v′′ ,

where g′e′e′′ is the degeneracy factor.

37

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41

CHAPTER – 3

ELECTRONIC STRUCTURE AND

SPECTROSCOPIC PROPERTIES OF THE SiC

RADICAL

3.1. Introduction

The simple diatomic SiC radical is known to be an important component of the carbon

star. It is also present in interstellar regions of space.1 Though astrophysically important,

the spectroscopy of this radical was not observed in laboratory for a long time. The major

difficulty to study the silicon-carbon compounds was that it needed a very high temperature

to vaporize these elements. Bondybey2 and Michalopoulos et al.3 attempted experiments

using laser vaporization of silicon carbide rod. But the detection of the SiC radical was

unsuccessful. Although Si2 and C2 are spectroscopically well known species, the spectro-

scopic identification of SiC has been made much later. However, ab initio calculations of

SiC are performed before the experimental detection. Lutz and Ryan4 have performed the

configuration interaction (CI) calculations and found that the ground state of the mixed first

row-second row diatomic is 3Π. Bruna et al.5 have also carried out large scale CI calculations

for the potential curves of the isovalent series of diatomic species, CN+, Si2, SiC, CP+, and

SiN+ in their low-lying states. The results of these calculations for the SiC radical have

agreed well with those of the previous calculations.4 Rohlfing and Martin6 have studied the

structure and spectroscopic properties of the isovalent diatomic molecules such as C2, Si2,

and SiC. These authors have used Moller-Plesset perturbation theory based on UHF refer-

ence function as well as externally contracted CI based on a multireference function of the

complete-active-space type for determining the spectroscopic constants of a few low-lying

states. Meanwhile, low-lying electronic states of SiC− and electron affinity of SiC have been

studied by Anglada et al.7 from large scale CI calculations. Dohman et al.8 have made

a comparison among various isoelectronic radicals possessing eight valence electrons. The

CASSCF and contracted CI calculations have been performed by Larsson9 to study the po-

tential curves of X3Π, B3Σ+, and C3Π states of the SiC molecule. The author has predicted

transitions between the ground state and the B and C states to occur in the wave length

range 4000-6000 A. Bauschlicher and Langhoff10 have performed CI calculations at various

levels of electron correlation to compute the spectroscopic constants of X3Π and A3Σ− states

of SiC. Their best estimates of re, ωe, and De for the ground state of the radical were 1.719 A,

962 cm−1, and 4.4 eV, respectively.

The SiC radical was first observed11 by high-resolution Fourier transform emission spec-

troscopy from a composite wall hollow cathode. The 0-0 band of the d1Σ+-b1Π system of

SiC has been observed near 6100 cm−1. This has been confirmed by ab initio calculations

42

performed at different level of accuracy. Molecular constants of several low-lying states,

namely X3Π, A3Σ−, a1Σ+, b1Π, c1∆, and d1Σ+ of SiC are predicted from these calculations.

The results are found to be comparable with the predictions of Bauschlicher and Langhoff.10

The ground state of SiC has been characterized from the microwave transition observed by

Cernicharo et al.12 The A3Σ−-X3Π system of SiC is analogous to the Ballik-Ramsay system

of C2. Brazier et al.13 have observed the 0-0 band of this system in emission near 4500 cm−1

and the reported the bond lengths are 1.81356 and 1.72187 A for A3Σ− and X3Π states, re-

spectively. The A-X band was found to be weak because of the difficulty in making the SiC

radical. Multireference CI calculations have been performed by Langhoff and Bauschlicher14

to study the A3Σ−-X3Π infrared transition in the radical. The 0-0 band of the A-X transi-

tion has also been reassigned in another theoretical study.15 The dipole moment functions

of A3Σ− and X3Π, and the transition moment functions as well as radiative lifetimes of the

A-X transition have also been reported. Martin et al.16 have computed three lowest states,

X3Π, A3Σ−, and a1Σ+ of SiC using augmented coupled cluster methods and different basis

sets. Thermochemistry of the radical has also been reported by these authors. Butenhoff

and Rohlfing17 have studied the C3Π-X3Π band system of the jet-cooled SiC radical using

a laser induced fluorescence (LIF) spectroscopy. The vibrational energies and rotational

constants for the lowest few vibrational levels of both C3Π and X3Π states of SiC have been

determined by these authors. Almost at the same time, Ebben et al.18 have measured seven

rovibronic bands belonging to the C3Π-X3Π transition in SiC produced by the laser vapor-

ization in combination with supersonic cooling. The radiative lifetimes of the C3Π state

were found to vary from 2886 ns to 499 ns in the lowest seven vibrational levels. Singles and

doubles CI calculations from single SCF configuration have been carried out by McLean et

al.19 on a series of diatomic species including SiC and SiC−. Some spectroscopic information

of X3Π, A3Σ−, b1Π, and c1∆ states of SiC and the 2Π state of SiC− have been reported.

The millimeter-wave rotational spectra with hyperfine structure of two stable isotopes with

nuclear spin, namely 29SiC and Si13C, were detected in the ground state.20,21 The vertical

and adiabatic ionization energies and electron affinities of SinC and SinO (n=1-3) molecules

have been reported by Boldyrev et al.22 from large-scale ab initio calculations at different

levels of correlation.

Grutter et al.23 have identified the electronic absorption spectra of SiC− and SiC in 5K

neon matrices using mass-selected deposition. The neutralization of the anion leads to the

observation of a new band system B3Σ+←X3Π of SiC in addition to the known systems,

43

namely A3Σ−←X3Π and C3Π←X3Π. The B-X band has the origin at 11 749 cm−1, while

the ωe value for the B3Σ+ state of SiC in neon matrix has been reported to be 1178 cm−1.

However, no gas phase data is available for this state. Recently24, the infrared emission

spectrum of the A3Σ−-X3Π electronic transition of the SiC radical in the gas phase has been

observed using a high resolution Fourier transform spectrometer. Three bands, 0-1, 0-0, and

1-0 of this system are found in 2770, 3723, and 4578 cm−1, respectively.

In this chapter we report potential energy curves of 32 Λ–S states of singlet, triplet, and

quintet spin multiplicities. Spectroscopic constants (re, Te, and ωe) of 23 states within 6 eV

are determined42 and compared with the existing data. Effects of the spin-coupling on the

spectroscopic properties of these states have been studied. Potential energy curves of several

low-lying Ω states of SiC have also been constructed. The radiative lifetimes of some of the

excited states have been predicted.

3.2. Computational details

3.2.1 RECPs and basis sets

The RECPs of Pacios and Christiansen25 are used to replace the 1s22s22p6 core electrons

of the Si atom and 3s23p2 valence electrons are kept available for the CI calculations. For

the carbon atom, the RECPs of the same authors have been employed to replace inner 1s2

electrons. The total number of active electrons in the CI space is, therefore, eight. The 4s4p

Gaussian basis sets of Pacios and Christiansen25 for Si are augmented with some diffuse and

polarization functions. Three s functions (ξs = 0.04525, 0.02715, and 0.0163 a−20 ), two p

functions (ξp = 0.06911 and 0.02499 a−20 ), five d functions (ξd = 4.04168, 1.46155, 0.52852,

0.19112, and 0.06911 a−20 ), and two f functions (ξf = 0.19112 and 0.06911 a−2

0 ) are added.26

The first two sets of d functions are contracted using coefficients of 0.054268 and 0.06973.

Similarly, the two f functions are contracted using the coefficients of 0.29301 and 0.536102.

The final basis set for Si used in the present CI calculations is (7s6p5d2f/7s6p4d1f). For the

carbon atom, the (4s4p) basis set of Pacios and Christiansen25 has been enhanced by adding

two sets of d functions of exponents 1.2 and 0.35 a−20 .

3.2.2 SCF MOs and CI

At each internuclear distance of SiC, we have performed self-consistent-field (SCF) calcula-

44

tions for the (σ2σ2π2)3Σ− state using the above mentioned basis sets. The entire calculations

are carried out in the C2v subgroup keeping Si at the origin and C in the +z axis. The sym-

metry adapted SCF-MOs are subsequently used for the generation of configurations in the

CI calculations. Throughout the calculations we have employed the MRDCI methodology

of Buenker and coworkers27−33 which uses perturbative correction and energy extrapolation

techniques. The table-CI algorithm is used to handle open shell configurations which appear

due to the excitation process. The details of CI methodologies are reviewed in chapter 2.

For each of the four irreducible representations of C2v with a given spin multiplicity, we

have chosen a set of reference configurations. Table 3.1 shows the summary of the MRDCI

calculations. A maximum of eight roots for singlets and triplets, and four roots for quintets

Table 3.1 Details of the configuration interaction calculations of SiC

Symmetry Nref./Nroot Total no. of Max. no. of ΣC2m

generated configs. selected configs.§ (%)1A1 95/5 2 138 650 42 603 901A2 118/8 2 982 377 74 932 893A1 120/7 4 341 820 72 786 893A2 100/7 4 496 112 85 208 905A1 80/4 3 322 297 47 318 905A2 88/4 3 045 164 52 539 881B1 153/8 3 378 566 81 426 89

3B1/3B2 151/7 6 118 182 68 444 905B1/5B2 125/3 4 386 236 40 900 90

§ Configuration selection threshold = 1.0 µhartree

are optimized. The total number of generated configurations easily exceeds several million.

We have used a configuration selection-threshold of 10−6 hartree so that the number of

selected configurations remains within 200 000. The table direct CI version34 of the MRDCI

code has been used throughout the calculation. The sums of squares of the coefficients of the

reference configurations for each root remain around 0.9. The energy extrapolation method

has been used to estimate energies at zero threshold. The higher excitations are taken into

account by the multireference analogue of the Davidson’s correction35,36 which improves the

total energy of the ground and excited states.

45


The spin-orbit interaction is included in the calculation by two-step variational calculations.37

We have allowed all the spin components of low-lying Λ-S states of SiC to interact. The

spin-orbit operators, which are compatible with the RECPs are taken from Pacios and

Christiansen.25 The sizes of the secular equations of A1, A2, and B1/B2 blocks in the C22v

double group are 41×41, 41×41, and 40×40, respectively for some selected number of roots

of Λ–S symmetries of the molecule. The details of the spin-orbit CI calculations are already

discussed.

3.3. Results and discussion

3.3.1 Spectroscopic constants and potential energy curves of Λ–S states

The ground states of both C and Si belong to the 3Pg symmetry. Table 3.2 shows dissoci-

ation correlation between the Λ–S states and atomic states leading to lowest five asymptotes.

The lowest dissociation limit, Si(3Pg)+C(3Pg) of SiC correlates with Σ+(2), Σ−, Π(2), and

∆ symmetries, a total of 18 Λ-S states. The second dissociation limit, Si(1Dg)+C(3Pg) cor-

relates with triplets, namely 3Σ+, 3Σ−(2), 3Π(3), 3∆(2), and 3Φ. The relative energy of this

limit computed from the CI calculation of the SiC molecule at a very large bond distance is

about 7000 cm−1 which is 800-900 cm−1 higher than the J-averaged observed value.38 The

Table 3.2 Dissociation correlation between the molecular and atomic states of SiC

Λ-S states Atomic states Relative energy / cm−1

Si + C Expt.a Calc.1Σ+(2), 1Σ−, 1Π(2), 1∆, 3Pg + 3Pg 0 03Σ+(2), 3Σ−, 3Π(2), 3∆,5Σ+(2), 5Σ−, 5Π(2), 5∆3Σ+, 3Σ−(2), 3Π(3), 3∆(2), 3Φ 1Dg + 3Pg 6124 70003Σ+, 3Σ−(2), 3Π(3), 3∆(2), 3Φ 3Pg + 1Dg 10 1583Σ−, 3Π 1Sg + 3Pg 15 2191Σ+(3), 1Σ−(2), 1Π(4), 1∆(3), 1Dg + 1Dg 16 282 18 8001Φ(2), 1Γ

a Averaged over J; Ref. 38

46

next two dissociation limits also correlate only with the excited triplets of SiC. The combi-

nation of Si(1Dg) and C(1Dg) generates a set of 15 excited singlet states of SiC. The relative

energy of these states at the dissociation limit is about 16 300 cm−1.

Potential energy curves of singlet, triplet, and quintet states of SiC without spin-orbit

coupling are shown in Figs. 3.1a-c. All the states correlating with the lowest dissociation

limit and a few excited states dissociating into higher asymptotes are studied here. The

computed spectroscopic constants (re, ωe, Te) of 23 Λ–S states are tabulated in Table 3.3.

The computed bond length of the SiC radical in the ground state (X3Π) is 1.74 A which

is nearly 0.02 A longer than the observed data. The ground-state bond length obtained

in earlier calculations at various level of theories varies between 1.721 and 1.75 A. The

vibrational frequency (ωe) of the ground state obtained in the present calculation is 930 cm−1

which is about 35 cm−1 smaller than the observed value. A comparison of the present data

with other theoretical results is shown in Table 3.3. The ground state of the SiC radical

is dominated by the σ21σ2π

31(80%) configuration at re (Table 3.4). The σ1 MO is mainly a

bonding combination of s and pz orbitals of Si and C atoms, while σ2 has the antibonding

character and is mostly localized on the silicon atom. The π1 MO is a strongly bonding,

comprising the px/y orbitals of the two atoms. The ground-state dissociation energy (De) of

SiC computed here is 4.05 eV, which is somewhat lower than the thermochemical D00 value

of 4.64 eV.39,40 The MRD-CI calculations41 with DZP+bond functions have reported a De

value of 4.3 eV.

The A3Σ− state is the first excited state of SiC. Its computed transition energy (Te) is

3985 cm−1 which compares well with the experimental or other theoretical data shown in

Table 3.3. The observed 0-0 band of the A3Σ−-X3Π system has been found near 4500 cm−1

from which a T00 value of 4578 cm−1 is obtained.13 However, Langhoff and Bauschlicher14

suggested an alternative assignment of the band to a 1-0 transition. These authors have

established the 0-0 band at 3700+200 cm−1 from the theoretical calculations using atomic

natural orbitals. The equilibrium bond length of the A3Σ− state estimated here is about

1.82 A which agrees well with the available values as listed in Table 3.3. The ωe value of this

state, obtained in the present calculations, is 857 cm−1 which is very close to the values found

in other calculations.14,15 Around the potential minimum, the A3Σ− state is dominated by

the σ21σ

22π

21(88%) configuration in which the π1 MO has a similar bonding characteristic as

in the ground state.

47

48

2 3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

50000

60000

Si + C

(a)

3ΦE3Π

1Dg + 3P

g

3Pg + 3P

g

33Σ

-

23Σ

-

A3Σ

-

33∆

C3Π

X3Π

23∆

33Σ

+

23Σ

+

D3∆B3Σ

+Ene

rgy

/ cm

-1

Bond Length / bohr

2 3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

50000

60000

Si + C

(b)

1Φ

1Dg + 1D

g

31Σ-

X3Π

21Π

b1Π

21Σ-

1Σ-

21∆

31Σ+

3Pg + 3P

g

c1∆

d1Σ+

a1Σ+

Ene

rgy

/ cm

-1

Bond Length / bohr

2 3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

50000

60000

Si + C

(c)

3Pg + 3P

g

X3Π

25Π

5Π

35Σ-

25Σ-

5Σ-

5∆

25Σ+

5Σ+

En

erg

y / c

m-1

Bond Length / bohr

Fig. 3.1 Λ-S states of SiC: for (a) triplet, (b) singlet, and (c) quintet spin

multiplicities

The lowest singlet state of SiC is a1Σ+ which originates from the σ2 → π1 transition.

The dominant closed shell configuration describing the state is σ21π

41(70%). The state is very

strongly bound with a binding energy of about 3.33 eV. Its estimated transition energy is

5325 cm−1 with an equilibrium bond length of 1.68 A which is shorter than that of the

ground state. The computed vibrational frequency (ωe) of the state is 975 cm−1. In general,

the spectroscopic parameters of the a1Σ+ state computed here agree well with the set of

data obtained from other calculations.11

Table 3.3 Spectroscopic constants of low-lying Λ-S states of SiC

State Te/cm−1 re/A ωe/cm−1 µe/D

Expt. Calc. Expt. Calc. Expt. Calc.

X3Π 0 [1.72187]a 1.74 [965.16]d 930 1.62

[1.7182]d (1.721)b (957)b

(1.732)h (954.2)h

(1.722)i (927)i

(1.724)j (978.7)j

A3Σ− [4500]a 3985 [1.81356]a 1.82 857 2.55

[3773.31]g (3700 [1.802]g,i (1.788)e (862)f

±200)b

(3619)i (1.82)b (865)b

(3831)j (855)j

a1Σ+ 5325 1.68 975 2.14

(5079)i (1.677)i (955)i

(4355)j (1053)j

b1Π 6725 1.75 930 1.81

(7259)i (1.713)e (963)i

c1∆ 9135 1.85 790 2.31

(9094)i (1.81)e (855)c

(9306)j

d1Σ+ 12 705 1.84 880 2.25

(12 338)i (1.794)i (980)j

(11 614)j

49

Table 3.3 ...continued


Expt. Calc. Expt. Calc. Expt. Calc.5Π 14 460 1.97 635 0.97

B3Σ+ 19 800 1.68 890 1.89

(18 954)h (1.669)h (913)h

C3Π [22 830.4]d 21 915 [1.919]d 1.95 [615.7]d 580 0.91

[22 829.46]e (22 768)h (1.908)h [618.85]e (615.8)h

1Σ− 23 245 2.18 490 0.97

D3∆ 24 485 2.18 508 0.94

E3Π 25 875 1.94 600 1.32

21Π 27 415 2.02 480 1.173Φ 28 465 2.02 570 0.935Σ+ 33 355 1.78 745 0.951Φ 33 750 2.01 585 1.05

31Σ+ 35 125 1.72 920 1.09

21Σ− 37 175 1.84 800 1.42

21∆ 38 940 1.82 710 1.555∆ 42 005 1.73 963 1.36

25Σ+ 45 760 2.00 790 0.70

25Σ− 46 200 1.83 600

35Σ− 48 575 1.79 1040

a Ref. 13, b Ref. 14, c Ref. 5, d Ref. 17, e Ref. 18, f Ref. 15, g Ref. 23, h Ref. 9,

i Ref. 11, j Ref. 16

There exists a close-lying b1Π state with an estimated Te of 6725 cm−1 which is some-

what smaller than the value predicted from the HF+SD calculation. The equilibrium bond

length as well as the vibrational frequency of b1Π are comparable with those of its triplet

counterpart, namely the ground state. As expected, the leading configuration characterizing

the b1Π state is the same as that of the ground state, which is indicated in Table 3.4. The

σ21σ

22π

21 configuration not only generates the first excited state as discussed above but also

two strongly bound singlets, namely c1∆ and d1Σ+. The estimated transition energy of c1∆

50

is 9135 cm−1 which is comparable with the value reported in the previous calculations.11,16

However, the calculation of Bruna et al.5 predicted its transition energy above 10 500 cm−1.

The ωe value of the c1∆ state estimated here is 790 cm−1 which is 65 cm−1 smaller than the

value reported in earlier calculations. The d1Σ+ state is the second root of the 1Σ+ symme-

try. At the equilibrium bond length, the state is dominated by an open shell configuration,

σ21σ

22π

21(60%), while there is at least 17% contribution of a closed shell configuration, σ2

1π41.

The computed transition energy of this state (Te=12 705 cm−1) agrees well, while the re

value reported here is 0.05 A longer than the value reported earlier.11 The vibrational fre-

quency of d1Σ+ at re is about 880 cm−1 which is 100 cm−1 smaller than the value estimated

in the earlier study.15

There is a low-lying 5Π state which has not been identified before. The state originates

from a σ21σ2π

21π2 (82%) configuration in which π2 is mostly antibonding MO comprising

the px/y orbitals of both Si and C. The computed transition energy of the state is about

14 460 cm−1 with ωe=635 cm−1 and re=1.97 A. The state is also strongly bound with an

estimated binding energy of 2.22 eV. Although 5Π is not a very important state from the

spectroscopic point of view, its spin components with Ω=0+, 0−, 1, 2, and 3 may influence

the lower states to some extent. Several other excited quintet bound states of the SiC radical

lie above 30 000 cm−1.

The next two important states of SiC are B3Σ+ and C3Π. The B state is the lowest root

of the 3Σ+ symmetry. Ab initio based CASSCF-CCI calculations of Larsson9 have predicted

the B3Σ+ state to lie around 18 954 cm−1 with a bond length of 1.67 A. In the present

calculation, the estimated Te of this state is about 19 800 cm−1. The computed bond length

is 0.01 A longer, while the calculated vibrational frequency of this state agrees well with

the value reported in the earlier calculation of Larsson.9 However, spectroscopic constants of

the B3Σ+ state obtained from both the theoretical calculations disagree with those derived

from the absorption spectrum of the B3Σ+←X3Π transition in a 5K neon matrix.23 The

origin of the observed band has been reported to be at 11 749 cm−1. The observed ωe of

1178 cm−1 also disagrees completely with the present value of 890 cm−1. The spectrum of

the B3Σ+-X3Π transition of SiC may need further experimental study. However, it is quite

clear that there is no other lower state except the ground one to which the transition from

B3Σ+ may take place. Theoretically, the B-X transition is expected to carry a reasonably

large intensity. Around the potential minimum, the B3Σ+ state is characterized by two

important configurations, σ1σ2π41 (45%) and σ2

1π31π2(30%) as shown in Table 3.4.

51

Table 3.4 Composition of Λ-S states of SiC at equilibrium bond length

State Configuration (% contribution)

X3Π σ21σ2π

31(80), σ1σ

22π

31(3)

A3Σ− σ21σ

22π

21(88)

a1Σ+ σ21π

41(70), σ2

2π41(2), σ2

1σ22π

21(6), σ1σ2π

31π2(5)

b1Π σ21σ2π

31(81), σ2

1σ2π1π22(2)

c1∆ σ21σ

22π

21(85)

d1Σ+ σ21σ

22π

21(60), σ2

1π41(16), σ2

2π41(3)

5Π σ21σ2π

21π2(82), σ2

1σ2π1π22(2), σ1σ

22π1π

22(2)

B3Σ+ σ1σ2π41(45), σ2

1π31π2(30), σ1σ2π

31π2(4), σ1σ2π

21π

22(2)

C3Π σ21σ2π

21π2(76), σ2

1σ2π31(3), σ1σ

22π

31(3)

1Σ− σ21σ

22π1π2(84)

D3∆ σ21σ

22π1π2(84)

E3Π σ21σ2π

21π2(78), σ1σ

22π

31(5)

21Π σ21σ2π

21π2(82), σ1σ

22π

31(2)

3Φ σ21σ2π

21π2(86)

5Σ+ σ1σ2π31π2(66), σ2

1π21π

22(13), σ1σ2π1π

32(4), σ1σ2π

21π

22(3), σ2

2π21π

22(2)

1Φ σ21σ2π

21π2(83)

31Σ+ σ1σ2π41(37), σ2

1π31π2(23), σ1σ2π

31π2(6), σ2

2π41(6), σ2

1π41(3)

21Σ− σ21π

31π2(78), σ2

2π31π2(4)

21∆ σ21π

31π2(78)

5∆ σ1σ2π31π2(82)

25Σ+ σ1σ2π31π2(42), σ2

1π21π

22(37)

25Σ− σ21σ2σ3π

21(47), σ1σ2π

31π2(18), σ2

1σ2σ5π21(10)

35Σ− σ21σ2σ3π

21(37), σ1σ2π

31π2(25), σ2

1σ2σ4π21(10)

The C3Π state is the second root of the ground-state symmetry and it originates predom-

inantly from the same configuration, σ21σ2π

21π2 that generates the lowest 5Π. This configu-

ration also generates eight more states of Π and Φ symmetries. The transition energy (Te)

of C3Π calculated here is about 21 915 cm−1 which is somewhat smaller than the observed

value17 of 22 830.4(9) cm−1 determined from the C3Π–X3Π band system of the SiC radical

produced by laser vaporization. In another experimental study18, analysis of seven rovibronic

52

bands involving the C3Π(v′=0-6)–X3Π(v′′=0) transition have resulted Te and ωe values of

22 829.46 cm−1 and 618.85 cm−1, respectively. The Si-C bond in the C3Π state is nearly

0.21 A longer than that in the ground state. However, the calculated re is 0.03 A longer than

the experimentally determined value of 1.919 A. The fitted vibrational frequency is about

580 cm−1, as compared to the observed value of 615.7(8) cm−1. The molecular constants

computed by Larsson9 are in better agreement with the observed data as shown in Table 3.3.

No other state beyond C3Π has been identified so far. We have predicted 1Σ− and 3∆ states

with their potential minima located almost at the same bond length of 2.18 A. The estimated

transition energies of 1Σ− and 3∆ states are 23 245 and 24 485 cm−1, respectively. Potential

energy curves of these states are a bit shallow compared to those of the other low-lying

states already discussed. This has been reflected in their smaller ωe values. Both the states

originate from the same configuration, σ21σ

22π1π2 with the same dominance at equilibrium.

We may therefore, expect a 3∆-X3Π transition to take place around 24 500 cm−1. Although

such a transition has not yet been observed, we have labeled 3∆ as the D state because it is

next to C3Π. There is another excited 3Π, denoted as E3Π which lies just above D3∆. The

computed transition energy of the E3Π state is about 25 875 cm−1. The state originates from

σ21σ2π

21π2(78%) which has also created the lower lying C3Π and 5Π states. The re and ωe

values of the E3Π state are very similar to those of C3Π. The E3Π state, however, dissociates

into the higher asymptote. The E-X transition is expected to be strong one, though no such

transition is experimentally reported so far.

The second root of 1Π, which is also characterized by the σ21σ2π

21π2 configuration, is weakly

bound. The potential minimum of the 21Π state is located around the bond length of 2.02 A

with an estimated transition energy of 27 415 cm−1 at equilibrium. Both the lowest singlet

and triplet Φ states are strongly bound and have similar spectroscopic properties. The energy

separation between them is about 5300 cm−1 with the 3Φ state lying lower. These states are

also predominantly characterized by the same configuration that generates 5Π, C3Π, E3Π,

and 21Π. The 3Φ–X3Π transition may be of some interest from the spectroscopic point of

view through their dipolar components only.

Potential energy curves of three lowest states of the 3Σ+ symmetry undergo avoided cross-

ings as seen in Fig. 3.1a. Analyzing the compositions of these states in the CI calculations, it

has been found that three important configurations, namely σ1σ2π41, σ2

1π31π2, and σ2

1σ22π1π2

mix up. An avoided curve crossing between the first and second root of 3Σ+ takes place

around 3.9 a0, while another crossing is noted between the second and third root in the

53

range 3.2-3.4 a0. The contributions of important configurations of all three 3Σ+ states over

a certain range of bond distances have been displayed in Table 3.5.

Two excited 3Σ− states, namely 23Σ− and 33Σ− also undergo an avoided crossing around

the bond length of 3.6 a0. This is reflected in the potential energy curves of these states

shown in Fig. 3.1a. The diabatic coupling is considerably large. In the bond length region

below 3.6 a0, the lower root is dominated by σ21π

31π2, while in the other region it is mainly

characterized by the σ21σ

22π1π2 configuration. The diabatic curve of the 23Σ− state is pre-

dicted to have a minimum around 2.2 A with an estimated transition energy of 33 100 cm−1.

We also expect the diabatic curve of 33Σ− to have a potential minimum around 1.8 A with

a predicted transition energy of about 36 200 cm−1.

Table 3.5 Compositions of the lowest three roots of 3Σ+ of SiC in the avoided crossing region

r/a0 B3Σ+ 23Σ+ 33Σ+

2.7 σ1σ2π41(70), σ2

1π31π2(8) σ2

1π31π2(69), σ1σ2π

41(9), σ1σ2π

31π2(80), σ1σ2π

41(6)

σ22π

31π2(4)

2.9 σ1σ2π41(63), σ2

1π31π2(13), σ2

1π31π2(64), σ1σ2π

41(14) σ2

1σ22π1π2(82)

σ1σ2π31π2(6)

3.1 σ1σ2π41(52), σ2

1π31π2(22), σ2

1π31π2(54), σ1σ2π

41(23) σ2

1σ22π1π2(82)

σ1σ2π31π2(6)

3.2 σ1σ2π41(45), σ2

1π31π2(29), σ2

1π31π2(45), σ1σ2π

41(27) σ2

1σ22π1π2(80), σ2

1π31π2(4)

σ1σ2π31π2(4)

3.5 σ21π

31π2(53), σ1σ2π

41(20) σ2

1σ22π1π2(80) σ1σ2π

41(45), σ2

1π31π2(25)

3.6 σ21π

31π2(57), σ1σ2π

41(13) σ2

1σ22π1π2(80) σ1σ2π

41(51), σ2

1π31π2(19)

3.8 σ21π

31π2(49), σ2

1σ22π1π2(27) σ2

1σ22π1π2(60), σ2

1π31π2(17) σ1σ2π

41(54), σ2

1π31π2(11)

3.9 σ21σ

22π1π2(55), σ2

1π31π2(26) σ2

1π31π2(41), σ2

1σ22π1π2(30) σ1σ2π

41(53), σ2

1π31π2(8),

σ1σ2π31π2(8)

4.2 σ21σ

22π1π2(74), σ2

1π31π2(11) σ2

1π31π2(59), σ2

1σ22π1π2(11) σ1σ2π

41(38), σ1σ2π

31π2(12),

σ21σ2σ6π

21(7), σ1σ2π

21π

22(5)

Both the 5Σ+ states reported here are weakly bound and dissociate into the lowest

limit through different channel. The potential minimum of 5Σ+ is located at 1.78 A with

ωe=745 cm−1 and Te=33 355 cm−1. Around the equilibrium bond length, the 5Σ+ state is

54

characterized predominantly by σ1σ2π31π2, while another configuration, σ2

1π21π

22 dominates at

the longer bond distances. This has resulted a predissociation of the 5Σ+ state through a

barrier of height 0.22 eV. A strong avoided crossing between the potential curves of the two

roots of 5Σ+ has created an apparent minimum in the potential curve of 25Σ+. Since the

coupling between these two roots is very strong, we have fitted the adiabatic curve and the

minimum is located at 2.0 A with ωe=790 cm−1 and Te=45 760 cm−1. The nature of the

potential energy curve of 25Σ+ reveals that there is a potential barrier of height 0.4 eV at

4.4 a0 beyond which the state dissociates through a repulsive path.

Three more excited singlets, namely 31Σ+, 21Σ−, and 21∆ of the SiC radical are strongly

bound. They correlate with the higher asymptote, Si(1Dg)+C(1Dg) as shown in Table 3.2.

The predicted transition energy of the 31Σ+ state is about 35 125 cm−1 at re=1.72 A.

The estimated vibrational frequency of the state is close to that of a1Σ+. A transition

of the type 31Σ+–a1Σ+ is expected to occur around 29 000 cm−1 with a large transition

probability. Both 21Σ− and 21∆ states, in the Franck-Condon region, are characterized

mainly by the σ21π

31π2 configuration. The potential energy curve of 21∆ shows an avoided

crossing near the bond distance of 4.0 a0. The estimated res of 21Σ− and 21∆ are 1.84

and 1.82 A, respectively. Analyzing the compositions of these states above 4.0 a0, a strong

mixing with other configurations is predicted. The vibrational frequency of the 21∆ state is

nearly 100 cm−1 smaller than that of 21Σ−. The computed transition energies of these two

excited singlets lie in the range 37 000-39 000 cm−1.

The lowest 5∆ state has a deep potential minimum at re=1.73 A with Te=42 005 cm−1

and it is mainly described by the σ1σ2π31π2 (82%) configuration. Potential energy curves in

Fig. 3.1c show that there is an avoided crossing around 4.1 a0 with a repulsive curve of an

excited 5∆ which is dominated by the σ21σ2σ6π1π2 configuration. This has resulted a predis-

sociation of 5∆ to the lowest dissociation limit through a large barrier of 1.3 eV. However,

the fitted potential curve in the Franck-Condon region has predicted ωe=963 cm−1. The

second and third root of 5Σ− have bound potential energy curves. The compositions of these

roots at different bond distances reveal that there is a strong mixing between (σ1σ2π31π2)5Σ−

and (σ21σ2σ3π

21)5Σ− in the range 3.4-3.5 a0. As a result, the potential minimum of 25Σ− is

shallow, while the adiabatic curve of 35Σ− is very sharp with a large ωe of 1040 cm−1. The

estimated re of the 35Σ− state is about 1.79 A, while the minimum of the adiabatic curve of

25Σ− is located around the bond length of 1.83 A with ωe=600 cm−1.

