Molspec Combined

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Molecular Spectroscopy and Structure

by

Peter F. Bernath

Departments of Chemistry and PhysicsUniversity of Waterloo

Waterloo, OntarioCanada N2L 3G1


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15.1 INTRODUCTION

15.2 ROTATIONAL SPECTROSCOPY

15.2.1 Diatomics15.2.2 Linear Molecules15.2.3 Symmetric Tops15.2.4 Asymmetric Tops15.2.5 Spherical Tops

15.3 VIBRATIONAL SPECTROSCOPY

15.3.1 Diatomics15.3.2 Linear Molecules

15.3.3 Symmetric Tops15.3.4 Asymmetric Tops15.3.5 Spherical Tops15.3.6 Raman Spectroscopy

15.4 ELECTRONIC SPECTROSCOPY

15.4.1 Diatomics15.4.2 Polyatomics

15.5 STRUCTURE DETERMINATION

15.6 REFERENCES


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15.1 INTRODUCTION

Our understanding of rotational-vibrational-electronic (rovibronic) spectra of molecules is

based on the non-relativistic Schrdinger equation [1],

(15.1)

The Born-Oppenheimer approximation is used to separate electronic and nuclear motion and thenthe nuclear motion is further assumed to be separable into vibrational and rotational motion, leadingto the simple equations

(15.2)

and

(15.3)

For molecules with net electronic spin and net electronic orbital angular momentum, additional termssuch as spin-orbit coupling need to be added to the Hamiltonian of equation (15.1).

The manifold of energy levels described by equation (15.2) are connected by transitions asdetermined by selection rules. More generally [2], an absorption line between energy levels E1 andE0 is represented by Beers law

(15.4)

where I0 is the initial radiation intensity,F is the cross-section (in m2), N0-N1 is the population

density difference (m-3) and R is the path length (m). The intrinsic line strength of a transition is thusmeasured by a cross-section, which is proportional to the square of a transition moment integral, i.e.

(15.5)

where p is a transition moment operator and A is the Einstein A factor for emission. Selection

rules and line strengths are obtained by a detailed examination of equation (15.5).

15.2 ROTATIONAL SPECTROSCOPY

All gas phase molecules have quantized rotational energy levels and pure rotationaltransitions are possible. A molecule can, in general, rotate about three geometric axes and can havethree different moments of inertia relative to these axes. The moment of inertia about an axis is


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defined as

(15.6)

where mi is the mass of the atom i and riz is the perpendicular (shortest) distance between this atomand the axis. The internal axis system of a molecule is chosen to have its origin at the center of massand is rotated so that the moment-of-inertia tensor is diagonal [2]. This is the principal-axis systemfor a rigid molecule. The three moments of inertia Ix, Iy and Izcan be used to classify molecules intofour different types of tops:

1. Linear molecule (including diatomics), Ix = Iy ; Iz = 0, e.g. CO, HCCH.

2. Spherical top, Ix = Iy = Iz, e.g. CH4, SF6.

3. Symmetric top, Ix = Iy Iz, e.g. BF3, CH3Cl.

4. Asymmetric top, Ix Iy Iz, e.g. H2O, CH3OH.

The internal molecular axes x, y, and z are labeled according to a certain set of rules based onmolecular symmetry [3]. An additional labeling scheme is also used that is based on the size of themoments of inertia. In particular, the axis labels A, B, and C are chosen to make the inequality IA#IB# IC true. Thus molecular symmetry determines the x, y and z labels but it is the size of themoments of inertia that set the A, B and C labels. In terms of the A, B and C labels, it isconventional to classify molecules into five categories:

1. Linear molecules, IA = 0, IB = IC.

2. Spherical tops, IA = IB = IC.

3. Prolate symmetric tops, IA


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15.2.1 Diatomics

For a rigid diatomic molecule in a 1E+ electronic state (no net spin or orbital angular

momentum) the rotational energy levels are given by

, (15.7)

where B is the rotational constant and J, the rotational quantum number, has values 0, 1, 2, ... Theunits of (15.7) are determined by the units chosen for B, which are generally cm-1, MHz or (rarely)J (joules).

Various equations for B are:

B (joules) = (15.8)

B (MHz) = 10-6 = , (15.9)

B (cm-1) = = (15.10)

in which I, the moment of inertia, is defined by

(15.11)

and the masses mA and mB are separated by a distancer. The reduced mass : of the AB molecule

(15.12)

is conventionally calculated using atomic (not nuclear) masses [5]. The use of a single symbol Bfor three separate physical quantities (energy, frequency and wavenumber) is clearly confusing butis the spectroscopic custom. Thus spectroscopists talk about energy levels but locate them usingcm-1 units.

A real molecule is not a rigid rotor because the bond between atoms A and B can stretch at


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the same time as the molecule rotates. As rotation increases, the centrifugal force stretches the bond,increasing r and decreasing the effective B value. The bond length also depends, in an average sense,on the vibrational state v. The non-rigid rotor energy level equation for vibrational state v is,

Fv(J) = BvJ(J + 1) - Dv[J(J + 1)]2 + Hv[J(J + 1)]

3 + Lv[J(J + 1)]4+ ... (15.13)

where Dv, Hv and Lv are centrifugal distortion constants. The vibrational dependence of the rotationand distortion constants is parameterized by

, (15.14)

, (15.15)

where "e, $e and (e are vibration-rotation interaction constants, and the Be and De values refer to theextrapolated equilibrium values at the bottom of the potential energy curve. Much of theconventional notation for spectroscopic constants is based upon Herzbergs three books [6-8], thework of Mulliken [3] and recent updates [5].

