DEVELOPMENT OF PREDICTIVE ELECTRONIC STRUCTURE METHODS

AND THEIR APPLICATION TO ATMOSPHERIC CHEMISTRY, COMBUSTION,

AND BIOLOGICALLY RELEVANT SYSTEMS

by

Evgeny Epifanovskiy

A Dissertation Presented to theFACULTY OF THE USC GRADUATE SCHOOLUNIVERSITY OF SOUTHERN CALIFORNIA

In Partial Fulfillment of theRequirements for the Degree

DOCTOR OF PHILOSOPHY(CHEMISTRY)

May 2011

Copyright 2011 Evgeny Epifanovskiy

Table of contents

List of tables iv

List of figures vii

Abstract x

Chapter 1: Introduction and overview 11.1 Potential energy surfaces and their intersections . . . . . . . . . . . . . 1

1.1.1 Methods for the ground electronic state . . . . . . . . . . . . . 41.1.2 Methods for excited electronic states . . . . . . . . . . . . . . . 101.1.3 Density functional theory methods . . . . . . . . . . . . . . . . 18

1.2 Previous studies of the green fluorescent protein chromophore . . . . . 191.3 Previous studies of the excited states of uracil . . . . . . . . . . . . . . 261.4 Chapter 1 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 2: Electronic properties of the green fluorescent protein chromophore 422.1 Chapter 2 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.1 Continuum solvation models . . . . . . . . . . . . . . . . . . . 442.2.2 Effective fragment potential-based QM/MM method . . . . . . 45

2.3 Anionic form of HBDI . . . . . . . . . . . . . . . . . . . . . . . . . . 462.3.1 Molecular orbital framework . . . . . . . . . . . . . . . . . . . 462.3.2 Vertical electron detachment energy of the HBDI anion . . . . . 482.3.3 Vertical excitation energy and electronic properties of the singlet

and triplet ππ∗ transitions of the HBDI anion . . . . . . . . . . 502.3.4 Cis-trans isomerization pathway of anionic HBDI . . . . . . . . 61

2.4 Chapter 2 conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.5 Chapter 2 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Chapter 3: The effect of oxidation on the electronic structure of the greenfluorescent protein chromophore 81

3.1 Chapter 3 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 81

ii

3.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 893.4 Chapter 3 conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 Chapter 3 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 4: Electronically excited states of uracil 1074.1 Chapter 4 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.3.1 Structure and electronic states of uracil . . . . . . . . . . . . . . 1104.3.2 Effect of the one-electron basis set . . . . . . . . . . . . . . . . 1144.3.3 Effects due to including higher excitations . . . . . . . . . . . . 1184.3.4 MRCI results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.5 Solvent effects . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.3.6 Best theoretical estimate of vertical excitation energies in uracil 125

4.4 Chapter 4 conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.5 Chapter 4 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Chapter 5: Direct location of the minimum point on intersection seams ofpotential energy surfaces with equation-of-motion coupled-clustermethods 134

5.1 Chapter 5 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.1 Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . 1375.2.2 Crossing location and minimization . . . . . . . . . . . . . . . 140

5.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.4 Benchmarks and examples . . . . . . . . . . . . . . . . . . . . . . . . 143

5.4.1 Cyclic trinitrogen cation . . . . . . . . . . . . . . . . . . . . . 1435.4.2 Nitrogen dioxide . . . . . . . . . . . . . . . . . . . . . . . . . 1465.4.3 para-Benzyne diradical . . . . . . . . . . . . . . . . . . . . . . 149

5.5 Discussion: Performance of the algorithm and details of implementation 1535.6 Chapter 5 conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.7 Chapter 5 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Chapter 6: Future work 1606.1 Chapter 6 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Bibliography 170

Appendix: Tensor algebra library 187

iii

List of tables

1.1 Selected theoretical estimates of the S0-S1 vertical excitation energy(∆E) and the corresponding wavelengths for the gas phase GFP chro-mophore. The experimental value is also given. . . . . . . . . . . . . . 22

2.1 The interaction of the bright ππ∗ state with the electron-detached contin-uum as reflected by the diminishing oscillator strength ( fL) in CIS andTD-DFT/BNL calculations. Also see Fig. 2.4. . . . . . . . . . . . . . 54

2.2 Vertical ππ∗ electronic excitation energies (∆E, in eV), correspondingwavelengths (nm) and oscillator strength ( fL) for the gas phase GFPchromophore (Fig. 1). The reference experimental value is 2.59 eV(479 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.3 Vertical triplet ππ∗ electronic excitation energies (∆E, in eV) of theHBDI anion in the gas phase. . . . . . . . . . . . . . . . . . . . . . . 60

2.4 Geometry parameters (distances in A, angles in degrees) at the stationarypoints for the gas-phase cis-trans chromophore isomerization. . . . . . 64

2.5 Cumulative natural charges on the fragments of the HBDI molecule: thephenyl and dimethylimidazolin rings and the CH bridge calculated withCASSCF(12/11) and DFT (PBE0 functional). . . . . . . . . . . . . . . 67

2.6 Cis-trans energy difference ∆Etc and the energy barrier Ea for the cis-trans isomerization of the HBDI anion calculated at various levels oftheory for the chromophore molecule and solvent. . . . . . . . . . . . 72

3.1 Vertical excitation energies (eV) and oscillator strengths of the ππ∗ statesof the doublet HBDI radical. The values of 〈S2〉 are also shown. . . . . 89

3.2 Vertical excitation energies (eV) and oscillator strengths of the excitedstates of the HBDI cation at its equilibrium geometry. . . . . . . . . . . 92

iv

3.3 Standard oxidation potentials corresponding to different one-electronoxidation transitions computed using gas-phase detachment and ioniza-tion energies (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.4 Selected geometric parameters (bond lengths in A, angles in degrees) ofthe HBDI anion, and its singly and doubly oxidized forms. . . . . . . . 98

4.1 Equilibrium geometry of uracil. The calculated structure is optimizedwith B3LYP/6-311G(2df,2pd). Experimental values are obtained byaveraging dimensions found in crystal structuresa. Atomic labels aredefined in Fig. 4.1, distances in A, angles in degrees. . . . . . . . . . . 113

4.2 Leading electronic configurations in the EOM-CCSD/aug-cc-pVDZ wavefunction. r is the weight of a configuration, R2

1 is the norm of the EOMsingles amplitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.3 Vertical EOM-CCSD excitation energies of uracil (eV). Ground stateCCSD energies are shown in hartree, oscillator strengths are given inparentheses. The singlet CCSD reference wave function was used forboth singlet and triplet EOM calculations. . . . . . . . . . . . . . . . . 116

4.4 Effect of triple excitations on vertical excitation energies. Shift in exci-tation energies due to triple excitations is shown in parentheses. Allenergies are in eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.5 Norms of the amplitudes Rm,k from Eq. (1.15) in the EOM-CC(2,3)/6-31G(d) wave function of uracil. The (10,10) active space is used. Totalnorm is not unity in virtue of biorthogonal properties of EOM-CC. . . . 120

4.6 MRCI excitation energies (eV) of the four lowest singlet excited statesof uracil. Oscillator strengths are given in parentheses. . . . . . . . . . 122

4.7 Vertical excitation energies (eV) of uracil in the gas phase and watersolution calculated with QM/MM. Energy shifts due to the solvent areshown in parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.8 Best estimates of the vertical excitation energies (eV) of the lowest sin-glet and triplet states compared with previously published results. . . . 126

5.1 Geometries and total energies of the intersection seam minima and pointsof triple degeneracy in N+

3 calculated by EOM-CCSD/6-31G. All bondlengths are given in angstroms, all angles are given in degrees, all ener-gies are given in a.u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

v

5.2 Results of energy minimization of the 21A2/11B1 intersection seam inN+

3 calculated by EOM-CCSD/6-31G. All bond lengths are given inangstroms, all angles are given in degrees. . . . . . . . . . . . . . . . . 146

5.3 Points of the C2v constrained seam of the X2A1/A2B2 intersection innitrogen dioxide. All bond lengths are given in angstroms, all angles aregiven in degrees, all energies are given in a.u. . . . . . . . . . . . . . . 148

5.4 Results of C2v-constrained energy minimization of the X2A1/A2B2 inter-section seam in NO2 calculated by EOMIP-CCSD/6-31G. All bond lengthsare given in angstroms, all angles are given in degrees. . . . . . . . . . 149

5.5 Equilibrium geometries and the MECP of the 1Ag and 3B1u electronicstates calculated by EOMSF-CCSD/6-31G*. Geometrical parametersare defined in Fig. 5.5. All bond lengths are given in angstroms, allangles are given in degrees. . . . . . . . . . . . . . . . . . . . . . . . . 151

vi

List of figures

1.1 Calculated adiabatic potential energy curves of the O2 molecule. Ateach internuclear distance, the lowest-energy solution of the electronicSchrodinger equation yields the ground state energy. Higher-energysolutions describe electronically excited states. . . . . . . . . . . . . . . 3

1.2 Action of different flavors of the EOM excitation operator forms differ-ent sets of target states giving rise to a suite of EOM-CC methods. . . . 12

1.3 The chemical structure and atomic labels of the anionic form of themodel GFP chromophore, 4’-hydroxybenzylidene-2,3-dimethylimidazolinone(HBDI) in the cis-conformation. . . . . . . . . . . . . . . . . . . . . . 20

2.1 The chemical structure and atomic labels of the anionic form of themodel GFP chromophore, 4’-hydroxybenzylidene-2,3-dimethylimidazolinone(HBDI) in the cis- (top) and trans- (bottom) conformations. . . . . . . 43

2.2 Two resonance structures of the HBDI anion. . . . . . . . . . . . . . . 46

2.3 Two molecular orbitals of the HBDI anion giving rise to the ππ∗ state.The character of the orbitals can be explained by considering linear com-bination of two localized π-bonding orbitals. . . . . . . . . . . . . . . . 47

2.4 Effect of increasing the number of diffuse functions in the basis set onthe density of states and the convergence of the lowest excited and thebright ππ∗ state. The calculations were performed with CIS (top) andTD-DFT/BNL (bottom). The basis set was varied from 6-311G(2pd,2df)to 6-311(4+,2+)G(2pd,2df). . . . . . . . . . . . . . . . . . . . . . . . 53

2.5 Computed energy profiles of cis-trans ground-state isomerization of theHBDI anion in the gas phase: (1) IRC calculated with DFT(PBE0)/6-31+G(d,p); (2) MEP calculated with CASSCF(12/11)/cc-pVDZ; (3) MEPcalculated with SOS-MP2/cc-pVDZ. . . . . . . . . . . . . . . . . . . . 62

2.6 Frontier valence molecular orbitals of the HBDI anion in the cis-form(left) and at the transition state of the cis-trans isomerization path (right). 66

vii

2.7 Cis-trans isomerization of the chromophore inside a cluster of watermolecules. Relative energies of the stationary points are computed withQM(CASSCF(12/11)/cc-pVDZ)/EFP/MM(TIP3P). . . . . . . . . . . . 71

3.1 Structures of deprotonated HBDI (HBDI anion, top) and the cation (bot-tom). The structure of the radical is similar to that of the cation, with theunpaired electron being on the carbon that hosts positive charge in thecation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2 Relevant molecular orbitals of deprotonated HBDI (HF/6-311G*). Inthe ground state of the anion, both π1 and π2 are doubly occupied, andthe bright state is derived by the π1→ π∗ excitation. Oxidized forms arederived by removing the electrons from π1. . . . . . . . . . . . . . . . 90

3.3 Energy levels and electronic states of the HBDI anion and radical. Thebright absorbing π1π∗ state at 2.62 eV is a resonance state embeddedin a photodetachment continuum beginning at 2.54 eV. The respectiveMOs are shown in Fig. 3.2. . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4 Electronic states of the doublet HBDI radical. The respective MOs areshown in Fig. 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.5 Electronic states of the doubly ionized HBDI anion. The respective MOsare shown in Fig. 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.6 Energy diagram of the relevant electronic states of deprotonated HBDI. 93

3.7 Infrared spectra of the anionic and cationic forms of deprotonated HBDIcomputed with RI-MP2/cc-pVTZ. . . . . . . . . . . . . . . . . . . . . 99