55


The spin-orbit coupling splits the ground-state dissociation limit, Si(3Pg)+C(3Pg) into

nine very closely spaced asymptotes as shown in Table 3.6. The largest splitting is only

267 cm−1 as obtained from the observed atomic spectral data.38 These asymptotes correlate

with 50 Ω states of the SiC radical. In the present calculations, we allow all these states to

Table 3.6 Dissociation correlation between Ω and atomic states of SiC

Ω States† Atomic states Relative energy/

Si + C cm−1

0+ 3P0 + 3P0 0

0−, 1 3P0 + 3P1 16

0+, 1, 2 3P0 + 3P2 43

0−, 1 3P1 + 3P0 77

0+(2), 0−, 1(2), 2 3P1 + 3P1 94

0+, 0−(2), 1(3), 2(2), 3 3P1 + 3P2 121

0+, 1, 2 3P2 + 3P0 223

0+, 0−(2), 1(3), 2(2), 3 3P2 + 3P1 240

0+(3), 0−(2), 1(4), 2(3), 3(2), 4 3P2 + 3P2 267

† Values in parenthesis are the corresponding number of states

interact. Potential energy curves of a few low-lying states of 0+, 0−, 1, 2 and 3 symmetries

of SiC are shown in Figs. 3.2a-d. These show several number of curve crossing phenomena.

Since both the atoms are light, the spin-orbit effects are expectedly small. The potential

energy curves show no great changes and the avoided crossings are very sharp. The ground

state of SiC splits in an inverted order with X3Π2 being the lowest spin-orbit component.

All four components of X3Π lie within 100 cm−1. The computed spectroscopic constants of

the low-lying Ω states up to 22 000 cm−1 of energy are shown in Table 3.7. Obviously, the

spectroscopic constants are obtained by fitting the diabatic curves. The spin-orbit mixing

does not change the spectroscopic constants much. The two components of A3Σ− are almost

inseparable. The transition energies are increased only by 70-75 cm−1 due to the spin-orbit

coupling. The diabatic curve of A3Σ−0+ cuts the similar component of a1Σ+ just beyond the

equilibrium point of the latter (3.2 a0). Thus, the potential energy curves of second and

56

57

2 3 4 5 6 7 8 9

0

5000

10000

15000

20000

25000

30000

35000

40000(a)

E3Π0 +

C3Π0+

5Π0+

d1Σ+

0 +

a1Σ+0+

A3Σ-0+

X3Π0 +

Ene

rgy

/ cm

-1

Bond Length / bohr

2 3 4 5 6 7 8 9

0

5000

10000

15000

20000

25000

30000

35000

40000

1Σ

-

0-

E3Π0-

C3Π0-

B3Σ+

0-

(b)

5Π0-

X3Π0-

En

erg

y / c

m-1

Bond Length / bohr

2 3 4 5 6 7 8 9

0

5000

10000

15000

20000

25000

30000

35000

40000

D3∆ 1

D3∆

3

Bond Length / bohr

Ene

rgy

/ cm

-1

E3Π1

C3Π

1B3

Σ+

1

5Π

-1,1,3

b1Π

1

A3Σ

-

1

X3Π 1

(c)

2 3 4 5 6 7 8 9

0

5000

10000

15000

20000

25000

30000

35000

40000

E3Π 2

D3∆ 2

C3Π 2

5Π 2

c1∆ 2

X3Π 2

(d)

Bond Length / bohr

Fig. 3.2 Ω states of SiC: for (a) 0+, (b) 0-, (c) 1, 3, and (d) 2 symmetries

third root of 0+ look somewhat different as compared to the corresponding Λ-S states. All

the singlet components, a1Σ+0+ , b1Π1, c1∆2, and d1Σ+

0+ remain almost unchanged in their

spectroscopic properties. The spin-orbit components of 5Π split in a regular order with a

maximum separation of 135 cm−1. The components of C3Π, however, split in an inverted

order as displayed in Table 3.7. Spectroscopic properties of the 0− and 1 components of

B3Σ+ are nearly the same.

Table 3.7 Spectroscopic constants of low-lying Ω states of SiC

State Te/cm−1 re/A ωe/cm−1

X3Π2 0 1.74 930

(0)a

X3Π1 60 1.74 925

(37.3)a

X3Π0− 95 1.74 933

(74.6)a

X3Π0+ 100 1.74 933

(74.6)a

A3Σ−1 4055 1.82 854

A3Σ−0+ 4060 1.82 855

a1Σ+0+ 5370 1.68 975

b1Π1 6770 1.76 931

c1∆2 9185 1.85 800

d1Σ+0+ 12 745 1.84 882

5Π−1 14 450 1.97 6255Π0− 14 480 1.97 6255Π0+ 14 485 1.97 6255Π1 14 515 1.97 6255Π2 14 550 1.97 6255Π3 14 585 1.97 625

B3Σ+1 19 880 1.69 871

B3Σ+0− 19 885 1.69 871

58



C3Π2 21 935 1.95 581

C3Π1 21 965 1.95 581

C3Π0− 21 995 1.95 581

C3Π0+ 22 000 1.95 582

a Ref. 16

3.3.3 Dipole moments and transition properties

In Table 3.3, we have reported the computed dipole moments (µe) of most of the low-

lying states at their equilibrium bond lengths. The ground-state µe is estimated to be about

1.62 D with Si+C− polarity. The first excited state (A3Σ−) has a larger µe value of 2.55 D.

All other excited states have the same sense of polarity as that of the ground state. The

variation of the dipole moment for the low-lying states of SiC as a function of bond distance

are plotted in Fig. 3.3a. The ground-state dipole moment function has a maximum at 4.6 a0.

After a short increase upto 3.2 a0, the dipole moment function of A3Σ− smoothly decreases

with increase in bond length. The situation is same for c1∆, while those of a1Σ+, b1Π, and

c1∆ have another similar type of variation as shown in the Fig. 3.3a. At very short bond

distance (2.5 a0), C3Π has the highest dipole moment of 1.17 ea0, but it suddenly falls with

increase in bond length. Dipole moment function of 5Π looks similar to that of the ground

state. Strikingly, all the functions tend to zero at very long bond distances.

Several electric dipole allowed transitions involving the low-lying singlets and triplets of

SiC are studied here. Transition probabilities of six triplet-triplet transitions, each of which

has the ground state as the lowest one, have been computed. Transition dipole moments

of these transitions as a function of bond distance are shown in Fig. 3.3b. The transition-

moment curve of A3Σ−-X3Π is smoothly decreasing to zero at the longer bond distance. In

the Franck-Condon region, the magnitude of the transition dipole moment of A-X is about

0.3 ea0. The computed partial radiative lifetime for this transition at v′=0 is about 125 µs and

it decreases considerably with the higher v′. Table 3.8 lists the partial radiative lifetimes for

eleven transitions at the lowest three vibrational levels. The B3Σ+-X3Π transition moments

in the Franck-Condon region are at least 10 times smaller than those of A-X. But the

computed lifetime for the B-X transition is lower due to larger energy differences. As

59

60

2 3 4 5 6 7-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2(a)

Bond Length / a0

E3Π

D3∆

1Σ -

C3Π

B3Σ

+

5Π

d1Σ + c1∆

b1Π

a1Σ +

A3Σ-

X3Π

Dip

ole

Mom

ent /

ea 0

3 4 5 6 7-0.25

0.00

0.25

0.50

0.75(b)

C3Π-A3

Σ-

E3Π-X3

ΠA3Σ

--X3Π

B3Σ

+-X3Π

C3Π-X3

Π

D3∆-X3Π

Tran

sitio

n M

omen

t / e

a 0

Bond Length / a0

3 4 5 6 7-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6(c)

31Σ+-d1Σ+

31Σ+-b1Π

d1Σ+-a1Σ+

21Π-a1Σ+

31Σ+-a1Σ+

Tran

sitio

n M

omen

t / e

a 0

Bond Length / a0

Fig. 3.3 Computed (a) dipole and transition moment functions involving (b) triplets & (c) singlets of SiC

expected, transitions from the excited Π states to the ground state are relatively strong.

Transition moment curves for both C3Π-X3Π and E3Π-X3Π look similar (Fig. 3.3b). These

two transitions are predicted to be more probable than either the A-X or B-X transition. The

radiative lifetime for the C3Π-X3Π transition at the lowest vibrational level is about 5.15 µs.

The computed partial lifetime for another transition, C3Π-A3Σ− is 216 µs at v′=0. The total

radiative lifetime of the C3Π state, computed after adding the transition probabilities of C-X

and C-A transitions is about 5.03 µs. However, this value is nearly 1.75 times longer than

the experimentally determined18 value of 2.886 µs. The transition probability of another

possible transition, C3Π-B3Σ+ is very low as the energy difference is only about 2000 cm−1.

The E3Π-X3Π transition is found to be the strongest of all triplet-triplet transitions and it

is expected to appear in the range 25 000-26 000 cm−1. The computed radiative lifetime for

this transition is estimated to be 1.1 µs in the lowest vibrational level.

Table 3.8 Radiative lifetimes (µs) of some excited states of SiC

Transition Lifetimes of the upper state at

υ′=0 υ′=1 υ′=2

A3Σ−-X3Π 125 105 91

(107.6)a (82.3)a (67.0)a

B3Σ+-X3Π 28.7 28.9 26.3

C3Π-X3Π 5.15 2.78 1.88

C3Π-A3Σ− 216 188 156

D3∆-X3Π 1250 970 760

E3Π-X3Π 1.10 1.17 1.25

31Σ+-a1Σ+ 0.51 0.47 0.43

31Σ+-b1Π 485 195 128

31Σ+-d1Σ+ 5.58 5.06 4.34

d1Σ+-a1Σ+ 9110 3800 2100

21Π-a1Σ+ 303 307 350

a Ref. 14

Transition probabilities of five singlet-singlet transitions are computed. The transition

dipole moment functions of these transitions are shown in Fig. 3.3c. Three transitions from

61

the 31Σ+ state, namely 31Σ+-a1Σ+, 31Σ+-b1Π, and 31Σ+-d1Σ+ are studied. The 31Σ+-

a1Σ+ transition has the largest transition probability. The computed radiative lifetime for

this transition is about 510 ns at v′=0. Adding the transition probabilities of these three

transitions, the total radiative lifetime of the 31Σ+ state is estimated to be 467 ns at the

lowest vibrational level. However, no transition from the 31Σ+ state of SiC is experimentally

known yet. Among the other transitions reported here, the 21Π-a1Σ+ transition is predicted

to be stronger than d1Σ+-a1Σ+. The computed lifetime of the former transition is about

300 µs.

3.4. Summary

Low-lying electronic states of the SiC radical have been studied by using ab initio based

MRDCI calculations which include pseudo potentials of both Si and C atoms. We have

compared the computed spectroscopic constants with the observed and previously calculated

data. At least 23 states of Λ-S symmetries have been reported within 6 eV of energy. Besides

the ground state, the triplets such as A3Σ−, B3Σ+, and C3Π are experimentally well studied.

Four singlets, a1Σ+, b1Π, c1∆, and d1Σ+ are located in between 5000 and 13 000 cm−1. The

lowest bound state of the quintet spin multiplicity is 5Π which is located around 14 460 cm−1.

Potential energy curves of all the low-lying states are constructed. The excited E3Π state

is not previously known. We have predicted that the transition energy of the E3Π state

is about 25 875 cm−1. Its re and ωe are 1.94 A and 600 cm−1, respectively. A number of

singlet and quintet states exist in between 30 000 and 50 000 cm−1. Some of these singlet

and quintet states are strongly bound. The ground-state dipole moment (µe) is calculated

to be 1.62 D with a Si+C− polarity. All the other excited states have the same sense of

polarity. The largest dipole moment is reported for the A3Σ− state. The spin-orbit effects

on the low-lying states of SiC are not significant to change their spectroscopic properties.

The largest splitting among the four spin components of X3Π is only 100 cm−1 with X3Π2

being the lowest one. The two components of both A3Σ− and B3Σ+ are inseparable. Several

electric dipole allowed triplet-triplet and singlet-singlet transitions at the Λ-S level have

been studied. The strongest triplet-triplet transition is predicted to be E3Π-X3Π in the

range 25 000-26 000 cm−1. The computed radiative lifetime of this transition at v′=0 is

1.1 µs. However, such a transition has not been observed yet. The computed radiative

lifetime of C3Π is 5.03 µs compared to the experimentally determined value of 2.886 µs.

62

The lifetimes for A-X and B-X transitions are predicted to be 125 and 28.7 µs, respectively.

Of three transitions from the excited 31Σ+ state, the 31Σ+-a1Σ+ transition has the largest

transition probability and it has the shortest lifetime (τ=510 ns at v′=0) among all the

transitions studied here.

63

3.5 References

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4 B.L. Lutz, J.A. Ryan, Astrophys. J. 194, 753 (1974).

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6 C.M. Rohlfing, R.L. Martin, J. Phys. Chem. 90, 2043 (1986).

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9 M. Larsson, J. Phys. B: At. Mol. Phys. 19, L261 (1986).

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11 P.F. Bernath, S.A. Rogers, L.C. O′Brien, C.R. Brazier, Phys. Rev. Lett. 60, 197 (1988).

12 J. Cernicharo, C.A. Gottlieb, M. Guelin, P. Thaddeus, J.M. Vrtilek, Astrophys. J. Lett.

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13 C.R. Brazier, L.C. O′Brien, P.F. Bernath, J. Chem. Phys. 91, 7384 (1989).

14 S.R. Langhoff, C. Bauschlicher Jr., J. Chem. Phys. 93, 42 (1990).

15 F.L. Sefyani, J. Schamps, Astrophys. J. 434, 816 (1994).

16 J.M.L. Martin, J.P. Francois, R. Gijbels, J. Chem. Phys. 92, 6655 (1990).

17 T.J. Butenhoff, E.A. Rohlfing, J. Chem. Phys. 95, 3939 (1991).

18 M. Ebben, M. Drabbels, J.J. ter Meulen, J. Chem. Phys. 45, 2292 (1991).

19 A.D. McLean, B. Liu, G.S. Chandler, J. Chem. Phys. 97, 8459 (1992).

20 R. Mollaaghababa, C.A. Gottlieb, J.M. Vrtilek, P. Thaddens, Astrophys. J. Lett. 352,

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22 A.I. Boldyrev, J. Simons, V.G. Zahrzewski, W. Von Niessen, J. Phys. Chem. 98, 1427

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64

23 M. Grutter, P. Freivogel, J.P. Mair, J. Phys. Chem. A 101, 275 (1997).

24 M.N. Deo, K.J. Kawaguchi, Mol. Spectrosc. 228, 76 (2004).


26 J.M.O. Matos, V. Kello, B.O. Ross, A.J. Sadlej, J. Chem. Phys. 89, 423 (1988).

27 R.J. Buenker, S.D. Peyerimhoff, Theo. Chim. Acta 35, 33 (1974).


29 R.J. Buenker, Int. J. Quantum Chem. 29, 435 (1986).

30 R.J. Buenker, in: P. Burton (Ed.), Proc. Workshop on Quantum Chemistry and Molecular

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31 R.J. Buenker, in: R. Carbo (Ed.), Studies in Physical and Theoretical Chemistry, vol. 21,

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32 R.J. Buenker, S.D. Peyerimhoff, W. Butscher, Mol. Phys. 35, 771 (1978).

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65

CHAPTER – 4


SPECTROSCOPIC PROPERTIES OF SiC+ AND SiC-

4.1. Introduction

In chapter 3 we have discussed the astrophysical importance of the small molecules of

silicon and carbon. The simple SiC radical has been identified in interstellar clouds and

stellar atmospheres.1−2 Although in recent years the radical has been widely studied both

experimentally as well as theoretically3−11, data on the electronic states of the positive and

negative ions of SiC are not much available in the literature. No photoelectron spectrum of

the SiC radical has been reported as yet. First theoretical calculations of SiC+ have been

carried out by Bruna et al.9 using ab initio MRDCI method. Potential energy curves of

12 Λ-S states and spectroscopic constants of six bound states of SiC+ have been reported

by these authors. Ionization potentials and dissociation energies have also been determined.

They have also calculated potential energy curves of the isovalent Si+2 ion. Boldyrev et al.10

have computed ionization energies (vertical as well as adiabatic) and electron affinities of a

series of SinC and SinO (n=1-3) molecules using four different abinitio based methods. The

optimized bond lengths and vibrational frequencies of the SiC+ ion in its 4Σ+, 2Πi, and 2Σ+

states have been estimated.

The negative ion, SiC− is isovalent with C−2 for which the electronic spectrum is known

in detail both experimentally and theoretically. Moreover, the ionic species, C− and Si− are

known to exist in different electronic states. It is therefore, expected to have many stable

low-lying electronic states of SiC−. Grutter et al.12 identified SiC− in 5K neon matrices for

the first time using mass selected deposition. Two transitions, namely A2Π-X2Σ+ and B2Σ+-

X2Σ+ of SiC− with the absorption band origins at 3538 and 21 683 cm−1, respectively were

observed. Spectroscopic constants of X2Σ+, A2Π, and B2Σ+ states were determined from

the experimental data. The detachment of electrons from the SiC− ion in the B2Σ+ state

embedded in the neon matrix was established to take place between 3.5 and 4.0 eV by means

of wavelength-selected irradiation. Anglada et al.13 carried out the first theoretical study

on the low-lying electronic states of SiC− using multireference CI method. These authors

reported two bound states, X2Σ+ and A2Π in contrast to the results for isovalent C−2 in

which a third species like 2Σ+u is known. The calculated electron affinity of SiC obtained by

these authors has been found to be 1.98 eV. Singles and doubles CI calculations at the CISD

and CISD+Q levels from single SCF configuration were performed by McLean et al.14 on a

number of diatomic species including SiC and SiC−. Spectroscopic aspects of the 2Π state of

SiC− were also reported by these authors. Hunsicker and Jones15 made a density functional

66

study with simulated anealing on SixCy and SixC−y species with x+y ≤ 8. The ground-state

bond length of SiC− has been found to be 1.677 A. Recently, Cai and Francois16 have studied

potential energy curves and spectroscopic constants of the X2Σ+, A2Π, and B2Σ+ states of

SiC− using internally contracted multireference CI calculations. They have also computed

adiabatic and vertical electron affinities of SiC to the A2Π and B2Σ+ states of SiC−, and the

electronic transition moment functions for both B2Σ+-X2Σ+ and B2Σ+-A2Π transitions.

In this chapter, we report the electronic states and spectroscopic properties of SiC+ and

SiC− from a large scale MRDCI calculation. Potential energy curves of 14 low-lying Λ-S

states of the cation and 21 Λ-S states of the anion are constructed from the computed CI

energies. Spectroscopic constants of all these states are estimated by fitting the potential

curves. Effects of the spin-orbit coupling on the spectroscopic properties of the Ω states

of SiC+ are studied. Transition moments of the electric dipole allowed and spin-forbidden

transitions are computed. Subsequently, the partial radiative lifetimes for some of these

transitions are also estimated. The computed vertical as well as adiabatic ionization energies

and electron affinities of SiC are reported and compared with the available data.



The whole calculations of SiC+ are performed using similar pseudo potentials and basis

sets of Pacios and Christiansen17,18 for Si and C as those used in case of SiC (sec. 3.2.1). This

is also important to determine the ionization potential of SiC. In case of SiC−, the RECPs

are same but an additional set of p polarization functions of an exponent of 0.034 a−20 and

another set of d polarization functions of an exponent of 0.15 a−20 are included in the basis

set of carbon. This essentially improves the correlation due to one extra electron. In order

to calculate the electron affinity of SiC, we have performed the ground state CI calculation

of SiC with these additional basis sets.


Self-consistent-field (SCF) calculations have been performed for the (σπ2)4Σ− state of

SiC+ and (σπ4)2Σ+ state of SiC− at different internuclear distances in the range 2.5-20 a0

using the AREP and basis sets mentioned above. The calculations are carried out in the

67

C2v subgroup keeping Si at the origin and C along the +z axis. The SCF-MOs are used

as one electron basis functions for the generation of configurations in the CI calculations.

We have employed the MRDCI methodology of Buenker and coworkers19−25 throughout the

present calculations. A set of reference configurations, ranging between 81-139, is chosen for

a few low-lying states of each spatial and spin symmetry of the C2v group. As displayed in

Table 4.1, a maximum of eight roots for doublets and four roots for quartets of SiC+ are

Table 4.1 Details of the configuration interaction calculations of SiC+


generated configs. selected configs.§ (%)2A1 139/8 2 507 035 61 488 932A2 82/7 2 176 838 54 510 934A1 81/3 1 781 560 31 712 914A2 116/4 2 402 094 46 470 94

2B1/2B2 128/7 2 627 429 53 175 934B1/4B2 123/4 2 602 264 30 917 93


Table 4.2 Details of the configuration interaction calculations of SiC−


generated configs. selected configs.§ (%)2A1 153/4 8 276 092 97 418 862A2 136/3 10 366 698 78 660 874A1 187/5 13 466 095 100 068 864A2 126/3 8 563 183 73 622 876A1 144/3 8 736 676 75 067 876A2 132/3 8 245 884 60 171 88

2B1/2B2 152/5 9 212 169 96 068 874B1/4B2 208/4 12 739 597 99 705 886B1/6B2 130/3 7 485 576 57 134 87


68

optimized. All single and double excitations are carried out from the reference configurations

generating a large number of configurations of the order of 2-3 millions. In case of SiC−, 4-5

roots are calculated for doublets and quartets, while the number of lowest roots optimized

for sextets is three. Table 4.2 shows that, the similar type of configuration interactions from

130-208 reference configurations of SiC− generate 7-12 million of configurations. However, a

configuration selection with a threshold of 1.0 µhartree has been made to reduce the number

of selected configurations for the CI below 200 000. The table direct CI version26 of the

MRDCI codes has been used here. The sums of the squares of the CI coefficients remain

around 0.90 for both the species. The energy extrapolation method has been used to estimate

CI energies at zero threshold. The multireference analogue of the Davidson’s correction27,28

takes care of the higher excitations. The full CI energies of the lowest few roots in each

symmetry for both SiC+ and SiC− are thus estimated.


The spin-orbit coupling has been included by two-step variational calculations. The spin-

independent CI wave functions are multiplied with appropriate spin functions which trans-

form according to the symmetry. The diagonals of the spin-included Hamiltonian matrix

consist of the estimated full CI energies and the off-diagonals are the spin-orbit matrix ele-

ments which are calculated by the Wigner-Eckart theorem and spin-projection method. The

1/2, 3/2, and 5/2 states of SiC+ are obtained in two degenerate representations, E1 and E2

of C22v group. Potential energy curves of the low-lying bound states are fitted to estimate

the spectroscopic constants (re, Te, ωe), vibrational energies, and vibrational wave functions.

In the subsequent calculations, transition dipole moments of important dipole allowed and

spin-forbidden transitions and their partial radiative lifetimes are computed.



A. SiC+

Six doublets and six quartets of Σ+, Σ−(2), Π(2), and ∆ symmetries of SiC+ correlate

with the lowest dissociation limit, Si+(2Pu)+C(3Pg). The second asymptote comprising the

ground-state Si+ and the first excited state (1Dg) of the carbon atom correlates with nine

69

doublets, 2Σ+(2), 2Σ−, 2Π(3), 2∆(2), and 2Φ. The lowest two dissociation limits of SiC+

are separated by 10 158 cm−1.29 Our MRDCI calculation gives an overestimated value of

11 755 cm−1 with an error of 16%. Only two excited electronic states, 2Σ+ and 2Π dissociate

into the third asymptote, Si+(2Pu)+C(1Sg) which is nearly 21 600 cm−1 above the lowest

one. The calculated value again exceeds it by 2442 cm−1 as shown in Table 4.3.

Table 4.3 Dissociation correlation between the molecular and atomic states of SiC+


Expt.a Calc.2Σ+, 2Σ−(2), 2Π(2), 2∆, Si+(2Pu) + C(3Pg) 0 04Σ+, 4Σ−(2), 4Π(2), 4∆2Σ+(2), 2Σ−, 2Π(3) Si+(2Pu) + C(1Dg) 10 158 11 7552∆(2), 2Φ2Σ+, 2Π Si+(2Pu) + C(1Sg) 21 613 24 055

a Averaged over J; Ref. 29

Using the MRDCI methodology mentioned in the previous section, we have constructed

potential energy curves of all 12 Λ-S states which correlate with the lowest dissociation

limit and two excited doublets, namely 22Σ+ and 22∆. Potential energy curves of quartets

and doublets are shown in Figs. 4.1a and 4.1b, respectively. The computed spectroscopic

constants of all 14 Λ-S bound states of SiC+ are listed in Table 4.4.

The ground state of SiC+ is of the X4Σ− symmetry. Its equilibrium bond length computed

here is 1.83 A which is 0.01 A lower than the value reported by Bruna et al.9 because of the

use of larger CI space. However, Moller-Plesset perturbation calculations10 have predicted

a somewhat smaller value of 1.804 A. The computed ground-state vibrational frequency is

817 cm−1 which is somewhat smaller than the perturbative value of 890 cm−1 but improved

over the earlier MRDCI result. Experimental data for even the ground state of SiC+ are not

yet known. The ground state is dominated by the σ21σ2π

21 (c2=0.84) configuration in which

the singly occupied σ2 MO is strongly bonding comprising s and pz atomic orbitals of Si

and C, while π1 is also a strongly bonding combination of px/y AOs of the two constituting

atoms. The doubly occupied σ1 MO is weakly antibonding and mostly localized on Si.

The ground-state dissociation energy (De) of SiC+ computed in the present investigation is

3.32 eV, which agrees well with the experimental value of (3.4-3.8)±0.6 eV estimated from

70

71

2 3 4 5 6 7 8 9 10 11 12

0

10000

20000

30000

40000

50000(a)

Si+(2Pu)+C(3Pg)4Σ

+

4∆

24Π

4Π

24Σ

-

X4Σ

-

Ene

rgy

/ cm

-1

Bond Length / a0

2 3 4 5 6 7 8 9 10 11 12

0

10000

20000

30000

40000

50000

Si+(2Pu)+C(1D

g)

(b)

22Σ+

22∆

Si+(2Pu)+C(3P

g)

22Σ -

22Π

2Σ+

2Σ-

2Π2∆

X4Σ -

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 4.1 Λ-S states of SiC+: for (a) quartet and (b) doublet spin multiplicities

D0(CSi+)=D0(CSi)+I.P.(Si)-I.P.(CSi). The calculations of Bruna et al.9 have reported D0

values of 3.71 and 3.22 eV using basis with and without bond functions, respectively.

Table 4.4 Spectroscopic constants of low-lying

Λ-S states of SiC+


X4Σ− 0 1.83 817

1.804a 890a

1.84b 797b

2∆ 10 266 1.88 723

10 160b 1.89b 702b

2Π 10 696 1.99 480, 700c

10 536b (1.75-2.01)b 741b

2Σ− 11 492 1.86 759

11 451b 1.88b 731b

2Σ+ 13 666 1.91 651

13 064b 1.98b 586b

22Π 14 311 1.87 1013

14 192b 1.89b 1017b

4∆ 21 173 2.46 2854Σ+ 21 473 2.48 281

22Σ− 23 723 2.34 4024Π 24 464 1.70 875

24Σ− 27 447 1.89 537

22∆ 30 315 2.45 310

22Σ+ 30 873 2.45 309

24Π 35 254 1.85 965

a Ref. 10, b Ref. 9

c An estimate from the diabatic curve

A set of five doublet states exist in the energy range 10 000-15 000 cm−1. The first

excited state of SiC+ is 2∆ with a computed transition energy of 10 266 cm−1. It is very

72

strongly bound with a binding energy of about 2.05 eV. The computed re and ωe of the 2∆

state match well with the earlier results of Bruna et al.9 At equilibrium, the state is mainly

characterized by the same dominant configuration as that of the ground state. However, two

other configurations, σ21σ2π1π2 (c2=0.07) and σ1σ

22π

21 (c2=0.04) also contribute to 2∆. The

two roots of 2Π interact strongly, as is evident from the avoided crossing between the two

potential curves in Fig. 4.1b. The curve-crossing point is located around 3.47 a0. Analysing

the CI wave functions for these two roots of 2Π symmetry, it is revealed that at bond lengths

shorter than 3.47 a0, the lower root is dominated by σ21π

31 and σ1σ2π

31 configurations, while

the higher one is described mainly by σ21σ

22π1. The situation is reversed after the avoided

crossing. In Table 4.5, we have tabulated the important configurations of these two roots at

six representative bond distances. The adiabatic potential energy curves of both 2Π and

Table 4.5 Composition (% contribution) of the lowest two roots of 2Π state of SiC+

in the bond length range 2.8-4.0 a0

r(a0) 2Π 22Π

2.8 σ21π

31(42), σ1σ2π

31(32), σ1σ2π

21π2(11) σ2

1σ22π1(73), σ1σ2π

31(8)

3.1 σ21π

31(43), σ1σ2π

31(25), σ1σ2π

21π2(12) σ2

1σ22π1(80), σ1σ2π

31(4)

3.3 σ21π

31(42), σ1σ2π

31(17), σ2

1σ22π1(12), σ2

1σ22π1(72), σ1σ2π

31(7)

σ1σ2π21π2(11)

3.5 σ21σ

22π1(50), σ2

1π31(24), σ1σ2π

21π2(6), σ2

1σ22π1(33), σ2

1π31(21), σ1σ2π

31(15),

σ1σ2π31(5) σ1σ2π

21π2(9), σ2

1π21π2(6)

3.7 σ21σ

22π1(74), σ2

1π31(8) σ2

1π31(35), σ1σ2π

31(15), σ1σ2π

21π2(12),

σ21π

21π2(11), σ2

1σ22π1(9)

4.0 σ21σ

22π1(79), σ2

1π31(4) σ2

1π31(37), σ2

1π21π2(18), σ1σ2π

21π2(13),

σ1σ2π31(11), σ2

1σ22π1(3)

22Π are fitted for estimating the spectroscopic constants. The potential energy curve of2Π is seen to have a double well. Although the inner well is not prominent, the minimum

lies around 1.77 A and does not hold any vibrational levels. On the other hand, the long-

distant minimum is found to be near 1.99 A. The corresponding Te and ωe are computed

to be 10 696 cm−1 and 480 cm−1, respectively. The vibrational frequency for the 2Π state

reported by Bruna et al.9 is 741 cm−1 which may be due to the fitting of the diabatic curve.

73

The corresponding ωe of the diabatic curve in our calculations is estimated to be about

700 cm−1. The interaction between the 2Π and 22Π states causes the 22Π state to have a

large ωe value of 1013 cm−1, which compares well to the previously calculated value. The

potential minimum of 22Π is located around 1.87 A. The energy gap between the minima of

the two roots of 2Π is about 3600 cm−1.

Potential minima of the next two doublets belong to 2Σ− and 2Σ+ symmetries. Both

states are dominated by the same configuration, σ21σ2π

21, as that in the ground state and like

2∆ state, two other excited configurations namely, σ1σ22π

21 and σ2

1σ2π1π2 contribute to a small

extent (Table 4.6). It may be noted that all four states of SiC+, X4Σ−, 2∆, 2Σ−, and 2Σ+ lie

within 14 000 cm−1. The computed Te of the 2Σ− state is about 11 492 cm−1 with re=1.86 A

and ωe=759 cm−1. The separation between 2Σ+ and 2Σ− is computed to be 2174 cm−1

which is larger than the value of 1613 cm−1 predicted in the earlier CI calculations.9 The

bond length of the 2Σ+ state is predicted to be shortened and the vibrational frequency is

predicted to be increased, relative to the previous calculation.