Each vibration state, v, has an effective internuclear separation, rv, defined by the equations

(15.16)

and (15.17)

There is a useful relationship (due to Kratzer [6]) for estimating the centrifugal distortion constant:

(15.18)

where Te is the equilibrium vibrational constant, eq. (15.60).

A typical potential energy curve is often approximated (for semi-quantitative work) as thatfor a Morse oscillator, viz.

, (15.19)


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in which De is the equilibrium dissociation energy, [not the equilibrium centrifugal distortionconstant appearing in equation (15.14)]. For the Morse oscillator, the main vibration-rotationinteraction term, "e, is given by the Pekeris relationship [6],

(15.20)

with the vibrational constants Teand Texe given by equation (15.60).

The selection rules for a pure rotational transition are )J = 1 so that the frequencies for aJ + 1 7 J transition are given by

vJ+17J = 2Bv(J + 1) - (4Dv - 2Hv)(J + 1)3 + 6Hv(J + 1)

5 + ... (15.21)

Thus the pure rotational transitions are a series of lines separated by approximately 2B. Excitedvibrational levels create a similar series of vibrational satellites near the main transitions for thev=0 vibrational level.

The cross-section for an absorption transition E17 E0 is given by [2]

(15.22)

and the Einstein A coefficient for emission is

(15.23a)

(15.23b)

where g0 is the permittivity of the vacuum, M10 is the transition dipole moment and g(


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, (15.25)

where )


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to the usual energy level expression. This is a second-order Stark effect because it depends upon thesquare of the electric field, E, and the energy level expression is derived using second-orderperturbation theory. The measurement of Stark splittings in rotational transitions is one of the

primary methods for measuring dipole moments.

We have ignored the possibility of a net electron spin and/or a net orbital angular momentum.If either one is present, all of the energy level expressions are modified and each rotational transitionwill have fine structure. For example, if a molecule has a single unpaired electron, S= (2E+ state),then all of the energy levels and transitions will be doubled [6].

The presence of nuclear spins in a molecule will also split the energy levels into components

and hyperfine structure will appear in the rotational transitions. In general, a nuclear spin will

vector couple with the rotational angular moment , viz.

, (15.31)

to give a total angular moment . The hyperfine structure can split a line into a maximum of 2I +1 components (J $ I) each labeled by an F value. The study of the fine and hyperfine structure ofrotational transitions is often a complicated, but well understood, task [9,10,11].

15.2.2 LINEAR MOLECULES

The rotational energy level expressions for diatomic molecules apply directly to linearmolecules in 1E+ electronic states. The only change is that each subscript v, e.g. in Bv, is to be

interpreted as a collection of vibrational quantum numbers and there is the possibility of new effectsfor the excited vibrational states. Such an effect is R-type doubling [12], which adds a term

(15.32)

to the energy level expression in the case of doubly-degenerate bending vibrational levels. The R-type doubling constant q measures the splitting of the rotational line into two R-doublet components.

It is the presence of vibrational angular momentum that is responsible for this effect.

The 3N-5 vibrational modes in a linear molecule also modify the vibration-rotationinteraction terms and the expression for Bv becomes

, (15.33)

where di is the degeneracy of the ith vibrational mode.


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15.2.3 SYMMETRIC TOPS

Symmetric top molecules have an additional rotational quantum number K which measures

the component of along the top (molecular symmetry) axis. Thus the rotational angularmomentum vector has components along the top axis and along the laboratory z-axis.

The rotational energy level expression for a rigid molecule is given by

EJK = BJ(J + 1) + (A - B)Ka2 (prolate top) (15.34)

EJK = BJ(J + 1) + (C - B)Kc2 (oblate top) (15.35)

with subscripts a and c added to K in order to distinguish the prolate and oblate cases. The rotationalconstants A, B and C are defined by analogy with eq. (15.8),

(15.36)

(15.37)

(15.38)

in energy units. The numerical formulae for B (equations 15.9 and 15.10) are also applicable for Aand C. Each energy level defined by the quantum numbers J and K has a (2J + 1)-fold MJ-degeneracy and a 2-fold K-degeneracy (K>0), in the absence of electric or magnetic fields. Bycustom the quantum number K is positive, with the symbol k occasionally being used when a signedquantum number is needed (i.e., K = |k|, k = -J, ..,0,..J).

For a non-rigid symmetric top the energy level expression becomes

Fv(J,K) = BJ(J + 1) - DJ [J(J + 1)]2 + (A - B) K2 - DKK

4 - DJK J(J + 1) K2 + ...(15.39)

for the prolate case and C replaces A in the oblate case. The centrifugal distortion constants DJ, DK,and DJKand the rotational constants, A, B and C also depend on the vibrational state by analogy withthe diatomic case.

The intensity of a pure rotational transition is proportional to the square of the permanentdipole moment, which (by symmetry) can only lie along the symmetry axis of the top. The selection


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rules are )K = 0 and )J = 1, which result in rotational transitions spaced by approximately 2B.For the non-rigid molecule the rotational transition frequencies for the transition J+1, K7 J, K aregiven by

. (15.40)

Thus, centrifugal distortion causes each line to split into J + 1 components for K = 0, 1, .... J. Sinceeach set of constants applies to a particular vibrational level, vibrational satellites can also arise frompopulation in an excited vibrational level.