4.1 Structure of uracil defining atom labels referred to in Table 4.1. . . . . 111

4.2 Frontier molecular orbitals of uracil at the ground state equilibrium geom-etry. Orbital energies are calculated with RHF/aug-cc-pVDZ. . . . . . . 112

4.3 Excitation energies (upper panel) and oscillator strengths (lower panel)of the lowest excited states of uracil calculated with EOM-CCSD andvarious one-electron basis sets. . . . . . . . . . . . . . . . . . . . . . . 115

4.4 Absorption spectrum of uracil with theoretical estimations of electronictransitions calculated with EOM-CCSD/aug-ANO-DZ and CR-EOM-CCSD(T)/aug-ANO-DZ. . . . . . . . . . . . . . . . . . . . . . . . . . 127

viii

5.1 The potential energy surface slices for the ground and the first excitedstates of a substituted ethylene. There is an energy barrier that preventsthe molecule from rotating freely around the double bond. The top ofthe barrier on the ground state PES corresponds to the minimum on theexcited state PES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2 D3h-constrained scans of adiabatic PESs of N+3 around the points of

intersections calculated by EOM-CCSD/6-31G. . . . . . . . . . . . . . 147

5.3 The seam of the X2A1/A2B2 intersection in nitrogen dioxide. At everyN–O distance, the PESs of both states were scanned to find the O–N–Oangle at which the states are degenerate. . . . . . . . . . . . . . . . . . 148

5.4 The singlet and the Ms=0 triplet target states of the para-benzyne dirad-ical derived from the high-spin triplet reference by spin-flip. . . . . . . 150

5.5 Geometrical parameters used to describe the para-benzyne diradical inTable 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.6 Convergence of the 1Ag/3B1u intersection optimization in the para-benzynediradical by EOMSF-CCSD/6-31G*. In the upper part, the relative ener-gies of both states are plotted for each iteration. The correspondingenergy gaps are shown in the lower part. Convergence is achieved inabout 50 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.7 Relative energies of the 1Ag and 3B1u states of para-benzyne diradicalat their equilibrium structures and MECP. Both at the singlet and tripletequilibrium geometry, the singlet is below the triplet. The two statesintersect 0.65 eV above the ground state minimum. . . . . . . . . . . . 153

6.1 Large biological systems such as this active center of GFP require treat-ment of the QM core and the MM environment using hybrid QM/MMmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.1 Components of the tensor algebra library. . . . . . . . . . . . . . . . . 189

ix

Abstract

This work demonstrates electronic structure techniques that enable predictive modeling

of the properties of biologically relevant species. Chapters 2 and 3 present studies of the

electronically excited and detached states of the chromophore of the green fluorescent

protein, the mechanism of its cis-trans isomerization, and the effect of oxidation. The

bright excited ππ∗ state of the chromophore in the gas phase located at 2.6 eV is found

to have an autoionizing resonance nature as it lies above the electron detachment level

at 2.4 eV. The calculation of the barrier for the ground-state cis-trans isomerization

of the chromophore yields 14.8 kcal/mol, which agrees with an experimental value of

15.4 kcal/mol; the electronic correlation and solvent stabilization are shown to have

an important effect. In Chapter 3, a one-photon two-electron mechanism is proposed to

explain the experimentally observed oxidative reddening of the chromophore. Chapter 4

considers the excited states of uracil. It demonstrates the role of the one-electron basis

set and triples excitations in obtaining the converged values of the excitation energies of

the nπ∗ and ππ∗ states. The effects of the solvent and protein environment are included

in some of the models.

Chapter 5 describes an implementation of the algorithm for locating and exploring

intersection seams between potential energy surfaces. The theory is illustrated with

examples from atmospheric and combustion chemistry.

x

Chapter 1: Introduction and overview

1.1 Potential energy surfaces and their intersections

In electronic structure theory, the full Hamiltonian in Schrodinger’s equation is sepa-

rated into the nuclear and electronic parts

H = Hnuc +Hel =−12

∇2R +Hel (1.1)

This decomposition known as Born–Oppenheimer, or adiabatic, approximation1 is made

possible by the separation of the nuclear and electronic degrees of freedom with the

assumption that the electrons in the system adjust instantaneously to any change of the

nuclei’s positions. Because the nuclei move infinitely slower than the electrons, the

electronic part Hel can be solved at fixed nuclear configurations. The solutions of the

electronic Schrodinger equation

HelΨi = EiΨi (1.2)

are the eigenfunctions Ψi of the electronic Hamiltonian Hel and are called electronic

wavefunctions. Their corresponding eigenvalues Ei are state energies. The set of all

the eigenpairs {(Ei,Ψi)}, where each pair is identified by a nonnegative integer i, estab-

lishes the universe of all possible electronic states of the system at the given nuclear

1

configuration. Electronic states with the smallest value of energy Ei are called ground

electronic states. States with the same value of energy are called degenerate states. It

is possible to have degenerate ground electronic states, however in this work we are

mainly concerned with the class of systems that have only one ground electronic state.

Each electronic wave function Ψi belongs to one of the irreducible representations

of the molecule’s point group symmetry. The corresponding state is labeled with that

irrep, which is called the “state symmetry.” In addition, each wave function Ψi is an

eigenfunction of the spin-squared operator S2, and the respective eigenvalue is the “state

spin” or “state multiplicity.” States of the same symmetry and spin form manifolds that

are uncoupled from each other. This makes it possible to solve Eq. 1.2 with appropriate

constraints for each irrep and multiplicity independently.

Let us introduce an order of the electronic states by their energies such that

Ei ≤ E j ∀i, j : i < j (1.3)

Then (E0,Ψ0) is the ground state.

Solving the electronic Schrodinger equation at every nuclear configuration R ∈ R3N

(N is the number of atoms, and 3N is the total number of three-dimensional Cartesian

coordinates) and then ordering the solutions as prescribed by Eq. 1.3 allows us to define

functions Vi : R3N → R that yield the energy of i-th state at the given positions of the

nuclei R

Ei|R =Vi(R) (1.4)

The function Vi(R) is called the adiabatic potential energy surface (PES) of i-th elec-

tronic state. Examples are shown in Fig. 1.1 (Reproduced from Ref. 2).

2

Figure 1 displays a complete collection of the lower lying valence electronic states of O2

and a number of the adiabatic potential energy curves which result from Rydberg-valenceinteractions. The states with the vibrational levels indicated by tick marks were calculated bythe RKR method. The dotted curves are the theoretical potentials generated by Saxon & Liu[1977] and the dashed curves are from Partridge et al. [1991] or Partridge [Private Comm.]. Thedot-dash curves are diabatic representations of the nl�g Rydberg states with n � 5.

Figure 1.1: Calculated adiabatic potential energy curves of the O2 molecule. Ateach internuclear distance, the lowest-energy solution of the electronic Schrodingerequation yields the ground state energy. Higher-energy solutions describe electron-ically excited states.

Consider a situation in which two or more states are degenerate over a set of nuclear

configurations

Vi(R) =Vj(R) ∀R ∈ S ⊆ R3N i 6= j (1.5)

Set S can be all possible nuclear positions S = R3N , but that case is not of particular

interest. Let us look at the case where S ⊂ R3N . The set S defines the subspace of

3

degeneracy of the states, and we can define a function Ui : S → R, which we call the

intersection seam.

The sets S and R3N are not completely independent. The Neumann–Wigner theo-

rem3 (the noncrossing rule) states that two conditions have to be imposed on the Hamil-

tonian in order for two states of the same symmetry to be degenerate. In diatomic

molecules with only one internal degree of freedom, PES of the same symmetry may not

intersect (hence “noncrossing rule”). In larger molecules the rule limits the dimension-

ality of the intersection hyperline to M−2 where M is the number of internal degrees of

freedom. For states of different symmetry or spin, there is only one condition (degener-

acy), and therefore such crossings are possible even in diatomic molecules. Section 5.2

presents more details on the theory of PES intersections.

PES crossings occur often in real systems4. Because they allow the system to start

on one state’s PES and continue on another’s, they facilitate radiationless transitions

between electronic states. In Chapter 5 we present an implementation of algorithms for

locating and optimizing intersection seams and demonstrate their work with examples.

1.1.1 Methods for the ground electronic state

The ground state is of particular interest in electronic structure theory because chemical

reactions mostly occur on the lowest-energy PES. One of the conventional approaches to

approximately solve for the ground state energy is to use Hartree–Fock method, which

employs variational principle and is based on the mean-field description of electron–

electron interactions. Using a set of one-electron basis set functions, one solves a system

of Roothaan equations self-consistently to obtain molecular orbitals and their energies.

Hartree–Fock method can be roughly described as taking the following steps. The

reader is encouraged to consult electronic structure texts5, 6 for more details.

4

1. A one-electron basis set is selected for each atom of the molecule. In this work we

use Pople’s split-valence7–9 and Dunning’s10 collections of basis sets, which are

based on the Cartesian Gaussian representation of Slater-type orbitals. Using the

basis set, orbitals of each type (s, p, d, etc.) are created. They are called atomic

orbitals φµ.

2. Initial molecular orbital matrix C and Fock matrix F are formed using core Hamil-

tonian, superposition of atomic densities, Huckel theory or another similar strat-

egy. The orbital overlap matrix S is computed.

3. Roothaan equation

FC = SCE

is solved for the matrix C and the diagonal matrix E. Because Fock matrix F itself

depends on C, the equation has to be solved self-consistently. Iterations continue

until the convergence criterion (usually |C(i)−C(i−1)|< ε) is met.

Resulting from this procedure, the C matrix contains in its rows the expansion coef-

ficients for obtaining molecular orbitals ψi in terms of atomic orbitals

ψi = ∑µ

ciµφµ (1.6)

and the diagonal matrix E contains corresponding molecular orbital energies.

Hartree–Fock theory does not include electronic correlation, but provides a starting

point for correlated methods.

5

Coupled-clusters theory

Coupled-clusters (CC) theory offers a size-extensive approach to include electronic cor-

relation into the description of the ground state. Similar to the configuration interaction

(CI) method, CC theory represents the wave function of the ground state as an expansion

in terms of the reference determinant, then all determinants with single substitutions,

followed by double substitutions, and so on. Unlike CI, which uses a linear excitation

operator to generate the expansion, CC theory introduces an exponential anzatz:

|ΨCC〉= exp(T )|Φ0〉

where |Φ0〉 is called the reference determinant (usually Hartree–Fock wavefunction).

The cluster operator T is a series of n-tuple excitation operators:

T = T1 +T2 +T3 + . . .

Each term in the series can be written in the second quantization form:

T1 = ∑ia

tai a†

aai T2 =14 ∑

i jabtabi j a†

aa†ba jai T3 =

16 ∑

i jkabcT abc

i jk a†aa†

ba†caka jai

and so on. These equations follow the conventional notation: i, j, . . . stand for occupied

in the reference determinant |Φ0〉 spin-orbitals, a,b, . . . designate virtual (unoccupied)

spin-orbitals. The creation and annihilation operators a†p and ap create or remove an

electron from the spin-orbital |p〉.

6

The wave function and energy are obtained by solving the system of CC equations

for the T amplitudes:

〈Φ0|exp(−T )H exp(T )|Φ0〉= ECC

〈Φ0|exp(−T )H exp(T )−ECC|Φi〉= 0

When all the terms of the expanded T operator are included in the CC model, it

becomes completely equivalent to the full CI series as all possible excited determinants

are present in the expansion. The CC energy in that case contains 100% of the electronic

correlation energy within the given basis set.

Using the full expansion of the cluster operator T , however, is impractical because

the computational complexity of the method grows exponentially with the size of the

system. Full CI results have only been reported for di- and triatomic molecules in rather

small basis sets.

To reduce the scaling of the CC method, the T operator is truncated. Limiting the

expansion of T to the first two terms, for example, gives rise to coupled-clusters with

single and double excitations (CCSD) method11

T = T1 +T2

Because of the exponential form of the wave function ansatz, CC theory retains size-

extensivity even with the truncated cluster operator. This is unlike CI, in which only the

full expansion is size-extensive. In addition, as evident from the Taylor expansion of

7

exp(T ), the CCSD wavefunction effectively includes contributions from higher than

single and double excitations:

exp(T1 +T2) = 1+T1 +T2 +12(T1 +T2)

2 + . . .= 1+T1 +T2 +T1T2 +T 2

12

+T 2

22

+ . . .