Table 4.6 Composition of Λ-S states of SiC+ at equilibrium bond length


X4Σ− σ21σ2π

21(84)

2∆ σ21σ2π

21(72), σ2

1σ2π1π2(7)), σ1σ22π

21(4)

2Π σ21σ

22π1(75), σ2

1π31(8)

2Σ− σ21σ2π

21(70), σ1σ

22π

21(7), σ2

1σ2π1π2(6))2Σ+ σ2

1σ2π21(65), σ2

1σ2π1π2(11), σ1σ22π

21(5), σ1σ

22π1π2(4))

22Π σ21π

31(28), σ2

1σ22π1(20), σ1σ2π

31(16), σ1σ2π

21π2(10), σ2

1π21π2(10)

4∆ σ21σ2π1π2(72), σ2

1σ3π1π2(8), σ1σ22π1π2(6)

4Σ+ σ21σ2π1π2(72), σ2

1σ3π1π2(8), σ1σ22π1π2(6)

22Σ− σ21σ2π1π2(64), σ2

1σ3π1π2(10), σ21σ2π

21(7)

4Π σ1σ2π31(52), σ1σ2π

21π2(33), σ2

1π21π2(5)

24Σ− σ1σ22π

21(32), σ1σ

22π1π2(25), σ2

1σ2π1π2(23), σ21σ2π

21(6)

22∆ σ21σ2π1π2(65), σ1σ

22π1π2(8), σ2

1σ3π1π2(7), σ21σ2π

21(4)

22Σ+ σ21σ2π1π2(67), σ1σ

22π1π2(9), σ2

1σ3π1π2(6), σ21σ2π

21(3)

24Π σ1σ2π21π2(30), σ2

1π21π2(28), σ1σ2π

31(26)

The next two quartet states, 4∆ and 4Σ+ have transition energies of 21 173 and 21 473 cm−1,

74

respectively. The potential energy curves (Fig. 4.1a) of these two states have similar char-

acteristics. Both of them dissociate into the lowest limit and are characterized by σ21σ2π1π2

(c2=0.72), σ21σ3π1π2 (c2=0.08), and σ1σ

22π1π2 (c2=0.06) configurations. The re and ωe of

these two states are very similar and their potential minima are shallow and shifted towards

the longer bond distance region. Neither of these two states is important from the spectro-

scopic point of view as their transitions to the ground state are forbidden. The 22Σ− state,

which originates from the same σ21σ2π1π2 configuration as the previous two quartets, also has

a shallow minimum at re=2.34 A with a smaller ωe of 402 cm−1. The computed transition

energy of the 22Σ− state at re is about 23 723 cm−1. The lowest two roots of 4Π also interact

strongly. The avoided crossing between the two 4Π roots has made the potential minimum

of the lower 4Π state very sharp but with a small binding energy of 0.29 eV. As a result of

the strong perturbation, the 4Π state has a shorter bond length (re=1.70 A) and larger ωe

(=875 cm−1). The upper state, 24Π has a deeper potential well with ωe=965 cm−1 and the

minimum is shifted at 1.85 A around which the avoided crossing takes place. Energetically,

the potential minimum of 24Π lies more than 10 000 cm−1 above 4Π. The analysis of CI

wave functions reveal that the electronic configurations, which contribute in the region of the

avoided crossing, are σ1σ2π31, σ1σ2π

21π2, σ2

1π21π2 and the compositions of both the 4Π states

are complex in nature. Two more excited doublets, 22∆ and 22Σ+ look very similar to the

corresponding lowest two quartet state. Potential energy curves and even the compositions

of these doublet states resemble those of the quartets. The computed transition energies of

22∆ and 22Σ+ states are estimated to be 30 315 and 30 873 cm−1, respectively with almost

the same re and ωes.

B. SiC−

Doublets, quartets, and sextets of Σ+ and Π symmetries of SiC− correlate with the lowest

asymptote, Si−(4Su)+C(3Pg). The second dissociation limit, Si(3Pg)+C−(4Su), which lies

only 0.12 eV away, correlates with another six excited Λ-S states of the same symmetries as

the previous one. A set of doublets and quartets of Σ+(2), Σ−, Π(3), ∆(2), and Φ symmetries

dissociate into Si−(2Du)+C(3Pg) with the observed atomic splitting (2Du-4Su) of 6952 cm−1

31 for Si−. The dissociation correlation between the atomic states and the corresponding

molecular states is shown in the Table 4.7. Potential energy curves of 21 Λ-S states of

doublet, quartet, and sextet spin multiplicities are shown in Figs. 4.2a-c, respectively, while

their spectroscopic constants (Te, re, ωe) are given in Table 4.8.

75

76

2 3 4 5 6 7 8

0

10000

20000

30000

40000

50000

60000(a)

2∆

2Σ-

D2ΠC2Π

B2Σ

+

X3Π

(SiC)

A2Π

X2Σ+

Ene

rgy

/ cm

-1

Bond Length / a0

2 3 4 5 6 7 8

0

10000

20000

30000

40000

50000

60000(b)

4Σ

-24Σ+

24∆

24Π

X3Π

(SiC)

X2Σ+

4Π4∆

4Σ

+

Ene

rgy

/ cm

-1

Bond Length / a0

2 3 4 5 6 7 8

0

10000

20000

30000

40000

50000

60000

70000(c)

26Σ

+

36Π

6Σ-

26Π6∆ 6

Π

X3Π

(SiC)

X2Σ

+

6Σ+

Bond Length / a0

Ene

rgy

/ cm

-1

Fig. 4.2 Λ-S states of SiC-: for (a) doublet, (b) quartet, and (c) sextet spin

multiplicities

Table 4.7 Dissociation correlation between the molecular and atomic states of SiC−

Λ-S states Atomic states Relative energya / cm−1

2Σ+, 2Π, 4Σ+, 4Π, 6Σ+, 6Π Si−(4Su) + C(3Pg) 02Σ+, 2Π, 4Σ+, 4Π, 6Σ+, 6Π Si(3Pg) + C−(4Su) 9682Σ+(2), 2Σ−, 2Π(3), 2∆(2), 2Φ, Si−(2Du) + C(3Pg) 69524Σ+(2), 4Σ−, 4Π(3), 4∆(2), 4Φ

a Averaged over J; Ref. 29, 31

The bond length of SiC− in the ground state (X2Σ+) is 1.69 A which compares well

with the previously calculated values. The CMRCI calculations of Cai and Francois16 at

various level predicted the ground-state bond length in the range 1.6794-1.7286 A. Their

best estimated ωe value of 1026.3 cm−1 agrees well with the present value of 1056 cm−1. The

second order Moller-Plesset perturbation (MP2) level calculations10 have resulted in a longer

bond length and a larger frequency for the ground state of SiC−. Experimental molecular

constants of SiC− in the ground state are not yet known. The ground state is dominated

by a pair of configurations, σ21σ2π

41 (c2=0.34) and σ1σ

22π

41(c2=0.28). The σ1 MO is strongly

antibonding comprising s and pz AOs of both Si and C atoms, while σ2 mainly consists of

their bonding combination. The π1 MO is a strongly bonding combination of px/y AOs of

the two constituting atoms. The computed ground-state dissociation energy (De) of SiC− is

about 5.14 eV which compares well with the previous value of 5.29 eV predicted from the

CMRCI calculations.16

The first excited state of the anion is A2Π, which lies close to the ground state. Its

computed transition energy is larger than the experimental value of 3556 cm−1 by 330 cm−1

only. Although the experimental bond length of A2Π is not known, the present re value

of 1.77 A supports the earlier theoretical prediction. The observed A2Π←X2Σ+ electronic

transition has a band origin at 3538 A in 5K neon matrices using mass selected deposition.12

The reported ωe of A2Π matches well with the value obtained in the present study (Table 4.8).

The state is dominated by σ21σ

22π

31 with three other open-shell configurations contributing

to a smaller extent as shown in Table 4.9. The next two quartets, 4Σ+ and 4∆, which

originate from the same set of configurations, are strongly bound. The 4∆ state has a

bond length shorter than 4Σ+ by about 0.05 A. The two states are separated by about

4500 cm−1 with their ωes lying in the range 810-850 cm−1. Both the quartets are dominated

77

by σ2π31π2 (c2=0.5) which also gives rise to the third quartet state, namely 4Σ−. In the

present calculations, the 4Σ− state, however, lies well above 4Σ+ and 4∆. The computed

transition energy (Te) of the 4Σ− state is about 23 040 cm−1, while its re and ωe are predicted

to be 1.78 A and 1022 cm−1, respectively.

Table 4.8 Spectroscopic constants of low-lying Λ-S states

of SiC−


X2Σ+ 0 1.69 1056

1.759b 1127b

1.70c 976c

1.6794d 1026.3d

1.677e

A2Π 3885 [3556]a 1.77 950 [941]a

3226b 1.710b 951b

3000c 1.77c 949c

3474.5d 1.7449d 945.7d

4Σ+ 14 865 1.84 8154∆ 19 327 1.79 845

B2Σ+ 20 960 [21 813±20]a 1.76 830 [717±25]a

21 278.5d 1.7858d 693.5d

4Π 20 640 1.92 702

24Π 21 687 1.73 1126

C2Π 22 297 1.74 10434Σ− 23 040 1.78 1022

D2Π 23 555 1.93 6502Σ− 25 530 1.79 9856Σ+ 26 360 2.09 5302∆ 27 300 1.80 830

24Σ+ 29 860 1.74 1010

24∆ 30 705 1.75 9656Π 36 554 1.95 598

26Π 40 963 2.01 615

78



6∆ 44 015 1.95 6306Σ− 44 680 1.94 695

26Σ+ 45 090 1.95 625

36Π 47 380 1.93 745

a Ref. 12, b Ref. 10, c Ref. 13, d Ref. 16, e Ref. 15

The B2Σ+ state of SiC− is the most important one from the spectroscopic point of view,

as the B2Σ+←X2Σ+ transition has been observed in absorption with a band origin around

21 680 cm−1. The computed transition energy of B2Σ+ is 20 960 cm−1 with a bond length

of 1.76 A. The present ωe of this state is at least 100 cm−1 larger than the value determined

from the absorption study in neon matrix. Like the lowest two doublets, the B2Σ+ state of

SiC− has a multiconfiguration character with a dominant configuration, σ2π31π2 (c2=0.44).

Two roots of 4Π are closely spaced with a separation of about 1000 cm−1. The lowest 4Π state

has a longer equilibrium bond length and a smaller ωe than those of the higher root. The

avoided crossing between 4Π and 24Π curves takes place around 1.78 A. Both the 4Π states

originate from the same dominant configuration, ...π21π2. However, the greater complexity

of the 24Π state may be noted from the composition given in Table 4.9. The next two 2Π

states, designated as C and D, are located just above 4Π with a gap of 1250 cm−1 in between

them. The avoided crossing in the potential curves of C2Π and D2Π is found to be around

3.5 a0. The estimated equilibrium bond length of the higher energy state, D2Π, is about

0.2 A longer than that of C2Π. The vibrational frequency of C2Π is comparable to that of the

ground state. A strong transition, C2Π←X2Σ+ is expected to take place around 22 300 cm−1.

Another transition, D2Π-X2Σ+ may not be so strong as the Franck-Condon overlap factor

is small due to a longer re of the excited state than the ground state. Around the potential

minimum, the C2Π state is composed of a number of open shell configurations. The dominant

configuration, σ21σ2σ4π

31 has only 23% contribution. The D2Π state is, however, less complex

with a dominant configuration of σ21σ

22π

21π2(c2=0.62). Two more doublets, 2Σ− and 2∆

are predicted to be strongly bound. Their computed transition energies are 25 530 and

27 300 cm−1, respectively. Although the equilibrium bond lengths of these states are almost

same, their ωes differ significantly. These two states show multiconfiguration character.

79

Table 4.9 Composition of Λ-S states of SiC− at equilibrium bond length


X2Σ+ σ21σ2π

41(34), σ1σ

22π

41(28), σ2

1σ2π31π2(8), σ1σ

22π

31π2(2)

A2Π σ21σ

22π

31(58), σ2

1σ2σ5π31(9), σ2

1σ22π

21π2(9) σ2

1σ2σ7π31(3),

4Σ+ σ21σ2π

31π2(50), σ1σ

22π

31π2(23), σ2

1σ2π21π

22(4)

4∆ σ21σ2π

31π2(50), σ1σ

22π

31π2(23), σ2

1σ2π21π

22(4)

B2Σ+ σ21σ2π

31π2(44), σ1σ

22π

31π2(15), σ1σ

22π

41(7), σ2

1σ2π41(5), σ2

1σ2π21π

22(3)

4Π σ21σ

22π

21π2(72), σ2

1σ2σ4π21π2(10)

24Π σ21σ

22π

21π2(35), σ2

1σ2σ4π31(14), σ1σ

22σ4π

31(10), σ2

1σ2σ3π31(8), σ1σ

22σ3π

31(7)

C2Π σ21σ2σ4π

31(23), σ1σ

22σ4π

31(18), σ2

1σ2σ3π31(15), σ1σ

22σ3π

31(11),

σ21σ2σ6π

31(5), σ1σ

22σ6π

31(3), σ2

1σ22π

31(2)

4Σ− σ21σ2π

31π2(43), σ1σ

22π

31π2(25), σ2

1σ22σ3π

21(3), σ2

1σ22σ4π

21(3)

D2Π σ21σ

22π

21π2(62), σ2

1σ2σ4π21π2(8), σ2

1σ2σ3π31(4), σ1σ

22σ3π

31(2)

2Σ− σ21σ

22σ4π

21(39), σ2

1σ22σ3π

21(33), σ2

1σ22σ6π

21(8), σ2

1σ2σ3σ4π21(2), σ2

1σ2σ4σ6π21(2)

6Σ+ σ21σ2π

21π

22(69), σ1σ

22π

21π

22(13), σ2

1σ5π21π

22(2)

2∆ σ21σ2π

31π2(47), σ1σ

22π

31π2(25)

24Σ+ σ21σ2π

31π3(46), σ1σ

22π

31π3(31)

24∆ σ21σ2π

31π3(44), σ1σ

22π

31π3(29), σ2

1σ2π31π2(2)

6Π σ21σ2σ3π

21π2(51), σ1σ

22σ3π

21π2(15), σ2

1σ2σ6π21π2(9), σ2

1σ2σ4π21π2(5)

26Π σ21σ2σ4π

21π2(31), σ2

1σ2σ5π21π2(20), σ2

1σ2σ6π21π2(9), σ2

1σ2σ3π21π2(7),

σ1σ22σ4π

21π2(5), σ1σ

22σ5π

21π2(5), σ1σ

22σ6π

21π2(2), σ1σ

22σ3π

21π2(2)

6∆ σ21σ2π

21π2π3(64), σ1σ

22π

21π2π3(18), σ1σ

22π1π

22π3(2)

6Σ− σ21σ2π

21π2π3(64), σ1σ

22π

21π2π3(18), σ1σ

22π1π

22π3(2)

26Σ+ σ21σ2π

21π2π3(62), σ1σ

22π

21π2π3(16), σ2

1σ2π21π

23(2)

36Π σ21σ2σ5π

21π2(33), σ2

1σ2σ6π21π2(20), σ1σ

22σ5π

21π2(9), σ1σ

22σ6π

21π2(7),

σ21σ2σ3π

21π2(4), σ2

1σ2σ4π21π2(4)

The potential minima of two closely spaced excited quartets, 24Σ+ and 24∆ are located

around 29 900 and 30 700 cm−1, respectively, and their re and ωe values are comparable.

Both the states are characterized by two almost equally dominant configurations, σ21σ2π

31π3

and σ1σ22π

31π3, in which the excited π3 MO is primarily a localized Si(px/y) orbital. At least

seven bound sextets of the SiC− anion have been predicted for the first time in the present

80

study. The lowest sextet, 6Σ+ is mainly characterized by σ21σ2π

21π

22. Its potential energy

curve looks flat in the longer bond length region (Fig. 4.2c). At equilibrium, the bond length

of SiC− in the 6Σ+ state is estimated to be 2.09 A with a vibrational frequency of 530 cm−1.

Spectroscopic features of the remaining sextets lying in the energy range 36 500-47 500 cm−1

are more or less similar. At the longer bond distances, three higher roots of 6Π show avoided

crossings as seen in Fig. 4.2c.


In the presence of the spin-orbit coupling all 14 Λ-S states of SiC+ are allowed to interact.

The lowest dissociation limit, Si+(2Pu)+C(3Pg) splits into six closely spaced asymptotes

with a maximum separation of only 331 cm−1. The effects of the spin-orbit coupling on the

spectroscopic constants are, therefore, expected to be small. Six asymptotes correlate with

27 Ω states of SiC+. Potential energy curves of some of the low-lying 1/2, 3/2, and 5/2

states are shown in Figs. 4.3a-c, while Table 4.10 displays the spectroscopic constants of the

Table 4.10 Spectroscopic constants of low-

lying Ω states of SiC+

State Te(cm−1) re(A) ωe(cm−1)

X4Σ−1/2 0 1.827 825

X4Σ−3/2 0 1.827 8252∆5/2 10 278 1.878 7282∆3/2 10 365 1.876 7202Π1/2 10 681 1.995 4852Π3/2 10 735 1.995 4762Σ−1/2 11 498 1.855 7602Σ+

1/2 13 675 1.911 652

22Π3/2 14 313 1.874 999

22Π1/2 14 343 1.872 10044Π5/2 24 440 1.697 8794Π3/2 24 458 1.698 8754Π1/2(I) 24 478 1.699 8774Π1/2(II) 24 494 1.698 867

81

82

2 3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000(a)

4Π1/2

22Π1/2

2Σ+

1/22Π1/2 2Σ-

1/2

X4Σ-

1/2

Ene

rgy

/ cm

-1

Bond Length / a0

2 3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000

Ene

rgy

/ cm

-1

(b)

4Π3/2

22Π

3/22∆

3/22Π3/2

X4Σ

-3/2

Bond Length / a0

2 3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000

(c)

4∆5/2

4Π5/2

2∆5/2

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 4.3 Ω states of SiC+: for (a) 1/2, (b) 3/2, and (c) 5/2 symmetries

bound states of SiC+. There is no splitting between the components of the ground state,

while the spin-orbit splitting between the components of 2∆ is only 87 cm−1 with no sig-

nificant changes in re and ωe values. The two components of the lowest 2Π state split in

a regular order with a separation of 54 cm−1. The spectroscopic parameters and potential

energy curves of 2Σ−1/2 and 2Σ+1/2 do not change much due to the spin-orbit coupling. The

spin components of the adiabatic 22Π state have similar potential curves with a splitting

of 30 cm−1 only. We have also predicted spectroscopic constants of four spin components

of 4Π with a maximum separation of 54 cm−1. It should be mentioned here that, for the

neutral species also (discussed in chapter 3) the spin-orbit mixing is very poor and almost

insignificant. For this reason no such effort is taken for SiC−.


A. SiC+

Transition probabilities of three electric dipole transitions, 4Π-X4Σ−, 24Π-X4Σ−, and

24Σ−-X4Σ− are large for SiC+. Fig. 4.4a shows the transition moments for these transitions

as a function of the bond length. Table 4.11 reports the computed radiative lifetimes for

these transitions. Because of the avoided crossings, the potential well of the 4Π state holds

only two vibrational levels. The computed radiative lifetime for the 4Π-X4Σ− transition at

the lowest two vibrational levels are 6.06 and 9.72 µs, respectively. The present calculations

predict that the 24Π-X4Σ− transition is more probable and the estimated lifetimes at the

Table 4.11 Radiative lifetime (s) of some excited states of SiC+

Transition Partial lifetime of the upper state ata

υ′=0 υ′=1 υ′=24Π-X4Σ− 6.06(-6) 9.72(-6)

24Π-X4Σ− 110(-9) 114(-9) 117(-9)

24Σ−-X4Σ− 4.31(-6) 1.80(-6)4Π1/2-X4Σ−3/2 0.39(-4) 1.48(-4)4Π1/2-X4Σ−1/2 1.32(-4) 3.94(-4)4Π3/2-X4Σ−1/2 1.29(-4) 3.76(-4)4Π5/2-X4Σ−3/2 1.29(-4) 3.50(-4)

a Values in parenthesis are power to base 10

83

84

2 3 4 5 6 7 8

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8(a)

4Π-X4Σ-

24Π-X4

Σ-

24Σ

--X4Σ

-

Tra

nsi

tio

n M

om

ent

/ ea 0

Bond Length / a0

2 3 4 5 6 7-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2(b)D2Π-X2Σ+

B2Σ+-X2Σ+

B2Σ

+-A2Π

A2Π-X2Σ+

C2Π-X2Σ+

Tran

sitio

n M

omen

t / e

a 0

Bond Length / a0

Fig. 4.4 Computed transition moment functions involving Λ-S states of (a) SiC+ and (b) SiC-

lowest few vibrational levels are of the order of one hundred nanoseconds. The radiative

lifetime for the 24Σ−-X4Σ− transition at v′=0 is 4.31 µs, while it is lowered to 1.8 µs in the

next vibrational level. With the inclusion of the spin-orbit coupling four transitions involving

quartet states have partial radiative lifetimes of the order of a few hundred microseconds

(Table 4.11). All the spin-forbidden transitions such as 22Π1/2-X4Σ−1/2, 22Π3/2-X4Σ−1/2 etc. are

very weak and are not expected to have sufficient intensities to be observed experimentally.

B. SiC−

Five doublet-doublet transitions are found to be significant for the anion. The computed

transition moments for these transitions are shown in Fig. 4.4b as a function of the bond

length. Transition moments of the A2Π-X2Σ+ transition decrease monotonically along the

bond distance, while other transition-moment curves look differently. In Table 4.12, we have

reported the computed radiative lifetimes for these transitions in the lowest four vibrational

levels. The predicted lifetime for the A-X transition is about 63 µs at v′=0, which reduces

further with v′. Transition probabilities of three transitions, B2Σ+-X2Σ+, C2Π-X2Σ+, and

D2Π-X2Σ+ are significantly high. The radiative lifetime for the C-X transition at v′=0 is

less than hundred nanoseconds, while for B-X and D-X, the computed lifetimes are 475

and 173 ns, respectively. However, the contracted CI calculations of Cai and Francois16

predicted somewhat smaller radiative lifetimes for the B2Σ+ state of the anion. The B2Σ+-

A2Π transition is also expected to be highly probable and the calculated lifetime for this

transition at v′=0 is 28 µs compared to the previously estimated value of 10 µs.

Table 4.12 Radiative lifetime (s) of some excited Λ-S states of SiC−

Transition Partial lifetime of the upper state ata

υ′=0 υ′=1 υ′=2 υ′=3

A2Π-X2Σ+ 6.30(-5) 5.10(-5) 4.30(-5) 3.80(-5)

B2Σ+-X2Σ+ 4.75(-7) 4.88(-7) 5.10(-7) 5.37(-7)

[3.06(-7)]b [3.11(-7)]b [3.17(-7)]b [3.24(-7)]b

C2Π-X2Σ+ 8.10(-8) 7.80(-8) 7.50(-8) 7.20(-8)

D2Π-X2Σ+ 1.73(-7) 3.32(-7) 1.75(-6) 2.69(-6)

B2Σ+-A2Π 2.80(-5) 2.70(-5) 2.60(-5) 2.50(-5)

[1.00(-5)]b [1.10(-5)]b [1.60(-5)]b [2.10(-5)]b

a Values in parenthesis are power to base 10, b Ref. 16

85

4.3.4 Dipole moments, ionization potentials and electron affinities

Table 4.13 shows the dipole moments (µe) of the ground and low-lying excited states of

SiC+. While calculating µes, the origin has been kept at the center of mass. The effect

of the spin-orbit coupling on the dipole moments is very small. The variations of dipole

moment as a function of r for some low-lying states of SiC+ are also shown in Fig. 4.5a. The

vertical ionization energies (VIE) are calculated from the differences in the CI energies at

the ground-state bond length of 1.74 A for SiC. The VIE of SiC to the ground state of the

ion is 8.84 eV compared to 8.71 eV computed by Bruna et al.9 The corresponding adiabatic

ionization energy (AIE) is smaller by only 0.08 eV. The appearance potential30 of the SiC+

ion has been found to be in the range (9.0-9.2)±0.4 eV. The computed ionization energies

for ionizations of the ground-state SiC radical to other excited states of the ion are reported

in Table 4.13.

Table 4.13 Ionization energies of SiC to some

low-lying states of SiC+ and their µes

State µe(D)a VIE(eV)b AIE(eV)

X4Σ− 1.19 8.84 8.76

(8.71)c (8.51, 8.62)c

2∆ 0.95 10.20 10.042Σ− 1.45 10.31 10.172Σ+ 0.73 10.70 10.454∆ -0.14 13.33 11.374Σ+ 0.13 13.44 11.42

22Σ− 0.22 13.41 11.704Π 2.05 11.81 11.78

24Σ− 2.03 12.29 12.16

22∆ 0.13 13.55 12.51

22Σ+ 0.02 13.65 12.59

24Π 1.27 13.33 13.11

a Origin at the center of mass

b At re=3.30 a0 of X3Π of SiC

c Ref. 30

86

87

2 3 4 5 6 7 8-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50(a)

22Π

2Π

2Σ

-

2Σ

+

2∆

24Π

4Π

24Σ-

4Σ

+4∆

X4Σ

-

Dip

ole

Mom

ent /

ea 0

Bond Length / a0

2 3 4 5 6 7 8 9 10 11-2

-1

0

1

2

3

4

5(b)

24Π

4Π

6Σ

+

4∆

4Σ+

D2Π

C2ΠA2

Π

B2Σ+

X2Σ

+

Dip

ole

Mom

ent /

ea 0

Bond Length / a0

Fig. 4.5 Computed dipole moment functions of low-lying Λ-S states of (a) SiC+ and (b) SiC-

Dipole moments (µe) of the ground and low-lying states of SiC− are given in Table 4.14.

These have been calculated by keeping the origin at the center of mass. The ground-state

dipole moment of the anion is reported here as 1.95 D, while it has the highest dipole moment

in its 4Σ− state. Fig. 4.5b shows how the dipole moment functions for few low-lying states

of SiC− vary with bond distances. The vertical electron affinities (EAvert) are calculated

from the differences in the CI energies at the ground-state bond length of 1.74 A for SiC at

the same level of calculation. The EAvert of SiC to the ground state of the anion is 2.24 eV,

which is smaller than the MP2 calculated data10 by 0.8 eV. A better agreement is noted

for the corresponding adiabatic electron affinities (EAad). The electron affinities of other

excited states of SiC− are also reported in Table 4.14. Another form of electron affinity may

be defined as the vertical detachment energy (EAV DE) of the anion. EAV DE=E(neutral at

optimized anion geometry)- E(optimized anion). The calculated EAV DE of SiC− is found to

Table 4.14 Electron affinities of SiC to some

low-lying states of SiC− and their µes

State µe(D)a EAad(eV) EAvert(eV)b

X2Σ+ 1.95 2.28 2.24

(2.25)c (2.32)c

(1.98)d

A2Π 2.41 1.79 1.79

(1.71)c (1.73)c

4Σ+ 0.24 0.44 0.354∆ 0.29 -0.11 -0.14

B2Σ+ 1.63 -0.27 -0.274Π 1.41 -0.28 -0.44

24Π 1.86 -0.345 -0.53

C2Π 2.46 -0.49 -0.494Σ− 3.17 -0.58 -0.59

D2Π 2.55 -0.64 -0.84


b At re=3.30 a0 of X3Π of SiC

c Ref. 10, d Ref. 13

88

be about 2.33 eV. The computed electron affinities follow the desired trend: EAvert < EAad

< EAV DE.

4.4. Summary

The electronic states of the SiC+ and SiC− ions have been investigated using ab initio

based MRDCI calculations which include pseudo potentials and the compatible Gaussian

basis functions of both Si and C atoms. There are at least 14 bound states of SiC+ and

21 electronic states of SiC− within 6 eV of energy.32,33

The spectroscopic constants of the ground state (X4Σ−) of SiC+ are improved over the

previous calculations of Bruna et al.9 The computed dissociation energy (D0) of SiC+ is

3.32 eV which matches well with the observed value. The strong interaction between the

lowest two 2Π states has created a shallow asymmetric double well in the potential energy

curve of 2Π. Consequently, the adiabatic potential curve of 22Π has a deeper potential well

with ωe=1013 cm−1. The energy separation between these two roots is about 3600 cm−1. Like

the doublets, the lowest two 4Π states undergo avoided crossing. Their strong interaction has

reduced the binding energy of the 4Π state, while the potential energy curve of 24Π shows

a sharp minimum. The spin-orbit interaction has almost no influence on the spectroscopic

properties of SiC+. The 24Π-X4Σ− transition is predicted to be the strongest transition

with a partial radiative lifetimes of about hundred nanoseconds at the lowest few vibrational

levels. Two other transitions, 4Π-X4Σ− and 24Σ−-X4Σ− are also expected to have sufficient

intensities for experimental observation. Transition dipole moments of none of the spin-

forbidden transitions involving quartet and doublet spin states are significant. All such

transitions are predicted to be very weak. The computed vertical and adiabatic ionization

energies for the ionization to the ground-state SiC+ ion compare well with the available data.

Only three states of the anion, namely X2Σ+, A2Π, and B2Σ+ have been studied before.

The computed ground-state dissociation energy of SiC− is 5.14 eV which compares well with

the previous results. Two new doublet Π states, denoted as C and D, are important from

the spectroscopic point of view. The re and ωe values of these two states differ significantly.

A number of hitherto unknown quartets and sextets of SiC− are predicted. The lowest

bound state of the quartet spin multiplicity is 4Σ+, which lies nearly 15 000 cm−1 above

the ground state. Like the doublets, spectroscopic features of the two closely spaced 4Π and

89

24Π states are quite different. The 4Π state has a longer equilibrium bond distance than the

higher root. Potential energy curve of the lowest sextet, 6Σ+ looks flat in the longer bond

length region. Three bound 6Π states exist within a gap of 11 000 cm−1. A2Π-X2Σ+ and

B2Σ+-A2Π transitions are not predicted to be very strong. Their partial radiative lifetimes

are estimated to be less than hundred microseconds. Three strong transitions, B-X, C-X,

and D-X are expected to be observed in the range 21 000-24 000 cm−1. The C-X and D-X

transitions are predicted for the first time in the present study. The radiative lifetimes for

these two transitions at v′=0 are expected to be around 80 and 170 ns, respectively. Vertical,

and adiabatic electron affinities of SiC to the ground and nine low-lying states of SiC− along

with their dipole moments are also computed.

90

4.5. References

1 A. Suzuki, Prog. Theor. Phys. 62, 936 (1979).

2 J. Cernicharo, C.A. Gottlieb, M. Guelin, P. Thaddeus, J.M. Vrtilek, Astrophys. J. Lett.

341, L25 (1989).

3 B.L. Lutz, J.A. Ryan, Astrophys. J. 194, 753 (1974).

4 P.J. Bruna, S.D. Peyerimhoff, R.J. Buenker, J. Chem. Phys. 72, 5437 (1980).

5 C.M. Rohlfing, R.L. Martin, J. Phys. Chem. 90, 2043 (1986).

6 H. Dohman, P.J. Bruna, S.D. Peyrimhoff, R.J. Buenker, Mol. Phys. 51, 1109 (1984).

7 M. Larsson, J. Phys. B: At. Mol. Phys. 19, L261 (1986).

8 P.F. Bernath, S.A. Rogers, L.C. O′Brien, C.R. Brazier, Phys. Rev. Lett. 60, 197 (1988).

9 P.J. Bruna, C. Petrongolo, R.J. Buenker, S.D. Peyerimhoff, J. Chem. Phys. 74, 4611

(1981).

10 A.I. Boldyrev, J. Simons, V.G. Zahrzewski, W. Von Niessen, J. Phys. Chem. 98, 1427

(1994).


12 M. Grutter, P. Freivogel, J.P. Maier, J. Phys. Chem. 101, 275 (1997).

13 J. Anglada, P.J. Bruna, S.D. Peyerimhoff, R.J. Buenker, J. Phys. B: At. Mol. Phys. 16,

2469 (1983).

14 A.D. McLean, B. Liu, G.S. Chaudler, J. Chem. Phys. 97, 8459 (1992).

15 S. Hunsicker, R.O. Jones, J. Chem. Phys. 105, 5048 (1996).

16 J.-L. Cai, J.P. Francois, J. Phys. Chem. A 103, 1007 (1999).


18 J.M.O. Matos, V. Kello, B.O. Roos, A. J. Sadlej, J. Chem. Phys. 89, 423 (1988).




22 R.J. Buenker, Int. J. Quant. Chem. 29, 435 (1986).


91


24 R.J. Buenker, in: R.Carbo (Ed.), Studies in Physical and Theoretical Chemistry, Current

Aspects of Quantum Chemistry, vol. 21, Elsevier, Amsterdam, p.17, 1982.

25 R.J. Buenker, R.A. Phillips, J. Mol. Struc. (Theochem) 123, 291 (1985).



Reidel, Dordrecht, The Netherlands, 1974.

28 G. Hirsch, P.J. Bruna, S.D. Peyerimhoff, R.J. Buenker, Chem. Phys. lett. 52, 442 (1977).

29 C.E. Moore, Tables of Atomic Energy Levels: vols. I-III, US National Bureau of Standards,

Washington, DC, 1971.