The application of an electric field results in a lifting of the MJ degeneracy with an additionalenergy level term E(1), to first order:

. (15.41)

The 2-fold K-degeneracy for K>0 results in a first order Stark effect with energy splittings directlyproportional to the product of the electric field strength, E, and the dipole moment, :, for smallfields. A good collection of dipole moments of molecules as determined by the Stark effect andother methods can be found in the Handbook of Chemistry and Physics [13].

15.2.4 ASYMMETRIC TOPS

There is no general energy level formula available for an asymmetric top molecule. The rigidrotor Hamiltonian

(15.42)

commutes with 2 and z in the laboratory coordinates system so that J and MJ remain good quantum

numbers. The Hamiltonian , however, does not commute with the components of (a, b, c) inthe molecular coordinate system. This means that the 2-fold K-degeneracy of the symmetric top islifted and each J splits into 2J + 1 components. In the asymmetric top case each of these levels is

labeled by an index J = J, ..., 0, ... -J (cf. k for a symmetric top) in order of decreasing energy. Theenergy levels of the asymmetric top are most easily derived using symmetric top basis functions, first

deriving the matrix elements of and then diagonalizing to find the energy eigenvalues andeigenvectors. For each value of J, a (2J + 1) x (2J + 1) matrix results and the 2J + 1 eigenvalues areeasily labeled by J.

The J labeling scheme, however, is not as popular as one based on a correlation diagrambetween the energy levels of a prolate symmetric top and an oblate symmetric top. The degree of


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asymmetry of a top is quantified by the asymmetry parameter6 with

. (15.43)

Values of6 range from -1 for a prolate top (B = C) and +1 for an oblate top (A = B) with values near0 for highly asymmetric tops. The correlation diagram is displayed as Figure 15.1 with prolate tops(J, Ka) on the left and oblate tops (J, Kc) on the right. The parameter6 can be viewed as a continuousparameter with all asymmetric tops lying between these two extreme cases. The non-crossing ruleis used to connect the energy levels, and the two limiting quantum numbers Ka and Kc serve to labeleach level. The relationship between theJJand system of labels is J = Ka - Kc. Notice that Ka

+ Kc = J or J + 1.

The energy levels for the rigid asymmetric top have exact analytical solutions for low valuesof J. A list of these energy levels for J = 0, 1, 2 and 3 are provided in Table 15.1. The energy levelsfor real molecules, however, require the addition of centrifugal distortion terms to the molecularHamiltonian and are obtained by numerical solution of the resulting Hamiltonian matrices [10,11].Fortunately, many asymmetric top molecules are close to either the oblate or prolate limits.

The selection rules for asymmetric tops depend on the components (:a, :b, :c) of thepermanent dipole moment vector, , along the a, b and c principal molecular axes. The selection

rules can be divided into three general cases:

1. a-type transitions when :a 0, )Ka = 0 ( 2, 4, ... ), )Kc = 1 (3, 5, .... )2. b-type transitions when :b 0, )Ka = 1 (3, ... ), )Kc = 1 (3, ... )

3. c-type transitions when :c 0, )Ka = 1 (3, 5, .... ), )Kc = 0 (2, 4 .... )

The transitions in brackets are weaker than the main transitions. A molecule of low symmetry canhave :a:b:c 0 so that transitions of all three types can be found in its rotational spectrum. Forvery asymmetric tops, the pure rotational spectra have a very irregular appearance. We have ignoredthe complications of fine and hyperfine structure as well as internal rotor structure.

Each vibrational level has a set of rotational constants whose vibrational dependence isparameterized by

(15.44)


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(15.45)

(15.46)

with di the degeneracy of the ith mode.

15.2.5 SPHERICAL TOPS

The energy levels of a rigid spherical rotor are given by

F(J) = BJ(J + 1) (15.47)Although this expression is identical to that for a rigid diatomic molecule, the degeneracy of eachlevel is (2J + 1)2 rather than 2J + 1. There is a (2J + 1)-fold MJ-degeneracy and a (2J + 1)-fold K-degeneracy because the spherical top molecule is quantized, like a symmetric top, in both thelaboratory and the molecular coordinate systems. For a non-rigid spherical top, the K-degeneracy canbe partly lifted so that cluster splittings of the energy levels can be seen [12]. Because its highdegree of symmetry, a spherical top has no permanent dipole moment and thus no allowed purerotational transitions.

15.3 VIBRATIONAL SPECTROSCOPY

A molecule must have a permanent dipole moment in order to have allowed pure rotationaltransitions. By contrast, a molecule needs to have its dipole moment change as it vibrates in orderto have an allowed vibrational spectrum, i.e., vibrational spectra depend upon dipole momentderivatives. This condition is much less restrictive so that all molecules apart from homonuclear

diatomics (which have both = 0 and ) have allowed vibrational spectra. For a polyatomic

molecule, however, not all of the 3N-6 vibrational modes (or 3N-5 for a linear molecule) willnecessarily be infrared active. For a molecule of sufficiently low symmetry, such as water, the 3vibrations all appear in the infrared spectrum. The classification of molecules using a set of symmetry

operations requires the application of group theory [2,4,14] and cannot be succinctly summarized.We will use the results of this symmetry classification as labels for vibrational and electronic states[3]. It can be shown that these labels, which are based on the irreducible representations of theappropriate molecular point group, can be applied to the individual rotational, vibrational andelectronic wavefunctions as well as to the overall product wavefunction,

. (15.48)


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It is not customary to use these labels for rotational energy levels and hence we have not done so.