This argument is applicable to any level of truncation of the T operator.

These two qualities make the truncated CC models superior to their CI counterparts.

Among all iterative CC methods, CCSD provides the most attractive balance between

the computational cost and accuracy. When chemical accuracy is desired (orders of

a few kcal/mol), the energy can be further improved using a non-iterative CCSD(T)

correction via perturbation theory12.

In practice the CC equations are solved numerically using the Jacobi method for the

system of linear equations in the matrix form. The vector solution is the CC amplitudes

and the update procedure is formulated as a recurring expression for the T vector13.

To improve the convergence of the scheme, the direct inverse of the iterative subspace

(DIIS)14 method is applied.

Multireference Møller–Plesset perturbation theory (MRMP2)

The MRMP2 model15 is a special single-state case of multiconfigurational quasidegen-

erate second order perturbation theory (MCQDPT2)16 implemented in GAMESS(US)17

and PC GAMESS18. The zeroth-order (reference) wave functions are state-averaged

CASSCF wave functions (ΨCASα ) for the target state α. The unperturbed Hamiltonian

operator is a sum of one-electron Fock-like operators, in which the occupation numbers

are replaced by the diagonal elements of the CASSCF density matrix. Beyond zeroth-

order multiconfigurational functions, the complementary eigenfunctions of the complete

8

active space configuration interaction Hamiltonian, as well as the wave functions gen-

erated by one- and two-electron excitations from the reference functions (S-space), are

considered via perturbation theory. The second-order corrections to the energy are given

by:

E(2)α = ∑

j

〈ΨCASα |V Φ j〉〈Φ jV |ΨCAS

α 〉E0

α−E(0)j

(1.7)

where functions Φ j belong to the S-space of uncontracted determinants. In a simi-

lar approach, CASPT219, the S-space consists of the contracted configurations. Both

MRMP2 and CASPT2, diagonalize-then-perturb methods, are widely used to calcu-

late the excitation energies of organic chromophores. A comprehensive benchmark

study20 reports excellent agreement between zeroth-order-corrected CASPT221 and an

accurate approximation to the coupled-cluster with singles, doubles, and triples method

(CC3)22, 23. Similar accuracy is observed in the recent computational studies employing

the MRMP2 methodology16, 24, 25.

Scaled opposite-spin MP2

Spin-component-scaled MP2 (SCS-MP2) is a semi-empirical approach based on scaling

different spin contributions to the MP2 correction as proposed by Grimme26:

Ec ≈ Ec,SCS−MP2 = pssE(2)ss + posE

(2)os (1.8)

The parallel-spin component E(2)ss and the antiparallel-spin component E(2)

os are

scaled to correct for their unbalanced contributions to the MP2 correlation energy.

Empirically found optimal values of the coefficients are pss =13 and pos =

65 . SCS-

MP2 provides an improved description for many systems, in which the ground state has

a single-reference character.

9

As the opposite-spin contribution is the major one, Jung et al.27 further simplified

the model by dropping the same-spin term altogether and scaling the opposite-spin con-

tribution up. Along with the RI technique, SOS-MP2 offers a significant improvement

in computational performance compared to the original MP2. The scaling coefficient —

the only empirical parameters in the method — can be optimized for a wide variety of

systems28.

1.1.2 Methods for excited electronic states

Equation-of-motion method

Applying the general linear excitation operator R on the ground state CC wave function

yields the excited state wave function

|Ψ(m)〉= R(m)|ΨCC〉= R(m) exp(T )|Φ0〉 (1.9)

(the integer m identifies the excited state). The structure of the excitation operator R is

the same as in CI. It can be expanded as

R = R1 +R2 +R3 + . . . (1.10)

where R1 generates all possible singly excited with respect to the reference determinants,

R2 generates all double excited determinants, and so on.

In the limit of the zero cluster operator T = 0, this approach becomes regular CI as

the excitation operator R acts directly on the reference determinant. If the series for R

is not truncated, the operator generates all possible excitations. In this case the full CI

result is recovered, no matter what the cluster operator T is.

10

The idea can be reformulated and generalized by introducing a similarity transfor-

mation for the electronic Hamiltonian H using the cluster operator:

H = exp(−T )H exp(T ) (1.11)

The similarity transformed Hamiltonian H has the same spectrum as the original Hamil-

tonian H. That allows us to “fold” the electronic correlation in the form of the clus-

ter operator into the Hamiltonian and find the eigenvalues (target state energies) and

eigenvectors (target state wave functions) in various subspaces of the Fock space. This

approach is called equation-of-motion coupled-clusters (EOM-CC) method29–37.

If the excitation operator R promotes one or more electrons from occupied to virtual

orbitals, it can be written in the second quantization form as

R(m)0 = r0 R(m)

1 = ∑ia

rai a†

aai R(m)2 =

14 ∑

i jabrab

i j a†aa†

ba jai (1.12)

This form of R preserves the number and spin of electrons. It is used in EOM-CC

for excitation energies (EOMEE-CC), which yields the energies and wave functions for

electronically excited states.

Operator R can be designed to remove or attach electrons, change their spin

(Fig. 1.2). This gives rise to a suite of EOM-CC methods: for ionization energies

(EOMIP-CC), for attachment energies (EOMEA-CC), spin-flip (EOMSF-CC). It is a

powerful toolkit that allows one to start with a convenient well-behaved reference state,

and form target states by applying the different flavors of the operator R38.

11

EOMEE-CC (excitation energies)

EOMIP-CC (ionization potential)

EOMEA-CC (electron attachment)

EOMSF-CC (spin-flip)

Figure 1.2: Action of different flavors of the EOM excitation operator forms dif-ferent sets of target states giving rise to a suite of EOM-CC methods.

12

EOM operator R is written in the tensor form in the basis of molecular orbitals. The

elements of this tensor (rai , rab

i j in Eq. 1.12) are called EOM amplitudes. They are found

by solving the eigenproblem

〈Φµ|H−E(m)|R(m)Φ0〉= 0 (1.13)

where 〈Φµ| are all µ-tuply excited with respect to the reference determinants.

Just like the cluster operator T in CC, the operator R can be truncated at different

excitation levels. In EOM-CCSD31, 32, 39, 40, both T and R include terms up to T2 and

R2. In the EOM-CC method with single, double, and full triple excitations (EOM-

CCSDT)41–44, the series for T and R are truncated after T3 and R3. It provides superb

quality of wave functions, but remains prohibitively expensive for all but tiny systems.

The EOM-CC(2,3)45, 46 approximation is obtained when R is truncated after R3, and T

is the same as in EOM-CCSD. The CC2 and CC3 methods are iterative approximations

to full EOM-CCSD and EOM-CCSDT, respectively47.

Explicit inclusion of triples, as in EOM-CCSDT or EOM-CC(2,3), results in the

O(N8) scaling versus the O(N6) scaling of EOM-CCSD, where N is the number of one-

electron basis set functions. Computational cost can be reduced by using a restricted

subset of triple excitations via an active space, where the three-body operators T3 and R3

are replaced by operators t3 and r3, which are required to include at least one excitation

within the active space41, 46, 48. CC2 and CC3 scale as O(N5) and O(N7), respectively.

In the active-space CCSDt and EOM-CCSDt methods49–60, the cluster operator T t

for CCSDt and the excitation operator for the m-th electronic state Rt for EOM-CCSDt

are given by

T t = T1 +T2 + t3 (1.14)

13

Rt = R0 +R1 +R2 + r3 (1.15)

Operators t3 and r3 are

t3(I) = ∑i≤ j≤K

∑A≤b≤c

t i jKAbca†

Aa†ba†

caKa jai, (1.16)

r3(I) = ∑i≤ j≤K

∑A≤b≤c

ri jKm,Abca†

Aa†ba†

caKa jai (1.17)

Capital letter indices denote orbitals from the active space. This form of the t3 and

r3 operators defines variant I of CCSDt and EOM-CCSDt, or CCSDt(I) and EOM-

CCSDt(I). The size of the operators can be reduced further by constraining two pairs

of spin-orbitals to the active space

t3(II) = ∑i≤J≤K,A≤B≤c

t iJKABca†

Aa†Ba†

caKaJai, (1.18)

r3(II) = ∑i≤J≤K,A≤B≤c

riJKABca†

Aa†Ba†

caKaJai, (1.19)

giving rise to the CCSDt(II) and the EOM-CCSDt(II) models. In CCSDt(III) and EOM-

CCSDt(III), all the spin-orbitals in t3 and r3 belong to the active space, which results in

the lowest memory and time requirements.

The active-space variant of EOM-CC(2,3), the EOM-CC(2,3) method, is based on

the CCSD reference and includes internal and semi-internal triple excitations via the r3

operator, as given by Eqs. (1.15) and (1.17).

The EOM-CCSD methods give an error in the range of 0.1–0.3 eV for excitation

energies, and including triples reduces it to 0.01–0.02 eV, as was shown for small

molecules61. A compilation of benchmarks by Schreiber et al.20 estimates errors in

the excitation energies of singlet states for CC2, EOM-CCSD, CC3, and EOM-CCSDT:

14

mean absolute errors are 0.486 eV, 0.12 eV, 0.016 eV, 0.029 eV, and maximum errors

are 1.08 eV, 0.23 eV, 0.047 eV, and 0.083 eV for the four methods, respectively.

Full-space EOM-CC(2,3) results closely follow those obtained with EOM-

CCSDT45, 46. However, the EOM-CC(2,3) method for excitation energies, EOMEE-

CC(2,3), is not size-intensive, which may result in increased error bars for larger

molecules. Slipchenko and Krylov46 report encouraging results for the active-space

EOM-CC(2,3), e.g., they observe errors on the order of 0.01 eV and 0.1 eV for singly

and doubly excited states, respectively. Kowalski et al.59 show that for states with the

singly excited character in small benchmark systems, EOM-CCSDt(I), (II), and (III)

give almost the same accuracy with the difference between EOM-CCSDt(I) and EOM-

CCSDt(III) excitation energies not exceeding 0.1 eV. The active space for EOM-CC

calculations with large basis sets has to be chosen with care, as diffuse orbitals often

appear below the frontier valence orbitals.

Non-iterative triples corrected EOM-CCSD(T)

A calculation of accurate excitation energies requires a proper account for both non-

dynamical and dynamical correlation. Methods that include active-space triple exci-

tations like EOM-CC(2,3) or EOM-CCSDt are capable of recovering non-dynamical

correlation arising from the doubly-excited character of a target state or a multi-

configurational reference. A small magnitude of this correction indicates that the

dynamical correlation component dominates in the overall effect. That can be accounted

for accurately by including a complete set of higher excitations or less expensively

through a non-iterative perturbative triples correction22, 47, 62–71.

The completely renormalized EOM-CCSD with non-iterative triples [CR-EOM-

CCSD(T)]68, 69 is based on the methods of moments of coupled-cluster equations

15

(MMCC)72, 73. It includes an approximate treatment of triple excitations and has the

O(N7) scaling like CCSD(T)12.

The CR-EOM-CCSD(T) correction δm for the m-th electronic state is:

δm =〈Ψm(3) |Mm,3(2)|Φ0〉〈Ψm(3) |ΨEOM

m 〉, (1.20)

where the Mm,3(2) operator corresponds to triply excited moments of the EOM-CCSD

equations for the m-th state,72∣∣ΨEOM

m⟩

and |Ψm(3)〉 are the EOM-CCSD and trial wave

functions, respectively. The latter takes the form:

|Ψm(3)〉= (P+Q1 +Q2 +Q3)(Rm,0 +Rm,1 +Rm,2 + Rm,3

)eT1+T2 |Φ0〉 , (1.21)

where T1, T2, Rm,0, Rm,1, Rm,2 are the cluster and excitation operators from the CCSD

and EOM-CCSD wave functions, and the Rm,3 operator approximates the exact EOM-

CCSDT Rm,3 operator. The P and Qm (m = 1,2,3) operators are projection opera-

tors onto the reference Hartree–Fock determinant |Φ0〉 and the subspace of all m-tuply

(m= 1,2,3) excited configurations, respectively. The amplitudes defining the Rm,3 oper-

ator are expressed in terms of the triply excited moments68, 69. As demonstrated previ-

ously74, 75, good estimates of vertical excitation energies are obtained by adding these

corrections to corresponding EOM-CCSD excitation energies.