30 J. Drowart, C. De Maria, M.G. Inghram, J. Chem. Phys. 29, 1015 (1958).

31 M.W. Chase Jr., C.A. Davis, J.R. Downey Jr., D.J. Frurip, R.A. McDonald, A.N. Syverud

JANAF Thermochemical Tables, 3rd ed, American Chemical Society and the American

Institute of Physics (for National Bureau of Standards), New York, 1986.

32 A. Pramanik, S. Chakrabarti, K.K. Das, Chem. Phys. Lett. 450, 221 (2008).

33 A. Pramanik, A. Banerjee, K.K. Das, Chem. Phys. Lett. 468, 124 (2009).

92

CHAPTER – 5


SPECTROSCOPIC PROPERTIES OF SnC AND SnC+

5.1. Introduction

Molecules of group 14 elements have drawn a special attention in recent years due to

their possible applications in catalysis, sensor films and new cluster materials.1,2 Most of

the intragroup 14 heteronuclear diatomic molecules such as SiC, GeC, SnC, GeSi etc. were

energetically characterized by using high temperature Knudsen effusion mass spectroscopic

experiments.3,4 Since the simple diatomic molecules are the building unit of cluster materials,

the knowledge of the chemical bonds in different electronic states of the molecules is essential.

In chapter 3 we have introduced the diatomic carbide SiC and in the subsequent chapter,

the spectroscopic properties of its cation and anion have been discussed. In this section, we

report the spectral behavior of SnC and its monopositive ion.

Goodfriend5 used empirical relationships for calculating vibrational constants of diatomic

molecules of group 14 elements. The predicted vibrational constant of SnC was reported

to be 1021 cm−1. The upper limit to the atomization enthalpy for SnC was determined as

452±14 kJ mol−1 from the Knudsen effusion mass spectroscopic method.6 The bond length

of SnC was estimated to be 1.97±0.08 A which is in consistent with the the trends in C2,

SiC and GeC bond lengths. In analogy with the SiC radical, it was assumed that the ground

state of SnC would be 3Π. Schmude Jr. and Gingerich6 also predicted excited states and

respective energies used for SnC as 3Σ (3300 cm−1), 1Σ (7200 cm−1), 1Π (7400 cm−1) and1∆ (7300 cm−1) based on theoretical work of Martin et al.7 for SiC and by Shim et al.8 for

GeC.

The structural parameters, optical constants and enthalpy of formation for cubic GeC and

SnC alloys were computed by Pandey et al.9 using methods based on a generalised gradient

approximation (GGA) to the density functional theory (DFT). Benzair et al.10 reported a

theoretical study of the ground-state and electronic properties of group 14 Zink-blende-like

GeC, SnC, and SiC compounds employing full-potential linearized augmented plane wave

(FP-LAPW) approach within the DFT in the local spin density approximation (LSDA)

including GGA. It was suggested from the distribution of the valence charge density that

the bond in GeC and SnC are more ionic than that in SiC. Another study was carried

out to see the effects of different forms of the correlation energy functional.11 Khenata et

al.12 made a complete analysis of the structural and electronic properties of GeC, SnC and

GeSn using FP-LAPW method. Very recently Li and Wang13 carried out DFT calculations

of the tin-doped carbon clusters SnCn/SnC+n /SnC−n (n=1-10) using B3LYP method with

93

TZP+ basis set. All neutral SnCn (n=1-10) clusters have Sn-terminated linear equilibrium

structures. The ground-state symmetry of SnC is 3Π, while that of other n-odd membered

clusters is 3Σ. Except for SnC2 and SnC10, the ground states of n-even numbered clusters

belong to 1Σ. They also predicted 2Π as the electronic ground state for linear SnC+n and

SnC−n , except for SnC/SnC+/SnC−, SnC2/SnC+2 , SnC+

4 , SnC+6 , and SnC10/SnC+

10. SnC+

has a 4Σ ground state with re=2.115 A as predicted by these authors.13 Another excited 2Π

state was proposed in the same calculation having a relative energy of 25 kJ/mol.

Extensive theoretical and experimental research about the ground and excited states

of homonuclear diatomic molecules of group 14 elements like C2, Si2, Ge2, Sn2, and Pb2

have been carried out successfully in past decades. But similar studies of the intragroup

14 diatomic molecules are not many, which may be due to the experimental difficulties in

isolating such heterodiatomic single molecules in the gas phase. Most of the previous studies

on SnC were performed in solid phase. Large scale MRDCI studies of Pramanik et al.14,15

on SiC and SiC+ have proposed a number of spectroscopically important states as well

as radiative lifetimes in the excited states. Although almost insignificant, they also have

performed spin-orbit interactions among the low-lying states of both the species. No such

experimental/theretical studies have so far been carried out on the SnC or SnC+ in the gas

phase. Spectroscopic constants and potential energy curves of the ground and low-lying

excited states are not known yet.

In this chapter we have presented an extensive theoretical study on SnC and SnC+ using

multireference singles and doubles configuration interaction (MRDCI) calculations. Potential

energy curves of a large number of electronic states have been constructed. Effects of spin-

orbit coupling on the spectroscopic constants of low-lying states are investigated. Transition

probabilities of some electric dipole allowed and spin-forbidden transitions are calculated.

Hence the radiative lifetimes of excited states are estimated from MRDCI wave functions.

At the same time we have computed the vertical and adiabatic ionization potential of SnC.

Dipole moment functions of some few states as well as transition moments involving ground

and low-lying excited states of the neutral and the cationic species are also investigated in

the present studies.

94



The full core RECPs of Sn have been taken from LaJohn et al.16 in which 5s25p2 electrons

of the atom are kept in the valence space, while the remaining inner electrons are described

by means of pseudo potentials. Therefore, the number of active electrons for Sn is 4 only. For

the carbon atom, the RECPs of Pacios and Christiansen17, which include 2s22p2 electrons

in the valence space have been used. The total number of electrons used for generating the

configuration space is 8. The 3s3p4d primitive Gaussian basis set for Sn is taken from LaJohn

et al.16 The first two d functions are contracted with coefficients 0.333845 and 0.474286. The

4s4p basis function of Pacios and Christiansen17 for the carbon atom have been augmented

with two sets of d functions of exponents 1.2 and 0.35 a−20 .


SCF calculations have been carried out for the (σ2σ2π2)3Σ− state of SnC and (σ2σπ2)4Σ−

state of SnC+ at different internuclear distances in the range 3.0-15.0 a0 using the above

mentioned RECPs and basis sets. It generates reasonably good optimized symmetry adopted

molecular orbitals which are used for MRDCI calculations. In both cases the molecule/ion

has been placed along the +z axis with Sn at the origin. The computations are carried out

in the C2v symmetry. The preliminary studies show that the 4d10 electrons of Sn do not

participate in the formation of low-lying electronic states of the molecule like SnC. Therefore,

in the present calculations, 4d10 electrons are kept within the full core RECP. The d electrons

do not affect the structural and spectroscopic features to a large extent.

In the next step, the MRDCI methodology of Buenker and coworkers18−24 have been

employed. For a given spin and spatial symmetry, a set of reference configurations is chosen as

shown in Table 5.2 and 5.2, respectively. All possible single and double excitations generate

a large number of configurations. A configuration-selection threshold of 0.5 µhartree is used

to make the total number of selected configurations below 200 000. The sums of the square

of coefficients of the reference configurations for lowest few roots remain in the range 0.90-

0.94. The details of the CI calculations for SnC and SnC+ are tabulated in Table 5.1 and

Table 5.2, respectively. As mentioned previously, Davidson correction25,26 has been made

and which ultimately leads to estimated full-CI energies of different states of SnC and SnC+.

95

Table 5.1 Details of the configuration interaction calculations of SnC


generated configs. selected configs.§ (%)1A1 106/8 751 622 56 811 911A2 78/8 870 693 57 267 913A1 81/8 1 372 987 69 347 903A2 83/8 1 616 996 73 619 915A1 69/4 1 158 747 40 461 925A2 45/3 658 327 37 393 901B1 59/8 942 678 56 303 90

3B1/3B2 97/8 1 645 731 67 109 905B1/5B2 70/3 1 136 265 33 246 92


Table 5.2 Details of the configuration interaction calculations of SnC+


generated configs. selected configs.§ (%)2A1 138/8 1 056 712 50 248 932A2 105/7 999 569 43 363 944A1 49/3 514 136 30 034 924A2 85/4 699 812 38 361 94

2B1/2B2 175/8 1 311 341 47 475 934B1/4B2 134/3 1 091 606 26 953 92



All the components of low-lying Λ-S states correlating with the ground limit and few

excited dissociation limits are allowed to interact in the spin-orbit CI calculations. The

spin-orbit operators compatible with RECPs are taken from LaJohn et al.16 and Pacios

and Christiansen.17 The Ω components of 0+, 0−, 1, 2, 3, and 4 symmetries belong to A1,

A2, and B1/B2 representations of C22v in case of SnC, while those of 1/2, 3/2, 5/2, and

96

7/2 symmetries belong to E1/E2 representations for SnC+. The dimensions of the secular

equations of A1, A2, and B1 blocks are 49, 48, 48 respectively for some selective number of

roots of Λ-S symmetries of the SnC molecule. On the other hand E1/E2 blocks of SnC+ have

the dimension of 57.

Potential energy curves for both spin-independent and spin-included low-lying states of

SnC/SnC+ are constructed. Spectroscopic constants are then determined by fitting these

potential energy curves. Transition dipole moments for the pair of vibrational functions in a

particular transition are computed. Einstein spontaneous emission coefficients and transition

probabilities are then calculated.



A. SnC

The ground state of both Sn and C belong to 3Pg and 18 Λ-S states of SnC correlate with

them. The interaction between the first excited state (1Dg) of Sn and the ground state of C

results in a set of nine excited triplets. Similarly, the next dissociation limit, Sn(3Pg)+C(1Dg)

which lies 10 000 cm−1 above the lowest limit, correlates with another set of nine triplets of

same symmetry as the previous one. As shown in Table 5.3, a set of fifteen singlets correlates

Table 5.3 Dissociation correlation between the molecular and atomic states of SnC


Sn + C Expt.a Calc.1Σ+(2), 1Σ−, 1Π(2), 1∆, 3Pg + 3Pg 0 03Σ+(2), 3Σ−, 3Π(2), 3∆,5Σ+(2), 5Σ−, 5Π(2), 5∆3Σ+, 3Σ−(2), 3Π(3), 3∆(2), 3Φ 1Dg + 3Pg 5764 73963Σ+, 3Σ−(2), 3Π(3), 3∆(2), 3Φ 3Pg + 1Dg 10 159 11 6663Σ−, 3Π 1Sg + 3Pg 14 3141Σ+(3), 1Σ−(2), 1Π(4), 1∆(3), 1Dg + 1Dg 15 923 19 1451Φ(2), 1Γ

a Ref. 27

97

with the limit at 15 923 cm−1 comprising 1Dg states of both the atoms. The MRDCI

estimated value, however, overestimates it by about 3200 cm−1.

The computed potential energy curves of a large number of triplet, singlet, and quintet

states of SnC without spin-orbit coupling are shown in Figs. 5.1a-c. Spectroscopic constants

of 31 bound states of SnC are given in Table 5.4. Like the two lighter carbides, namely

SiC and GeC, the ground state of SnC is X3Π. The computed ground-state bond length of

the molecule is 2.023 A with a vibrational frequency of 646 cm−1. Although the molecular

constants of the ground state of SnC are not known, the computed equilibrium bond length

follows the desired trend: re(SiC) < re(GeC) < re(SnC). The ground state is dominated by

σ21σ2π

31 (73%) at re. The σ1 MO is a strongly a bonding combination of the s orbitals of both

Sn and C, while σ2 is weakly antibonding and mainly localized on the Sn atom. The π1 MO

is strongly bonding comprising the px/y orbitals of the two atoms. The computed ground-

state dissociation energy of SnC is about 3.06 eV. Neither any experimental study such as

Knudsen effusion mass spectrometric nor any theoretical calculation have been reported so

far for SnC. The present De value is smaller than that of either SiC or GeC which is expected

due to the heavier mass of SnC.

Analogous to SiC and GeC, the first excited state of SnC is predicted to be 3Σ−, which

is designated as A3Σ−. As expected, the state is slightly more stabilized compared to that

of the lighter carbides. The computed transition energy (Te) of the state is 3775 cm−1 and

the potential minimum is shifted towards the longer bond distance by about 0.09 A from

the ground-state re. The predicted vibrational frequency is 590 cm−1. Though not observed

as yet, the A3Σ−←X3Π transition in SnC is predicted to take place around 3800 cm−1.

Comparing the previous results of SiC, it is predicted that the calculated transition energy

of A3Σ− is underestimated by about 500 cm−1. In an analogous situation, the observed

0-0 band of the A-X system in SiC has been found to be near 4500 cm−1. The CI wave

functions show that the A3Σ− state of SnC is characterized mainly by σ21σ

22π

21 with c2=0.84

at re, where the MOs have similar bonding characteristics as the ground state. The state

is reasonably strongly bound with a binding energy of 2.6 eV estimated from the MRDCI

study.

The σ2→π1 excitation in the ground state generates the lowest singlet state belonging

to the 1Σ+ symmetry. The contribution of the closed shell leading configuration, σ21π

41 at

equilibrium is about 65%. We designate the state as a1Σ+ whose estimated transition energy

is 6505 cm−1. Its equilibrium bond length (1.942 A) is shorter than that of the ground state,

98

99

2 3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

50000(a)

43∆

73Π

63Π

33∆43

Σ-

3Pg + 1Dg

1Dg + 3Pg

3Pg + 3Pg

Sn + C

33Σ

-

33Σ

+

53Π

23∆

43Π

23Σ

-

3Φ

33Π23Σ+

C3∆

3Σ+

B3Π

A3Σ

-

X3Π

Ene

rgy

/ cm

-1

Bond Length / a0

2 3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

50000(b)

41∆

41Σ

+

31∆

21∆

1Dg + 1Dg

3Pg + 3Pg

Sn + C

31Σ+31

Π

21Σ

-

1Φ

21Π

1Σ

-

d1Σ

+

c1∆b1

Πa1Σ+

X3Π

En

erg

y / c

m-1

Bond Length / a0

2 3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

50000

(c)

3Pg + 3P

g

Sn + C

25Σ

-

5∆

25Σ

+

5Σ

- 25Π

5Σ

+

5Π

X3Π

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 5.1 Λ-S states of SnC: for (a) triplet, (b) singlet, and (c) quintet spin

multiplicities

while their ωe values are comparable. The σ21σ2π

31 configuration, which is the dominant one

for the ground state, also generates the singlet counterpart, b1Π. The calculated binding

energy of the b1Π state is about 2.18 eV. Its equilibrium bond length is only 0.01 A longer

than the ground-state re, while its ωe is comparable.

Table 5.4 Spectroscopic constants of low-lying Λ-S states of SnC


X3Π 0 2.023 646 2.44

A3Σ− 3775 2.117 590 2.97

a1Σ+ 6505 1.942 635 2.78

b1Π 7100 2.032 635 2.48

c1∆ 7705(8394)a 2.158(2.124)a 545 2.475Π 9620 2.241 475 2.13

d1Σ+ 11 560 2.092 710 2.401Σ− 17 275 2.446 365 1.55

B3Π 17 315 2.250 450 1.81

C3∆ 18 315 2.443 405 1.603Σ+ 18 530[18 460]b 2.717[2.434]b

23Σ+ 20 865 2.325 485 1.44

33Π 21 130 2.278 422 2.25

21Π 21 824 2.432 274 1.863Φ 23 174 2.282 438 2.00

23Σ− 26 064 2.515 315 2.95

43Π 27 654 2.332 376 1.78

23∆ 27 970 2.178 480 1.991Φ 28 223 2.275 452 2.53

33Σ+ 32 845 1.992 1012 2.16

53Π 33 010 2.097 652 3.15

21Σ− 33 050 2.132 537 2.74

31Π 33 070 2.352 400 2.34

33Σ− 33 138 2.221 520 2.15

31Σ+ 35 937 2.016 550 2.72

100



31∆ 38 478 2.285 576 3.44

25Σ+ 40 639 2.208 556 1.46

41Σ+ 41 225 2.426 390 3.87

41∆ 43 051 2.660 693 -0.435∆ 43 529 2.000 651 2.34

25Σ− 48 191 2.000 659 2.46

a Ref. 13, b Second minimum

Besides A3Σ−, the σ21σ

22π

21 configuration generates two strongly bound singlets, c1∆ and

d1Σ+. Although none of these states of SnC is experimentally known yet, we have designated

them according to the labels given in SiC.14 The c1∆ state is energetically more stable than

d1Σ+. The estimated Te of c1∆ is about 7705 cm−1, which is somewhat smaller than the value

predicted by Li and Wang13 from B3LYP/TZVP+ calculations. The re and ωe of the state

predicted here are 2.158 A and 545 cm−1, respectively. The previously calculated re of the 1∆

state of SnC was reported to be 2.124 A. The d1Σ+ state, which lies 11 560 cm−1 above the

ground state, is strongly bound. Analyzing the CI wave functions at different bond distances

it is found that the d1Σ+ state strongly interacts with the lower state, a1Σ+. At equilibrium,

there is at least 19% contribution of the closed shell configuration σ21π

41 (Table 5.5). As a

result of the strong interaction, the vibrational frequency of d1Σ+ is predicted to be larger

than that of both the ground and a1Σ+ state. The estimated equilibrium bond length in the

d1Σ+ state is 2.092 A.

The lowest quintet state, 5Π originates from a σ21σ2π

21π2(83%) configuration in which π2

mainly consists of antibonding combination of px/y orbitals of both the atoms. The computed

transition energy of this strongly bound state is about 9620 cm−1 with ωe=475 cm−1 and

re=2.241 A. The binding energy of 5Π is predicted to be 1.84 eV. Although such a low-lying

quintet state may not have much influence on the spectroscopy of the molecule, the spin

components may result in many spin-forbidden transitions. The next important state of

SnC is B3Π which is the second root of the ground-state symmetry. It is predominantly

described by σ21σ2π

21π2, the same one that generates the lowest 5Π state. It may be noted

that this configuration yields another eight states of Π and Φ symmetries. The B3Π-X3Π

101

transition is predicted to take place around 17 300 cm−1. The computed re of the B3Π state is

at least 0.2 A longer than the ground-state equilibrium bond length. As a result, the Franck-

Condon overlap factor for the B-X transition should be much less than what is expected.

The fitted vibrational frequency of the B3Π state is 450 cm−1. In an analogous situation

such a transition is experimentally observed around 21 915 cm−1 for the SiC molecule. In

the present calculations, we have predicted 1Σ− and 3∆ states with their potential minima

located almost at the same bond length around 2.445 A. The former is more stable than

the latter by about 1040 cm−1. Potential energy curves of both the states are shallow

compared to those of other low-lying states. However, a weak 3∆-X3Π transition may take

place around 18 315 cm−1. Though not observed, the 3∆ state is labeled as C because it

lies next to B3Π. 1Σ− and C3∆ states arise from the configurations, σ21σ

22π1π2(77%) and

σ21σ

22π

21(76%), respectively.

Table 5.5 Composition of Λ-S states of SnC at equilibrium bond length


X3Π σ21σ2π

31(73), σ2

1σ2π21π2(5), σ2

1σ2π1π22(5), σ1σ

22π

31(3)

A3Σ− σ21σ

22π

21(84), σ2

1σ22π

22(2)

a1Σ+ σ21π

41(65), σ2

1σ22π

21(7), σ2

1π21π

22(6), σ1σ2π

31π2(4)

b1Π σ21σ2π

31(78), σ2

1σ2π21π2(5), σ2

1σ2π1π22(2)

c1∆ σ21σ

22π

21(79), σ2

1σ22π

22(5)

5Π σ21σ2π

21π2(83)

d1Σ+ σ21σ

22π

21(52), σ2

1π41(19), σ2

1π21π

22(5), σ2

1σ22π

22(3), σ2

2π41(2)

B3Π σ21σ2π

21π2(77), σ2

1σ2π31(6)

1Σ− σ21σ

22π1π2(77), σ2

1σ2σ3π1π2(7)

C3∆ σ21σ

22π

21(76), σ2

1σ2σ3π1π2(7)3Σ+ σ2

1π31π2(63), σ2

1σ22π1π2(8), σ1σ2π

41(5), σ2

1π1π32(4), σ1σ2π

21π

22(3)

23Σ+ σ21π

31π2(41), σ2

1σ22π1π2(29), σ2

1π1π32(7), σ2

1π21π

22(4)

33Π σ21σ2π

21π2(82)

21Π σ21σ2π

21π2(77), σ2

1σ2π1π22(3)

3Φ σ21σ2π

21π2(84)

23Σ− σ21σ

22π1π2(42), σ2

1σ22π

21(21), σ2

1σ2σ3π21(8), σ2

1π31π2(6), σ2

1σ2σ3π1π2(5)

43Π σ21σ2π

21π2(80)

102



23∆ σ21π

31π2(74), σ2

1σ22π1π2(5), σ2

1π1π32(3)

1Φ σ21σ2π

21π2(83)

33Σ+ σ1σ2π41(48), σ2

1π31π2(17), σ1σ2π

31π2(7), σ2

1σ22π1π2(5), σ1σ2π

21π

22(5)

21Σ− σ21π

31π2(80)

53Π σ21σ2π

21π2(41), σ1σ

22π

31(34), σ2

1σ2π32(2), σ1σ

22π

21π2(6)

31Π σ21σ2π

21π2(80)

33Σ− σ21π

31π2(62), σ2

1σ22π1π2(12), σ2

1σ2σ3π21(5)

31Σ+ σ1σ2π41(24), σ2

1π31π2(16), σ2

1π21π

22(13), σ1σ2π

31π2(10), σ2

1π41(5), σ1σ2π

21π

22(3)

31∆ σ21π

31π2(37), σ2

1σ22π1π2(34), σ2

1σ2σ3π21(8), σ2

1σ22π

22(2)

25Σ+ σ1σ2π31π2(53), σ2

1π21π

22(19), σ1σ2π1π

32(5), σ1σ2π

21π

22(5)

41Σ+ σ21σ

22π1π2(43), σ2

1σ22π

21(26), σ2

1σ2σ3π21(10)

41∆ σ21π

31π2(52), σ2

1π21π

22(17), σ2

1σ22π

21(8), σ2

1σ22π1π2(2)

5∆ σ1σ2π31π2(80), σ1σ2π

21π

22(4), σ1σ2π1π

32(3)

25Σ− σ21σ2σ3π

21(77), σ2

1σ2σ3π1π2(3), σ21σ2σ6π

21(3)

The potential energy curve of the 3Σ+ state of SnC shows a very shallow double minima

at 4.0 and 4.6 a0 (Fig. 5.1a) as a result of an avoided crossing between two close-lying roots

of 3Σ+ symmetry. The barrier height is estimated to be only 200 cm−1. Transition energies

at the two minima are located at 18 530 and 18 460 cm−1, respectively. The CI coefficients

show that the state at the shorter-distant minimum is characterized by σ21π

31π2, while at

the longer distant minimum, it is described by σ21σ

22π1π2. In Table 5.5, we have only given

the detailed composition of the 3Σ+ state at the short distant minimum. A minimum is

created in the potential energy curve of the second root of 3Σ+ due to avoided crossing. It

lies 20 865 cm−1 above the ground state. We have designated it as 23Σ+ and the fitted re is

about 2.325 A with ωe=485 cm−1. As expected, the composition of 23Σ+ at the potential

minimum consists of two important configurations, namely σ21π

31π2 and σ2

1σ22π1π2. A second

avoided crossing takes place between the second and third root of 3Σ+ in the range 3.4-3.8 a0.

The potential minimum in the adiabatic curve of 33Σ+ is predicted to be at 1.992 A, where

σ1σ2π41 and σ2

1π31π2 configurations dominate. The estimated transition energy of the 33Σ+

is 32 845 cm−1 with a large vibrational frequency of 1012 cm−1. The 33Σ+ state potential

103

curve predissociates into Sn(1Dg)+C(3Pg) through a barrier of about 0.85 eV as a result of

an avoided crossing with a repulsive state.

Three excited 3Π states, 33Π, 43Π and 53Π are found to be weakly bound and dissociate

into Sn(1Dg)+C(3Pg). They originate mainly from the π1→π2 excitation of the ground state.

At equilibrium, there is a strong contribution of the σ1σ22π

31 configuration in the 53Π state.

On the contrary, 33Π and 43Π states are relatively pure. Fig. 5.1a shows a low barrier of

only 0.2 eV in the potential energy curve of 53Π. Transitions from these excited 3Π states

to the ground state are expected to take place in the range 21 000-33 000 cm−1, though

none of them is experimentally observed. The higher excited state, 53Π has bond length and

vibrational frequency, comparable to those of the ground state.

The 21Π state, which lies just above 33Π, is also characterized predominantly by σ21σ2π

21π2.

The state is weakly bound with a much longer equilibrium bond length. Both 3Φ and 1Φ

state of SnC are bound and originate from σ21σ2π

21π2 configuration. The computed re and ωe

of these two states are very similar. However, the 3Φ state dissociates into Sn(1Dg)+C(3Pg),

while the singlet counterpart correlates with a higher asymptote. The triplet state is more

stable than the singlet one by about 5050 cm−1. The potential minima of the second and

third root of 3Σ− are separated by 7075 cm−1. The lower state, 23Σ− has a longer bond

length of 2.515 A, while for the higher state the Sn-C bond is relatively short (re=2.221 A).

Fig. 5.1a shows that the predissociation of the state to Sn(1Dg)+C(3Pg) may take place with

an estimated barrier of 0.4 eV. This is due to another curve crossing with a repulsive state

of the same symmetry.

The lowest 5Σ+ state is not bound and dissociates into the lowest limit. The nature of

the potential energy curves of 5Σ+ and its higher root, 25Σ+ show a very strong mixing.

As a result, a potential well is created at r=2.21 A in the potential curve of 25Σ+ with a

small barrier. It predissociates rapidly into Sn(3Pg)+C(3Pg). Two configuration, namely

σ1σ2π31π2 and σ2

1π21π

22 mainly dominate in the lowest two 5Σ+ states. The composition of

25Σ+ at 2.208 A is given in Table 5.5. Two other quintets, 5∆ and 25Σ− have potential

minima around 2.0 A having almost same vibrational frequencies. As shown in Fig. 5.1c,

the 5∆ state predissociates almost in the similar way as in 25Σ+, but with a larger barrier.

Of the remaining excited triplets, 23∆ is bound with a transition energy of 27 970 cm−1 at

equilibrium.

At least six more excited singlets which dissociate into Sn(1Dg)+C(1Dg) are bound. The

104

predicted transition energy of 21Σ− is 33 050 cm−1 at re=2.132 A. Within 45 000 cm−1

of energy, 31Π, 31Σ+, 31∆, 41Σ+, and 41∆ states of SnC are reported to be bound. As a

result the strong interaction between 21∆ and its higher root, the adiabatic potential curve

of the lowest one is flat in the bond length region 4.0-4.8 a0. This is confirmed from the

compositions of the two states in this region. A minimum appears in the potential energy

curve of 31∆ at 2.285 A. Two configurations, namely σ21π

31π2 and σ2

1σ22π1π2 are dominant in

characterizing 21∆ and 31∆ states of SnC. The minimum in the potential curve of 41∆ is

also due to an avoided crossing with its lower root. The 31Π state, which originates from

σ21σ2π

21π2 is strongly bound having a binding energy of 1.33 eV. The computed vibrational

frequency of the state at equilibrium is 400 cm−1. Two excited weakly bound 1Σ+ states are

also reported in the present study. The spectroscopic properties of these two states differ

largely.

B. SnC+

First dissociation limit of SnC+, Sn+(2Pu)+C(3Pg) correlates with a set of six doublets

and six quartets. Like SiC+ 14, there are nine doublets of Σ+(2), Σ−, Π(3), ∆(2), and Φ

symmetries, all of which dissociate into the second asymptote lying 10 159 cm−1 above the

first one. The computed value of 10 660 cm−1 matches nicely with the observed one.27 Only

two doublets correlate with the third dissociation limit, Sn+(2Pu)+C(1Sg) at 21 614 cm−1

as shown in Table 5.6. The 5Su state of C combines with the ground state of Sn+ (2Pu)

to generate two quartets and two sextets which belong to Σ− and Π symmetries. We have

computed the quartets only which dissociate at 33 701 cm−1. The potential energy curves

Table 5.6 Dissociation correlation between molecular and atomic states of SnC+


Expt.a Calc.2Σ+, 2Σ−(2), 2Π(2), 2∆, Sn+(2Pu) + C(3Pg) 0 04Σ+, 4Σ−(2), 4Π(2), 4∆2Σ+(2), 2Σ−, 2Π(3), 2∆(2), 2Φ Sn+(2Pu) + C(1Dg) 10 159 10 6602Σ+, 2Π Sn+(2Pu) + C(1Sg) 21 614 22 6874Σ−, 4Π, 6Σ−, 6Π Sn+(2Pu) + C(5Su) 33 701 32 428

a Averaged over J, Ref. 27

105

106

2 3 4 5 6 7 8 9 10 11 12

0

10000

20000

30000

40000

50000

60000(a)

Sn+(2Pu) + C(5Su)

Sn+(2Pu) + C(3Pg)

34Π

34Σ-

24Π

24Σ -4Π

4Σ+

4∆

X4Σ -

En

erg

y / c

m-1

Bond Length / a0

2 3 4 5 6 7 8 9 10 11 12

0

10000

20000

30000

40000

50000

60000(b)

52Π

42∆

Sn+(2Pu) + C( 1S

g)

Sn+(2Pu) + C( 1D

g)

Sn+(2Pu) + C(3P

g)

42Σ+

2Φ32∆

32Σ+

32Σ-

22Σ +

22∆42Π

32Π22Σ-

22Π

2Σ+

2Σ- 2∆2Π

X4Σ-

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 5.2 Λ-S states of SnC+: for (a) quartet and (b) doublet spin multiplicities

are plotted in Figs. 5.2a, b, while the spectroscopic constants of the bound states are tabu-

lated in Table 5.7.


Λ-S states of SnC+


X4Σ− 0 2.112 591

2.115a

2Π 8965 2.330 423

8744a

2∆ 10 032 2.173 4852Σ− 11 459 2.156 4952Σ+ 12 797 2.233 410

22Π 14 184 2.132 6454∆ 16 366 2.773 2164Σ+ 16 622 2.795 207

22Σ− 19 546 2.664 269

32Π 24 714 2.889 251

22∆ 25 864 2.733 271

22Σ+ 26 723 2.738 258

32Σ− 27 551 3.011 226

42Π 29 941 2.488 226

24Π 35 330 2.032 785

34Σ− 39 614 2.345 395

42∆ 49 440 2.384 391

a Ref. 13

Like the lighter homologue, the ground state of SnC+ has a 4Σ− symmetry with an

equilibrium bond length of 2.112 A. The ionization involves the removal of a π1 bonding

electron from SnC. Our calculated re agrees well to the earlier value calculated by Li et

al.13 The estimated vibrational frequency of the ground state is 591 cm−1. The σ21σ2π

21

configuration dominates throughout the potential energy curve of the ground state, where

107

σ1 is mostly antibonding comprising s orbitals of Sn and C, σ2 is a bonding MO involving s

and pz orbitals of the constituting atoms, and π1 is weakly bonding, centering on C atom.

As a periodic trend the dissociation energy of the ground state is reduced from 3.32 eV for

SiC+ to 2.47 eV for SnC+.

Two strongly interacting 2Π states create an adiabatic potential minimum at 2.33 A with

a transition energy of 8965 cm−1. At equilibrium, it has been characterized by σ21σ

22π1(79%)

with 4% contribution from σ21π

31. The equilibrium vibrational frequency of the state is

computed here as 423 cm−1. The second 2Π state has a very sharp potential well with

re=2.132 A and ωe=645 cm−1. It is situated at 5219 cm−1 above 2Π. The composition of

22Π is multiconfigurational in nature, σ21π

31 being the highest contributing configuration with

c2=0.27. As shown in Table 5.8, σ21σ

22π1 also makes a large contribution (c2=0.24) to it.