15.3.1 DIATOMICS

The potential energy curve for nuclear motion of a diatomic molecule can be approximatedby a harmonic oscillator near the minimum, i.e. with the potential energy

E = V(r) = k(r-re)2 , (15.49)

with k the force constant. The resulting classical vibrational frequency is

, (15.50)

where : is the reduced mass (not the dipole moment!). The quantum mechanical energy levels arethus given as

E = h


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appearing in equation (15.53) are not [9]. In equations (15.53) and (15.54) all of the spectroscopicconstants, including the Morse $ parameter, are in cm-1 and the fundamental constants are in SI units.A Morse oscillator also has the dissociation energy De = Te

2/(4 Texe). In the above equations, the

symbol De is customarily used for both the dissociation energy and the equilibrium centrifugaldistortion constant, relying on the context to distinguish between the two. Here in equation (15.53),the subscript e has been deleted for clarity.

While large number of potential energy functions have been proposed, the Dunham form

, (15.56)

with , (15.57)

and , (15.58)

is the most widely used. The vibration-rotation energy levels of the Dunham potential are given bythe double sum

E(v, J) = EYjk (v + )j [J(J + 1)]k. (15.59)

The Dunham coefficients Yjkcan be related back to the Dunham potential parameters ai [9] and to thecustomary spectroscopic constants as follows:

Y10 = Te Y20 = -Texe Y30 = Teye Y40 = Teze

Y01 = Be Y11 = -"e Y21 = (e

Y02 = -De Y12 = -$e

Y03 = He

The customary vibrational energy level expression is

G(v) = Te(v + ) - Texe(v + )2 + Teye(v + )

3 + Teze(v + )4 + .... (15.60)

The relationship between potential parameters and spectroscopic constants contains some correctionterms, first derived by Dunham, and additional terms are needed to account for the breakdown of theBorn-Oppenheimer approximation [15]. Notice that the use of Dunham Yik constants avoids theconfusion created by constants with the same symbol such as the equilibrium centrifugal distortion


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constant and the dissociation energy (both De), as well as the customary negative signs in front ofTexe, "e and De. The best collection of spectroscopic constants for diatomics remains the book byHuber and Herzberg [16].

A harmonic oscillator has selection rules)v = 1, which leads to the fundamental vibrationalband v = 1 : 0 plus various hot bands, coresponding to v = 2 :1, 3: 2, .... By definition, a hot bandoccurs between two excited vibrational levels. Vibrational bands can appear in absorption v = 1 70 or emission v = 1 6 0; it is customary to put the excited state quantum number first.

Real molecules are anharmonic oscillators because the potential energy function containscubic and higher order terms (15.56), and because the dipole moment is not simply a linear functionof the internuclear separation, but rather has the form

(15.61)

A real diatomic molecule is thus both mechanically and electrically anharmonic. This anharmonicityallows overtone transitions to appear with )v = 2, 3, ... Each increase in)v results in a decreasein intensity of an order of magnitude (or more). In terms of the vibrational constants, a fundamentalband occurs at

)G =


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It is customary to label the upper energy level with a single prime and the lower with a double

prime, and to write the upper level first. The fundamental 1-0 vibrational band of HCl thus has

energy levels

E(vN = 1, JN) =


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, (15.72)

where di is the degeneracy of the ith mode and the sum is over the distinct vibrational frequencies.For example, CO2 has 4 normal modes but the bending modes are doubly degenerate so that there areonly 3 fundamental frequencies corresponding to the symmetric stretch


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(15.75)

in which the six xijs and the g2 term account for the anharmonic behavior, and vibrational resonancesare ignored. Indeed most molecules display interactions between vibrational modes that shift someof the energy levels away from the values predicted by eq. (15.75). These vibrational perturbationshave been classified into various types such as Fermi resonance and Coriolis resonance interactions[12]. There is no convenient collection of vibrational frequencies of stable molecules other than theolder work of Shimanouchi [17]. Jacox has compiled vibrational constants for transient polyatomics[18].

Linear molecules have two basic types of vibrational motion: parallel to the linear axis (z-axis)and perpendicular to the linear axis (in the xy-plane). Those parallel to the axis involve bondingstretching motions while those perpendicular involve bending motions. Stretching modes are thuscalled parallel bands and bending modes are called perpendicular bands because of the direction ofthe oscillating dipole moment. Parallel bands like the


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modes. For example, anharmonicity allows the


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for K + 1 7K (+ sign) and K - 1 7 K (- sign) for the prolate top. The K sub-bands are thusapproximately spaced by 2[A(1-.)-B]. The usual expressions (15.68), (15.69) and (15.76) for P, Qand R branch lines hold approximately for each sub-band. Depending on the magnitudes of A, B and

. these perpendicular bands can have well-separated sub-bands or a massively congested appearancewith strongly overlapping sub-bands.