Second-order perturbation treatment of the similarity transformed Hamiltonian H

yields two variants of non-iterative triples energy corrections: EOM-CCSD(dT) and

EOM-CCSD(fT), diagonal and Fock triples, respectively76. The corrections are intro-

duced for EOMEE-CCSD and EOMSF-CCSD methods, and the spin-flip variant is size-

intensive.

16

Multi-reference configuration interaction

In MRCI methods, the wave function is expanded as:

∣∣ΨMRCI⟩= NCSF

∑L=1

cL |ψL〉 . (1.22)

The basis functions of the expansion {|ψL〉} are configuration state functions (CSFs).

The CSFs, which are linear combinations of Slater determinants, are eigenfunctions of

spin operators and have a correct spatial symmetry77. The MRCI energies and wave

functions are obtained by diagonalizing the Hamiltonian in the basis of the CSFs.

MRCI calculations begin by determining zero-order wave functions and MOs in a

CASSCF calculation. The CASSCF wave function includes configurations created by

all possible excitations within an active orbital space. The coefficients of the expan-

sion (1.22), cL, and the MOs are variationally optimized. Dynamical correlation is

then described by MRCI by including single and double excitations from the refer-

ence CASSCF configurations. The MRCI method is very accurate, provided that all the

important configurations are included in the expansion. This requirement is easily satis-

fied for small molecules, but as the size of the system increases, the expansion becomes

prohibitively large, and truncations are necessary. The Davidson correction78 provides

a simple formula for evaluating the contribution of the missing quadruple excitations to

the energy computed with configuration interaction with up to double excitations.

Scaled opposite-spin CIS(D)

In the same manner as SOS-MP2 is introduced for correcting the ground state energy,

SOS-CIS(D) is designed for excitation energies79. The computational scaling of SOS-

CIS(D) is only O(N4), which is a significant improvement over the O(N5) scaling of

17

CIS(D). The accuracy of SOS-CIS(D) is very similar to that of CIS(D) for valence

states, whereas the performance for the Rydberg states is improved. Based on a set of

over forty various excited states in over twenty organic molecules, the mean signed error

in the SOS-CIS(D) vertical excitation energy is 0.02 eV for valence states and−0.08 eV

for Rydberg transitions. Limitations of SOS-MP2 and SOS-CIS(D) are the same as MP2

and CIS(D), respectively. For example, these methods fail when the ground-state wave

function acquires a significant multiconfigurational character, as at a cis-trans isomer-

ization transition state, and for excited states with considerable doubly excited character.

Open-shell (e.g., doublet) states can also cause difficulties due to spin-contamination.

1.1.3 Density functional theory methods

Long-range-corrected density functionals

In long-range-corrected functionals, a range-separated representation of the Coulomb

operator80, 81 is used to mitigate the effects of the self-interaction error. The contribution

from the short-range part is described by a local functional, whereas the long-range part

is described using the exact Hartree-Fock exchange. The separation depends on a param-

eter γ. In the BNL approach82, γ is optimized for each system using Koopmans-like argu-

ments: γ is adjusted such that the HOMO energy equals the difference between the total

BNL energies of the N- and N-electron systems. Initial benchmarks82, 83 demonstrated

an encouraging performance for excited states, and even such challenging systems as

ionized dimers. In ωPB97X84, γ and other parameters are optimized using standard

training sets. Benchmark results have demonstrated consistently improved performance

relative to non-long-range-corrected functionals.

18

1.2 Previous studies of the green fluorescent protein

chromophore

Unique electronic properties of the green fluorescent protein (GFP) whose natural func-

tion is to convert blue light to green light have motivated a number of experimental and

theoretical studies85, 86 and have been exploited in numerous practical applications87–89.

Due to their fundamental and practical importance, studies of the structure and prop-

erties of photoreceptor proteins and their denatured chromophores constitute an impor-

tant field of modern research. Moreover, GFP can be considered as a model for other

fluorogenic unsymmetric methine dyes90–95, and is of interest to organic photovoltaic

materials. For example, the fluorescent protein motif has already inspired the creation

of new organic phototovoltaic sensitizers96 and other optoelectronic materials97.

From the theoretical perspective, characterization of the electronic structure of iso-

lated chromophores is the first step towards understanding their photochemical and pho-

tobiological properties in realistic environments. Modeling isolated species involves

calculating the molecular parameters of the chromophores in the gas phase and in solu-

tion using quantum chemistry methods. This work presents accurate calculations of

the properties of biological chromophores with ab initio methods using the model GFP

chromophore, 4’-hydroxybenzylidene-2,3-dimethylimidazolinone (HBDI) anion, as a

benchmark system (Fig. 1.3). We also study the vertical excitation and electron detach-

ment energies, discuss the electronic properties of the excited and detached states, the

cis-trans isomerization of the HBDI anion in the ground electronic state.

19

Figure 1.3: The chemical structure and atomic labels of the anionic form ofthe model GFP chromophore, 4’-hydroxybenzylidene-2,3-dimethylimidazolinone(HBDI) in the cis-conformation.

Studies of excited states of HBDI

Earlier experimental studies have characterized the absorption of HBDI in the native

protein environment98. The spectrum of the wild-type GFP has two broad absorp-

tion bands at 396 nm (3.13 eV) and 476 nm (2.60 eV) assigned to the neutral and

anionic forms of the chromophore, respectively. The spectra in aqueous solution99, 100

reveal strong pH sensitivity: the absorption maximum at neutral pH is at 370 nm

(3.35 eV), whereas at pH=13 and pH=1 it is shifted to 426 nm (2.91 eV) and 396 nm

(3.13 eV), respectively. The absorption of denatured wild-type proteins exhibit similar

pH-dependence101. The shifts were attributed to different protonation and deprotona-

tion forms, as well as strong interaction with water. The latter is consistent with a large

change (about 7 D) of the dipole moment upon excitation determined in Stark effect

measurements taken in a buffered at pH=6.5 glycerol solution at 77 K102. Therefore,

the protonation and deprotonation of HBDI and solvent effects, which strongly affect

its electronic properties, need to be accounted for in theoretical models. However, it is

important to characterize the chromophore in the gas phase first in order to quantify the

solvent effect separately.

20

Recently, a gas-phase action spectrum of anionic HBDI (as well as other protonated

forms) using photodestruction spectroscopy of mass-selected ions injected into an elec-

trostatic ion-storage ring was reported providing an important reference for theory99, 100.

The spectrum shows an absorption band centered at 2.59 eV (479 nm), which extends

from 2.4–2.8 eV (440–520 nm), as well as a minor peak around 2.3 eV (540 nm). The

authors emphasized a striking similarity between the absorption bands in the gas phase

and in the protein and suggested that the protein environment shields the chromophore

from water and that the absorption in the protein is an intrinsic property of HBDI. While

the role of the protein still needs to be investigated, this measurement facilitates more

direct comparison between the gas-phase calculations of vertical excitation energies and

the experimental absorption for benchmarking theoretical methods.

The large absorption band in the gas phase spectrum of the HBDI anion has been

assigned as the ππ∗ transition, however, the nature of the minor feature at 2.3 eV has not

been discussed.

A variety of electronic structure techniques ranging from simple semiempirical

approximations to high-level ab initio methods have been applied to simulate the prop-

erties of the cis-anionic form of HBDI103–109. Selected representative results are sum-

marized in Table 1.1.

These studies have identified the absorbing state of the HBDI anion as the S1 state

derived from a HOMO-LUMO excitation of the ππ∗ character. LUMO, however, is a

valence π∗-like orbital only in relatively small basis sets; including diffuse functions

increases the number of molecular orbitals between the HOMO and the lowest π∗-like

orbital. Though the bright state retains its ππ∗ character, it is not, strictly speaking, a

HOMO-LUMO transition in a realistic basis set.

21

Tabl

e1.

1:Se

lect

edth

eore

tical

estim

ates

ofth

eS 0

-S1

vert

ical

exci

tatio

nen

ergy

(∆E

)and

the

corr

espo

ndin

gw

avel

engt

hsfo

rth

ega

spha

seG

FPch

rom

opho

re.T

heex

peri

men

talv

alue

isal

sogi

ven.

Met

hod

toco

mpu

teth

eex

cita

tion

ener

gyM

etho

dto

optim

ize

grou

nd-

stat

ege

omet

ry∆

E(e

V)

Wav

e-le

ngth

(nm

)R

ef.

IND

O-C

IPM

32.

7744

8R

ef.1

03a

CIS

/6-3

1G(d

)R

HF/

6-31

G(d

)4.

3728

4R

ef.1

05T

D-D

FT(B

P86)

/6-3

1++G

(d,p

)B

3LY

P/6-

31++

G(d

,p)

2.94

421

Ref

.108

b

CA

SPT

2/6-

31G

(d)

CA

SSC

F(12

/11)

/6-3

1G(d

)2.

6746

5R

ef.1

07SA

C-C

I/D

ZV

B3L

YP/

6-31

G(d

)2.

2255

8R

ef.1

06M

RM

P2ba

sed

onSA

-CA

SSC

F(16

/14)

/(au

g)-c

c-pV

DZ

PBE

0/au

g-cc

-pV

DZ

2.47

501

Ref

.109

aug-

MC

QD

PT2

base

don

SA-C

ASS

CF(

16/1

4)/(

aug)

-cc-

pVD

ZPB

E0/

aug-

cc-p

VD

Z2.

5448

9R

ef.1

09

Exp

erim

ent

2.59

479

Ref

.99,

100

aA

clos

eva

lue

(444

nm)w

asob

tain

edin

the

late

rsem

iem

piri

calc

alcu

latio

nsof

the

ND

DO

type

(Ref

.104

).b

Fora

nov

ervi

ewof

prev

ious

TD

-DFT

calc

ulat

ions

usin

gdi

ffer

entf

unct

iona

lsan

dba

sis

sets

,see

Tabl

e1

ofR

ef.1

08;o

nly

the

valu

ecl

oses

tto

the

expe

rim

enta

lexc

itatio

nen

ergy

ispr

esen

ted

here

.

22

The first theoretical studies on the chromophore, dating back to the late 1990s103, 104,

employed semiempirical methods based on the neglect-of-differential overlap approxi-

mation. They placed the ππ∗ state at 2.77 eV, or 0.18 eV above the experimental absorp-

tion maximum. Ab initio calculations using configuration interaction with single excita-

tions (CIS) in a small basis set105 grossly overestimated the excitation energy. These and

other results from Table 1.1 reveal that accurate calculations of the vertical excitation

energy for the bright ππ∗ transition are challenging for quantum chemistry.

Time-dependent density functional theory (TD-DFT), which can be applied to very

large systems, gave rise to high expectations in the field of photochemistry. However,

the notorious self-interaction error110–112 results in an unphysical description of charge-

transfer (CT) states113, which are common in large molecules. In addition to artificially

low excitation energies of real CT states, spurious CT states appear, spoiling the descrip-

tion of other states. The number of these false states increases steeply with the system

size114. In the case of fluorescent protein chromophores, TD-DFT has been reported to

perform quite modestly, as discussed in Ref.108. In that study, which examined var-

ious functionals, the best agreement with the experiment was obtained for the BP86

functional115, 116. This value (2.94 eV or 421 nm), which is listed in Table 1.1, is still

0.35 eV (60 nm) away from the experiment. Overall, TD-DFT excitation energies range

from 2.94 to 3.23 eV108, the popular B3LYP functional117, 118 yields 3.05 eV.