Table 5.8 Composition of Λ-S states of SnC+ at equilibrium bond length


X4Σ− σ21σ2π

21(84)

2Π σ21σ

22π1(79), σ2

1π31(4)

2∆ σ21σ2π

21(69), σ2

1σ2π1π2(12)), σ1σ22π

21(3), σ1σ

22π1π2(3)

2Σ− σ21σ2π

21(71), σ2

1σ2π1π2(8), σ1σ22π

21(5), σ1σ

22π1π2(3)

2Σ+ σ21σ2π

21(58), σ2

1σ2π1π2(19), σ1σ22π1π2(4), σ1σ

22π

21(3)

22Π σ21π

31(27), σ2

1σ22π1(24), σ2

1π21π2(13), σ1σ2π

21π2(11), σ1σ2π

31(10)

4∆ σ21σ2π1π2(58), σ2

1σ3π1π2(28)4Σ+ σ2

1σ2π1π2(58), σ21σ3π1π2(28)

22Σ− σ21σ2π1π2(47), σ2

1σ2π21(24), σ2

1σ3π1π2(8), σ1σ22π1π2(4)

32Π σ21σ2σ3π2(38), σ2

1σ22π2(16), σ2

1σ23π2(14), σ2

1π21π2(10),

σ21σ2σ3π1(2), σ2

1σ22π1(2)

22∆ σ21σ2π1π2(55), σ2

1σ3π1π2(14), σ21σ2π

21(13), σ1σ

22π1π2(4)

22Σ+ σ21σ2π1π2(60), σ2

1σ3π1π2(15), σ21σ2π

21(6), σ1σ

22π1π2(5)

32Σ− σ21σ3π1π2(40), σ2

1σ2π1π2(36), σ21σ2π

21(6), σ2

1σ3π21(4)

42Π σ21π

21π2(52), σ1σ2π

21π2(16), σ2

1σ22π2(9), σ2

1π31(6)

24Π σ1σ2π31(37), σ1σ2π

21π2(31), σ2

1π21π2(16)

34Σ− σ1σ22π

21(32), σ2

1σ3π21(26), σ1σ2σ3π

21(12), σ1σ

22π

21(8)

42∆ σ1σ22π

21(26), σ1σ

22π1π2(19), σ2

1σ3π21(15), σ1σ2σ3π

21(9),

σ21σ2π1π2(6), σ1σ2σ3π1π2(4)

108

The generating configuration of the ground state, σ21σ2π

21 also gives rise to a set of three

doublets namely, 2∆, 2Σ−, and 2Σ+ having bond length of about 2.20 A. The estimated ωe

of these states vary between 410 and 495 cm−1. However, these are spectroscopically less

important as the relative population in the lowest doublet (2Π) is expectedly low. But after

including the spin-orbit coupling, the corresponding Ω components may be of great interest,

from the spectroscopic point of view. One more 2Σ− state having very shallow potential

well holding three vibrational levels also dissociates into the first dissociation limit. In the

longer bond length region (> 5 a0), it strongly interacts with the lower 2Σ−. Te of the state

is estimated to be 19 546 cm−1.

A set of closely lying 4∆ and 4Σ+ arises at around 16 500 cm−1 with considerably large

bond distances of 2.773 and 2.795 A, respectively. Their potential energy curves look flat-

tened with small binding energy of 0.44 eV only. Both the states are generated from an open

shell σ2π1π2 configuration. However, due to symmetry forbidness they are spectroscopically

unimportant. Unlike SiC+, the next 4Π and 24Σ− states of SnC+ are repulsive in nature.

It may be mentioned here that SiC+ showed two transitions namely, 4Π–X4Σ− and 24Σ−–

X4Σ− with band origins at 24 464 and 27 447 cm−1, respectively. Although no experimental

data is available, they have been predicted to have high intensities. In case of SnC+, both

the transitions are absent. The 24Π–X4Σ− transition is expected for SnC+ also, with an

excitation energy of 35 330 cm−1. The computed lifetime in the ground vibrational level of

24Π is reported to be 84 ns. This state strongly couples with another root of 4Π which is

strongly repulsive. Thus, the 24Π state predissociates through this repulsive channel crossing

a potential barrier of 0.41 eV.

All the bound doublets dissociating into second asymptote have bond length ≥2.73 A and

their vibrational frequencies are within 275 cm−1. The only exception is for 42Π. A relatively

shorter bond length (2.488 A) is due to the avoided crossing between the third and fourth

roots of 2Π. The state is about 21 000 cm−1 above the lowest doublet, 2Π. At equilibrium,

42Π is dominated by σ21π

21π2 (52%), while the composition of 32Π is complicated. The

potential energy curves of 32Σ+, 32∆, 2Φ, and 52Π are repulsive in nature. We also predict

two more excited bound states, 34Σ− and 42∆ at 39 614 and 49 440 cm−1, respectively. The

first one dissociates into Sn+(2Pu)+C(5Su), while the dissociation of 42∆ is not confirmed.

The equilibrium composition of 34Σ− is multiconfigurational in nature (Table 5.8), however,

the absorption may be confirmed by the experiment at 252 nm.

109


A. SnC

The ground-state limit, Sn(3Pg)+C(3Pg) splits into nine closely spaced levels under the

influence of the spin-orbit coupling. The observed atomic spectral data shows a largest

splitting of 3471 cm−1. 50 Ω states of 0+, 0−, 1, 2, 3, and 4 symmetries, which originate

from 18 Λ-S states, correlate with these nine asymptotes as shown in Table 5.9. Potential

energy curves of these Ω state of SnC are shown in Figs. 5.3a-d. Since one of the two atoms

Table 5.9 Dissociation correlation between Ω and atomic states of SnC

Ω States† Atomic states Relative energy / cm−1

Sn + C Expt.a Cal.

0+ 3P0 + 3P0 0 0

0−, 1 3P0 + 3P1 16 18

0+, 1, 2 3P0 + 3P2 43 52

0−, 1 3P1 + 3P0 1692 1104

0+(2), 0−, 1(2), 2 3P1 + 3P1 1708 1122

0+, 0−(2), 1(3), 2(2), 3 3P1 + 3P2 1735 1170

0+, 1, 2 3P2 + 3P0 3428 2969

0+, 0−(2), 1(3), 2(2), 3 3P2 + 3P1 3444 2993

0+(3), 0−(2), 1(4), 2(3), 3(2), 4 3P2 + 3P2 3471 3034

a Moore’s Table, Ref. 27


is relatively heavy, we expect a considerable spin-orbit effects on the spectroscopic features

of the molecule. The ground state of SnC splits in an inverted order with an overall splitting

of 1063 cm−1. The spectroscopic constants of the lowest component, X3Π2 change only

marginally. Table 5.10 shows the computed spectroscopic constants and compositions of the

low-lying Ω states up to 25 000 cm−1 of energy. The 3Π2 and 3Π0− components remain almost

pure X3Π, while some mixing with other states are noted for 3Π1 and X3Π0+ . Although the

computed spin-orbit splitting of the two components of A3Σ− is about 50 cm−1, they play an

important role in influencing some of the nearby states. There is a substantial contribution

of the X3Π0+ component in A3Σ−0+ state. Transition energies of b1Π1 and c1∆2 components

110

111

2 3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000(a)

3P2+3P

0,1,2

3P1+3P

1,2

3P0+3P

0,2

Sn + C

33Π0+

B3Π0+

d1Σ+

0+5Π

0+

a1Σ

+

0+

A3Σ

-

0+

X3Π0+X3Π

2

E

nerg

y / c

m-1

Bond Length / a0

2 3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000

0-(V)

(b)

23Σ

+

0-

33Π

0-

B3Π0-

3P2 + 3P

1,2

3P1 + 3P0,1,2

Sn + C

3P0 + 3P1

1Σ-

0-

5Π0-

X3Π0-X3Π

2

En

erg

y / c

m-1

Bond Length / a0

2 3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000(c)

3Φ3

C3∆3

5Π3

3P2 + 3P0,1,2

3P1 + 3P

0,1,2

3P0 + 3P

1,2

Sn + C

5Π1

5Π-1

b1Π1

A3Σ-

1

X3Π1X3Π

2

Ene

rgy

/ cm

-1

Bond Length / a0

2 3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000(d)

3Φ4

3P2 + 3P

0,1,2

3P1 + 3P1,2

3P0 + 3P

2

Sn + C3Φ

2

33Π2

C3∆2

B3Π2

5Π2

c1∆2

X3Π2

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 5.3 Ω states of SnC: for (a) 0+, (b) 0-, (c) 1, 3, & (d) 2, 4 symmetries

Table 5.10 Spectroscopic constants and composition of low-lying Ω states of SnC

State Te/cm−1 re/A ωe/cm−1 Contribution of Λ-S states / (%)

X3Π2 0 2.025 644 X3Π(99), c1∆(1)

X3Π1 469 2.027 637 X3Π(97), A3Σ−(2), b1Π(1)

X3Π0+ 808 2.027 626 X3Π(93), A3Σ−(5), 5Π(1)

X3Π0− 1063 2.019 662 X3Π(99)

A3Σ−1 4299 2.117 588 A3Σ−(92), X3Π(4), 5Π(2), b1Π(1)

A3Σ−0+ 4347 2.112 604 A3Σ−(88), X3Π(7), a1Σ+(3), 5Π(1)

a1Σ+0+ 7306 1.934 671 a1Σ+(94), A3Σ−(4), X3Π(1)

b1Π1 7778 2.037 631 b1Π(97), A3Σ−(2), X3Π(1)

c1∆2 8248 2.159 543 c1∆(98), X3Π(1)5Π−1 9455 2.238 475 5Π(56), b1Π(39), A3Σ−(4)5Π0− 9838 2.239 476 5Π(98), X3Π(2)5Π0+ 9983 2.235 472 5Π(93), A3Σ−(4), a1Σ+(3)5Π1 10 353 2.236 476 5Π(96), X3Π(1), b1Π(1), A3Σ−(1)5Π2 10 763 2.238 473 5Π(98), X3Π(1)5Π3 11 183 2.240 469 5Π(99)

d1Σ+0+ 12 353 2.092 716 d1Σ+(97), X3Π(2), A3Σ−(1)

1Σ−0− 17 155 2.431 317 1Σ−(72), 3Σ+(20), X3Π(3), B3Π(5)

B3Π2 17 518 2.244 447 B3Π(98), c1∆(1), 5Π(1)

B3Π0+ 18 423 2.246 432 B3Π(98), d1Σ+(1)

B3Π0− 18 713 2.319 365 B3Π(77), 3Σ+(18), 1Σ−(5)

C3∆2 19 083 2.442 406 C3∆(94), c1∆(2), B3Π(2), 3Φ(1)

0−(V) 19 660 2.350 330 3Σ+(66), 1Σ−(26), 23Σ+(4), B3Π(3)

C3∆3 20 032 2.453 395 C3∆(98), 3Φ(1), 5Π(1)

33Π0+ 20 935 2.276 431 33Π(98)

33Π0− 20 953 2.284 435 33Π(95), 1Σ−(2), 3Σ+(1)

23Σ+0− 21 848 2.331 486 23Σ+(87), B3Π(8), 1Σ−(4)

33Π2 22 518 2.273 405 33Π(98), 5Σ+(1), C3∆(1)3Φ2 23 023 2.282 437 3Φ(95), C3∆(3), c1∆(1)3Φ3 23 703 2.286 431 3Φ(94), 1Φ(3), C3∆(2)3Φ4 24 704 2.281 423 3Φ(100)

112

are increased by 500-700 cm−1 due to the spin-orbit coupling, while other spectroscopic

constants remain unchanged.

Six spin components of 5Π split in a regular order with 5Π−1 lying at the lowest. The

overall spin-orbit splitting is about 1728 cm−1. However, the equilibrium bond lengths and

vibrational frequencies of these components vary only slightly. The wave function of 5Π−1

at equilibrium shows about 39% contribution from 1Π (Table 5.10). Potential energy curves

of 5Π−1 and 5Π1 components undergo avoided curve crossings with that of b1Π1 due to their

larger equilibrium bond lengths. Except for 5Π0− , the spectroscopic parameters of other five

components of 5Π (reported in Table 5.10) are obtained from their respective diabatic curves.

The transition energy of d1Σ+0+ is increased by about 800 cm−1 due to the spin-orbit coupling.

Among the four components of the excited B3Π state, B3Π2 and B3Π0+ components have

distinct potential minima, while other two components involve several avoided crossings

making them difficult to fit for the calculations of the spectroscopic constants. However, the

calculations show that the spin-orbit splitting of the components of B3Π takes place in the

inverted order. The magnitude of splitting between the 0+ and 2 components of it is about

900 cm−1. The changes in the other spectroscopic constants are not significant. It should be

mentioned here that, while fitting the diabatic curve of B3Π0− , which has contribution from

B3Π by more than 75% at re (=2.319 A), gives an estimated value of Te=18 713 cm−1. The

relatively longer bond length of it is due to significant mixing with the similar components

of 3Σ+ and 1Σ−. Thus the overall splitting among the components of B3Π exceeds 1000

cm−1. Unlike B3Π, the spin components of the next higher root split in a regular order. The

potential energy curves of 33Π0+ , 33Π0− , and 33Π2 are fitted adiabatically. As the remaining

component of 1 undergoes strong spin mixing with the nearby components, the minimum

of it could not be located. The present results show that the separation between 0+ and

2 components is about 1585 cm−1. Like all other spin mixed states, re and ωe of these

components do not differ much. Spectroscopic constants of C3∆2, C3∆3, 3Φ2, 3Φ3, and 3Φ4

components are also reported in Table 5.10.

B. SnC+

After spin-orbit coupling the ground state dissociation limit splits into six sublevels. As

the spin-orbit splitting of 3P state of C is only 43 cm−1, the spin-orbit effect due to C is

negligible. Thus, the overall splitting among the Ω components of the SnC+ ion (4294 cm−1)

resembles the separation between 2P1/2 and 2P3/2 states of Sn+(Fig. 5.4). However, the

113

114

2 3 4 5 6 7 8 9 10 11 12

0

5000

10000

15000

20000

25000

30000

Sn+ + C

(a)

4∆

3/2

4Σ+

3/2

2P

3/2 +

3P

0, 1, 2

2P1/2

+ 3P1, 2

22Π3/2

2∆

3/22Π

3/2

X4Σ

-

3/2

Ene

rgy

/ cm

-1

Bond Length / a0

2 4 6 8 10 12

0

5000

10000

15000

20000

25000

30000

Sn+ + C

(b)

4Π1/2

22Σ

-

1/2

4Π1/2

4Σ+1/2

2P3/2

+ 3P0, 1, 2

2P1/2

+ 3P0, 1, 2

22Π

1/2

2Σ

+

1/2

2Σ

-

1/22Π

1/2

X4Σ-1/2

X4Σ -3/2

Ene

rgy

/ cm

-1

Bond Length / a0

2 3 4 5 6 7 8 9 10 11 12

0

5000

10000

15000

20000

25000

30000

Sn+ + C

(c)

2P3/2

+ 3P1, 2

2P

1/2 +

3P

0, 2

4∆

7/2

4∆5/2

2∆5/2

X4Σ

-

3/2

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 5.4 Ω states of SnC+: for (a) 3/2, (b) 1/2, and (c) 5/2 symmetries

calculated value of 4230 cm−1 (Table 5.11) compares it almost accurately. The ground state

splits into two almost inseparable Ω components, X4Σ−1/2 and X4Σ−3/2 with a separation of

only 21 cm−1 with similar re and ωe values. Two components of 2Π are shifted downwards

by 375 and 31 cm−1, respectively. The spin-orbit interaction does not change their re and ωe

much. 4∆ contributes to both of them by 1-2%. 2Π3/2 mixes with the similar component of2∆ by an amount of 8%. However, the spin-orbit coupling allows the 2Π–X4Σ− transition to

take place through their respective dipolar components. The predicted lifetime in the υ′=0

state of 2Π3/2 is of the order of 400 µs.

Table 5.11 Dissociation correlation between Ω and atomic states of SnC+

Ω states Atomic states Relative energy / cm−1

Sn+ + C Expt.a Cal.

1/2 2P1/2+3P0 0 0

1/2(2), 3/2 2P1/2+3P1 16 25

1/2(2), 3/2(2), 5/2 2P1/2+3P2 43 55

1/2, 3/2 2P3/2+3P0 4251 4185

1/2(3), 3/2(2), 5/2 2P3/2+3P1 4267 4205

1/2(4), 3/2(3), 5/2(2), 7/2 2P3/2+3P2 4294 4230


Ω=3/2 and 5/2 components of 2∆ split in an inverted order with a separation of 363 cm−1.

The re of 2∆5/2 is increased by 0.018 A, while its ωe is reduced by 20 cm−1. The contribution

of 4∆ in the 2∆5/2 state is 1% only. On the other hand, 2∆3/2 is strongly perturbed by the

component of 2Π. Its equilibrium bond length and vibrational frequency are changed only

marginally. Spin-orbit interactions do not change the spectroscopic constants of 2Σ−1/2 by

a considerable amount, only Te is increased by 134 cm−1. But in the longer bond length

region it is strongly perturbed by 4∆1/2. The next root of Ω=1/2 is dominated by 2Σ+. The

state is shifted upward by 237 cm−1, while re and ωe remain almost unchanged. However,

it interacts with the component of 2Σ− to a large extent and dissociates into the third

asymptote (Fig. 5.4b). Unlike 2Π, 22Π splits in an inverted pattern placing 22Π3/2 at least

400 cm−1 below 22Π1/2. Both the states mix to some extent with the similar components

of 4∆, 4Π, and 4Σ+. The extent of mixing is maximum at around 4.6 a0 which leads the

dissociation of both the states following complicated adiabatic pathway. We have fitted

115

the smooth adiabatic curves near equilibrium and the results are tabulated in Table 5.12.

The spin-orbit effect slightly increases the re of 22Π3/2, while it decreases the same for the

other component, 22Π1/2. However, for both the states the ωe values do not change in a

considerable amount.

Table 5.12 Spectroscopic constants and composition of low-lying Ω states of SnC+


X4Σ−3/2 0 2.111 587 X4Σ−(99), 2Π(1)

X4Σ−1/2 21 2.113 588 X4Σ−(99)2Π1/2 8590 2.333 421 2Π(96), 2Σ+(2), 4∆(1)2Π3/2 8934 2.327 393 2Π(89), 2∆(8), 4∆(1), 4Σ−(1)2∆5/2 10 075 2.188 465 2∆(95), 4∆(1)2∆3/2 10 438 2.185 473 2∆(73), 2Π(23), 22Π(4)2Σ−1/2 11 325 2.161 474 2Σ−(93), 2Σ+(4), 2Π(2)2Σ+

1/2 13 034 2.237 423 2Σ+(90), 2Σ−(5), a2Π(3), 2Π(1)

22Π3/2 14 166 2.148 609 22Π(92), 2∆(5), 4Π(1), X4Σ−(1)

22Π1/2 14 608 2.115 634 22Π(94), 2Σ−(3), 2Σ+(2), 4∆(1)4∆3/2 16 545 2.649 365 4∆(73), 2∆(15)4Σ+

3/2 16 858 2.744 247 4Σ+(71), 4∆(17), 2Π(5), 2∆(1)4Σ+

1/2 16 870 2.645 275 4Σ+(62), 22Σ−(13), 2Σ−(10), 22Π(6), 4Π(5)4∆5/2 17 340 2.691 285 4∆(88), 2∆(11)4∆7/2 17 914 2.765 213 4∆(100)4Π1/2(I) 18 711 3.029 150 4Π(33), 4Σ+(23), 2Σ−(22), 24Σ−(9), 22Σ−(6),

X4Σ−(3), 2Σ+(1)

22Σ−1/2 20 440 2.610 290 22Σ−(38), 4Π(34), 22Π(13), 24Σ−(10), 22Σ+(1)4Π1/2(II) 21 298 2.649 185 4Π(65), 22Π(13), 24Σ−(10), 22Σ−(8)

The overall splitting of 4∆ into its omega components is in regular order, but the Ω=1/2

component of it suffers many avoided crossings and hence its minimum could not be located.

However, the components are separated by more than 1200 cm−1. The diabatic curve of4∆3/2 shows relatively high ωe of 365 cm−1 compared to the other components. 2∆ makes a

large contribution to it. The transition energy of 4∆3/2 is estimated to be 16 545 cm−1 with

an equilibrium bond length of 2.649 A. The vibrational frequency of the 4∆5/2 component

116

is also somewhat larger while the re is shortened by 0.08 A because of spin-orbit mixing.

The 4∆7/2 remains almost pure with almost no changes in the spectroscopic constants. Two

components of 4Σ+ are almost inseparable. Both the components are energetically shifted

upward by more than 200 cm−1. Their potential minima are separated by 0.099 A, both of

which are lower than that of 4Σ+, however. 4∆ makes a large contribution (17%) to 4Σ+3/2,

while the components of lowest two roots of 2Σ− contribute to 4Σ+1/2 by more than 20%. The

potential minima of three more components with Ω=1/2 are located at 18 711, 20 440, and

21 298 cm−1, respectively. Two of them are dominated by 4Π. It may be mentioned here that

in the absence of any spin-orbit coupling 4Π was purely repulsive (see Fig. 5.2a). For example,

the eighth root of 1/2 is mainly characterized by 4Π(33%), 4Σ+(23%), and 2Σ−(22%) at

equilibrium. The shallow potential well has a characteristic frequency of 150 cm−1 only. A

large bond distance of 3.029 A makes the state spectroscopically insignificant. The next

root of 1/2 is characterized by 22Σ− (38%) with more than 30% contribution from 4Π. The

highest contribution of 4Π (c2=0.65) is noted at the tenth root of 1/2. The minimum of the

state is located at 2.649 A with a vibrational frequency of 185 cm−1. The above two states

are also spectroscopically unimportant due to very low Franck-Condon overlap factors.


A. SnC

As the ground state of SnC belongs to 3Π, there are several dipole allowed triplet-triplet

transitions. In addition, three transitions involving low-lying singlets are also studied here.

Transition probabilities of these transitions are calculated from transition moment data.

Fig. 5.5a shows transition moment curves of all nine transitions as a function of the bond

distance. For all the curves, transition moments smoothly decrease to zero at the longer

Sn-C bond distance. The partial radiative lifetimes at the lower few vibrational levels for

all nine transitions are tabulated in Table 5.13. As seen in Fig. 5.5a, transition moment of

A-X transition monotonically decreases with the increase in bond length. Its value around

the equilibrium bond length of the upper state is nearly 0.25 ea0. The computed lifetime

for the A-X transition at υ′=0 is about 220 µs, which decreases with υ′ due to longer re of

the A3Σ− compared to that of the ground state. Although transition dipole moments and

the energy differences for the B3Π-X3Π transition are sufficiently large, the Franck-Condon

overlap factors are small. As a result, the B-X transition is expected to be weak. The partial

lifetime at v′=0 for this transition is 6.5 ms. Four transitions from the excited triplets,

117

118

2 3 4 5 6 7 8 9 10-0.60

-0.45

-0.30

-0.15

0.00

0.15

0.30

0.45

0.60

0.75(a)

31Σ+- b1Π

31Σ+- a1Σ+

31Σ+- d1Σ+

33Σ+- X3Π

23∆- X3Π

53Π - X3Π

33Π - X3Π

B3Π- X3Π

A3Σ -- X3Π

Tran

sitio

n M

omen

t / e

a 0

Bond Length / a0

2 3 4 5 6 7 8 9 10 11-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50(b)

24Π - X4Σ-

42Π - 2Π

34Σ

- - X4Σ

-

42∆ - 2

∆

Tran

sitio

n M

omen

t / e

a 0

Bond Length / a0

Fig. 5.5 Computed transition moment functions involving Λ-S states of (a) SnC and (b) SnC+

Table 5.13 Radiative lifetime (s) of some of the excited states of SnC

Transition Partial lifetimes of the upper state ata Total lifetime

υ′=0 υ′=1 υ′=2 υ′=3 at υ′=0

A3Σ−–X3Π 2.20(-4) 1.90(-4) 1.70(-4) 1.45(-4)

B3Π–X3Π 6.50(-3) 5.42(-4) 1.35(-4) 1.01(-4)

33Π–X3Π 3.70(-5) 1.30(-5) 8.50(-6) 6.90(-6)

53Π–X3Π 6.57(-7) 8.11(-7) 2.24(-5)

23∆–X3Π 1.85(-6) 2.09(-6) 3.13(-6) 3.85(-6)

33Σ+–X3Π 1.27(-7) 1.35(-7) 1.72(-7) 2.10(-7)

31Σ+–a1Σ+ 2.15(-6) 1.81(-6) 1.36(-6) 6.50(-7)

31Σ+–b1Π 3.40(-5) 1.90(-5) 1.30(-5) 8.76(-6)

31Σ+–d1Σ+ 2.46(-6) 1.35(-6) 8.11(-7) 5.50(-7) τ31Σ+=1.11(-6)

(A3Σ−0+–X3Π0+)‖ 7.67(-3) 6.75(-3) 5.95(-3) 5.27(-3)

(A3Σ−0+–X3Π1)⊥ 1.97(-4) 1.71(-4) 1.52(-4) 1.40(-4) τA3Σ−0+

=1.92(-4)

(A3Σ−1 –X3Π1)‖ 1.11(-2) 9.75(-3) 8.60(-3) 7.72(-3)

(A3Σ−1 –X3Π0+)⊥ 5.45(-4) 4.50(-4) 3.83(-4) 3.33(-4)

(A3Σ−1 –X3Π0−)⊥ 8.35(-4) 6.44(-4) 5.31(-4) 4.57(-4)

(A3Σ−1 –X3Π2)⊥ 3.00(-4) 2.65(-4) 2.36(-4) 2.16(-4) τA3Σ−1=1.54(-4)

(a1Σ+0+–X3Π0+)‖ 1.65(-3) 1.60(-3) 1.89(-3) 2.36(-3)

(b1Π1–X3Π1)‖ 4.97(-2) 7.00(-2)

(b1Π1–X3Π0+)⊥ 3.78(-3) 4.45(-3) 6.43(-3)

(b1Π1–X3Π0−)⊥ 2.13(-3) 1.62(-3)

(b1Π1–X3Π2)⊥ 5.70(-4) 5.00(-4) τb1Π1=3.98(-4)

(c1∆2–X3Π2)‖ 2.47(-2) 8.93(-3) 5.53(-3) 4.00(-3)

(5Π0+–X3Π0+)‖ 1.80(-3) 1.37(-3) 1.23(-3) 1.03(-3)

(5Π0−–X3Π0−)‖ 2.42(-3) 2.03(-3) 1.80(-3) 1.56(-3)

(5Π2–X3Π2)‖ 1.56(-3) 1.30(-3) 1.15(-3) 1.04(-3)

(d1Σ+0+–X3Π0+)‖ 8.86(-4) 7.53(-4) 6.49(-4) 5.83(-4)


namely 33Π, 53Π, 23∆, and 33Σ+ to the ground state are predicted to have larger transition

probabilities. The 33Σ+-X3Π transition, which should take place around 32 845 cm−1, may

119

be quite strong. The estimated radiative lifetime for this transition at the lowest vibrational

level is about 127 ns which increases with υ′. The 53Π-X3Π transition is found to be more

probable than the transition from either B3Π or 33Π state. The lifetime of the 23∆-X3Π

transition is of the order of microsecond. Transition probabilities of three singlet-singlet

transitions involving the excited 31Σ+ state are studied here. The computed transition

dipole moment functions of 31Σ+-a1Σ+, 31Σ+-d1Σ+, and 31Σ+-b1Π transitions are similar

in nature. First two transitions are more probable than the third one. The partial lifetime

for these three transitions are estimated to be 2.15, 2.46, and 34 µs, respectively at v′=0.

Summing up the transition probabilities, the total radiative lifetime of 31Σ+ is found to be

1.11 µs at the lowest vibrational level.

Transition probabilities of many transitions involving the X3Π2, X3Π1, and X3Π0+ com-

ponents of SnC are calculated. The A3Σ−1 -X3Π0+ transition is predicted to have larger

transition probabilities than A3Σ−0+-X3Π0+ . The computed partial lifetime for 0+-0+ tran-

sition is 7.67 ms, while for the 1-0+ transition it is less than a millisecond. The lifetimes

of A3Σ−1 –X3Π0− and A3Σ−1 –X3Π2 transitions at υ′=0 are found to be 0.84 and 0.30 ms,

respectively. Several weak spin-forbidden transitions are also reported in Table 5.13. At

the lowest vibrational level, the computed partial lifetimes of a1Σ+0+ and d1Σ+

0+ are 1.65

and 0.89 ms, respectively. Another important transition b1Π1-X3Π0+ is expected to carry

sufficient intensity. However, b1Π1-X3Π1 and b1Π1-X3Π2 transitions are predicted to have

larger transition probabilities. The spin-forbidden transitions originating from different spin

components of 5Π are weak and the partial radiative lifetimes of some of these transitions

are listed in Table 5.13.

B. SnC+

As mentioned earlier, only two spin allowed transitions from the ground state of SnC+

are expected to occur. The transition moment functions for both 24Π–X4Σ− and 34Σ−–

X4Σ− decrease sharply with the bond distance in the Franck-Condon region. The MRDCI

calculations predict a lifetime of 84 ns in the υ′=0 level of 24Π. The lifetime suddenly

increases for the next vibrational level as shown in Table 5.14. 34Σ−–X4Σ− transition is also

expected to have sufficient intensity. The radiative lifetime of the upper state (34Σ−) is 256

ns in its lowest vibrational level. It is much lower (111 ns) in its υ′=1 level. However, the

lifetime value increases for the next vibrational level possibly due to the avoided crossing

interaction between 34Σ− and its neighboring root. Two doublet-doublet transitions namely,

120

42Π–2Π and 42∆–2∆ also have comparatively high transition moments as shown in Fig. 5.5b.

Although the lowest doublet (2Π) is situated above the ground state by more than 1 eV,

transitions from these two states to 2Π and 2∆ are expected to occur due to large energy

gaps and high Franck-Condon factors. The total radiative lifetime in the ground vibrational

level of 42∆ is about 135 ns. 42Π state has a radiative lifetime of the order of 2-3 µs.

Table 5.14 Radiative lifetime (s) of some excited states of SnC+

Transition Partial lifetime of the upper state ata Total lifetime

υ′=0 υ′=1 υ′=2 υ′=3 at υ′=0

24Π–X4Σ− 8.37(-8) 2.07(-7) 2.15(-7) 5.48(-7)

34Σ−–X4Σ− 2.56(-7) 1.11(-7) 9.23(-7) 8.43(-7)

42Π–2Π 2.52(-6) 2.10(-6) 2.26(-6) 2.25(-6)

42∆–2∆ 2.14(-7) 9.66(-7) 8.10(-7) 8.00(-7)

42∆–2Π 3.63(-7) 3.86(-7) 4.06(-7) 4.21(-7) τ42∆=1.35(-7)

(2Π3/2–X4Σ−3/2)‖ 4.16(-4) 4.72(-4) 5.43(-4) 6.31(-4)

(2∆3/2–X4Σ−3/2)‖ 1.14(-3) 1.26(-3) 1.49(-3) 3.41(-3)

(2∆3/2–X4Σ−1/2)⊥ 6.01(-3) 6.02(-3) 6.25(-3) 7.28(-3) τ2∆3/2=9.58(-4)

(2∆5/2–X4Σ−3/2)⊥ 1.92(-3) 1.94(-3) 1.97(-3) 1.99(-3)

(2Σ−1/2–X4Σ−1/2)‖ 3.01(-3) 1.65(-3) 1.95(-3) 2.11(-3)

(2Σ−1/2–X4Σ−3/2)⊥ 4.32(-3) 3.79(-3) 3.84(-3) 5.30(-3) τ2Σ−1/2

=1.77(-3)

(2Σ+1/2–X4Σ−1/2)⊥ 4.68(-5) 4.77(-5) 6.59(-5) 1.34(-4)

(22Π1/2–X4Σ−1/2)‖ 8.51(-4) 7.80(-4) 8.42(-4)

(22Π3/2–X4Σ−3/2)‖ 1.72(-2) 2.83(-2) 4.20(-2) 4.65(-2)

(22Π3/2–X4Σ−1/2)⊥ 1.02(-3) 1.38(-3) 4.05(-3) τ22Π3/2=9.63(-4)


Many spin forbidden transitions through the respective dipolar components are reported

in Table 5.14. Their transition moment values are within 0.16 ea0. The parallel component

of 2Σ+1/2–X4Σ−1/2 has the highest transition moment value of 0.158 ea0 at 4.95 a0. Due to low

Franck-Condon overlap factor the computed lifetime is of the order of one tenth of a second.