The absorption line intensities are given by the general expression eq. (15.22) but the

transition moment factor needs to be evaluated for the symmetric top. In particular,

, (15.80)

where SJKis a rotational line strength factor called a Hnl-London factor and :10 is to be interpretedas a vibrational transition dipole moment, rather than the permanent dipole moment as in the pure

rotational case. A collection of infrared band strengths of molecules can be found in the book editedby Rao and Weber [19]. The Hnl-London factors SJKare provided in Table 15.2 for the 9 differentJN KN7 JO KO cases. The absorption coefficient " for the E17E0 transition is then given by

, (15.81)

where the degeneracy factors in the upper (d1) and lower (d0) states need to be included together withthe line shape function g(


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branch lines.

15.3.4 ASYMMETRIC TOPS

The vibrational energy levels of asymmetric tops are given by the general energy levelexpression (15.74). The symmetry of asymmetric tops is sufficiently low that degenerate vibrationallevels and a resulting vibrational angular momentum does not occur. The energy level expressionthen simplifies to

(15.83)

There are thus 3N-6 distinct vibrational frequencies for asymmetric rotor molecules. These modes

are allowed if they have an oscillating dipole moment.

The rotational energy levels and line strengths of the asymmetric rotor are not given byanalytical formulae. Energy levels and intensities are thus computed numerically by diagonalizingthe asymmetric rotor Hamiltonian using symmetric top basis functions. Line intensities are computednumerically, based on the known Hnl-London factors that describe transition moment matrixelements between the symmetric top basis functions.

The bands of an asymmetric top are classified by whether they have oscillating dipolemoments in the a, b or c directions. Bands are a-type, b-type or c-type if they have transition dipolemoments:a,:b or:c. These transition dipole moments are proportional to dipole moment derivatives

in the a, b and c directions for the particular motion described by the normal mode in question. Forexample, in H2O the symmetry axis is the b-axis so the symmetric stretching motion and the bendingmotion are both b-type bands. The antisymmetric stretching motion, in contrast, gives an oscillatingdipole moment along the a-axis which lies in the plane of the molecule and is perpendicular to thesymmetry axis. The rotational selection rules for a -, b - and c-type transitions have already beengiven. In general, a molecule with sufficiently low symmetry can have a mode that has an oscillatingdipole moment with components in a, b and c directions and thus all three types of transitions willbe found in this band. In general, asymmetric rotor bands have a very complex appearance.However, if the molecule is close to a symmetric rotor (i.e., *6*.1) then the bands will look likeparallel or perpendicular (with .=0) bands of the corresponding symmetric top.

15.3.5 SPHERICAL TOPS

Because of their high symmetry, spherical tops always have some degenerate vibrationalmodes and the full vibrational energy level expression (15.74) applies. Only triply degeneratefundamental vibrational modes are infrared active. For example, CH4 has 4 vibrational frequencies:


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Only


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and different intensities for the vibrational modes.

The vibrational frequencies of molecules are mainly deduced from infrared and Raman spectra

as well as from vibronic transitions (section 15.4). Vibrational frequencies vary from molecule tomolecule, but certain regularities are obvious. In particular, certain types of chemical bonds andfunctional groups have characteristic stretching and bending frequencies. These characteristic groupfrequencies (Table 15.3[2,13]) are widely used for qualitative analysis in organic [24] and inorganic[25] chemical spectroscopy. The analysis of materials is the main practical application of vibrationalspectroscopy.

15.4 ELECTRONIC SPECTRA

The electronic transitions of molecules show the greatest variety of all of the different typesof spectra. This is largely because the spin and orbital angular momenta often change and the ground

and excited states can have different geometries. In fact, transitions that change the symmetry of amolecule, such as linear to bent, are not uncommon. Changes in geometry, in addition to thenecessity for considering effects such as spin-orbit and spin-rotation coupling, make the study ofelectronic spectra particularly complicated.

Electronic transitions have associated vibrational and rotational structure. A particularrovibronic transition occurs at a line position < with

< = Te + GN(vN) + FN(JN) - GO (vO) - FO(JO) (15.87)

where G(v) and F(J) are the vibrational and rotational energy level expressions already discussed, and

Te is the equilibrium transition energy between the states. This expression assumes, of course, thatthe electronic, vibrational and rotational energies can be separated and that the states involved are notsubject to some sort of fast dynamical process. If one or both of the states is, for example,predissociated or preionized then the rotational or even the vibrational structure may be intrinsicallyunresolvable. If a state participates in a fast dynamical process (time scale ~)t) then the linewidth()E or)


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nuclear rotational , electron orbital and electron spin momenta, viz.

. (15.90)

The components of along the A-B axis are called7, E and S, respectively. Each stateis identified by the term symbol 2S+17. The degeneracy of the 2S+17 term is (2S + 1)2 because of the2S + 1 different possible values ofE = S, S-1, ..... - S and the two fold *7* orbital degeneracy. Forstates with 7 = 0, i.e. 2S+1E+ or2S+1EG states, the degeneracy is 2S + 1. The superscripts + or - areadded in this case to distinguish between electronic states that are symmetric (+) or antisymmetric (-)with respect to reflection in the symmetry plane containing the nuclei. For homonuclear moleculesan additional right subscript is added to the term symbol, 2S+17g or

2S+17u. If the electronicwavefunction is symmetric with respect to inversion of the electrons through the center of symmetrythen g is used, while u identifies the antisymmetric case. Notice that the g or u symmetry of an

electronic state applies only to homonuclear molecules and is not to be confused with the total parityof a rovibronic state. Total parity is the symmetry associated with the inversion of all particles(electrons and nuclei) through the origin in the laboratory coordinate system. This is a symmetryoperation for all molecules because the energy levels depend only on the relative positions of theparticles, which are unchanged by this operation [2].