CASPT2, the complete active space self-consistent field (CASSCF) method with

second-order perturbation theory (PT2) correction19, has been applied to model pho-

tochemical properties of organic chromophores in various media107, 119–128. The

CASPT2/6-31G(d) result107 from Table 1.1 agrees fairly well with the experimental

value; however, the CAS distribution of 12 electrons over 11 orbitals is a truncation

of the entire π-orbital active space, which requires the CASSCF(16/14) wave function

23

for the GFP chromophore. Bravaya et al.109 performed calculations of the vertical

ππ∗ transition energies for various forms of HBDI using a very expensive and highly

correlated approach, i.e. state-averaged CASSCF(16/14) wave functions constructed

in the full π-orbital active space augmented by perturbative corrections: multireference

second-order Møller-Plesset perturbation theory (MRMP2)15 and an extended version of

the multiconfigurational quasidegenerate perturbation theory (aug-MCQDPT2)16, 24, 25

(using ground-state equilibrium geometries optimized using DFT with the PBE0 func-

tional129, 130 and the (aug)-cc-pVDZ basis set:10 aug-cc-pVDZ on oxygen atoms and

cc-pVDZ on all other atoms). The MRMP2 and aug-MCQDPT2 results are within 0.12

and only 0.05 eV from the experimental value, respectively (Table 1.1). These data sug-

gest that one can compute the positions of absorption bands of biological chromophores

with an accuracy of 10–20 nm (less than 0.1 eV) by applying perturbatively corrected

CASSCF-based approaches. These techniques, however, are computationally demand-

ing, and their execution requires advanced skills and extreme care, as the application of

the method involves: (i) a careful selection of a large number of active space orbitals

in fairly large basis sets; (ii) converging the state averaged CASSCF solutions corre-

sponding to the ππ∗ transition, especially in realistic basis sets; (iii) a careful and often

ambiguous treatment of perturbative corrections to the reference CASSCF solutions.

Moreover, gradient calculations are only available for bare CASSCF wave functions.

Thus, it is desirable to find a more robust approach of a comparable accuracy for wider

applications in modeling the properties of biological chromophores.

Contrarily to the bright singlet ππ∗ state, little is known about the triplet states of

the GFP chromophore. The fluorescence properties of GFP suggest that the intersystem

crossing is not efficient, at least in the chromophore in the native protein environment. In

24

the gas-phase photocycle, however, triplet states can play an important role. For exam-

ple, possible population trapping in the triplet has been suggested as a possible explana-

tion for observed millisecond-long lifetimes of the photoexcited ions in the ion-storage

ring experiments131. Spin-forbidden relaxation channels have also been considered in

the studies of GFP mutants132, 133.

Cis-trans isomerization of HBDI

The cis-trans isomerization (or Z/E diastereomerization) of the photoswitchable fluores-

cent chromophores plays an essential role in their photophysical properties. For exam-

ple, some photoconversion mechanisms involve cis-trans isomerization134, 135. Kindling

and blinking phenomena, as well as loss of fluorescence yield of bare chromophores in

solution, are also related to this process. In a broader context, the photophysics of GFP

is similar to that of other fluorogenic unsymmetric methine dyes90–95, and is of interest

with connection to organic photovoltaic materials.

The majority of the experimental and computational studies of this process focused

on the excited state dynamics. However, the ground electronic state potential energy

surface (PES) also needs to be considered for two main reasons. First, after photoi-

somerization has completed, the photoswitchable protein returns to its initial state to

prepare for the next cycle. The recovery apparently takes place on the ground state PES.

Second, the details of the ground-state isomerization can elucidate the chromophore’s

rearrangements along the more complicated excited-state route. Despite previous stud-

ies of the cis-trans photoisomerization of GFP-like chromophores136–141 using quantum

chemistry modeling103, 104, 126, 142–145, many questions remain unanswered.

The process of cis-trans isomerization of the HBDI anion in aqueous solution was

investigated experimentally by He et al.136 who estimated the free energy differences

25

and the activation energies using the NMR technique. The authors stressed that the

activation energy of 15.4 kcal/mol derived from their measurements in aqueous solu-

tion is in distinct disagreement with the results of calculations,103, 104 which estimated

that the barrier is above 21 kcal/mol. The relatively low value of the barrier (as com-

pared to other isomerization reactions involving exocyclic double bonds) has also been

emphasized in subsequent studies of the isomerization and several explanations have

been suggested137, 141. For example, thermal isomerization studies of model GFP-like

compounds137 suggested that different substituent groups may have a significant effect

on the activation energy by changing the interaction between two resonance structures

of the chromophore. Tolbert and coworkers considered mechanisms involving changes

in chemical structure of the chromophore, e.g., addition/elimination pathway141. No

ab initio calculations have been reported so far to resolve this disagreement between

experimental measurements and theoretical estimates and explain the low value of the

barrier.

1.3 Previous studies of the excited states of uracil

Natural nucleobases—adenine, guanine, thymine, cytosine, and uracil—combine with

residues of phosphoric acid and sugars to form nucleotides, the monomers of nucleic

acids. Being UV chromophores, the nucleobases define a large portion of the RNA

and DNA photochemistry and are used as model systems to study the properties of the

polymers146–149.

UV radiation is harmful to living organisms: when absorbed by the nucleic acid, it

initiates excited-state dynamics that can result in structural damage. The process, which

starts with an electronic excitation of a UV chromophore, can be quenched by radia-

tionless relaxation to the ground state. Photoexcited natural nucleobases have lifetimes

26

of less than one picosecond, and that is believed to be responsible for the remarkable

photostability of DNA. In a review, Crespo-Hernandez et al.146 discuss mechanisms,

pathways, and dynamics of the relaxation process.

The mechanism of the excited state population decay varies from nucleobase to

nucleobase and depends on the order of the lowest electronically excited states, which

correspond to a forbidden nπ∗ and an allowed ππ∗ transition. Areas of potential

energy surfaces (PESs) where the ground and the excited states are degenerate or near-

degenerate play a special role in the radiationless relaxation dynamics4, 150, 151. The

mechanism of the efficient radiationless decay through conical intersections in uracil

was initially investigated using the multi-reference configuration interaction (MRCI)

method by Matsika152. Later results obtained with other methods153–158 are in qualita-

tive agreement with MRCI.

The equilibrium structure of uracil is planar and has the Cs point group symmetry.

The first singlet excited state corresponds to the nπ∗ transition and belongs to the A′′

irreducible representation. The second singlet excited state, A′, arises from the ππ∗

transition. Only the second transition has oscillator strength by symmetry. The next two

valence singlet states are also of the nπ∗ and ππ∗ types. A Rydberg A′′ state is close in

energy to the second nπ∗ state.

Absorption spectra of uracil in the gas phase159 and in solution155 show a broad fea-

ture centered at 244 nm (5.08 eV) and 259 nm (4.79 eV), respectively. These values

have been interpreted as the vertical ππ∗ excitation energy. The assignment was sup-

ported by complete active space self-consistent field (CASSCF)160 calculations with the

perturbation theory correction (CASPT2) and time-dependent density functional theory

27

(TDDFT) with the PBE0 functional155, 161, which produce gas-phase vertical excita-

tions of 5.0 eV and 5.26 eV, respectively. However, more reliable MRCI152, and EOM-

CCSD20, 162 predict higher values of the vertical excitation energy for the bright ππ∗

state. More recent CASPT2 calculations20 employing an empirical parameter to correct

for “the known tendency of CASPT2 to slightly underestimate excitation energies”20,

yield a slightly higher excitation energy, which agrees better with EOM-CCSD, but

moves away from the experimental absorption maximum. Thus, the excellent agree-

ment between the earlier CASPT2 calculations160 and the experiment is rather coinci-

dental. The authors of two recent benchmark studies20, 162 note that EOM-CCSD ener-

gies are generally higher than CASPT2 and that inclusion of triple excitations through

the iterative CC3 method47 usually brings them down. For smaller molecules and a

non-augmented basis, an impressive correlation between CC3 and CASPT2 results has

been observed20. CC3 calculations162 of uracil in a polarized double-zeta quality basis

produce an excitation energy of the ππ∗ state 0.2 eV lower than EOM–CCSD and do

not reveal any significant doubly excited character of the state. Unfortunately, no CC3

results with larger bases have been reported.

28

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41

Chapter 2: Electronic properties of the

green fluorescent protein chromophore

2.1 Chapter 2 introduction

This chapter presents an approach to accurate calculations of the properties of bio-

logical chromophores with ab initio methods using the model GFP chromophore, 4’-

hydroxybenzylidene-2,3-dimethylimidazolinone (HBDI) anion, as a benchmark system

(Fig. 2.1). It discusses the character of the bright ππ∗ state of the HBDI anion.

Among the benchmarked computational methods are perturbation theory corrected

multiconfigurational approaches, EOM-CCSD, as well as TD-DFT with two range-

separated functionals, BNL1 and ωPB97X2. We also characterize the lowest ππ∗ triplet

and report the vertical electron detachment energy (VDE). Based on the calculated VDE,

we assign the minor feature as due to the photodetachment transition. This has impor-

tant implications on the character of the bright state: the ππ∗ transition is a resonance

state embedded in an ionization continuum.The triplet state, however, lies below VDE.

As the resonance state, the ππ∗ singlet has a finite lifetime and can undergo autoioniza-

tion due to coupling to the ionization continuum. Contrary to that, the triplet may have

much longer lifetime. Thus, population trapping in the triplet state in the gas-phase

photocycle seems to be required to explain millisecond kinetics of the fragments yield

42

Figure 2.1: The chemical structure and atomic labels of the anionic form ofthe model GFP chromophore, 4’-hydroxybenzylidene-2,3-dimethylimidazolinone(HBDI) in the cis- (top) and trans- (bottom) conformations.

in the ion-storage ring experiments3. Moreover, the resonance nature of the ππ∗ state in

the anionic GFP might be responsible for very different kinetics of the photofragment

yield of the anionic and protonated GFP3.

2.2 Computational details

The equilibrium geometries of the HBDI anion were optimized by DFT with the

PBE0 variant4 of the Perdew-Burke-Ernzerhof (PBE) hybrid functional5 and by

CASSCF(14/12). The cc-pVDZ basis set6 was used in both calculations. After observ-

ing noticeable differences in vertical excitation energies computed using these two

geometries, we reoptimized the equilibrium structure using MP2 with cc-pVTZ6, which

43

yields very accurate structures for well-behaved closed-shell molecules7. MP2 calcula-

tions employed the resolution-of-the-identity (RI) technique. The CASSCF and PBE0

structures are C1, whereas the RI-MP2 optimization produced a Cs minimum.

Vertical excitation energies were computed by MRMP2, TDDFT with the BNL and

ωPB97X functionals, CIS, SOS-CIS(D), and EOMEE-CCSD. The VDE was computed

at the CASSCF geometry as the energy of the Hartree–Fock HOMO (Koopmans’ theo-

rem), by EOMIP-CCSD, and by using the BNL HOMO energy (as described below, this

is equivalent to computing VDE as the difference between the total BNL energies of the

anion and the neutral radical). The ωPB97X and B3LYP Koopmans’ theorem and ∆E

values are also given for comparison.

The cis-trans isomerization pathways of the chromophore were studied with the DFT

and CASSCF methods. Solvent effects we estimated using continuum solvation models,

as well as by explicit treatment of water molecules in a QM/MM scheme.

MRMP2 and D-PCM calculations were carried out with the PC GAMESS version8

of the GAMESS(US) quantum chemistry package9. CIS, SOS-MP2, SOS-CIS(D),

EOM-CCSD, and BNL calculations were performed with Q-Chem10. GAMESS(US)9

was employed for C-PCM and SVPE computations. The QM/MM implementation is

based on PC GAMESS8 and the Tinker molecular mechanics package11.

2.2.1 Continuum solvation models

To simulate solvent effects on the chromophore’s cis-trans isomerization energy pro-

file in aqueous solution, we employ three versions of the continuum solvation model12:

D-PCM, C-PCM, and SVPE. In the simplest approach, the dielectric polarizable con-

tinuum model (D-PCM)13, the water solvent is treated as a continuous unstructured

dielectric with a dielectric constant of 78.39. C-PCM14 is a version of PCM that takes

44

into account certain corrections in the boundary conditions of the electrostatic problem

in accord with the conductor-like screening model. The surface and volume polariza-

tion for electrostatic (SVPE) approach15 employs the dielectric continuum model to

incorporate the effects of volume polarization that arise from the solute charge density

penetrating outside the solute cavity. This effect can be significant for reaction barriers,

especially for ionic solutes15.

2.2.2 Effective fragment potential-based QM/MM method

Solvent effects can be described explicitly in a combined quantum mechanical-

molecular mechanical (QM/MM) approach based on the effective fragment potential

(EFP) model16, 17. In this scheme, solvent molecules (water in our case) are represented

by effective fragments in the MM-part. The fragments affect the Hamiltonian of the

QM-part (HBDI anion) by their electrostatic potentials expanded up to octupole terms.