Inspite of lower transition moment lifetime of the corresponding perpendicular transition is

around 50 µs at the lower vibrational level. Transitions involving the components of 2∆,

121

2Π, 2Σ− and the corresponding ground-state component are studied here. Of these, the

spin forbidden transitions like, (2Π3/2–X4Σ−3/2)‖ is of great interest. The computed radiative

lifetime of the 2Π3/2 state at υ′=0 is 416 µs. The value increases monotonically with the

increase in the vibrational quantum number. The 22Π1/2 state has a lifetime of the order of

850 µs for the parallel component of the 22Π1/2–X4Σ−1/2 transition. The transition lifetime

for the Ω=3/2 component of 22Π is much higher (963 µs).

5.3.4 Dipole moments and ionization energies

The computed dipole moment of SnC is reported to be 2.44 D in its ground state. It

is much higher in A3Σ− with the same sense of polarity (Sn+C−). Table 5.4 shows the

equilibrium dipole moments (µe) of other excited states. The Spin-orbit coupling decreases

the ground-state dipole moment by 0.02 D for X3Π2 component. But it increases to some

extent for other three components as shown in Table 5.16. Both the components of A3Σ−

have slightly lesser dipole moment than the originating states. The effect is insignificant for

the next three states, but in case of 5Π−1 it is more prominent and µe is increased by 0.22 D.

The variation of dipole moment functions for some low-lying states of SnC are shown in

Fig. 5.6a. All the curves smoothly converge to zero dipole moment at longer bond distances.

The ground-state dipole moment of SnC+ is computed here as -0.847 D. As the dipole

moment is origin dependent, the origin is kept at the center of mass of SnC+. However, the

variation of dipole moments with the bond distance for some low-lying Λ-S states of the ion

are shown in Fig. 5.6b. The values shown in Fig. 5.6b are obtained keeping Sn at the origin.

Dipole moment functions of most of the states pass through maxima. The 4∆ and 4Σ+

states of SnC+ have very high maximum values of 1.237 and 1.232 ea0, respectively, both

at 3.9 a0. In Table 5.15 we have tabulated the equilibrium dipole moments of few low-lying

states following the same convention as that of the ground state. The ground state and a

few low-lying doublets have opposite sense of polarity. Spin-orbit effect does not change the

ground-state dipole moment to a large amount, but it affects those of the excited states.2Π1/2 has at least 0.10 D less dipole moment than its originating Λ-S state. The effect is

more prominent for the two spin components of 2∆. Comparatively large dipole moment of2∆3/2 may be due to mixing of it with the similar component of 2Π.

We have calculated the vertical ionization energies (VIE) of SnC from the difference in

estimated CI energies of the ground-state neutral molecule and the cation at the ground

state as well as some low-lying excited states. The estimated CI energies are taken from

122

123

2 3 4 5 6 7-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

B3Π

(a)

b 1Π

1Σ -

d 1Σ+

a1Σ+

c1

∆

23Σ

+

3Σ

+

3∆

A3Σ

-

5ΠX

3Π

Dip

ole

Mom

ent /

ea 0

Bond Length / a0

3 4 5 6 7 8

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3 (b)

22Π

2Π

2Σ

-

2∆

22Σ

+24Π

4∆

4Σ

+

X4Σ-

Dip

ole

Mom

ent /

ea 0

Bond Length / a0

Fig. 5.6 Computed dipole moment functions involving Λ-S states of (a) SnC and (b) SnC+

Table 5.15 Ionization energies of SnC to some

low-lying states of SnC+ and their µes


X4Σ− -0.847 7.70 7.662Π -0.496 9.08 8.772∆ -0.061 9.01 8.902Σ− -0.758 9.16 9.082Σ+ 0.439 9.43 9.24

22Π -0.711 9.51 9.424∆ 1.384 11.14 9.694Σ+ 1.392 11.22 9.72

22Σ− 0.588 11.42 10.08

32Π 2.147 11.65 10.73

22∆ 1.189 12.21 10.86

22Σ+ 1.533 12.31 10.97

32Σ− 0.976 13.23 11.07

42Π 0.699 12.92 11.37

24Π -0.699 12.04 12.04

34Σ− -0.336 13.17 12.56

42∆ 0.194 14.39 13.79


b At re=3.85 a0 of X3Π of SnC

the same level of MRDCI calculation at the ground-state equilibrium bond distance of SnC

(3.85 a0). Table 5.15 predicts that, minimum 7.70 eV energy is required to ionize the SnC

molecule. However, spin-orbit correction demands the requirement of somewhat greater

amount of energy (7.77 eV). Ionization of SnC to the first excited state of SnC+ (2Π) requires

9.08 eV of energy. As there are four more doublets in the close vicinity, the photoionization

of SnC within 9.51 eV may lead different photoionized products. Table 5.15 also shows the

adiabatic ionization energies of SnC, but no experimental data are available for comparison.

It should be noted here that, as ionizations of SnC lead to longer bond length, the adiabatic

ionization energies are always less than the corresponding vertical ionization energies. In

124

a general trend, the ionization energies of SnC are increased to some extent by spin-orbit

interactions which indicates that, the neutral molecule is energetically more stabilized by

spin-orbit coupling. Table 5.15 and Table 5.16 compare the ionization energies of SnC

without and with spin-orbit corrections, respectively.

Table 5.16 Spin-orbit corrected dipole moments and ionization energies of SnC

Molecule State µe(D) Ion State µe(D)a VIE(eV)b AIE(eV)

SnC X3Π2 2.42 SnC+ X4Σ−3/2 -0.843 7.767 7.724

X3Π1 2.45 X4Σ−1/2 -0.840 7.769 7.727

X3Π0+ 2.49 2Π1/2 -0.395 9.143 8.789

X3Π0− 2.45 2Π3/2 -0.451 9.068 8.832

A3Σ−1 2.93 2∆5/2 -0.029 9.104 8.973

A3Σ−0+ 2.88 2∆3/2 -0.263 9.143 9.018

a1Σ+0+ 2.78 2Σ−1/2 -0.690 9.210 9.128

b1Π1 2.49 2Σ+1/2 0.311 9.539 9.340

c1∆2 2.47 22Π3/2 -0.637 9.609 9.4815Π−1 2.35 22Π1/2 -0.693 9.596 9.530


b At re=3.85 a0 of X3Π2 state of SnC

5.4. Summary

MRDCI studies, which include RECPs and SO coupling, reveal the existence of a large

number of low-lying electronic states of SnC, none of which is observed yet. The ground

state of SnC belongs to the X3Π symmetry with re=2.023 A and ωe=646 cm−1. The dipole

moment of SnC in the ground state at re is about 2.44 D with a polarity of Sn+C−. It

is affected marginally by spin-orbit interactions. The dissociation energy of the molecule

is estimated to be 3.06 eV, which is smaller than that of either SiC or GeC. The ground

state of SnC is characterized by an open shell configuration, σ21σ2π

31. At least 30 excited

singlet, triplet and quintet states of Λ-S symmetries have been predicted within 6 eV of

energy. A weak transition, A3Σ−-X3Π is predicted around 3775 cm−1. Three triplet-triplet

transitions such as B3Π-X3Π, 33Π-X3Π, and 53Π-X3Π are expected to be strong. The lowest

125

quintet state, 5Π is strongly bound. A strong mixing between the lowest two 3Σ+ states has

created a double minima in the potential energy curve of the lowest root. The extent of the

spin-orbit coupling is significantly large due to the presence of the heavier Sn atom. The

largest splitting among the components of X3Π, which split in an inverted order, is more

than 1000 cm−1. However, the spin-orbit splitting between 0+ and 1 components of A3Σ− is

expected to be small. Many spin-forbidden transitions such as a1Σ+0+–X3Π0+ , d1Σ+

0+–X3Π0+ ,

b1Π1–X3Π0+ etc. are studied, the partial lifetime of which are in the millisecond order and

they are predicted to be weak.

The removal of a π electron from SnC generates the cation SnC+ which has 4Σ− ground-

state like the other homologous cations like SiC+, GeC+. It requires at least 7.77 eV of energy.

The computed dissociation energy of the ground-state cation is 2.47 eV which follows the

trend SiC+> GeC+> SnC+. The ground-state bond length and vibrational frequency of

SnC+ are estimated to be 2.112 A and 591 cm−1, respectively which are not much affected

by spin-orbit effects. Two important transitions from the ground state are expected to be

occurred within 4.91 eV. Excited 24Π and 34Σ− states have the radiative lifetimes of 84

and 256 ns, respectively at υ′=0. Total radiative lifetime of the lowest vibrational state

of 42∆ is also of the order of hundred nanosecond. Several spin-forbidden transitions are

also expected for SnC+. The perpendicular component of the 2Σ+1/2–X4Σ−1/2 transition has

a lifetime of 50 µs in the lowest few vibrational levels. The ground-state dipole moment of

the ion is computed here as -0.847 D.

126

5.5. References

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J. Menendez and T.S.T. Tsong, Appl. Phys. Lett. 89, 231924 (2006).

3 G. Gigli, G.M. Eloni, M. Carrozzino, J. Chem. Phys. 122, 14303 (2005).

4 A. Ciccioli, G. Gigli, G. Meloni, E. Testani, J. Chem. Phys. 127, 54303 (2007).

5 P.L. Goodfriend, Canad. J. Phys. 45, 3425 (1967).

6 R.W. Schmude Jr., K.A. Gingerich, J. Chem. Phys. 109, 3069 (1998).

7 J.M.L. Martin, J.P. Francois, R. Gijbels, J. Chem. Phys. 92, 6655 (1990).

8 I. Shim, M. Sai Baba, K.A. Gingerich, J. Phys. Chem. 102, 10763 (1998).

9 R. Pandey, M. Rerat, C, Darrigan, M. Causa, J. Appl. Phys. 88, 6462 (2000).

10 A. Benzair, B. Bouhafs, C. Mathien, H. Aourag, Phys. Lett. A 282, 299 (2001).

11 A. Benzair, H. Aourag, Phys. Stat. Solidi B 231, 411 (2002).

12 R. Khenata, H. Baltache, M. Sahnun, M. Driz, M. Rerat, B. Abbar, Physica B 336, 321

(2003).

13 G. Li, C. Wang, J. Mol. Struct. (THEOCHEM) 824, 48 (2007).


15 A. Pramanik, S. Chakrabarti, K.K. Das, Chem. Phys. Lett. 450, 221 (2008).

16 L.A. LaJohn, P.A. Christiansen, R.B. Ross, T. Atastroo, W.C. Ermler, J. Chem. Phys.

87, 2812 (1987).








127

23 R.J. Buenker, in Studies in Physical and Theoretical Chemistry, R. Carbo, Ed.; Elsevier:

Amsterdam, The Natherlands, Vol. 21 (Current Aspects of Quantum Chemistry), 1981.

24 R.J. Buenker, R.A. Phillips, J. Mol. Struct. (THEOCHEM) 123, 291 (1985).


Reidel, Dordrecht, The Netherlands, 1974.

26 G. Hirsch, P.J. Bruna, S.D. Peyreimhoff, R.J. Buenker, Chem. Phys. Lett. 52, 442 (1977).

27 C.E. Moore, Tables of Atomic Energy Levels, vols. I-III, US National Bureau of Standards,

Washington, DC, 1971.

28 Electronic Spectrum of SnC: A Theoretical Study,

A. Pramanik, K.K. Das (communicated).

29 Theoretical Investigation of Electronic States of SnC+,

A. Pramanik, K.K. Das (to be communicated).

128

CHAPTER – 6


SPECTROSCOPIC PROPERTIES OF PbC AND PbC+

6.1. Introduction

Energetic information of chemical bond involving permutation of all elements in the en-

tire periodic table have been collected over the years.1 Besides their applications, the simple

diatomic molecules draw special interest in contributing the information about their bond

length, bond energy etc. As mentioned in the earlier chapters, intragroup 14 heteronuclear

diatomic molecules have generated a special interest in recent years because of their possible

applications in catalysis, sensor films and mostly, they are the building blocks of cluster

materials.2−4 Seven out of ten intragroup 14 diatomics have been energetically character-

ized by Knudesen effusion mass spectroscopic technique.5,6 Ciccioli et al.6 have investigated

the thermodynamic properties of diatomics containing lead. They have also predicted the

unknown dissociation energies of SiSn and PbC.

The number of electrons and the relativistic effect are quite high, so theoretical calcula-

tions on the molecules containing lead are very difficult task. RECP based DFT studies on

the lead-doped carbon clusters, PbCn/PbC+n /PbC−n (n=1-10) have been carried out using

B3LYP method with both CEP-31G and TZP+ basis sets by Li et al.7 Their studies include

the structure, stability, ionization potentials (IPs), electron affinities (EAs), and fragmenta-

tion energies of the PbCn/PbC+n /PbC−n (n=1-10) clusters. They predicted the ground state

of neutral PbC molecule as 3Π with an equilibrium bond length of 2.063 A. Two excited

states, 5Π and 1∆ at the energies 21.6 and 22.9 kcal/mole, respectively are also reported.

However, Ciccioli et al.6 have performed back to back electronic structure calculations in

the CCSD(T) level of theory together with small core relativistic pseudopotential (aug-cc-

pVTZ) on intragroup 14 diatomics along with PbC to determine the molecular constants

(re, ωe), necessary for their data analysis. They also computed adiabatic ionization energies

(AIEs), adiabatic electron affinities (AEAs), term energies, and dissociation energies (D00).

According to their prediction, the D00 value for PbC is about 1.93 eV. The 4H0 value for

the reaction PbC(g) = Pb(g) + C(g) is 248.1 kJmole−1 as predicted by Ciccioli et al.6

Contradictorily, they predicted the ground state of PbC as X3Σ− with re and ωe, 2.191 A

and 565 cm−1, respectively.

On the other hand, ionization of the neutral carbide leads to the 4Σ− ground-state of the

ionic species, as predicted in a recent theoretical study of SiC+.13 The B3LYP/DFT studies7

also report the ground state of PbC+ as 4Σ with ...π2 valence electronic configuration. Its

equilibrium bond length is predicted to be 2.179 A which is longer than that of the neutral

129

molecule by 0.096 A. They have also predicted an excited 2∆ state with re=2.191 A for

PbC+.

The diatomic carbides of third to fifth row elements are extensively studied by many

authors.8−12 But similar results of PbC are not available in literature. This is for the first

time, we have carried out a large scale MRDCI study using RECP to study electronic

structure and spectroscopic properties of the ground as well as low-lying excited states of

PbC and PbC+ within 6 eV of energy. Spin-orbit interaction has also been incorporated to

show the changes in the potential energy curves and spectroscopic parameters in comparison

to their corresponding Λ-S states. Transition properties of some excited states with different

spin multiplicities have been computed. Their radiative lifetimes have also been predicted by

calculating the Einstein’s spontaneous emission coefficients. An attempt has been made to

determine the dipole moment of the neutral as well as cationic species. Ionization energies

for the ionization to the low-lying states of the cation from the ground state have been

calculated.



In the present study, the RECPs of Ross et al.14 replace the inner electrons of Pb, leaving

the remaining 6s26p2 electrons in the valence space. The valence (3s3p4d) Gaussian basis set

from the same reference14 is used for the calculation. For C atom, the RECPs of Pacios and

Christiansen15, which retain the outer 2s22p2 electrons in the valence space, are employed.

The optimized (4s4p) Gaussian basis set compatible with the above mentioned RECPs is

augmented with two sets of d functions of exponents 1.2 and 0.35 a−20 , respectively. The

total number of active electrons in the valence space is 8.


All the calculations are performed in the C2v subgroup of the main group C∞v, placing Pb

at origin and choosing z axis as the molecular axis. SCF calculations for the 3Σ−(σ21σ

22π

21)

state of PbC and 2Σ+(σ21σ2π

21) state of PbC+ are performed at different internuclear distances

between 3 to 15 a0. This generates 64 symmetry adapted molecular orbitals. Out of these

64 MOs, 2 MOs of very high energy are discarded, while the remaining 62 MOs are treated

130

Table 6.1 Details of the configuration interaction calculations of PbC


generated configs. selected configs.§ (%)1A1 77/4 924 542 33 942 921A2 68/3 988 646 32 301 913A1 121/8 2 595 174 95 042 913A2 131/8 3 011 089 98 771 935A1 98/4 2 194 717 53 630 925A2 81/3 1 432 123 49 471 901B1 130/3 1 582 196 33 855 93

3B1/3B2 203/8 4 011 903 77 444 935B1/5B2 91/3 1 716 528 42 572 92


Table 6.2 Details of the configuration interaction calculations of PbC+


generated configs. selected configs.§ (%)2A1 86/7 875 393 47 519 922A2 61/6 912 974 37 371 914A1 60/3 802 940 36 986 914A2 79/4 997 050 41 078 92

2B1/2B2 108/8 1 211 008 59 662 924B1/4B2 115/3 1 181 565 36 167 92


as basis for the consequent CI calculations. The MRDCI methodology of Buenker and co-

workers16−23 has been employed throughout the calculation. The details of the method are

already discussed. For a give spin symmetry, a set of reference configurations is chosen for

each of the four irreducible representations of C2v (Table 6.1). Millions of configurations

are generated during the optimization of 8 lowest roots of triplets and 4 roots of singlets

and quintets of PbC. For the PbC+ ion we have optimized similar number of doublets and

131

quartets (see Table 6.2). A configuration selection threshold of 0.5 µhartree has been used

such that sum of the square of the CI coefficients becomes more or less 0.90. The higher order

excitations from the reference configurations are taken care by the multireference analogue

of Davidson’s correction.24,25


In the subsequent steps, we have introduced the spin-orbit interaction through the spin-

orbit operators of Pb and C atoms. All the low-lying Λ-S states are allowed to interact.

In the C22v double group, the resulting Ω states of PbC belong to A1, A2, and B1/B2 and

those of PbC+ correspond to E1 and E2, respectively. The details are discussed in earlier

chapter.26

Spectroscopic constants (re, Te, ωe) are obtained by fitting PECs constructed from the

CI energies. The vibrational energies and wave functions are obtained from the numerical

solutions of one dimensional nuclear Schrodinger equation. Transition dipole moments in-

volving different Λ-S and Ω states of spin and/or dipole allowed transitions are computed.

These also give the estimate of the radiative lifetimes of the excited states.



A. PbC

Eighteen electronic states of Σ+(2),Σ−,Π(2), and ∆ symmetries of singlet, triplet and

quintet spin multiplicities correlate with the lowest dissociation limit, Pb(3Pg)+C(3Pg). In

the atomic level, the first excited state (1Dg) of Pb and that of C differ in their relative energy

by 0.2 eV only. Thus, the two limits, Pb(3Pg)+C(1Dg) and Pb(1Dg)+C(3Pg) are expected

to be very closed. Both of them correlate with the triplets of Σ+,Σ−(2),Π(3),∆(2), and

Φ symmetries. Two sets of 3Σ− and 3Π states dissociate into the fourth and fifth limits

(Table 6.3) at 19 761 and 21 614 cm−1, respectively above the first dissociation limit. The

first excited states (1Dg) of both Pb and C combine to form 15 singlets of different symmetries

which lie at 21 911 cm−1 above the lowest limit. However, the calculated value of 18 710

cm−1 is underestimated by 3200 cm−1 from the experimental observation.27

132

Table 6.3 Dissociation correlation between the molecular and atomic states of PbC


Pb + C Expt.a Calc.1Σ+(2), 1Σ−, 1Π(2), 1∆, 3Pg + 3Pg 0 03Σ+(2), 3Σ−, 3Π(2), 3∆,5Σ+(2), 5Σ−, 5Π(2), 5∆3Σ+, 3Σ−(2), 3Π(3), 3∆(2), 3Φ 3Pg + 1Dg 10 159 68753Σ+, 3Σ−(2), 3Π(3), 3∆(2), 3Φ 1Dg + 3Pg 11 752 11 7403Σ−, 3Π 1Sg + 3Pg 19 7613Σ−, 3Π 3Pg + 1Sg 21 6141Σ+(3), 1Σ−(2), 1Π(4), 1∆(3), 1Dg + 1Dg 21 911 18 7101Φ(2), 1Γ

a Ref. 27

Table 6.4 represents the spectroscopic constants of some of the low-lying Λ-S states of

PbC and their corresponding PECs are displayed in Figs. 6.1a-c. Like the lighter carbide,

the ground state of PbC is 3Π with re= 2.15 A and ωe= 579 cm−1. The state is dominated

by σ21σ2π

31 configuration, where σ1 is an antibonding combination of s orbitals of both Pb

and C, while σ2 is mostly bonding comprising s & pz atomic orbitals of the constituting

atoms. The π1 MO is purely bonding type. The B3LYP/TZVP+ calculation of Li et

al.7 reported a smaller bond length of 2.063 A. Contradictorily, Ciccioli et al.6, from their

CCSD(T) level of theory predicted X3Σ− as the ground state of PbC with bond length and

vibrational frequency of 2.191 A and 565 cm−1, respectively. The ground-state dissociation

energy is 2.46 eV at the Λ-S CI level which is in good agreement with the value of 2.58 eV

(248.1 kJ/mole) computed in CCSD(T) level of theory.6 However, with the inclusion of spin-

orbit coupling, the De value reduces significantly to 1.97 eV which however, compares quite

accurately with the best estimated value of 1.93 eV (185 kJ/mole).

The first excited state of PbC, A3Σ− lies very close to the ground state with a transition

energy of only 2136 cm−1. It has equilibrium bond length and vibrational frequency of 2.24

A and 534 cm−1, respectively (Table 6.4). The π1 → σ2 electronic transition is responsible

for the existence of this state. Due to a very low energy separation, the A-X band is expected

to appear with a very low intensity. The a1∆ state is the lowest singlet of PbC with a longer

133

134

3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

(a)

3Pg + 1D

g

1Dg + 3P

g

3Pg + 3P

g

Pb + C

33Σ

+

43Σ-

33∆

63Π53

Π 23Φ33Σ

-

23∆

43Π

23Σ

-

3Φ

33Π

23Σ

+

3Σ

+

23Π 3∆

A3Σ

-

X3Π

Ene

rgy

/ cm

-1

Bond Length / a0

3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

(b)

1Dg + 1D

g

3Pg + 3P

g

Pb + C

21∆

1Φ

21Π

1Σ

-

d1Σ

+

c1Π

b1Σ

+ a1∆

X3Π

Ene

rgy

/ cm

-1

Bond Length / a0

3 4 5 6 7 8 9 10

0

10000

20000

30000

40000

(c)

3Pg + 3P

g

Pb + C

25Σ

-

25Σ+

5∆

5Σ

-

5Σ

+

25Π

5Π

X3Π

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 6.1 Λ-S states of PbC: for (a) triplet, (b) singlet, and c) quintet spin

multiplicities

Table 6.4 Spectroscopic constants of low-lying Λ-S states of PbC


X3Π 0 2.15 579 3.00

0a 2.063a

A3Σ− 2136 2.24 534 3.28

0b 2.191b 565b

a1∆ 5952 2.29 475 2.83

8009a 2.182a

5Π 6651 2.37 431 2.75

7555a 2.335a

b1Σ+ 6683 2.17 353 2.86

c1Π 7025 2.16 548 2.97

d1Σ+ 10 731 2.20 645 2.801Σ− 13 572 2.63 316 1.833∆ 14 576 2.59 334 1.863Σ+ 14 851 2.64 309 1.91

23Π 15 286 2.40 370 2.17

33Π 18 477 2.44 368 2.66

23Σ+ 18 610 2.45 315 1.273Φ 20 334 2.42 389 2.34

23Σ− 21 042 2.65 303 2.94

43Π 24 460 2.51 325 1.871Φ 25 154 2.41 415 2.97

23∆ 25 534 2.36 340 1.96

21∆ 29 172 2.54 342 3.74

33Σ− 30 186 2.39 345 2.64

33Σ+ 38 070 2.05 620 2.95

25Σ− 38 270 2.25 556 1.11

a Ref. 7, b Ref. 6

equilibrium bond length of 2.29 A. However, Li et al.7 calculated a shorter bond length of

2.182 A but no experimental result is known for comparison. At equilibrium, it has the same

135

dominant configuration, σ21σ

22π

21 (77%) as that of A3Σ−. The longer bond length may be

attributed to the removal of an electron from strongly bonding π1 to the weaker bonding σ2.

The computed vibrational frequency of the state is 475 cm−1.

A strongly bound quintet is located around 6650 cm−1 above the ground state having a

longer bond length of 2.37 A. The state originates from the π1 → π2 electronic transition.

The antibonding character of π2 changes the bonding character of the molecule in this state.

As shown in Table 6.5, the same configuration (σ21σ2π

21π2) generates three more excited 3Π

and a pair of 1,3Φ states. It may be mentioned here that this configuration gives rise to four

more Π states, all of which are not computed here.

The lowest 1Σ+ state, designated as b, interacts strongly with the next root of the same

symmetry, namely d1Σ+. As a result a shallow double minima appear in the potential

energy curve of b1Σ+ (Fig. 6.1b). The energy barrier between the two minima of b1Σ+ is

extremely low and thus the adiabatic potential well on fitting gives an estimated re= 2.17 A

and ωe=353 cm−1. The ωe obtained by fitting the adiabatic curve of d1Σ+ is 645 cm−1.

The longer magnitude of ωe for this state is due to avoided crossing with its lower root.

However, at the potential minima, both the states b and d are described by two dominating

configurations, σ21σ

22π

21 and σ2

1π41 as shown in Table 6.5. The ground-state configuration

dominates in c1Π state having a similar bond length and vibrational frequency as those of

the ground state. The first dissociation limit correlates with two more singlets namely, 1Σ−

and 21Π. The 1Σ− state has a longer bond length and low binding energy of 0.78 eV, while

the 21Π state is repulsive.

Since the ground state of PbC is of 3Π symmetry and there exists another close lying 3Σ−,

the excited 3Π, 3Σ+, 3Σ−, and 3∆ states are spectroscopically important. The potential

minima of the lowest 3∆ and 3Σ+ states are at longer bond distances and thus, Franck-

Condon overlap factors are quite low for them. The second and third roots of 3Π can be

identified in emission spectroscopy at around 15 300 and 18 500 cm−1, respectively. The 33Π

state is more strongly bound than 23Π as evident from Fig. 6.1a. However, they dissociate

into two different asymptotes.

The 23Σ+ state interacts with the lowest root of 3Σ+. The state is spectroscopically less

important due to small Franck-Condon overlap factor. A 3Φ state is generated dominantly

by σ21σ2π

21π2 (80%) configuration with a binding energy of 0.794 eV. In the longer bond

length region (> 5.5 a0) it interacts with a higher root of 3Φ which is repulsive in nature.

136

Table 6.5 Composition of Λ-S states of PbC at equilibrium bond length


X3Π σ21σ2π

31(71), σ2

1σ2π21π2(7), σ2

1σ2π1π22(6)

A3Σ− σ21σ

22π

21(83)

a1∆ σ21σ

22π

21(77), σ2

1σ22π1π2(5), σ2

1σ22π

22(4)

5Π σ21σ2π

21π2(84)

b1Σ+ σ21σ

22π

21(42), σ2

1π41(33), σ2

1π21π

22(3)

c1Π σ21σ2π

31(76), σ2

1σ2π1π22(6), σ2

1σ2π21π2(4)

d1Σ+ σ21π

41(38), σ2

1σ22π

21(31), σ2

1π21π

22(8), σ2

1π31π2(4)

1Σ− σ21σ

22π1π2(63), σ2

1σ2σ4π1π2(15), σ21σ2σ7π1π2(6), σ2

1σ2σ3π1π2(3)3∆ σ2

1σ22π1π2(62), σ2

1σ2σ4π1π2(15), σ21σ2σ7π1π2(6), σ2

1σ2σ3π1π2(3)3Σ+ σ2

1σ22π1π2(54), σ2

1σ2σ4π1π2(15), σ21σ2σ7π1π2(6), σ2

1π31π2(5),

σ21π

21π

22(4), σ2

1σ2σ3π1π2(3)

23Π σ21σ2π

21π2(74), σ2

1σ2π31(4), σ2

1σ2π1π22(2)

33Π σ21σ2π

21π2(82), σ2

1σ2π31(2)

23Σ+ σ21π

31π2(47), σ2

1σ22π1π2(23), σ2

1π21π

22(8), σ2

1π1π32(6)

3Φ σ21σ2π

21π2(82)

23Σ− σ21σ

22π1π2(34), σ2

1σ22π

21(28), σ2

1σ2σ5π21(6), σ2

1π31π2(5)

43Π σ21σ2π

21π2(68), σ2

1σ2π1π22(7), σ2

1σ4π1π22(3), σ2

1σ4π21π2(2)

1Φ σ21σ2π

21π2(85)

23∆ σ21π

31π2(73), σ2

1π21π

22(5), σ2

1π1π32(4)

21∆ σ21σ

22π

21(37), σ2

1σ22π1π2(23), σ2

1π31π2(16), σ2

1π21π

22(3), σ2

1σ2σ4π1π2(3)

33Σ− σ21π

31π2(63), σ2

1π21π

22(8), σ2

1σ22π1π2(6)

33Σ+ σ1σ2π41(56), σ1σ2π

31π2(9), σ1σ2π

21π

22(6), σ2

1π31π2(5)

25Σ− σ21σ2σ3π

21(75), σ2

1σ2σ4π21(5), σ2

1σ2σ3π1π2(4)

Another 3Σ− state also exists at 21 042 cm−1 having bond length at least 0.50 A greater

than that of the ground state. Beyond the potential minimum, the state is flattened and

thus it has a low vibrational frequency of 303 cm−1. Transition to this state is also less

probable because of poor overlap region with the ground state. Among the bound 3Π states,

binding energy is the least for 43Π. It has a relatively longer re and hence spectroscopically

unimportant. However, it interacts with fifth and sixth root of the same symmetry in the

137

longer bond length region. Above 25 000 cm−1, there are two bound singlets designated as 1Φ

and 21∆, both of which dissociate into sixth asymptote, Pb(1Dg)+C(1Dg). Their potential

minima have been located at 2.41 and 2.54 A, respectively.

In the Λ-S level, 23∆ and 33Σ− are very important states of PbC from the spectroscopic

point of view. They are weakly bound with binding energies 0.19 and 0.14 eV, respectively.

Both the states are characterized by the σ21π

31π2 configuration, the origin of which is the

σ2 → π2 electronic transition. Like two other lighter carbides (SiC and SnC), the 33Σ+ state

of PbC arises dominantly from σ1σ2π41 configuration (c2=0.56). It is bound with an energy

barrier of 0.21 eV only. However, 33Σ+ may be isolated in the emission band at around

38 000 cm−1. All the quintets except 5Π and 25Σ− are repulsive. The latter one undergoes

predissociation at around 5.0 a0 through a potential barrier of 0.33 eV.

B. PbC+

The first dissociation limit of PbC+, comprising Pb+(2Pu) and C(3Pg), correlates with a

set of six doublets and six quartets of Σ+, Σ−(2), Π(2), and ∆ symmetries. The ground

state of Pb+ (2Pu) and the first excited state of C (1Dg) combine to form nine states of

Σ+(2), Σ−, Π(3), ∆(2), and Φ symmetries. All these states dissociate into the second

asymptote lying 11 782 cm−1 above the first one. The atomic spectral data27, however,

shows that the calculation is overestimated by about 1700 cm−1 with an error of 16%. Only

two doublets namely, 42Σ+ and 62Π correlate with the third asymptote, Pb+(2Pu) and C(1Sg)

which lies 22 782 cm−1 above the ground limit. Here also our estimated value exceeds the

experimental value, although by 5% only. Atomic combination of Pb+(2Pu)+C(5Su) around

Table 6.6 Dissociation correlation between the molecular and atomic states of PbC+


Expt.a Calc.2Σ+, 2Σ−(2), 2Π(2), 2∆, Pb+(2Pu) + C(3Pg) 0 04Σ+, 4Σ−(2), 4Π(2), 4∆2Σ+(2), 2Σ−, 2Π(3), 2∆(2), 2Φ Pb+(2Pu) + C(1Dg) 10 159 11 7822Σ+, 2Π Pb+(2Pu) + C(1Sg) 21 614 22 7824Σ−, 4Π, 6Σ−, 6Π Pb+(2Pu) + C(5Su) 33 701 33 580

a Averaged over J, Ref. 27

138

139

3 4 5 6 7 8 9 10 11 12

0

10000

20000

30000

40000

50000

60000(a)

Pb+(2Pu) + C(5S

u)

Pb+(2Pu) + C(3P

g)

34Π

34Σ

-

24Π

24Σ

-

4Π

4Σ

+4∆

X4Σ

-

Ene

rgy

/ cm

-1

Bond Length / a0

3 4 5 6 7 8 9 10 11 12

0

10000

20000

30000

40000

50000

60000(b)

Pb+(2Pu) + C(1S

g)

Pb+(2Pu) + C(1D

g)

Pb+(2Pu) + C(3P

g)

42Σ

+

32Σ

+

72Π

62Π

52Π

2Φ

32∆

42Π

32Σ-

22Σ

+22

∆ 32Π

22Σ-

22Π

2Σ

+

2Σ

-

2∆

2Π

X4Σ

-

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 6.2 Λ-S states of PbC+: for (a) quartet and (b) doublet spin multiplicities

33 580 cm−1 correlates with two quartet states, 34Σ− and 34Π. The MRDCI calculated value

agrees quite accurately with the experimental observation (Table 6.6). Potential energy

curves of most of the quartets and doublets are displayed in Figs. 6.2a and b, respectively.