Spin-orbit coupling is accounted for by considering the phenomenological spin-orbitHamiltonian,

SO = A , (15.91)

that causes an energy level splitting of

ESO = A 7E. (15.92)

The effect of spin-orbit coupling is thus to lift the (2S + 1)-fold spin degeneracy for7>0. Each spin-component of a 2S+17 term is then labeled with S which is written as a subscript, 2S+17S. A two-folddegeneracy remains to account for the *S* possibilities. Note that, by custom,7 andS are usuallynot signed but that G is signed to differentiate the 2S+1 spin components.

The selection rules for allowed one-photon electric dipole-allowed transitions are:

1. )7 = 0, 12. )S = 0 (light molecules only)3. )S = 0, 14. OnlyE+ - E+, EG - EG but A - E+ and A - EG are both allowed.5. Only g : u for homonuclear molecules.

Each electronic transition has associated vibrational and rotational structure. In the simplest


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approximation the transition vN, JN7 vO, JO occurs at

. (15.93)

The line absorption or emission intensities are given by the usual formulae, (15.22) and (15.23). Inthe case of singlet - singlet electronic transitions (i.e., 17N - 17O) the transition dipole moment is givenby

, (15.94)

in which is the square of the electronic transition dipole moment, SJ is the Hnl-London factor

(Table 15.2) and q is the Franck-Condon factor defined by

(15.95)

The Franck-Condon factor is the square of the overlap of the vibrational wavefunctions between thetwo electronic states. The Hnl-London factors are taken from Table 15.2, with7 used for K. Thus,a 1E+ - 1E+ electronic transition has the same rotational line strength factors as an infrared transitionof a diatomic and a 1A-1E+ electronic transition has the same Hnl-London factors as a perpendicularvibrational transition of a linear polyatomic molecule. Notice that the electronic and rotationaldegeneracy is also required in the expression for the absorption coefficient or for the Einstein Afactor.

The Franck-Condon factor q is a measure of how the electronic transition dipole moment isdivided up amongst the different vibrational bands. If the two electronic potential energy curves arevery similar in shape then only the diagonal ()v=0) vibrational bands are allowed because of theorthogonality of the vibrational wavefunctions, i.e. because

* < vN* vO > *2 = *ij . (15.96)

This is generally not the case, however, and off-diagonal vibrational bands with )v 0 are usuallyfound.

If the equilibrium bond length changes substantially then the rotational structure of eachvibrational band in an electronic transition will appear very different from a typical infraredvibrational band. For most vibration-rotation transitions BN.BO but for electronic transitions BN isoften very different from BO. This leads to the formation of band heads, where the rotational lines ina branch pile up and turn around [2,6].

The expressions for line positions in P, Q and R branches (ignoring centrifugal distortion) are


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such as eq. (15.102) are often not useful in a quantitative sense because of effects such as vibroniccoupling, which prevent the separation of electronic and vibrational motion.

Each vibrational band has associated rotational structure with energy levels described insimple cases by the formulae already discussed. The spectra of linear and symmetric tops cangenerally be classified as parallel and perpendicular depending upon the direction of the electronictransition dipole moments. For asymmetric tops the rotational selection rules can be classified as a-type, b-type and c-type depending upon the orientation of the electronic transition dipole moment.The electronic transition dipole moment can be evaluated using the electronic wavefunctions as

(15.103)

where the integration is over the electronic coordinates. The possibility of geometry changes coupled

with the large number of special effects such as vibronic coupling, Jahn-Teller effect and Renner-Teller effect, to say nothing of fine structure and hyperfine structure, make electronic spectroscopyof polatomics a fascinating and challenging area of study [8].

15.6 STRUCTURE DETERMINATION

There is a direct relationship between the three moments of inertia and the moleculargeometry, which may be expressed by the relations

(15.104)

(15.105)

(15.106)

where the ith atom of mass mi is located at (ai, bi, ci) in the principal axis system. High resolutionspectroscopy is thus one of the most reliable methods for determining molecular geometry [27]. Themain problem is that molecules have at most three moments of inertia, which is usually inadequateto determine the large number of bond angles and bond lengths. The solution is to record the spectraof isotopically substituted molecules and to assume that the geometry is invariant. In general, eachisotopomer provides an additional three moments of inertia so that spectra of a sufficient number ofisotopomers must be recorded in order to determine uniquely the unknown geometrical parameters.In this procedure, the center-of-mass equations

Emi = 0 (15.107)

with = (ai, bi, ci) for each isotopomer are required as constraints.


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Over the years, a number of techniques have been devised to determine molecular structurefrom moments of inertia [11,27]. Each method provides a slightly different set of bond lengths andbond angles. The three most important types of structures are designated as r0-, re- and rs-structures.

The best structure is considered to be the equilibrium or re-structure. Within the Born-Oppenheimer approximation it is only the re-structure that does not change with isotopic substitution.For a diatomic molecule, it is easy to obtain an re-structure because, eq. (15.14) only two Bv valuesare required in order to extrapolate to a Be value, from which the re value can then be calculated. Fora polyatomic molecule, it is more difficult to determine Ae, Be and Ce because, in general, 3N-6 "evalues are required for each rotational constant, see eqs. (15.44), (15.45) and (15.46). For largermolecules this is a very difficult task because a rotational analysis is required for each infraredfundamental (or equivalent information from a rotational or an electronic spectrum must beavailable).