The parameters of these one-electron electrostatic potentials, as well as contributions

from interactions between polarizable effective fragments and the QM-region, are com-

puted in preliminary ab initio calculations of the electronic densities of individual frag-

ments. The exchange-repulsion potentials, which are combined with the electrostatic

and polarizability terms, are obtained from preliminary ab initio calculations as well.

The original EFP approach16 treats interactions between solvent molecules as EFP-

EFP interactions. Studies of chemical reactions in aqueous solution18, 19 have shown

that replacing the EFP-EFP terms by the empirically calibrated TIP3P potential yields a

faster computational scheme for large water clusters around the solute.

45

2.3 Anionic form of HBDI

2.3.1 Molecular orbital framework

Fig. 2.2 shows two resonance structures of the HBDI anion. The interaction between

these two structures result in charge delocalization between the two oxygen atoms and

in scrambling CC bond orders, as discussed, for example in Ref. 40. Changes in bond-

orders in the bridge region due to the resonance are believed to be important in thermal

isomerization20.

Figure 2.2: Two resonance structures of the HBDI anion.

The Natural Bond Orbital (NBO)21, 22 charges on the phenoxy or the imidazolin

oxygens (computed using the RI-MP2 densities) are −0.65 and 0.66, respectively. The

almost equal values of the charges suggest significant contributions from both resonance

structures. As the result of the resonance interaction, the C1P-C1B bond gains double-

bond character, whereas the order of C1B-C1I bond is reduced, and the bridge moiety

acquires allylic character. It is interesting to compare the respective bond lengths with

the values for the double and single CC bonds between sp2 hybridized carbons, e.g.,

in a planar and twisted ethylene (see Ref. 24 for explanation regarding the choice of

the reference structures). Our best estimates of the C1P-C1B and C1B-C1I bond lengths

(RI-MP2/cc-pVTZ) are 1.394 A and 1.378 A, respectively. The lengths of the CC bond

in ethylene is 1.333 A at the planar geometry (where the formal bond order is 2), and

46

1.470 A23 at the twisted configuration, where the π-bond is completely broken24. Thus,

assuming a linear relationship between the bond order and bond length, one can assign

55% and 67% of a double-bond character to the C1P-C1B and C1B-C1I bonds, respec-

tively. The NBO analysis assigns 1.8 electrons to a slightly asymmetric allylic three-

center bond. The relative contributions of the C1P and C1I carbons are 38 and 62 %,

respectively, in a semi-quantitative agreement with the bond orders derived from the

bond lengths.

Figure 2.3: Two molecular orbitals of the HBDI anion giving rise to the ππ∗ state.The character of the orbitals can be explained by considering linear combinationof two localized π-bonding orbitals.

47

The resonance interaction is also reflected by the shape of MOs. Fig. 2.3 shows two

Hartree–Fock orbitals involved in the bright ππ∗ and in the photodetachment transitions

(HOMO and valence LUMO). These orbitals, which are traditionally referred to as π and

π∗, have quite complicated shapes and are delocalized over the entire molecule. Their

character in the bridge region can be explained by considering two interacting π orbitals,

as shown in Fig. 2.3. The HOMO can be described as an out-of-phase combination of

two localized π bonds, whereas the large electron density on C1B in the LUMO can be

derived from the in-phase combination. Of course, due to the delocalized character of

the orbitals, this picture is just an approximation, but it allows one to see the origin of the

large oscillator strengths and changes in charge distribution in the excited state, and also

provides a useful framework for explaining the character of the transition state along

cis-trans isomerization coordinate.

2.3.2 Vertical electron detachment energy of the HBDI anion

Since the HBDI anion is a closed-shell system, it is stable in the gas phase and has a

relatively large vertical electron detachment energy (VDE). However, as shown below, it

does not support bound electronically excited singlet states, and the lowest valence exci-

tation is embedded in an ionization continuum. Such resonance states are very common

in molecular anions25, and play an important role in dissociative electron attachment

processes26, 27. Thus, the broad character of the experimental action spectrum28, 29 is at

least partially due to the broadening of the resonance-like ππ∗ state by its interaction

with the ionization continuum. Finite lifetime and autoionization decay of this state

should be taken into account when considering photoinduced dynamics and lifetime

of the HBDI anion in the gas phase. In the condensed phase, solvent may stabilize the

anion such that its excited states become bound. However, one-photon photodetachment

48

channel may still be relevant for the anionic forms, especially when the production of

solvated electrons is considered30. The resonance character of the ππ∗ state also has

significant consequences in the electronic structure calculations25, 31 of excited states, as

described below.

The simplest estimate of VDE obtained by applying Koopmans’ theorem is

2.56–2.93 eV depending on the basis set, the largest value obtained with the 6-

311(2+,+)G(2df,2pd) basis. The EOMIP method, which includes electronic correlation,

can provide more reliable estimates of VDE. However, due to the size of the system,

we are limited to relatively modest basis sets. In the 6-31G* basis, Koopmans’ VDE is

2.56 eV, while EOMIP-CCSD yields 2.05 eV. The larger 6-31+G* basis increases the

energies to 2.91 and 2.48 eV, respectively. Thus, including correlation reduces VDE by

0.4–0.5 eV, and assuming the effect is consistent throughout basis sets, we estimate that

the target EOMIP-CCSD/6-311(2+,+)G(2df,2pd) energy is 2.4–2.5 eV.

B3LYP/cc-pVDZ energy difference calculations yield 2.46 eV, which is consider-

ably larger than the respective Koopmans value of 0.92 eV. The ωPB97X/cc-pVDZ

energy difference value is 2.39 eV, whereas the respective Koopmans IE is much higher

(2.83 eV).

By construction, the BNL energy difference VDE is equal to the respective HOMO

energy. Recent benchmarks32 demonstrated that BNL produces very accurate ionization

energies. VDE calculated with BNL in the small cc-pVDZ basis is 1.99 eV (γ = 0.250),

and it increases to 2.52 eV in the 6-311G(2+,+)G(d,p) basis, for which γ = 0.228. Addi-

tional sets of diffuse orbitals or more extensive polarization do not affect this value, e.g.

VDE calculated with BNL/6-311G(3+,2+)G(2df,2pd) is 2.53 eV.

The discrepancies between the Koopmans and ∆E values have important implica-

tions for the excited state calculations, as the former value defines the onset of the

49

ionization continuum in CIS/TDDFT calculations. Thus, with B3LYP, the continuum

begins 1.54 eV below its own VDE, whereas the situation with ωPB97X is reverse —

the continuum states appear 0.44 eV above the respective VDE. BNL, by constriction,

is internally consistent, and the continuum states in TDDFT calculations appear exactly

at the respective VDE.

Thus, our estimate of VDE is 2.4–2.5 eV, within 0.1 eV from the maximum of the

weak absorption feature at 2.3 eV. The remaining discrepancy between the two values

might be due to the uncertainties in equilibrium geometries or possibl vibrational exci-

tation of the molecules in the experiment.

Although correlation has significant effect on VDE, the ionized state has Koopmans-

like character, i.e. the leading EOM-IP amplitude corresponds to ionization from the

HOMO (see Fig. 2.3) and equals 0.96. Thus the Hartree–Fock HOMO is a good approx-

imation to a correlated Dyson orbital33.

2.3.3 Vertical excitation energy and electronic properties of the sin-

glet and triplet ππ∗ transitions of the HBDI anion

Singlet ππ∗ state: Benchmark results

Motivated by the discrepancies in previous theoretical estimates of the ππ∗ excitation

energy of the HBDI anion (Table 1.1), we set out to benchmark other electronic struc-

ture methods, with the purpose of identifying a rigorous yet fairly inexpensive quantum

chemistry approach that can be employed in condensed-phase applications. The experi-

mental maximum of absorption is at 2.59 eV (479 nm), and the band’s full width at half

maximum (FWHM) is 0.25 eV (45 nm)28, 29. Assuming that the absorption maximum

corresponds to the vertical transition of the lowest-energy isomer (which is not entirely

50

clear, as the temperature of the ions in the ring is unknown), one would like the computed

wavelength to fall within 2.47–2.72 eV (456–502 nm). However, due to the resonance

nature of the ππ∗ state, calculating the excitation energies and oscillator strengths, as

well as comparing them with the experimental spectrum, are not as straightforward as in

the case of excited states lying below the electron detachment energy. In the following,

we will use the term “detached states” to identify the electronic states that compose the

continuum instead of the usual “ionized states” as the initial species is anionic.

The ground-state equilibrium geometry was optimized with PBE0/cc-pVDZ,

CASSCF/cc-pVDZ, and RI-MP2/cc-pVTZ. Although the differences between the

geometries are small (maximum bond length deviation is 0.03 A and angles agree within

2 degrees), the ππ∗ excitation energy computed using wavefunction-based methods dif-

fers by about 0.1 eV. This effect of geometry is consistent with previous calculations

by Olsen34 for a similar system (p-hydroxybenzylidene-imidazolin-5-one, HBI). Of the

three structures, the RI-MP2/cc-pVTZ one is the most accurate7. Since most of the

changes in electron density occur in the bridge region, it is interesting to compare the

C1p-C1B and C1B-C1I bond lengths computed by different methods. The RI-MP2 val-

ues are 1.394 A and 1.378 A, which is very close to the PBE0 values of 1.404 A and

1.385 A. Due to the absence of dynamical correlation, CASSCF exaggerates the bond

alternation giving 1.406 A and 1.397 A.

Calculations in small basis sets create discrete states from the continuum and artifi-

cially exclude the detached states from the picture. Assuming that the character of the

resonance state of interest is well described, such calculations may provide a fairly good

estimate of the position of this state in the continuum, however, it is difficult to predict

how expanding the basis set will affect the resonance state25, 31. Moreover, one should

anticipate broadening of the resonance state due to the interaction with the detached

51

states. Increasing the size of the one-electron basis brings the continuum states down.

When the basis set is large enough to accommodate a detached electron, the lowest

excited state will correspond to the detached state (this fact has been exploited in pilot

implementations of EOMIP-CCSD based on the EOMEE-CCSD code by adding a very

diffuse orbital to the basis to describe the ionized electron35, 36). It can be shown for-

mally, that in the case of CIS and TD-DFT, the energy of the lowest of such CIS-IP states

equals the HOMO energy, that is, the Koopmans theorem can be proven by considering

configuration interaction of all singly detached determinants.

When the one-electron basis set is expanded with diffuse functions, the density of

states rapidly increases with both CIS and TD-DFT (BNL), as illustrated in Fig. 2.4. At

the same time, the lowest excited state converges to Koopmans’ VDE. Both methods

split the oscillator strength of the bright ππ∗ transition among multiple states, but BNL

does that to a larger degree, which results in three to four states with close oscillator

strengths (Table 2.1). That may be due to the remaining self-interaction error in BNL

calculations.

The lowest excited CIS and TD-DFT states shown in Fig. 2.4 fall below the Koop-

mans estimate in large basis sets due to additional stabilization by the interaction with

other CIS determinants that vanish once the detached electron is infinitely far from

the core. Indeed, if a separate large non-interacting orbital is used instead of diffuse

functions, the lowest excitation energy in such a system is exactly equal to the HOMO

energy.

As the density of low-lying states that approximate the continuum increases and the

oscillator strength gets redistributed, it becomes increasingly more difficult to compute

or even identify the resonance state. By following the evolution of the states with the

basis set increase, the limiting value of the resonance state can be extrapolated using the

52

Figure 2.4: Effect of increasing the number of diffuse functions in the basis set onthe density of states and the convergence of the lowest excited and the bright ππ∗

state. The calculations were performed with CIS (top) and TD-DFT/BNL (bottom).The basis set was varied from 6-311G(2pd,2df) to 6-311(4+,2+)G(2pd,2df).

53

Table 2.1: The interaction of the bright ππ∗ state with the electron-detached con-tinuum as reflected by the diminishing oscillator strength ( fL) in CIS and TD-DFT/BNL calculations. Also see Fig. 2.4.