The removal of one electron from the π1 bonding orbital of the ground-state PbC makes

the bond weaker in PbC+. The ground state X4Σ− has a longer bond length of 2.24 A. The

state has a multi configuration character with the dominant configuration σ21σ2π

21(58%). The

equilibrium vibrational frequency of the state is reported to be 543 cm−1 (Table 6.7), while

its dissociation energy is predicted to be 2.12 eV. No experimental data is available for

comparison.


Λ-S states of PbC+


X4Σ− 0 2.24 543

2.179a

2Π 8183 2.44 3922∆ 9475 2.33 416

4582a 2.191a

2Σ− 10 665 2.33 3902Σ+ 11 856 2.41 356

22Π 13 767 2.26 4764∆ 14 176 2.94 1954Σ+ 14 392 2.95 191

22Σ− 16 712 2.82 249

32Π 23 148 2.95 264

22∆ 23 595 2.80 285

22Σ+ 24 685 2.83 255

32Σ− 25 278 3.04 201

42Π 28 390 2.65 206

34Σ− 38 540 2.77 315

24Π 42 151 2.09 582

a Ref. 7

140

The lowest two roots of 2Π interact strongly minimizing the lower one. Thus unlike SiC+

but like SnC+, the first excited state of PbC+ is 2Π having a longer bond length of 2.44 A and

adiabatic transition energy of 8183 cm−1. The upper 2Π state is comparatively loosely bound

and it has comparable bond length as that of the ground state. After the incorporation of

the spin-orbit interaction this state may be spectroscopically important. The lowest 2Π is

characterized by σ21σ

22π1(58%), while the upper one is dominated by σ2

1π31(45%) with some

other open shell configurations as given in Table 6.8. In between these two states there

exist three doublets, 2∆, 2Σ−, and 2Σ+ all of which originate from the same dominant

configuration as the ground state. The 22Σ− state with a longer bond length of 2.82 A exists

at around 16 700 cm−1.

Table 6.8 Composition of Λ-S states of PbC+ at equilibrium bond length


X4Σ− σ21σ2π

21(58), σ2

1σ2π1π2(12), σ1σ22π

21(9), σ1σ

22π1π2(4)

2Π σ21σ

22π1(58), σ2

1σ22π2(11), σ2

1σ2σ4π1(7), σ21σ2σ4π2(7)

2∆ σ21σ2π

21(70), σ2

1σ2π1π2(9)), σ1σ22π

21(5), σ2

1σ2π22(3)

2Σ− σ21σ2π

21(59), σ2

1σ2π1π2(14), σ1σ22π

21(4)

2Σ+ σ21σ2π

21(60), σ2

1σ2π1π2(11), σ21σ2π

22(6)

22Π σ21π

31(45), σ2

1π21π2(7), σ1σ2π

31(7), σ2

1σ22π1(5),

σ1σ2π21π2(4), σ2

1π1π22(4)

4∆ σ21σ2π1π2(82), σ2

1σ2π1π5(4)4Σ+ σ2

1σ2π1π2(82), σ21σ2π1π5(4)

22Σ− σ21σ2π1π2(47), σ2

1σ3π22(12), σ2

1σ3π1π2(9), σ21σ2π

22(8),

σ21σ2π

21(5)

32Π σ21σ

22π1(56), σ1σ

22π2(17), σ2

1π1π22(6), σ2

1σ2σ3π2(5)

22∆ σ21σ2π1π2(71), σ2

1σ2π22(3), σ2

1σ3π22(3), σ2

1σ2π21(2)

22Σ+ σ21σ2π1π2(76), σ2

1σ2π21(4)

32Σ− σ21σ2π1π2(42), σ2

1σ2π21(15), σ2

1σ2π22(15), σ2

1σ3π22(8)

42Π σ21π

21π2(38), σ2

1π1π22(26), σ2

1π31(8)

34Σ− σ21σ3π1π2(19), σ2

1σ3π22(12), σ2

1σ2π22(8), σ2

1σ3π21(5)

24Π σ1σ2π31(48), σ2

1π31(8), σ1σ2π

21π2(5), σ1σ2π1π

22(5)

A one electron transition, π1 → π2 gives rise to 4∆ and 4Σ+ states with comparable re

141

and ωe values. Both the states are weakly bound with a maximum of 10 vibrational levels.

The remaining quartets which dissociate into the first dissociation limit, are repulsive in

nature. The only exception is for 24Π which has been isolated with a single vibrational level.

Unlike the lighter carbide ions such as SiC+, SnC+, transition to this state is not expected

to occur because of its shorter bond length and low binding property. At equilibrium, 24Π

is characterized mainly by σ1σ2π31 with three other configurations as shown Table 6.8. An

excited 4Σ− is located with a vibrational frequency of 315 cm−1. The calculated equilibrium

bond length of the state is large enough (2.77 A), yet an emission band may be expected at

around 38 500 cm−1.

Eleven doublets correlate with second and third dissociation limits. Of these seven states

have been isolated as bound or quasi-bound in nature. Most of them are dominated by

an open shell ...π1π2 configuration. Their equilibrium bond lengths are above 2.65 A and

vibrational frequencies vary between 200 and 285 cm−1. The remaining four states are

repulsive, one of which has been assigned as 2Φ.


A. PbC

The spin-orbit splitting is quite strong for PbC because of the heavy atom, Pb. The lowest

dissociation limit, Pb(3Pg)+C(3Pg) under the spin-orbit coupling splits into ten asymptotes.

The dissociation correlation between the Ω states of PbC and the corresponding atomic

combinations are shown in Table 6.9. There are 53 omega states which are allowed to mix

in the spin-orbit CI calculations. The spin-orbit splitting of the components of 3P0,1,2 of the

carbon atom is only 43 cm−1 and thus the states coming from them are almost inseparable.

However, the low-lying 0+, 0−, 1, 2, 3, and 4 components are plotted in Figs. 6.3a-d.

Spectroscopic constants of the bound Ω states are also shown in Table 6.10.

Because of the spin-orbit coupling, there are extensive changes in the spectroscopic prop-

erties of the low-lying Ω states of PbC. The ground state splits into four Ω components

having a very large energy separation. Both the X3Π and A3Σ− states contribute largely to

the lowest component of 0+. Its re is intermediate between those of the most contributing

states, while ωe has been lowered to 470 cm−1. 5Π0+ also significantly contribute to the low-

est 0+ component. The first Ω=2 state has been assigned to X3Π2 which is slightly mixed

up with the similar components of a1∆2 and 5Π2. However, the spectroscopic properties of

this state are not changed too much from its originating Λ-S state. The lowest Ω=1 state is

142

located 230 cm−1 above the ground state. It has a significant contribution from X3Π, A3Σ−

as well as 5Π. The re and ωe are completely different as compared to those of the contributing

Λ-S states. The situations are similar for the second and third roots of 1 also. The spin-orbit

coupling affects the X3Π0− to a large extent. The computed transition energy of this state

is 3295 cm−1. The re and ωe are also altered by 0.035 A and 74 cm−1, respectively.

Table 6.9 Dissociation correlation between Ω and atomic states of PbC

Ω States† Atomic states Relative energy / cm−1

Pb + C Expt.a Cal.

0+ 3P0 + 3P0 0 0

0−, 1 3P0 + 3P1 16 20

0+, 1, 2 3P0 + 3P2 43 55

0−, 1 3P1 + 3P0 7819 3540

0+(2), 0−, 1(2), 2 3P1 + 3P1 7835 3565

0+, 0−(2), 1(3), 2(2), 3 3P1 + 3P2 7862 3590

0+, 1, 2 3P0 + 1D2 10 193 7025

0+, 1, 2 3P2 + 3P0 10 650 10 390

0+, 0−(2), 1(3), 2(2), 3 3P2 + 3P1 10 666 10 405

0+(3), 0−(2), 1(4), 2(3), 3(2), 4 3P2 + 3P2 10 693 10 440



At equilibrium, 1(II) is composed of almost 30% of each of X3Π, A3Σ−, and 5Π states. The

minimum has been located at 4002 cm−1 with re and ωe, 2.258 A and 460 cm−1, respectively.

Like the ground-state, second root of 0+ is also contributed almost equally by X3Π and

A3Σ−. The computed re of the state is shorter than that of 0+(I), and ωe is more close to

that of X3Π. The X3Π state splits in an inverted order with Ω=2 lying lowest, while A3Σ−

splits in a regular pattern. Thus, 0+(II) has a longer contribution of the X3Π state. Due

to the spin-orbit mixing, the a1∆2 is energetically destabilized by an amount 1492 cm−1.

Its re is increased by 0.021 A, while ωe is decreased by 12 cm−1 only. Inspite of significant

mixing with other states, potential energy curves of all the five spin components of 5Π are

fitted. The spin-orbit components are separated in a regular pattern. The third root of 1 is

dominated by the 5Π−1 component. However, A3Σ−1 contributes in it by more than 30%,

143

144

3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000

33Π0

+

(a)

23Π0+

d1Σ

+

0+

0+(IV)

5Π

0+

0+(II)

0+

3P2 + 3P0, 1, 2

3P0 + 1D2

3P1 + 3P1, 2

3P0 + 3P0, 2

Pb + C

E

nerg

y / c

m-1

Bond Length / a0

3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000

0 -(VI)

0-(VI)

0-(V)0 -(IV)

(b)

3P2 + 3P

1, 2

3P1 + 3P

0, 1, 2

3P0 + 3P

0, 1

Pb + C

1Σ -

0-

5Π

0-

X3Π0-

0+

En

erg

y / c

m-1

Bond Length / a0

3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000

1(V)

(c)

3Φ3

3∆3

5Π3

3P2 + 3P0, 1, 2

3P0 + 1D

2

3P1 + 3P

0, 1, 2

3P0 + 3P0, 1, 2

Pb + C

3∆

1

c1Π1

1(III)

1(II)

10+

En

erg

y / c

m-1

Bond Length / a 0

3 4 5 6 7 8 9 10

0

5000

10000

15000

20000

25000

30000

35000

3Φ

2

(d)

3Φ

4

3P2 + 3P

0, 1, 2

3P0 + 1D2

3P1 + 3P

1, 2

3P0 + 3P0, 2

Pb + C

2(V)

2(IV)

5∆2

a1∆2

X3Π

2

0+

En

erg

y / c

m-1

Bond Length / a0

Fig. 6.3 Ω states of PbC: for (a) 0+, (b) 0-, (c) 1, 3, and (d) 2, 4 symmetries

Table 6.10 Spectroscopic constants and composition of low-lying Ω states of PbC


0+ 0 2.215 470 A3Σ−(40), X3Π(39), 5Π(17), b1Σ+(4)

X3Π2 125 2.163 550 X3Π(89), a1∆(6), 5Π(3)

1 230 2.198 495 X3Π(55), A3Σ−(29), 5Π(12), c1Π(3), 3Σ+(1)

X3Π0− 3295 2.185 505 X3Π(85), 1Σ−(5), 5Π(4)

1(II) 4002 2.258 460 X3Π(37), A3Σ−(30), 5Π(29)

0+(II) 5160 2.202 565 X3Π(41), A3Σ−(36), d1Σ+(17), b1Σ+(5)

a1∆2 7444 2.311 463 a1∆(76), 23Π(8), X3Π(5), 5Π(5), 3∆(3),3Φ(2)

1(III) 8130 2.307 525 5Π(49), A3Σ−(34), X3Π(5), c1Π(4), 23Σ+(3),3Σ+(1)

5Π0− 8956 2.310 515 5Π(67), X3Π(27), 23Σ+(4)5Π0+ 8765 2.302 485 5Π(58), b1Σ+(22), X3Π(11), d1Σ+(4),

23Σ−(2)

c1Π1 9580 2.256 360 c1Π(52), 5Π(32), A3Σ−(4), X3Π(3),3Σ+(2), 23Π(2), 33Π(2)

0+(IV) 10 960 2.267 505 b1Σ+(51), 5Π(23), A3Σ−(22)

1(V) 11 148 2.276 550 5Π(44), c1Π(38), X3Π(8), A3Σ−(6)5Π2 11 410 2.345 450 5Π(82), X3Π(10), 3∆(3), a1∆(3), 23Π(2)5Π3 12 310 2.376 405 5Π(95), 3∆(2)1Σ−0− 13 272 2.632 303 1Σ−(56), 3Σ+(34), 23Π(5), 5Π(1)3∆1 13 935 2.589 407 3∆(95), 5Π(1)

d1Σ+0+ 15 407 2.209 600 d1Σ+(77), X3Π(14), A3Σ−(4), 23Π(3)

3∆2 17 095 2.571 377 3∆(58), 23Π(26), 3Φ(8), 5Π(5)

23Π1 17 382 2.431 355 23Π(72), 21Π(15), 23Σ+(6), 33Π(2), 5Π(2)

33Σ+1 17 445 2.593 297 3Σ+(62), 23Π(18), 23Σ−(4), 5Π(4), 21Π(3),

33Π(2)

2(V) 17 846 2.450 475 23Π(35), 3∆(31), 3Φ(12), a1∆(6), 5Π(2)

33Π0+ 18 135 2.540 282 33Π(58), 23Π(20), d1Σ+(11), X3Π(5), 5Σ+(3)

0−(IV) 18 214 2.535 270 23Π(41), 33Π(34), 3Σ+(22), 1Σ−(1)

0−(V) 19 220 2.395 222 33Π(42), 23Π(37), 23Σ+(16), 1Σ−(3)

145


State Te/cm−1 re/A ωe/cm−1 Contribution of Λ-S states / (%)3∆3 19 565 2.618 285 3∆(71), 5Π(18), 3Φ(9)

1(IX) 20 270 2.398 398 33Π(42), 21Π(21), 3∆(14), 23Π(9), 23Σ+(3),5Π(2)

23Π0+ 20 512 2.380 390 23Π(65), 33Π(23), d1Σ+(6), 5Σ+(2), 23Σ−(1)

0−(VI) 20 575 2.492 365 3Σ+(28), 23Σ+(26), 1Σ−(23), 23Π(17)3Φ2 22 148 2.445 400 3Φ(65), 3∆(10), a1∆(8), 5Σ+(3), 23Π(3),

5Π(3)3Φ3 22 250 2.473 393 3Φ(62), 3∆(16), 1Φ(14), 5Π(4)

0−(VII) 23 925 2.500 363 23Σ+(56), 23Π(19), 1Σ−(10), 33Π(5), 5Π(3),3Σ+(1)

3Φ4 25 745 2.455 327 3Φ(96), 5∆(2)

which may have resulted an increase in ωe by 94 cm−1, and its re is shortened to 2.31 A. The

spin forbidden 5Π0−–X3Π0− transition is predicted to have sufficient intensity. The radiative

lifetime of 5Π0− is of the order of hundred microseconds. At 5.2 a0 the potential energy

curve of 5Π0− crosses that of the 1Σ−0− state and dissociates into the fourth asymptote (Fig.

6.3b). 5Π0+ interacts strongly with b1Σ+0+ and the resulting adiabatic potential well is fitted.

The second component of 5Π1 is strongly perturbed by c1Π1. It gives Te=11 148 cm−1 at

re=2.276 A with an equilibrium vibrational frequency of 550 cm−1. On the other hand, the

diabatic curve of c1Π1, which crosses 5Π−1 at around 4 a0 gives an estimated Te value of

9580 cm−1. 5Π2 and 5Π3 states are energetically shifted upward by 4500-5700 cm−1 due to

spin-orbit mixing. These two states are spectroscopically important as both of them can

emit to the X3Π2.

The potential minimum of the fourth root of 0+ is located at 2.267 A with a transition en-

ergy of10 960 cm−1. It is dominated mostly by b1Σ+, however, both 5Π and A3Σ− contribute

to a large extent. At around 13 300 cm−1 there exists a root of 0− which is dominated by1Σ−(56%) and 3Σ+(34%). The diabatically fitted re and ωe of this state are closer to those

of 1Σ−, thus it has been assigned as 1Σ−0− . The curve crossing between 1Σ−0− and 5Π0− is

confirmed from the analysis of CI wave functions in between 4.5-5.5 a0. The adiabatic curve

of 3∆1 has been fitted with the unchanged re (2.589 A). Due to small Franck-Condon factor,

146

the transition from this state is not expected to occur. A dipole allowed 0+-0+ transition

has been located at around 15 400 cm−1. The upper state is labeled as d1Σ+0+ with the

computed re and ωe of 2.209 A and 600 cm−1, respectively. It has significant mixing with

X3Π0+ . In the longer bond length region it interacts strongly with the 0+ components of 23Π

and 33Π. As a result of strong spin-orbit interaction the fourth root of Ω=2 looks flattened

in the equilibrium region. The diabatically fitted curve shows a potential minimum at 2.571

A with a transition energy of 17 095 cm−1. As 3∆ makes the major contribution (58%)

to it, it has been named as 3∆2. The next root of 2 has been fitted adiabatically. The

estimated Te of the state is reported to be 17 864 cm−1 at 2.450 A with ωe=475 cm−1. The3∆3 component is perturbed by two other components, namely 5Π3 and 3Φ3. The transition

energy of the state is computed to be 19 565 cm−1 which is 5000 cm−1 away from the 3∆

state itself. The ωe of the state is lowered by 49 cm−1. However, because of the smaller

Franck-Condon factor transitions from this state with 4Ω=0, 1 are expected to be weak.

Near 17 400 cm−1, two roots of Ω=1 cross each other at the bond distance of 4.8 a0.

Analysis of the wave functions reveals that they originate from 23Π and 3Σ+, respectively.

Fitting both the curves diabatically, the spectroscopic constants are displayed in Table 6.10.

All the four spin components of 23Π show potential minima at 17 382 17 864, 18 214, and

20 512 cm−1, respectively. The excited 33Π split in a regular order. A strong spin-orbit

mixing is indicated in the composition shown in Table 6.10. The 3Π2 does not have clear

potential minimum because of several avoided curve crossings around 22 000 cm−1 in the

4.8-5.6 a0 bond distance region. The Ω components of 3Φ mix up with these of 3∆ around

22 000 cm−1. The 3Φ4 state is almost pure, though its Te is increased by more than 5000

cm−1.

B. PbC+

The ground term of the ion Pb+ (2P) splits into J=1/2 and J=3/2 with a separation of

14 081 cm−1, while that of C (3P) splits into three sub-levels with a maximum separation

of 43 cm−1 only. Consequently, the ground-state dissociation limit of the molecular ion,

Pb+(2P)+C(3P) correlates with six sub levels with a maximum separation of 14 124 cm−1

as shown in Table 6.11. A set of two 1/2, two 3/2, and a 5/2 dissociates in the limit

Pb+(2P1/2)+C(1D2). Thus a total of 32 Ω states of 1/2, 3/2, 5/2, and 7/2 symmetries

correlate with three sets of sub-levels within 14 124 cm−1 of energy. The potential energy

curves and their spectroscopic constants are given in Figs. 6.4a-c and Table 6.12, respectively.

147

148

3 4 5 6 7 8 9 10 11

0

5000

10000

15000

20000

25000

30000

35000

3/2(VI)

22Π3/2

(a)

Pb+(2P3/2) + C(

3P0, 1, 2)

Pb+(2P1/2

) + C(1D2)

Pb+(2P1/2) + C(

3P1, 2)

4Σ

+

3/2

3/2(V)

3/2(III)

2Π3/2

X4Σ

-

3/2

Ene

rgy

/ cm

-1

Bond Length / a0

3 4 5 6 7 8 9 10 11

0

5000

10000

15000

20000

25000

30000

35000

1/2(VII)

(b)

Pb+(2P1/2) + C(1D2)

Pb+(2P3/2

) + C(3P0, 1, 2

)

Pb+(2P1/2) + C(3P0, 1, 2)

1/2(VI)

2Σ -

1/2 2Π

1/2

X4Σ-1/2

X4Σ-3/2

Ene

rgy

/ cm

-1

Bond Length / a0

3 4 5 6 7 8 9 10 11

0

5000

10000

15000

20000

25000

30000

35000(c)

Pb+(2P3/2

) + C(3P0, 1, 2

)

Pb+(2P1/2

) + C(1D2)

Pb+(2P1/2) + C(3P1, 2)

4∆7/2

4∆

5/2

2∆5/2

X4Σ-3/2

Ene

rgy

/ cm

-1

Bond Length / a0

Fig. 6.4 Ω states of PbC+: for (a) 3/2, (b) 1/2, (c) 5/7 & 7/2 symmetries

The bond length of ground-state spin component, X4Σ−3/2 is increased by 0.01 A, while its ωe

is lowered by 38 cm−1 due to the spin-orbit interaction. The spin-orbit zero-field splitting is

computed to be 286 cm−1. The composition of both the ground-state components are given

in Table 6.12.

Table 6.11 Dissociation correlation between Ω and atomic states of PbC+

Ω states Atomic states Relative energy / cm−1

Pb+ + C Expt.a Cal.

1/2 2P1/2+3P0 0 0

1/2(2), 3/2 2P1/2+3P1 16 23

1/2(2), 3/2(2), 5/2 2P1/2+3P2 43 54

1/2(2), 3/2(2), 5/2 2P1/2+1D2 10 193 11 360

1/2, 3/2 2P3/2+3P0 14 081 13 446

1/2(3), 3/2(2), 5/2 2P3/2+3P1 14 097 13 466

1/2(4), 3/2(3), 5/2(2), 7/2 2P3/2+3P2 14 124 13 496


The two components (1/2 and 3/2) of the first excited 2Π state are separated by about

623 cm−1. Their equilibrium bond lengths are elongated by 0.03-0.06 A, while their ωes are

reduced by about 75 cm−1 as a result of the spin-orbit mixing. Table 6.12 shows that 2Π1/2

mixes with the components of 4∆, 4Σ−, while the other component mixes strongly with2∆3/2, 4∆3/2, and 4Σ−3/2. The spin-orbit mixing permits the X4Σ− →2 Π transition through

their respective dipolar components. The Ω=3/2 component of 2∆ is very strongly perturbed

by the similar components of first and second root of 2Π. In absorption spectroscopy, the2∆3/2 state may be isolated at 10 430 cm−1. The situation is more complex for 2∆5/2. The

mixing contribution of 4∆ results its quasi bound nature with a very small potential barrier.

Due to its low binding energy, the emission from this state is not expected. The estimated

re and ωe of 2∆5/2 are 2.459 A and 249 cm−1, respectively.

The third root of 1/2 is dominated by 2Σ−(69%). Because of the interaction with the

Ω=1/2 component of repulsive 4Π, this state is also very poorly bound and consequently no

transition to this state is expected to be observed. The 2Σ+1/2 and 4Σ+

1/2 components also

contribute to this by 8% and 7%, respectively. Fourth and fifth root of 1/2 are repulsive in

149

Table 6.12 Spectroscopic constants and composition of low-lying Ω states of PbC+


X4Σ−3/2 0 2.251 505 X4Σ−(89), 2Π(6), 4∆(2), 4Π(1)

X4Σ−1/2 286 2.252 509 X4Σ−(91), 2Π(2), 2Σ+(3), 4Π(2)2Π1/2 6255 2.501 313 2Π(66), 4∆(19), X4Σ−(6), 2Σ+(2),

2Σ−(2), 4∆(2), 4Π(1)2Π3/2 6974 2.472 318 2Π(49), 2∆(25), 4∆(12), X4Σ−(9),

4Σ+(2), 4Π(1)2∆5/2 8902 2.459 249 2∆(85), 4∆(10), 4Π(3)2Σ−1/2 9622 2.468 246 2Σ−(69), 4Π(9), 2Σ+(8), 4Σ+(7)

3/2(III) 10 430 2.401 249 2∆(39), 2Π(23), 22Π(22), 4Σ+(4),

X4Σ−(4), 4∆(2), 4Π(2)

1/2(VI) 14 955 2.720 286 2Π(26), 4∆(25), 4Π(13), 22Π(12), X4Σ−(7),

22Σ−(5), 22Σ+(4), 4Σ+(4), 2Σ−(2)

22Π3/2 15 065 2.415 286 22Π(56), 2∆(14), 2Π(10), X4Σ−(9), 4Π(7)

3/2(V) 15 943 2.581 418 4∆(40), 2∆(29), 22∆(9), 4Σ+(7), 42Π(6),

X4Σ−(3), 24Π(2), 4Π(1)

1/2(VII) 16 415 2.549 375 4Π(32), 2Σ+(26), 22Π(23), X4Σ−(10), 2Σ−(3)4Σ+

3/2 17 533 2.676 301 4Σ+(52), 2Π(12), 22Π(10), X4Σ−(7),4∆(6), 22∆(5), 24Σ−(3), 4Π(1)

4∆5/2 18 569 2.683 248 4∆(64), 2∆(23), 2Φ(5), 22∆(3), 32∆(3)

1/2(VIII) 18 920 2.659 193 4Π(20), 2Σ−(18), 22Σ−(14), 22Σ+(10),

32Σ−(8), 22Π(8), 32Π(7), 4Σ+(7)

1/2(IX) 19 893 2.825 235 4Σ+(41), 2Σ−(31), 22Σ−(14), 22Σ+(4), 2Σ+(3)4∆7/2 20 106 2.913 199 4∆(99)

1/2(X) 20 605 2.909 241 32Π(30), 2Π(15), 22Σ−(8), 4∆(8), 4Σ+(7),

22Π(7), 22Σ+(6), 32Σ+(4), 4Π(3)

3/2(VII) 21 240 3.114 132 22∆(48), 4∆(18), 2∆(10), 2Π(9), 4Σ+(3)

1/2(XI) 22 543 2.834 190 22Π(41), X4Σ−(23), 4Π(17), 32Π(7),2Σ+(3), 2Σ−(2)

1/2(XII) 23 287 2.991 133 4Π(42), 22Π(30), 24Σ−(18), 22Σ−(3), 2Σ+(2)

nature. However, both the 1/2(IV) and 1/2(V) states dissociate into the Pb+(2P1/2)+C(1D2)

150

limit with a maximum contribution coming from 4Π. The equilibrium bond length of the

1/2(VI) state of PbC+ is 2.720 A and it is spectroscopically not significant due to poor

Franck-Condon overlap factor. Its potential minimum is located at 14 955 cm−1.

Above 15 000 cm−1, the purity of both the spin components Ω=1/2 and 3/2 is almost

destroyed. There occur several number of avoided curve crossing phenomena. Strong spin-

orbit interactions result enormous changes in the spectroscopic properties. The 22Π3/2 state

has been identified with re= 2.415 A and ωe = 286 cm−1. The X4Σ−3/2 → 22Π3/2 transition is

expected to occur with a band origin at 15 065 cm−1. The influence of the 3/2 components

of 2∆, 4Σ− and 4Π changes re and ωe to a large extent. The 11th root of 1/2 is characterized

by 22Π with a contribution of 41%. The adiabatic curve shows a potential minimum at

2.834 A with Te=22 543 cm−1 and a low vibrational frequency of 190 cm−1. The fifth root

of 3/2 at equilibrium is dominated by 4∆ (40%) with a transition energy of 15 943 cm−1.

However, 2∆ state also contributes largely. The equilibrium vibrational frequency of the

state is comparatively large (418 cm−1). Potential energy curve of 4∆1/2 suffers a large

number of avoided crossing and hence could not be fitted. But the other two components,4∆5/2 and 4∆7/2 are fitted. The 4∆5/2 state mixes with the components of 2∆ and 2Φ, while

the 4∆7/2 is almost pure. However, the four components of 4∆ split in a regular pattern.

The two components of 4Σ+ are shifted upward by 3141 and 5501 cm−1, respectively, by

spin-orbit effect. They split in an inverted manner, re of both the states are shortened, while

their vibrational frequencies are increased to a considerable amount. The adiabatic potential

minimum of tenth root of 1/2 is also complex with maximum contribution coming from 32Π

(30%). Very long bond distance makes it spectroscopically unimportant. A very shallow

minimum with a low vibrational frequency of 132 cm−1 is located at 21 240 cm−1 and the

state is assigned to be Ω=3/2 having 48% contribution from 22∆. Spectroscopic constants

of several other low-lying Ω states along with the equilibrium compositions are tabulated in

Table 6.12.


A. PbC

Many spin forbidden and dipole allowed transitions are studied for the neutral PbC

molecule. Transition moment functions for some selected transitions are plotted against

internuclear distance in Fig. 6.5a. The radiative lifetimes (partial as well as total) are re-

ported in Table 6.13. The 33Σ−–A3Σ− transition is predicted to be the strongest one. The

151

152

3 4 5 6 7 8-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6 (a)

A3Σ- - X3Π

33Σ

- - X3Π

23Π - A3Σ-

23∆ - X3

Π

33Π - A3

Σ-

33Σ

- - A3Σ

-

33Π - X3

Π

23Π - X3

Π

Tran

sitio

n M

omen

t / e

a 0

Bond Length / a0

3 4 5 6 7 8-0.05

0.00

0.05

0.10

0.15

0.20

0.25(b)

(22Π3/2-X4

Σ-

1/2)xy

(22Π3/2-X4

Σ-

3/2)z

(3/2(III)-X4Σ

-

1/2)xy

(2Π1/2-X4

Σ-

1/2)z

(2Π3/2

-X4Σ-

3/2)

z

(3/2(III)-X4Σ

-

3/2)

z

Tran

sitio

n M

omen

t / e

a 0

Bond Length / a0

Fig. 6.5 Computed transition moment functions involving few low-lying states of (a) PbC and (b) PbC+

corresponding transition moment curve has a minimum in Franck-Condon region. Total

radiative lifetime in the υ′=0 state of 33Σ− of PbC is about 320 ns. 33Σ+–X3Π is another

strong transition for PbC. The partial radiative lifetime in the ground vibrational level of

33Σ+ is 0.84 µs, but it becomes almost double in the next higher vibrational level. Both

the transition moment curves of 23Π–X3Π and 33Π–X3Π pass through maximum. In the

Franck-Condon region, the former transition has a greater transition moment and hence a

shorter radiative lifetime which again increases monotonically with the higher vibrational

levels. The situation is opposite for the transition from A3Σ− state. In that case, transition

to the upper 3Π is much more probable than that to the lower one. The 33Π has shorter total

radiative lifetime than that of 23Π. The transition moment of 23∆–X3Π has a maximum

at 3.9 a0 and the curve then slowly converge to zero. The partial radiative lifetime for

A–X transition is estimated to be of the millisecond order. The dipole allowed transitions

such as (0+(IV)–0+(I))‖, (1(V)–X3Π1)‖, (5Π2–X3Π2)‖, (d1Σ+0+–X3Π1)⊥ etc. have radiative

lifetime less than hundred microsecond. Transitions involving the components of 5Π and the

corresponding ground state components are highly probable due to strong spin-orbit mixing.

The 5Π2 and 1(V) states have the total radiative lifetimes of 58 and 43 µs, respectively.

A strong transition is indicated at around 15 400 cm−1, the upper state of which is the

component of d1Σ+. As the 0− component of the ground state is shifted upward by more

than 3000 cm−1, transitions from this state are relatively weaker. However, 5Π0− has a total

lifetime of 136 µs. The predicted lifetime in 5Π3 state is also in the order of few millisecond.

The 5Π3–X3Π2 transition is expected to appear at 12 310 cm−1.