If IAe

, IBe

and ICe

values are not available then structures may be calculated using thevibrationally- averaged moments of inertia derived from A0, B0 and C0 via eqs. (15.36), (15.37) and(15.38). This is unfortunate because rigid-body relationships such as

IC = IA + IB (15.108)

for a planar molecular are most nearly true for equilibrium moments of inertia. In fact, eq. (15.108)never holds exactly; it is used to define a moment of inertia defect via

) = IC - IA - IB (15.109)

for planar molecules. Empirically, planar molecules should all have small positive ) values. Anydeviations from the value of) expected empirically is taken as evidence for non-planarity or forfluxional behavior.

Most structures of polyatomic molecules are r0-structures. Even for diatomic molecules, r0values are reported if only a single B0 value is available. For a bond between two heavy atoms, r0 andre distances differ only slightly. For a hydrogen bond length, however, r0 and re values differsubstantially. Deuterium bond lengths are also shorter than corresponding H bond lengths because

of the significantly smaller zero point energy (by approximately ) for a deuterium atombonded to a heavy atom.

These problems with r0-structures have led over the years to numerous schemes (some of themempirical) to estimate re-structures. The most important of these methods is based on isotopicsubstitution and results in rs-structures (s for substitution). The basic ideas are due to Kraitchman andCostain [11]. By using the moments of inertia of a parent molecule and of the molecule with a singleisotopic substitution in Kraitchmans equations, the distance to the substituted atom from the centerof mass can be calculated. Thus, by repeated single isotopic substitution for each atom, a fullsubstitution (rs) structure is derived. The rs-structure is a better approximation to the re-structure and


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normally the bond lengths obey the inequality r0$ rs$ re. It is rare that a full substitution structureis determined because of the work involved in making and recording spectra of all possible singly-substituted isotopomers. In addition, some elements have only a single stable isotope. Generally, a

partial rs-structure is determined using Kraitchmans equations for some of the atoms and then theremaining geometrical parameters are derived using the moments-of-inertia equations, (15.104),(15.105) and (15.106), or the center-of-mass equation (15.107).

There are some convenient collections of molecular structures including the Handbook ofChemistry and Physics [13], the paper of Harmony et al. [28] and the Landolt-Bornstein series [29].More recently the MOGADOC database [30] has become available with structural data based mainlyon electron diffraction and microwave spectroscopy. Additional spectroscopic data are available fromvarious specialized databases including HITRAN [31], GEISA [32] and the JPL catalog [33].

Although bond lengths (and angles) like vibrational frequencies vary from molecule to

molecule, some regularities can be discerned. For example, the bond length between a carbon anda hydrogen atom is about 1.09 D in all molecules. Bond lengths are also inversely correlated withbond order, which is defined (approximately) as the number of electron pairs in the chemical bondholding two atoms together in a molecule. The concept of an average bond length is therefore usefuland a table of typical values [13, 34] is provided (Table 15.4). Actual molecules may have bondlengths that differ somewhat from the values reported in Table 15.4.


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15.6 REFERENCES

1. L. Cohen-Tannoudji, B. Diu and F. Lalo, Quantum Mechanics, vol. 1 and 2, (John-Wiley,

N.Y., 1977).

2. P.F. Bernath, Spectra of Atoms and Molecules, (Oxford UP, N.Y., 1995).

3. R.S. Mulliken,J. Chem. Phys.23, 1997 (1955).

4. P.R. Bunker and P. Jensen,Molecular Symmetry and Spectroscopy, 2nd edition (NRC Press,Ottawa, 1998).

5. I. Mills, T. Cvitas, K. Homann, N. Kallay and K. Kuchitsu, Quantities, Units and Symbols inPhysical Chemistry, 2nd edition, (Blackwell, Oxford, 1993).

6. G. Herzberg, Spectra of Diatomic Molecules, 2nd edition (Van Nostrand Reinhold, N.Y.,1950).

7. G. Herzberg,Infrared and Raman Spectra of Polyatomic Molecules, (Van Nostrand Reinhold,N.Y., 1945).

8. G. Herzberg,Electronic Spectra and Electronic Structure of Polyatomic Molecules, (VanNostrand Reinhold, N.Y., 1967).

9. C.H. Townes and A.L. Schawlow,Microwave Spectroscopy, (Dover, N.Y., 1975).

10. H.W. Kroto,Molecular Rotation Spectra (Dover, N.Y., 1992).

11. W. Gordy and R. Cook. Microwave Molecular Spectra, 3rd edition, (Wiley, N.Y., 1984).

12. D. Papousek and M.R. Aliev, Molecular Vibrational-Rotational Spectra, (Elsevier,Amsterdam, 1982).

13. D.R. Lide, editor,Handbook of Chemistry and Physics, 79th ed. (CRC Press, Boca Raton, FL,1998).

14. D.M. Bishop, Group Theory and Chemistry, (Dover, N.Y., 1993).

15. J.F. Ogilvie, The Vibrational and Rotational Spectrometry of Diatomic Molecules, (AcademicPress, San Diego, 1998).

16. K.P. Huber and G. Herzberg, Constants of Diatomic Molecules, (Van Nostrand Reinhold,N.Y., 1979).


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17. T. Shimanouchi,J. Phys. Chem. Ref. Data, 9, 1149 (1980) and references therein.