Basis set Number Energy, eV fL

CIS6-311G(2df,2pd) 1 3.92 1.586-311(+,+)G(2df,2pd) 2 3.81 1.506-311(2+,+)G(2df,2pd) 6 3.66 0.22

9 3.74 0.1410 3.85 1.14

6-311(2+,2+)G(2df,2pd) 9 3.60 0.0812 3.82 1.3517 4.02 0.06

6-311(3+,2+)G(2df,2pd) 37 3.73 0.4340 3.84 1.00

TD-DFT (BNL)6-311G(2df,2pd) 2 3.44 1.476-311(+,+)G(2df,2pd) 3 3.27 1.386-311(2+,+)G(2df,2pd) 7 3.18 0.54

9 3.24 0.3110 3.38 0.52

6-311(2+,2+)G(2df,2pd) 8 3.12 0.159 3.15 0.15

12 3.29 0.3913 3.29 0.63

6-311(3+,2+)G(2df,2pd) 35 3.23 0.4737 3.24 0.3939 3.37 0.1540 3.37 0.15

54

stabilization method by Taylor37, as has been done in the calculations of resonance elec-

tron attached states25, 31. From the data in Table 2.1, we can estimate that the excitation

energy converges to 3.8 eV with CIS and to 3.3 eV with TD-DFT/BNL. Overall, adding

sets of diffuse functions results in lowering the excitation energy of the bright state by

0.1 eV.

The CIS and TD-DFT/BNL calculations demonstrate that the resonance state is

embedded in an electron detached continuum, and the broad character of the experi-

mental spectrum can be due to the interaction with the continuum. Thus, small basis set

calculations of the vertical excitations energies of the ππ∗ state can only provide a rough

estimate of the position of the absorption maximum.

Introducing a non-iterative doubles correction via SOS-CIS(D) significantly low-

ers the energy of the ππ∗ transition, while the detached state energies are affected less.

We could not obtain SOS-CIS(D) results in heavily augmented bases sets due to the

lack of the corresponding auxiliary bases for the RI procedure, which becomes unsta-

ble upon adding second set of the diffuse functions. The magnitude of correction is

0.9–1.2 eV for the ππ∗ state and 0.1–0.3 eV for the detached states. MRMP2 based

on sa-CASSCF(14/12) in the modest cc-pVDZ basis yields 2.51–2.61 eV for the ππ∗

excitation energy. SOS-CIS(D)/cc-pVDZ gives 2.71–2.81 eV, which is consistent with

the more rigorous MR-MP2 estimates.

At the CASSCF-optimized geometry, EOM-CCSD/6-31G* yields an excitation

energy of 3.16 eV and an oscillator strength fL = 1.27. Including only one set of dif-

fuse functions lowers the energy to 3.04 eV ( fL = 1.25). The change in the oscillator

strength is consistent with the observed drop of the EOM amplitude corresponding to

the ππ∗ transition from 0.85 to 0.74. Benchmarks on the electronically excited states of

55

closed-shell molecules have shown that the combined effect of increasing the basis set

in EOM-CCSD calculations and the triples correction can be as large as 0.3–0.4 eV.

The computational cost of MRMP2 and EOM-CCSD does not allow us to use these

methods to fully explore the basis set effect on the ππ∗ excitation energy, e.g., via sta-

bilization method. We anticipate a noticeable effect of improving the basis set on the

energy of the ππ∗ state. For example, with SOS-CIS(D), the excitation energy with the

cc-pVTZ basis is lower by 0.13 eV relative to cc-pVDZ. The EOM-CCSD excitation

energy also drops by as much as 0.2 eV upon expanding the basis set from 6-31G* to 6-

31+G*. The oscillator strength calculated with CIS is about 10% lower when basis sets

with diffuse functions are used compared to those without, demonstrating the beginning

of the interactions with the continuum.

The ππ∗ excitation energy obtained with MRMP2 based on state-averaged

CASSCF(16/14), which spans the entire π-electron active space (Table 1), and a more

compact (14/12) active space (Table 2.2) agree equally well with the experimental

result: 2.47 eV (501 nm) and 2.52–2.61 eV (476–491 nm), respectively, versus 2.59 eV

(479 nm). The most expensive aug-MCQDPT2/CASSCF(16/14) approach (Table 1)

does not perform noticeably different than MRMP2/CASSCF(14/12). Therefore, one

can rely on the fairly practical MRMP2 approach based on a reduced active space in

the CASSCF wavefunction. Overall, the effect of contracting the active space is less

than 0.1 eV, which is smaller than the uncertainty due to the equilibrium geometry and

anticipated effects of extending the basis set beyond the double-zeta level.

In view of complexities associated with performing multireference perturbation

theory and underlying CASSCF calculations, we find the results of the inexpensive

56

Tabl

e2.2

:Ver

tical

ππ∗

elec

tron

icex

cita

tion

ener

gies

(∆E

,in

eV),

corr

espo

ndin

gw

avel

engt

hs(n

m)a

ndos

cilla

tors

tren

gth

(fL)f

orth

ega

spha

seG

FPch

rom

opho

re(F

ig.1

).T

here

fere

nce

expe

rim

enta

lval

ueis

2.59

eV(4

79nm

).

Met

hod

Gro

und-

stat

ege

omet

ryop

timiz

edw

ithPB

E0/

cc-p

VD

ZC

ASS

CF(

14/1

2)/c

c-pV

DZ

RI-

MP2

/cc-

pVT

Z

∆E

f Lnm

∆E

f Lnm

∆E

f Lnm

MR

MP2

base

don

sa-C

ASS

CF(

14/1

2)/c

c-pV

DZ

2.52

b49

1b2.

61c

476c

EO

M-C

CSD

/6-3

1G(d

)3.

081.

2640

23.

161.

2739

23.

1239

8E

OM

-CC

SD/6

-31+

G(d

)2.

971.

2441

83.

041.

2540

8

SOS-

CIS

(D)/

cc-p

VD

Z2.

711.

59a

457

2.81

1.61

a44

12.

751.

60a

451

SOS-

CIS

(D)/

aug-

cc-p

VD

Z2.

571.

45a

482

2.67

1.45

a46

42.

611.

45a

475

SOS-

CIS

(D)/

cc-p

VT

Z2.

581.

54a

480

2.68

1.56

a46

32.

621.

54a

473

SOS-

CIS

(D)/

aug-

cc-p

VT

Z2.

581.

37a

480

2.90

1.04

a42

7

TD

-DFT

/BN

L/c

c-pV

DZ

3.44

1.51

360

3.50

1.51

354

3.59

1.55

346

TD

-DFT

/BN

L/6

-311

(+,+

)G(2

df,2

pd)

3.22

1.38

385

3.27

1.38

379

TD

-DFT

/ωW

B97

x/cc

-pV

DZ

3.52

1.52

352

3.59

1.53

346

3.55

1.53

349

TD

-DFT

/ωW

B97

x/6-

311(

+,+)

G(2

df,2

pd)

3.38

1.45

367

3.44

1.46

360

aO

scill

ator

stre

ngth

calc

ulat

edw

ithC

IS.b

Att

heeq

uilib

rium

geom

etry

com

pute

dw

ithPB

E0/

(aug

)-cc

-pV

DZ

(diff

use

func

tions

only

onox

ygen

atom

s).c

Att

heeq

uilib

rium

geom

etry

com

pute

dw

ithC

ASS

CF(

12/1

1)/(

aug)

-cc-

pVD

Z(d

iffus

efu

nctio

nson

lyon

oxyg

enat

oms)

.

57

SOS-CIS(D) method very encouraging: at the PBE0-optimized geometry, the SOS-

CIS(D)/cc-pVTZ ππ∗ excitation energy of 2.58 eV is within 0.01 eV from the exper-

imental absorption maximum, and agrees very well with MRMP2 calculations.

The EOM-CCSD/6-31+G* excitation energy is 0.38 eV above the experimental

maximum, which is outside the EOM-CCSD error bars. Analysis of the EOM ampli-

tudes confirms singly excited character of the ππ∗ state, thus, with an adequate basis

set, EOM-CCSD error bars should not exceed 0.3 eV. In addition to anticipated basis

set effects and interactions with the continuum states, other factors such as discrepan-

cies due to equilibrium geometry, as well as the uncertainty in the experimental value

(FWHM of the experimental absorption is 0.25 eV), make it difficult to arrive at a def-

inite conclusion. More extensive calculations using larger basis sets (and ideally using

stabilization graphs) and estimates of triples corrections (to account for dynamical cor-

relation) are required to assess the performance of EOM-CCSD for this molecule. Basis

set and triples effects alone have been shown to account for as much as 0.3 eV in exci-

tation energies of ππ∗ character.

Finally, we present TD-DFT results computed with the range-separated functionals.

Using the BNL functional1 with the small cc-pVDZ basis set, the ππ∗ state appears the

second lowest, which is consistent with the respective VDE. At the CASSCF geome-

try, its excitation energy is 3.50 eV, and the oscillator strength is fL = 1.51. The 6-

311(+,+)G(2df,2pd) basis lowers the excitation energy to 3.27 eV, and the oscillator

strength to 1.38. ωPB97X2 gives similar values for the excitation energy and the oscil-

lator strengths, i.e., 3.59 eV ( fL = 1.51) and 3.44 ( fL = 1.46) with the cc-pVDZ and

6-311(+,+)G(2df,2pd), respectively. Because the Koopmans continuum with ωPB97X

begins at higher energies (see section 2.3.2), the detached states do not appear below the

excited states in these calculations.

58

Overall, it is unrealistic to expect accuracy better than 0.1 eV (20 nm) from computa-

tional protocols applicable to a molecule of this size even for non-resonance states, and

the observed discrepancies between different methods confirm that. Moreover, when

assessing the accuracy of computed values, one should keep in mind the finite width of

the experimental absorption band. Thus, more calculations are necessary to provide a

converged theoretical estimate, especially stabilization analysis. For practical applica-

tions, however, it is important that all the reliable theoretical methods agree with each

other in that the origin of the intensity in the resonance state is due to ππ∗ excitation.

SOS-CIS(D) offers an inexpensive alternative to more rigorous multireference methods

if single excitations are dominant in the wavefunction.

Changes in electronic density in the singlet ππ∗ state

As one may expect from the molecular orbital character (Fig. 2.3) and the large oscil-

lator strength, electronic excitation results in a significant redistribution of electronic

density. A convenient measure of charge distribution is the permanent dipole moment.

Although in charged system it is origin-dependent, the difference between the two dipole

moments, ∆µ = µgr− µex, is not. At the CIS level of theory, the value of |∆µ| is 0.6 D,

and its direction is in the molecular plain pointing towards the bridge carbon. This value

can be compared with the experimentally measured ∆µ, derived from Stark effect mea-

surements in a buffered at pH=6.5 glycerol solution at 77 K38. This work also reports the

angle between ∆µ and ∆µtr. Strikingly, the experimental value is 10 times larger than the

computed one. Since ∆µ is related to the changes in orbital occupations upon excitation,

it is dominated by contributions from the leading excitation amplitudes, and should be

reproduced fairly accurately at the CIS level. Thus, large discrepancy is likely to be due

to the solvent effect. Indeed, polar solvents result in the increased dipole moment of the

59

Table 2.3: Vertical triplet ππ∗ electronic excitation energies (∆E, in eV) of the HBDIanion in the gas phase.

Basis set CIS SOS-CIS(D)

cc-pVDZ 2.03 1.91aug-cc-pVDZ 2.02 1.88cc-pVTZ 2.03 1.86

solute. For example, dipole moment of water in bulk water is about 30% larger than in

the gas phase. More polar charge distribution in the ground state in solvent is clearly

seen from the respective NBO charges (see Table 2.5). Thus, for the difference of dipole

moments of two states one may anticipate an enhanced effect.

Triplet ππ∗ state

The vertical excitation energies at the RI-MP2/cc-pVTZ geometry of the lowest triplet

state are summarized in Table 2.3. The analysis of the wave function confirms that the

triplet is derived from the transitions between the same orbitals as the singlet (Fig. 2.3).

As expected, all methods consistently place the triplet considerably below the singlet.

The variations between the methods are smaller for the triplet state. Our best value

(SOS-CIS(D)/cc-pVTZ) is 1.86 eV. The 0.76 eV gap between the singlet and the triplet

does not suggest efficient intersystem crossing at this geometry. The triplet state is

0.3–0.4 eV below VDE and is, therefore, a bound electronic state. Thus, much longer

lifetime is expected for this state (as compared to the singlet), not only because the

radiationless relaxation to the ground state is a spin-forbidden process, but also because

the autoionization channel is absent.