Table 6.13 Radiative lifetime (s) of some of the excited states of PbC


υ′=0 υ′=1 υ′=2 υ′=3 υ′=4 at υ′=0

A3Σ−–X3Π 1.40(-3) 9.97(-4) 8.08(-4) 6.65(-4) 5.60(-4)

23Π–X3Π 1.66(-5) 1.67(-5) 1.69(-5) 1.89(-5) 2.77(-5)

23Π–A3Σ− 3.10(-3) 1.78(-3) 1.31(-3) 1.11(-3) 0.91(-3) τ23Π=1.65(-5)

33Π–X3Π 9.97(-5) 3.18(-5) 2.08(-5) 2.24(-5)

33Π–A3Σ− 1.04(-5) 1.03(-5) 1.02(-5) 1.02(-5) 1.01(-5) τ33Π=9.41(-6)

23∆–X3Π 2.33(-6) 2.80(-6) 3.65(-6) 4.14(-6) 4.06(-6)

33Σ−–X3Π 1.73(-6) 1.91(-6) 2.30(-6)

153



υ′=0 υ′=1 υ′=2 υ′=3 υ′=4 at υ′=0

33Σ−–A3Σ− 3.95(-7) 3.68(-7) 3.43(-7) τ33Σ−=3.21(-7)

33Σ+–X3Π 8.40(-7) 1.67(-6)

(0+(II)–0+)‖ 8.17(-3) 4.38(-3) 2.78(-3) 2.02(-3) 1.55(-3)

(0+(II)–1)⊥ 1.51(-4) 1.47(-4) 1.44(-4) 1.43(-4) 1.40(-4) τ0+(II)=1.48(-4)

(5Π0+–0+)‖ 1.16(-4) 1.53(-4) 1.77(-4) 1.99(-4) 2.16(-4)

(5Π0+–1)⊥ 5.95(-3) 2.60(-3) 1.86(-3) 1.53(-3) 1.55(-3) τ5Π0+=1.14(-4)

(0+(IV)–0+)‖ 9.56(-5) 7.78(-5) 6.76(-5) 6.19(-5) 5.80(-5)

(0+(IV)–1)⊥ 1.26(-4) 1.29(-4) 1.30(-4) 1.31(-4) 1.34(-4) τ0+(IV )=5.44(-5)

(d1Σ+0+–0+)‖ 1.48(-3) 4.99(-4) 2.30(-4) 2.16(-4)

(d1Σ+0+–1)⊥ 6.28(-5) 6.04(-5) 5.45(-5) 5.02(-5) τd1Σ+

0+=6.02(-5)

(5Π0−–X3Π0−)‖ 2.28(-4) 2.12(-4) 2.06(-4) 2.56(-4) 2.41(-3)

(5Π0−–1)⊥ 3.35(-4) 3.51(-4) 3.55(-4) 3.62(-4) 3.65(-4) τ5Π0−=1.36(-4)

(1(II)–1)‖ 7.43(-2) 2.64(-2) 1.87(-2) 1.59(-2) 1.46(-2)

(1(III)–1)‖ 1.28(-3) 1.06(-3) 8.89(-4) 8.60(-4) 8.45(-4)

(1(IV)–1)‖ 1.18(-3) 5.85(-4) 5.50(-4) 6.65(-4) 8.00(-4)

(1(V)–1)‖ 7.11(-5) 6.28(-5) 6.25(-5) 8.60(-5) 8.53(-5)

(1(V)–X3Π0−)⊥ 1.70(-2) 4.57(-2) 9.28(-2) 1.65(-1) 6.59(-2)

(1(V)–0+)⊥ 2.95(-4) 3.12(-4) 4.15(-4) 6.59(-4) 9.56(-4)

(1(V)–X3Π2)⊥ 1.68(-4) 2.55(-4) 3.45(-4) 3.53(-4) 4.28(-4) τ1(V )=4.26(-5)

(a1∆2–X3Π2)‖ 2.17(-4) 2.15(-4) 2.13(-4) 2.09(-4) 2.07(-4)

(a1∆2–1)⊥ 4.07(-4) 3.95(-4) 3.80(-4) 3.75(-4) 3.63(-4) τa1∆2=1.42(-4)

(5Π2–X3Π2)‖ 6.02(-5) 5.76(-5) 5.62(-5) 5.94(-5) 6.94(-5)

(5Π2–1)⊥ 1.31(-3) 1.42(-3) 1.47(-3) 1.36(-3) 1.35(-3) τ5Π2=5.76(-5)

(5Π3–X3Π2)⊥ 1.31(-3) 1.11(-3) 9.85(-4) 8.65(-4) 7.70(-4)


B. PbC+

No spin allowed transition is expected for PbC+ within 30 000 cm−1 of energy. The

excited 34Σ− and 24Π states are primarily capable of emitting to the ground state. The

154

24Π state predissociates, while 34Σ− has a large bond distance. The computed radiative

lifetime for the 34Σ−–X4Σ− transition is 25 µs at υ′=0. As Table 6.14 shows, the lifetime

decreases rapidly with the increase in vibrational quantum number. A similar kind of tran-

sition was also predicted for SnC+ (section 5.3.3) with a radiative lifetime of the order of

few nanosecond. Six dipole allowed transitions along with partial radiative lifetimes of the

upper states of PbC+ are displayed in Table 6.14. The parallel component of 22Π3/2–X4Σ−3/2

has been predicted to be the strongest one. We have plotted the transition moment val-

ues for parallel and perpendicular components of six transitions in Fig. 6.5b. The parallel

component transition moments look similar; all of them have a highest peak near the Franck-

Condon overlap zone and subsequently they tend to zero value as the internuclear distance

increases. In general, parallel transitions involving 3/2-3/2 components are stronger than

the perpendicular transitions involving the corresponding 3/2-1/2 components. Radiative

lifetimes in the 22Π3/2 state for the (22Π3/2–X4Σ−1/2)⊥ transition is roughly 10 times larger

than that for (22Π3/2–X4Σ−3/2)‖. This is because of the fact that in the Franck-Condon region

the transition moment values for the parallel component are almost three times than those

of the perpendicular one. The transition moment value for the parallel component of the

22Π3/2–X4Σ−3/2 transition is maximum (0.245 ea0) at r= 4.9 a0. The radiative lifetime in the2Π3/2 state is about 60 µs at υ′=0. Although a low-lying 2∆5/2 has been isolated around

8900 cm−1, the radiative lifetime in that state could not be measured because of its very low

binding property.

Table 6.14 Radiative lifetime (s) of some of the excited states of PbC+

Transition Partial lifetime of the upper state ata Total lifetime

υ′=0 υ′=1 υ′=2 υ′=3 at υ′=0

34Σ−–X4Σ− 2.50(-5) 3.10(-6) 8.08(-7) 3.33(-7)

(2Π1/2–X4Σ−1/2)‖ 1.61(-4) 1.82(-4) 1.92(-4) 1.96(-4)

(2Π3/2–X4Σ−3/2)‖ 6.25(-5) 7.30(-5) 7.97(-5) 9.70(-5)

(3/2(III)–X4Σ−3/2)‖ 4.37(-5)

(3/2(III)–X4Σ−1/2)⊥ 7.19(-4) τ3/2(III)=4.12(-5)

(22Π3/2–X4Σ−3/2)‖ 1.15(-5) 1.30(-5)

(22Π3/2–X4Σ−1/2)⊥ 1.08(-4) 1.09(-4) τ22Π3/2=1.04(-5)


155

156

3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6(a)

3∆

23Π

1Σ

-

3Σ

+

1∆

21Σ

+

1Σ

+

1Π

A3Σ

-5Π

X3Π

Dip

ole

Mom

ent /

ea 0

Bond Length / a0

3 4 5 6 7 8-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4(b)

2Σ+

2∆

2Σ

-

22Π

2Π

4Σ

+

4∆

X4Σ

-

Dip

ole

Mom

ent /

ea 0

Bond Length / a0

Fig. 6.6 Computed dipole moment functions of few low-lying states of (a) PbC and (b) PbC+

6.3.4 Dipole moments and ionization energies

The computed dipole moment as a function of bond length for different Λ-S states of

PbC are plotted in Fig. 6.6a. Their corresponding equilibrium values (µe) are also listed in

Table 6.4. The computed µe varies from 1.83 to 3.74 D for the low-lying Λ-S states. Fig. 6.6a

shows that dipole moment tends to zero at longer bond distances as expected for the neutral

molecules. The ground state dipole moment of PbC is reported to be 3.0 D with Pb+C−

polarity. The π1 → σ2 transition not only increases the bond length but also increases the

charge separation as σ2 is more localized on C atom. Consequently, the molecule is more

polar in its A3Σ− state with a greater dipole moment of 3.28 D. All the other excited states

except 21∆ have smaller dipole moment than the ground state. The 21∆ state has the highest

dipole moment of 3.74 D due to very long bond length of 2.54 A. Inspite of their longer re,

the states like 1Σ−, 3∆, 3Σ+ etc., which arise from π1 → π2 electronic transition, have the

dipole moments less than 2 D. The reason behind it is the localization of antibonding π2

more on Pb and thus a π back donation from C→Pb decreases the µes.

Fig. 6.6b represents the dipole moment functions of several low-lying states of PbC+. The

numerical values are computed keeping Pb at the origin of the Cartesian coordinate system.

The corresponding dipole moment function of the ground state (X4Σ−) passes through a

minimum, while those of 4∆, and 4Σ+ pass through maximum. Rest of the curves look

similar; they have a minimum followed by a maximum point, and all the curves converge to

zero value at very large bond distances. In Table 6.15, we have presented the equilibrium

dipole moments of some of the low-lying states of PbC+. Here the values have been calculated

by keeping the origin at the the center of mass of the ion. The ground-state dipole moment is

predicted to be -1.11 D, while that of the first excited state is -0.36 D. Inspite of an increase

in bond length the decrease in µe may be attributed to the electronic transfer from more

polarized π1 bonding MO to less polarized σ2 bonding orbital. The dipole moment of 4∆

and 4Σ+ are extremely large in consistent with the lighter homologue.13

Table 6.15 gives the calculated values of vertical as well as adiabatic ionization energies

to the low-lying states of the cation in the Λ-S level, whereas in Table 6.16 we have given

the same values after the spin-orbit correction. Ionization of PbC to the ground-state PbC+

requires the energy of 7.26 eV which is approximately three times the ground-state dissoci-

ation energy of PbC at the Λ-S level. As the cation has to some extent longer bond length,

the adiabatic energy separation between these two states is somewhat lass than the VIE.

With the inclusion of spin-orbit interaction the VIE is changed to be 7.40 eV. However, no

157

photoelectron measurement is available for comparison.

Table 6.15 Ionization energies of PbC to some

low-lying states of PbC+ and their µes


X4Σ− -1.11 7.26 7.202Π -0.36 8.59 8.212∆ -0.06 8.55 8.382Σ− -0.06 8.67 8.522Σ+ 0.64 8.94 8.67

22Π -0.51 8.99 8.914∆ 1.28 10.39 8.954Σ+ 1.29 10.47 8.98

22Σ− -0.04 10.65 9.27

32Π 2.33 11.20 10.08

22∆ 1.02 11.58 10.12

22Σ+ 1.51 11.68 10.26

32Σ− 1.31 12.34 10.33

42Π 0.60 11.37 10.72

34Σ− -1.33 12.92 11.99

24Π -0.71 12.43 12.42


b At re=4.05 a0 of X3Π of PbC

Spin-orbit effects change the dipole moments of both PbC and PbC+ to a large extent. For

the neutral molecule, the spin-orbit ground state (0+) has a greater dipole moment of 3.15 D

compared to that of the Λ-S ground-state. A strong coupling between the 0+ components of

X3Π and A3Σ− increases the µe of 0+. The spin-orbit coupling reduces the dipole moments

of both the spin components X4Σ−3/2 and X4Σ−1/2 of the cationic ground-state by an amount

0.20 D. The equilibrium dipole moments of a few lower roots of the PbC and PbC+ are given

in Table 6.16.

158

Table 6.16 Spin-orbit corrected dipole moments and ionization energies

Molecule State µe(D) Ion State µe(D)a VIE(eV)b AIE(eV)

PbC 0+ 3.15 PbC+ X4Σ−3/2 -0.89 7.40 7.39

X3Π2 2.90 X4Σ−1/2 -0.90 7.44 7.43

X3Π1 3.06 2Π1/2 0.15 8.41 8.17

X3Π0− 3.02 2Π3/2 0.04 8.45 8.25

1(II) 3.02 2∆5/2 0.67 8.63 8.49

0+(II) 2.93 2Σ−1/2 0.05 8.71 8.58

a1∆2 2.78 3/2(III) -0.02 8.75 8.68


b At re=4.20 a0 of the 0+ state of PbC

6.3.5 Comparison of some spectroscopic properties of MC and MC+ (M= Si, Sn,

Pb)

Though experimentally verified only for SiC, X3Π is the proposed ground state for all

the carbides of Group IVA elements. However, there is an excited 3Σ− state the relative

energy of which gradually decreases from SiC to PbC. Consequently, it increases the relative

population in the A3Σ− thereby decreasing the intensity of A–X band. Table 6.17 compares

the computed spectroscopic properties of some low-lying electronic states of MC (M= Si, Sn,

Pb) molecules. With increasing the molecular mass the equilibrium bond length increases

while ωe and De tend to decrease substantially. Spin-orbit effect rigorously changes the

ground-state dissociation energy for heavier species. Thus after the spin-orbit correction the

ground-state dissociation energy of PbC becomes almost half of that of SiC. After the spin-

orbit coupling the ground state splits in an inverted order, being X3Π2 the lowest component

both for SiC and SnC. But in case of PbC, with the additional mixing of A3Σ−0+ , 0+ becomes

the lowest root which has essentially equal contribution from X3Π and A3Σ−. The spin-orbit

coupling does not change the equilibrium geometries of the molecules, except for PbC. With

increasing bond length from 1.74 A (for SiC) to 2.15 for PbC the µe value increases from

1.62 D to 3.00 D indicating a greater charge separation in PbC. It also reflects the more ionic

character in PbC. It can be shown that SiC is 19.4% ionic in nature, whereas, percentage of

ionic character in Sn-C and Pb-C bonds are about 25 and 29, respectively. The first singlet

state of both SiC and SnC is reported to be 1Σ+, but it is of ∆ symmetry for PbC. This is

159

Table 6.17 Comparison of some spectroscopic properties of

MC (M=Si, Sn, Pb) molecules

State Propertya SiC SnC PbC

X3Π re/A 1.74 2.02 2.15

ωe/cm−1 930 646 579

De/eV 4.05 3.06 2.46

µe/D 1.62 2.44 3.00

(2.42)b (3.15)b

VIE/eV 8.84 7.70 7.26

(7.77)b (7.40)b

zfsc/cm−1 100 1063 >3000

A3Σ− Te/cm−1 3985 3775 2136

re/A 1.82 2.12 2.24

ωe/cm−1 857 590 5341Σ+ Te/cm−1 5325 6505 6683

re/A 1.68 1.94 2.17

ωe/cm−1 975 635 3535Π Te/cm−1 14 460 9620 6651

re/A 1.97 2.24 2.37

ωe/cm−1 635 475 431

A3Σ− τ (at υ′=0)/s 1.25(-4) 2.20(-4) 1.40(-3)

23Π τ (at υ′=0)/s 5.03(-6) 6.50(-3) 1.56(-5)

33Π τ (at υ′=0)/s 1.10(-6) 3.70(-5) 9.41(-6)

31Σ+ τ (at υ′=0)/s 4.67(-7) 1.06(-6) -5Π0+ τ (at υ′=0)/s - 1.80(-3) 1.14(-4)

a Subscript e refers to equilibrium property

b After spin-orbit correction

c Zero field splitting

because of the avoided crossing between the first two 1Σ+ states. Analysis of the CI wave

functions show that extent of mixing between these two states increases from SiC to PbC.

Thus their relative energy separation gradually decreases. As a result, the 1∆ state becomes

160

the lowest root of singlet, in case of PbC. The components of 5Π gradually become spec-

troscopically more important on going downwards the group. 5Π state is quickly stabilized

from SiC to PbC. On the other hand, with increasing the atomic mass of M (M=Si, Sn, Pb)

the extent of spin-orbit coupling increases enormously. Thus the spin forbidden transitions

involving the components of 5Π and the ground-state components become more and more

prominent. Not only 5Π, all other states coming from the π1 → π2 electronic transition

are stabilized due to a greater stabilization of the antibonding π2 MO. However, the total

radiative lifetime at the lowest vibrational level of 5Π0+ of PbC is about 15 times less than

that of SnC.

Table 6.18 Comparison of some spectroscopic properties of

MC+ (M=Si, Sn, Pb) ions

State Propertya SiC+ SnC+ PbC+

X4Σ− re/A 1.83 2.11 2.24

ωe/cm−1 817 591 543

De/eV 3.32 2.74 2.12

(2.15)b (1.25)b

µe/D 1.19 -0.85 -1.11

zfsc/cm −1 0 21 2862∆ Te/cm−1 10 266 10 032 9475

re/A 1.88 2.17 2.33

ωe/cm−1 723 485 4162Π Te/cm−1 10 695 8965 8183

re/A 1.99 2.33 2.44

ωe/cm−1 480 423 392

22Π Te/cm−1 14 311 14 184 13 767

re/A 1.87 2.13 2.26

ωe/cm−1 1013 645 476

4E (22Π–2Π) 3616 5219 55844Π Te/cm−1 24 464 24 501d 23 703d

re/A 1.70

ωe/cm−1 875

Be/eV 0.37

161


State Propertya SiC+ SnC+ PbC+

24Π Te/cm−1 35 254 35 330 42 151

re/A 1.85 2.03 2.09

ωe/cm−1 965 785 582

Be/eV 0.86 0.41 0.044Π τ (at υ′=0)/s 6.06(-6) - -

24Π τ (at υ′=0)/s 1.10(-7) 8.37(-8) -

34Σ− τ (at υ′=0)/s - 2.56(-7) 2.50(-5)

a Subscript e refers to equilibrium property

b After spin-orbit correction

c Zero field splitting

d Vertical excitation energy

Table 6.18 displays a comparative study of the spectroscopic constants of some of the low-

lying states of the monopositive ions, SiC+, SnC+, and PbC+. In each case, the ionization

involves the removal of a π1 bonding electron from the molecular ground state. Consequently,

the resulting cation has greater bond length than the corresponding neutral species. As a

periodic trend, the binding energy of the ground-state ions decreases monotonically down

the group. The re and ωe values of them also follow the expected trend. Although the zero

field splitting is very low for all the cations, the spin-orbit interaction largely affects the

dissociation energy of the ground state for SnC+ and PbC+. It is only 1.25 eV for X4Σ−3/2

of PbC+. The first excited state of SiC+ belongs to the 2∆ symmetry, whereas 2Π is the

first excited state for SnC+ and PbC+. The 2Π state has a double well adiabatic potential

for SiC+, the minimum of 22Π is situated just 3616 cm−1 above it. Now, with the increase

in the extent of the avoided crossing the separation between the two minima increases and2Π falls below 2∆ for the later two cations. The 4Π state of SiC+ is bound with a binding

energy of 0.37 eV only, but for other two cations the state is repulsive. On the other hand,

24Π is observed with potential minimum for all the three cations, however, the depth of the

potential well gradually decreases from SiC+ to PbC+.

162

6.4. Summary

Using relativistically corrected pseudo core potential, abinitio based MRDCI calculations

have been performed on the heaviest diatomic carbide of the group IVA and its monopositive

ion.28 Potential energy curves along with the spectroscopic properties of low-lying Λ-S states

are reported for the first time. The neutral PbC has X3Π ground-state and A3Σ− is lying

only 2136 cm−1 above it. On the other hand, due to the removal of one π1 electron from the

ground-state PbC, the resulting cation has 0.09 A longer bond length in its X4Σ− ground-

state. The spectroscopic constants of the states of PbC and PbC+ are largely affected by

spin-orbit interaction. The ground state of PbC splits into four Ω components with a large

equilibrium separation. Unlike other carbides in group IVA, it has the ground state of 0+

symmetry. Total spin-orbit splitting in the first dissociation limit matches well with the

experimental observation of 10 693 cm−1. The spin-orbit interaction splits Pb+(2P)+C(3P)

into six sub-levels which are overall 14 124 cm−1 apart. The computed spin-orbit corrected

IP of Pb is in good agreement with the experimental observation of 7.415 eV.

Several spin allowed or spin forbidden transitions are reported for PbC. Transitions to

23Π, 33Π, 23∆, and 33Σ− are of great interest. Dipole allowed transition lifetimes of the 5Π

components are noticeable, many of them are less than 100 µs. d1Σ+0+ also has total radiative

lifetime of 60 µs at its lowest vibrational level. Only one spin allowed transition is expected

for PbC+, but some dipole allowed transitions are important to be observed. Total radiative

lifetime at υ′=0 state of 2Π3/2 and 22Π3/2 are about 60 and 10 µs, respectively. Dipole and

transition dipole moment functions of several states of PbC and PbC+ are presented. The

equilibrium dipole moment of the ground-state PbC is predicted to be 3.00 D. The dipole

moments of some of the low-lying states of PbC+ are also reported. The spin-orbit coupling

increases the ground-state equilibrium dipole moment of PbC which is reported to be 3.15 D.

On the other hand similar effect reduces the µe of PbC+ from 1.11 D to 0.89 D. Though these

are origin dependent, they may be useful for the analysis of microwave spectra. The vertical

and adiabatic ionization energies of PbC to many low-lying states of PbC+ are reported, but

no experimental result is found for comparison. Inclusion of the spin-orbit effect predicts

the requirement of 7.40 eV of energy to ionize the molecule PbC.

163

6.5. References

1 K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV: Constants

of Diatomic Molecules, Van Nistrand Reinhold, New York, 1979.

2 A. Druzhinin, I. OstrovSkii, and I. Kogut, Mater. Sci. Semicond. Process. 9, 853 (2006).

3 J. Tolle, R. Roucka, V. D’costa, J. Menendez, A. Chizmeshya, and J. Kouvetakis, Mater.

Res. Sco. Symp. Proc. 891, 579 (2006).

4 J. Tolle, A.V.G. Chizmeshya, Y.-Y. Fang, J. Kouvetakis, V.R. D’costa, C.-W. Hu,

J. Menendez, and I.S.T. Tsong, Appl. Phys. Lett. 89, 231924 (2006).

5 G. Gigli, G. Meloni, M. Carrozzino, J. Chem. Phys. 122, 14303 (2005).

6 A. Ciccioli, G. Gigli, G. Meloni, E. Testani, J. Chem. Phys. 127, 54303 (2007).

7 G. Li, X. Xing, and Z. Tang, J. Chem. Phys. 118, 6884 (2003).

8 P.A. Denis and K. Balasubramanian, J. Chem. Phys. 123, 54318 (2005).

9 B. Suo and K. Balasubramanian, J. Chem. Phys. 126, 224305 (2007).

10 L.T. Ueno, L.R. Martin, A.D. Pino Jr., F.R. Ornellas, F.B.S. Machado, Chem. Phys.

Lett. 432, 11 (2006).


12 Electronic Spectrum of SnC: A Theoretical Study,


13 A. Pramanik, S. Chakrabarti, K.K. Das, Chem. Phys. Lett. 450, 221 (2008)

14 R.B. Ross, W.C. Ermler, P.A. Christiansen, J. Chem. Phys. 93, 6654 (1990).





19 R.J. Buenker, in: P. Burton (Ed.), Proc. Workshop on Quantum Chemistry and

Molecular Physics in Wollongong, Wollongong, Australia, 1980.

20 R.J. Buenker, in: R. Carbo (Ed.), Studies in Physical and Theoretical Chemistry, vol. 21,

Current Aspects of Quantum Chemistry, Elsevier, Amsterdam, p.17, 1982.

164


22 R.J. Buenker, R.A. Philips, J. Mol. Struct. (Theochem) 123, 291 (1985).



Reidel, Dordrecht, The Netherland, 1974.

25 G. Hirsch, P.J. Bruna, S.D. Peyerimhoff, R.J. Buenker, Chem. Phys. Lett. 52, 442

(1977).

26 A.B. Alekseyev, R.J. Buenker, H.-P. Lieberman, G. Hirsch, J. Chem. Phys. 100, 2989

(1994).

27 C.E. Moore, Tables of Atomic Energy Levels, vols. I–III, U.S. National Bureau of

Standards: Washington, DC, 1971.

28 Electronic Structure and Spectroscopic Properties of PbC and PbC+: an MRDCI Study,


165

CONCLUSION

The structural and spectroscopic aspects of intragroup IVA heteronuclear diatomic molec-

ules like SiC, SnC, PbC and some of their monopositive and mononegative ions have been

studied in the present thesis. It deals with the analysis of the results of abinitio based quan-

tum mechanical calculations on the above species. A high-level multireference singles and

doubles configuration interaction method, which incorporates relativistically corrected effec-

tive core potentials and spin-orbit coupling, has been employed throughout the calculations.

The diabatic as well as adiabatic potential energy curves are constructed for the ground and

low-lying electronic states. These provide different types of spin-forbidden and spin-allowed

transitions which are possible within experimental ranges. The dipole and transition dipole

moment functions are also constructed for these molecules and ions. The calculated ioniza-

tion energies and electron affinities match well with the available experimental and other

theoretical data.

Of all the species studied here, experimental results of SiC are available only. Although

SiC radical was first studied theoretically, later on many experimental spectroscopic studies

on the ground state (X3Π) as well as a few triplets like A3Σ−, B3Σ+, C3Π etc. were reported.

The calculated spectroscopic constants of several low-lying states of SiC agree well with the

experimental results. Besides the triplets, several bound states of singlet and quintet spin

multiplicities are computed here. The lowest quintet bound state 5Π is located around

14 460 cm−1. An excited 3Π namely E3Π is isolated for the first time in our calculation

with Te=25 875 cm−1. The ground-state dipole moment is calculated to be 1.62 D with a

Si+C− polarity. The molecule has the highest dipole moment of 2.55 D in its A3Σ− state.

The sense of polarity is same for other excited states also. The spin-orbit coupling is almost

insignificant for SiC. Due to the spin-orbit interaction, the largest splitting among the four

spin components of X3Π is only 100 cm−1. Many spin allowed transitions are studied for

the radical. The observed E–X transition is predicted to be the strongest one. A radiative

lifetime of 1.1 µs is estimated for this transition. 31Σ+–a1Σ+ is also an important transition

for SiC. It has a very short lifetime of the order of 450 ns in its lowest vibrational level.

Unlike the neutral species, a very few experimental data are available for the unicharged

ions of SiC. The computed ground state of SiC+ is X4Σ− while its dissociation energy of

3.32 eV matches well with the experimental data. An asymmetric double well is observed in

the potential energy curve of the lowest doublet state of SiC+ (2Π). Like the doublets, the

lowest two 4Π states undergo avoided crossing. Their strong interaction reduces the binding

energy of 4Π. On the other hand, 24Π has a sharp potential minimum. The potential energy

166

curves of SiC− are also of great interest. Only the ground (X2Σ+) and two other excited

states of the anion were studied before. We have computed two new doublets, namely C2Π

and D2Π which are spectroscopically important. A number of hitherto unknown quartets

and sextets of SiC− are also predicted. Like the SiC radical, spin-orbit interactions have

almost no influence on the spectroscopic properties of the ions. The 24Π-X4Σ− transition is

predicted to be the strongest one for SiC+ with a partial radiative lifetimes of about hundred

nanoseconds at the lowest few vibrational levels. In case of SiC−, A2Π–X2Σ+ and B2Σ+-A2Π

transitions are not predicted to be strong, but three strong transitions, B-X, C-X, and D-X

are expected to be observed in the range 21 000-24 000 cm−1. The last two transitions are

predicted for the first time in the present study. The present thesis also computes the dipole

moments of the ions, vertical and adiabatic ionization energies and electron affinities of SiC

which compare well with the available data.

The spectroscopic properties of SnC and SnC+ are not well known in literature. In

the present thesis, the CI calculations have been performed on these two species. The

ground state of SnC belongs to the X3Π symmetry with re=2.023 A and ωe=646 cm−1. It is

characterized by an open shell configuration, σ21σ2π

31. At least 30 excited singlet, triplet and

quintet states of Λ-S symmetries have been predicted within 6 eV of energy. The ground-

state dissociation energy of the molecule is estimated to be 3.06 eV which is reduced to

2.87 by spin-orbit interaction. The computed dipole moment of SnC in the ground state

is about 2.44 D with Sn+C− polarity. Three triplet-triplet transitions such as B3Π-X3Π,

33Π-X3Π, and 53Π-X3Π are expected to be strong for SnC. Besides these, transitions to

excited 31Σ+ from the lower singlets are expected to have enough intensities. Total radiative

lifetime of 31Σ+ in the lowest vibrational level is found to be 1.05 µs. Ionizing the molecule

by the energy of the order of 7.70 eV it reaches to 4Σ− ground-state of SnC+. Besides the

ground state, potential energy curves of 16 more excited states of doublet and quartet spin

multiplicities are constructed in the present thesis. Spectroscopic constants are not largely

affected by the spin-orbit coupling, but several spin forbidden transitions like 2Π3/2–X4Σ−3/2,

22Π1/2–X4Σ−1/2, 2Σ+1/2–X4Σ−1/2 etc. are expected to have significant intensities.

The spectral behaviors of the heaviest carbide of group IVA are somewhat different from

the lighter species due to a strong spin-orbit coupling. The ground state of PbC is computed

here as X3Π which is in analogy with the lighter carbides of group IVA and some other

intragroup IVA diatomics. The first excited state of the molecule (A3Σ−) lies only 2136

cm−1 above the ground state. The ground-state dissociation energy of PbC is reported to be

167

1.97 eV which matches well with the previously estimated value. The adiabatic separation

among the spin-orbit components of the ground state exceeds 3000 cm−1. The overall split-

ting in the dissociation limit agrees well with the experimental observation in the atomic

level. The computed ionization energy of Pb is also in good agreement to the experimentally

observed value. Many dipole allowed transitions of PbC are predicted for the first time.

Transitions involving the components of 5Π, d1Σ+, a1∆ etc. and the corresponding ground-

state components are found to be important. The PbC+ ion has X4Σ− ground-state with

excited 2Π and 2∆. Total radiative lifetime of the 22Π3/2 state of PbC+ at the ground vibra-

tional level is of the order of 10 µs. The spin-orbit corrected ground-state dipole moments

of PbC and PbC+ are 3.15 and -0.89 D, respectively. With the inclusion of the spin-orbit

interaction the vertical ionization potential of PbC is reported to be 7.40 eV.

Throughout the theses we have presented a detailed structural and spectroscopic infor-

mation of intragroup IVA heteronuclear diatomics from the state-of-the-art ab initio based

CI calculations. Emphasis has been given on the carbides of group IVA and some of their

monopositive and mononegative ions. We hope these spectroscopic data would be very

helpful to the experimentalist in future.

168

List of Publications

∗1. Electronic States and Spectroscopic Properties of SiTe and SiTe+.

Surya Chattopadhyaya, Anup Pramanik, Amartya Banerjee, and Kalyan Kumar Das,

J. Phys. Chem. A 110, 12303 (2006).

∗2. Ab initio configuration interaction study of the low-lying electronic states of InF.

Amartya Banerjee, Anup Pramanik, Kalyan Kumar Das,

Chem. Phys. Lett. 429, 62 (2006).

∗3. B2Σ+-X2Σ+ and C2Π-X2Σ+ transitions in InF++: A configuration interaction study.

Amartya Banerjee, Anup Pramanik, Kalyan Kumar Das,

Chem. Phys. Lett. 435, 208 (2007).

4. The electronic spectrum of the SiC radical: A theoretical study.

Anup Pramanik, Kalyan Kumar Das, J. Mol. Spectrosc. 244, 13 (2007).

5. Theoretical studies of the electronic spectrum of SiC+.

Anup Pramanik, Susmita Chakrabarti, Kalyan Kumar Das,

Chem. Phys. Lett. 450, 221 (2008).

∗6. MDRCI studies on the electronic states of InBr and InBr+.

Amartya Banerjee, Anup Pramanik, Susmita Chakrabarti, Kalyan Kumar Das,

J. Mol. Struc. (THEOCHEM) 893, 37 (2009).

7. Electronic states of SiC−: A theoretical study.

Anup Pramanik, Amartya Banerjee, Kalyan Kumar Das,

Chem. Phys. Lett. 468, 124 (2009).

8. Electronic spectrum of SnC: A theoretical study.


9. Theoretical investigation of electronic states of SnC+.


10. Electronic structure and spectroscopic properties of PbC and PbC+: an MRDCI study.

A. Pramanik, A. Banerjee, K.K. Das (to be communicated).

∗Papers are not included in the thesis

169

Documents

THEORETICAL STUDIES ON THE SPECTROSCOPY OF SOME INTRAGROUP IVA HETERONUCLEAR DIATOMIC MOLECULES AND THEIR IONS