18. M. Jacox, Vibrational and Electronic Energy Levels of Polyatomic Transient Molecules,J.

Phys. Chem. Ref. Data, Monograph No. 3, 1994; VEEL Database (NIST, Gaithersburg, MD).

19. M.A.H. Smith, C.P. Rinsland, V. Malathy Devi, L.S. Rothman and K.N. Rao, inSpectroscopyof the Earths Atmosphere and the Interstellar Medium, K.N. Rao and A. Weber, eds.(Academic Press, San Diego, 1992).

20. J.P. Champion, M. Lote and G. Pierre in Spectroscopy of the Earths Atmosphere and theInterstellar Medium, K.N. Rao and A. Weber, eds. (Academic Press, San Diego, 1992).

21. N.B. Colthup, L.H. Daly and S.E. Wilberley, Introduction to Infrared and RamanSpectroscopy, 3rd edition, (Academic Press, San Diego, 1990).

22. D.A. Long,Raman Spectroscopy, (McGraw-Hill, London, 1977).

23. W. Demtrder, Laser Spectroscopy, 2nd edition, (Springer, Berlin, 1996).

24. L.J. Bellamy, The Infrared Spectra of Complex Molecules, (Chapman and Hall, London,1975).

25. K. Nakamoto,Infrared and Raman Spectra of Inorganic and Coordination Compounds, 4th

edition, (Wiley, N.Y., 1986).

26. H. Lefebvre-Brion and R.W. Field, Perturbations in the Spectra of Diatomic Molecules,(Academic Press, Orlando, FL, 1986).

27. A. Domenicano and I. Hargittai,Accurate Molecular Structures, (Oxford UP, N.Y., 1992).

28. M.D. Harmony, V.W. Laurie, R.L. Kuczkowski, R.H. Schwendeman, D.A. Ramsay, F.J.Lovas, W.J. Lafferty and A.G. Maki,J. Phys. Chem. Ref. Data, 8, 619 (1979).

29. Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology,New Series, Group II (Springer, Berlin).

30. J. Vgt, MOGADOC Database, Sekt. Spektren and Strukturdokumentation, University ofUlm, D-89069, Ulm, Germany.

31. L.S. Rothman et al.,J. Quant. Spectrosc. Radiat. Transfer, 60, 665 (1998).

32. N. Jacquinet-Husson et al.,J. Quant. Spectrosc. Radiat. Transfer, 59, 511 (1998).


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33. H.M. Pickett, R.I. Poynter, E.A. Cohen, M.L. Delitsky, J.C. Pearson and H.S.P. Mller,J.Quant. Spectrosc. Radiat. Transfer, 60, 891 (1998).

34. S.R. Radel and M.H. Navidi, Chemistry, (West, St. Paul, MN, 1990).


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Table 15.1. Rigid Asymmetric Rotor Energy Levels for J = 0, 1, 2, 3

JJ F(JJ)

000 00 0

110 11 A + B

111 10 A + C

101 1-1 B + C

220 22 2A + 2B + 2C + 2[(B - C)2 + (A - C)(A - B)]

221 21 4A+ B +C

211 20 A + 4B + C

212 2-1 A + B + 4C

202 2-2 2A + 2B + 2C - 2[(B - C)2 + (A - C)(A - B)]

330 33 5A + 5B + 2C + 2[4(A - B)2 + (A - C)(B - C)]

331 32 5A + 2B + 5C + 2[4(A - C)2 - (A - B)(B - C)]

321 31 2A + 5B + 5C + 2[4(B - C)2 + (A - B)(A - C)]

322 30 4A + 4B + 4C

312 3-1 5A + 5B + 2C - 2[4(A -B)2 + (A - C)(B - C)]

313 3-2 5A + 2B + 5C - 2[4(A - C)2 - (A - B)(B - C)]

303 3-3 2A + 5B + 5C - 2[4(B - C)2 + (A - B)(A - C)]


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Table 15.2. Hnl-London Rotational Line Strength Factors

SJK )K = + 1 )K = 0 )K = -1

)J = 1

)J = 0

)J = -1


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Table 15.3 Infrared Group Wavenumbers

Group


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Table 15.4 Bond Lengths in

Single Bonds

H C N O S F Cl Br I

H 0.74 1.09 1.01 0.96 1.34 0.92 1.27 1.41 1.61

C 1.54 1.47 1.43 1.82 1.35 1.77 1.94 2.14

N 1.45 1.40 1.36 1.75 1.79 1.97

O 1.48 1.42 1.70 1.72 1.87

S 2.05 1.56 2.07 2.27

F 1.42 1.63 1.76 1.91

Cl 1.99 2.14 2.32

Br 2.28 2.47

I 2.67

Multiple bonds

Bond Length/

CCa 1.54

C=C 1.34C/C 1.20

CNa 1.47

C=N 1.28

C/N 1.16

COa 1.43

C=O 1.20

C/O 1.13

N/N 1.10

O=O 1.21a Single bonds, repeated for comparison purposes.


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FIGURE CAPTIONS

Figure 15.1

The correlation diagram and labeling of the rotational energy levels of an asymmetric top molecule.

Figure 15.2The vibration-rotation spectrum of a diatomic molecule such as HCl.

Figure 15.3Angular momenta in a diatomic molecule.

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