60

2.3.4 Cis-trans isomerization pathway of anionic HBDI

Gas phase

On the ground-state PES, conversion between the lower-energy cis-isomer and the

higher-energy trans-isomer takes place when the two rings are rotated along the C1I-

C1B bond as shown in Fig. 2.1. The dihedral angle N-C1I-C1B-H, which defines the

reaction coordinate, is denoted as τI .

The shape of the isomerization pathway can be characterized by: (i) the activation

energy Ea defined as the relative energy of the transition state (TS) with respect to the

cis-isomer; and (ii) the energy difference Etc between the trans- and cis-forms. We

began with gas-phase DFT calculations with the PBE0 functional and the 6-31+G(d,p)

basis set and located stationary points that correspond to the cis-, trans- and TS struc-

tures. The latter is characterized by a single imaginary frequency of 725i cm−1. The

intrinsic reaction coordinate (IRC) profile was computed by starting steepest descent

pathways from the TS point in both directions along the Hessian eigenvector that corre-

sponds to the imaginary frequency.

The results of this calculation are discouraging in two aspects: the cusp-like shape of

the profile (Fig. 2.5), which is consistent with the large value of the imaginary frequency

at the saddle point, and the value of the activation energy Ea(DFT) = 34.5 kcal/mol,

which is more than twice higher than the experimental estimate of 15.4 kcal/mol in

aqueous solution39. The computed energy difference between the trans- and cis- struc-

tures Etc(DFT) = 2.3 kcal/mol is in excellent agreement with the experimental estimate

in solution39, most likely due to error cancellation.

61

Figure 2.5: Computed energy profiles of cis-trans ground-state isomerization of theHBDI anion in the gas phase: (1) IRC calculated with DFT(PBE0)/6-31+G(d,p);(2) MEP calculated with CASSCF(12/11)/cc-pVDZ; (3) MEP calculated with SOS-MP2/cc-pVDZ.

In the CASSCF energy profile (Fig. 2.5), the stationary points were fully opti-

mized with CASSCF(12/11)/cc-pVDZ. The points in between representing the mini-

mum energy path (MEP) were computed varying the value of the dihedral angle τI

and minimizing the energy by relaxing all other degrees of freedom. Although this

curve does not represent the true IRC path, it is expected to be a good approximation

to it. Along with a smoother curvature in the vicinity of the saddle point (imaginary

frequency 77i cm−1), this profile yields a reasonable value of the trans-cis energy dif-

ference Etc(CASSCF) = 3.5 kcal/mol and a much lower (and closer to the experimental

estimate) value of the activation energy Ea(CASSCF) = 22.5 kcal/mol.

When calculating the energy profile in the CASSCF(12/11) approximation with a

fairly large active space, we performed careful selection of the orbitals in order to avoid

62

dubious solutions of the variational problem. To better understand changes in the elec-

tronic structure along the isomerization pathway, we also computed the energy pro-

file with the smallest active space CASSCF(2/2). Selection of active orbitals in this

approach was performed on the base of previously optimized orbitals at the TS point.

Then the descent in both directions towards minimum energy structures was easy to

accomplish. As expected, the corresponding value of the activation energy, 25 kcal/mol,

was slightly larger than that computed with CASSCF(12/11).

As discussed in more detail below, there is an increase in charge localization at the

TS relative to that the minimum energy points.

The transition state located with SOS-MP2/cc-pVDZ is characterized by an imagi-

nary frequency of 188i cm−1. Starting from that point, the MEP was taken to the cis-

and trans-configurations (Fig. 2.5). Both DFT and SOS-MP2 exhibit a cusp at the tran-

sition state, which is a manifestation of the multireference character of the ground-state

wavefunction. The composition of the CASSCF wavefunction in the region discussed

below confirms that.

Table 2.4 presents computed equilibrium geometry parameters in the bridging

region: the C1P-C1B and C1I-C1B bond lengths and the τP (C2P-C1P-C1B-H) and τI

(N-C1I-C1B-H) dihedral angles (Fig. 2.1). The parameters optimized at the PBE0/6-

31+G(d,p), CASSCF(12/11)/cc-pVDZ, and SOS-MP2/cc-pVDZ levels are in agreement

with those obtained by Olsen and Smith40 with SA3-CAS(4/3)/DZP, which stands for

CASSCF with the (4/3) active space averaged over three states computed with the DZP

basis set. Overall, the geometry parameters at the stationary points are rather insensitive

to the level of theory: DFT, large active space state-specific CASSCF, small active space

state-averaged CASSCF, and SOS-MP2 yield bond lengths that agree within 0.02 A, and

angles that agree within 2◦.

63

Tabl

e2.

4:G

eom

etry

para

met

ers

(dis

tanc

esin

A,a

ngle

sin

degr

ees)

atth

est

atio

nary

poin

tsfo

rth

ega

s-ph

ase

cis-

tran

sch

rom

opho

reis

omer

izat

ion.

Stru

ctur

eM

etho

dC

1P–C

1BC

1B–C

1Iτ

Pτ

I

Cis

-iso

mer

PBE

0/6-

31+G

(d,p

)1.

404

1.38

50

180

CA

SSC

F(12

/11)

/cc-

pVD

Z1.

406

1.38

30

180

SOS-

MP2

/cc-

pVD

Z1.

415

1.39

70

180

SA3-

CA

S(4,

3)/D

ZPa

1.40

81.

382

018

0

Tran

sitio

nst

ate

PBE

0/6-

31+G

(d,p

)1.

365

1.45

80.

388

.2C

ASS

CF(

12/1

1)/c

c-pV

DZ

1.36

21.

477

1.0

86.4

SOS-

MP2

/cc-

pVD

Z1.

374

1.47

60.

087

.6

Tran

s-is

omer

PBE

0/6-

31+G

(d,p

)1.

403

1.39

20

0C

ASS

CF(

12/1

1)/c

c-pV

DZ

1.40

21.

395

00

SOS-

MP2

/cc-

pVD

Z1.

414

1.40

50

0SA

3-C

AS(

4/3)

/DZ

Pa1.

407

1.39

00

0

aR

ef.4

0.

64

At both cis- and trans-equilibrium points, the molecule is essentially planar except

that the methyl groups naturally have out-of-plane atoms. Analysis of the structures

reveals that the difference between C1P-C1B and C1B-C1I is less than expected for the

structure shown in Fig. 2.1. This can be explained by considering two resonance struc-

tures of the HBDI anion, see Fig. 2.2. The interaction between these two structures

results in charge delocalization between two oxygen atoms and in scrambling CC bond

orders as discussed, for example, in Ref. 40. The C1P-C1B bond acquires a double-

bond character, whereas the order of C1B-C1I bond is reduced. Natural Bond Orbital

(NBO)21, 22 charges and bond orders are consistent with computed bond lengths. Over-

all, CASSCF slightly exaggerates bond alternation relative to DFT or MP2, in favor of

the canonical structure (Fig. 2.1).

At the TS, the two rings are nearly perpendicular to each other (τI = 87−88◦), which

disturbs the π system and breaks the resonance interaction. The bond alternation pattern

is reversed such that C1P-C1B becomes shorter than C1B-C1I suggesting that one of the

two resonance structures becomes dominant. All the methods agree on the magnitude of

the change relative to the equilibrium structures: C1P-C1B, which is longer at the cis- and

trans-geometries, becomes shorter by about 0.04 A at the TS, whereas the C1B-C1I bond

is about 0.07 A longer at the saddle point. NBO charges (Table 2.5) and the molecular

orbital picture (Fig. 2.6) reveal almost complete charge localization on the imidazolin

ring.

Despite the similarity of the computed geometry parameters and charge distributions

at the minima and TS computed with DFT and CASSCF, respective activation energies

differ by more than 10 kcal/mol (34.5 versus 22.5 kcal/mol). Note that the corresponding

cis-trans energy differences are very close: 2.3 and 3.5 kcal/mol, respectively. A close

inspection of the charges from Table 2.5 reveals slightly more polar charge distribution

65

Figu

re2.

6:Fr

ontie

rva

lenc

em

olec

ular

orbi

tals

ofth

eH

BD

Iani

onin

the

cis-

form

(left

)and

atth

etr

ansi

tion

stat

eof

the

cis-

tran

siso

mer

izat

ion

path

(rig

ht).

66

Table 2.5: Cumulative natural charges on the fragments of the HBDI molecule:the phenyl and dimethylimidazolin rings and the CH bridge calculated withCASSCF(12/11) and DFT (PBE0 functional).

CASSCF(12/11)/cc-pVDZ PBE0/6-31+G(d,p)

cis TS trans cis TS trans

Gas phasePhenyl −0.61 −0.20 −0.50 −0.59 −0.30 −0.57Bridge 0.13 0.13 0.11 0.10 0.20 0.09Imidazolin −0.52 −0.93 −0.61 −0.51 −0.90 −0.52

Solutiona

Phenyl −0.85 −0.09 −0.91 −0.63 −0.19 −0.70Bridge 0.14 0.08 0.11 0.13 0.16 0.13Imidazolin −0.29 −0.99 −0.20 −0.50 −0.97 −0.43

a Calculated with QM/MM: EFP for the solvent-QM part interaction and TIP3P for thewater-water interaction.

for the PBE0 density: the positive charge on the bridge moiety is 0.19, versus 0.13

at the CASSCF level. Because of the large energy penalty due to charge separation

in the gas phase, small differences in ionicity may produce a large effect. Another

notable difference is larger asymmetry in oxygen charges: at the CASSCF level the

imidazolin oxygen is by 0.22e more negative than the phenyl oxygen, whereas at the

DFT level this difference is reduced to 0.15. This suggests larger contribution of the

second resonance structure in the CASSCF wave function, which can also contribute to

the energy difference.

More ionic character of the PBE0 density and the cusp-like shape of the pro-

file (Fig. 2.5) are due to multiconfigurational character of the wave function at the

TS, which is not adequately described by DFT (or MP2). The CASSCF amplitudes

67

show almost equal weights of the two dominant configurations, . . .(HOMO)2 and

. . .(HOMO)1(LUMO)1 at TS. The HOMO and LUMO at the TS differ considerably

from those at the equilibrium geometry (shown in Fig. 2.3) — they become localized

on the imidazolin and phenyl rings, respectively. Thus, (HOMO)2 and (LUMO)2 cor-

respond to the two charge-localized configurations, and their interaction results in less

ionic electron distribution. The ionicity is also reduced by (HOMO)1(LUMO)1.

A stability analysis of the Hartree-Fock wavefunction at the TS shows an RHF-UHF

instability with a negative eigenvalue of −0.037. The DFT/PBE0 solution, however,

proves to be stable, which means that using the symmetry broken unrestricted solution

to achieve better description of the barrier will not be useful in this case. Therefore,

a multiconfigurational approach is necessary not only for describing excited-state iso-

merization of the GFP-like chromophores, but also for modeling isomerization in the

ground state.

Although the CASSCF wave function is capable of capturing the multiconfigura-

tional character of the wave function, it needs to be augmented by dynamical correlation

to provide accurate energy differences. We included dynamical correlation correction

via multireference perturbation theory (MRMP2) for the Etc and Ea energies (at the

CASSCF geometries). This yields an activation energy of Ea = 26.2 kcal/mol, which is

3.7 kcal/mol higher than the CASSCF result.

The CASSCF results represent an improvement over DFT, however, there is still

a considerable discrepancy between the theoretical (22–26 kcal/mol) and experimental

(15.4 kcal/mol) values for Ea, as noted by the authors of experimental studies20, 39, 41.

Below we demonstrate that this discrepancy is resolved when solvent effects are taken

into account.

68

Aqueous solution: Continuum solvation models

Continuum solvation models12 provide a reasonable starting point in modeling ground-

state isomerization in aqueous solution. At first, two versions of the polarized continuum

model (PCM), D-PCM13 and C-PCM14, were applied to optimize the equilibrium geom-

etry parameters of the cis- and trans-isomers and the TS configuration and to compute

the relative energies ∆Etc and Ea at the PBE0/6-31+G(d,p) level. Both models produced

similar results: ∆Etc = 2.1 (D-PCM) and 2.3 kcal/mol (C-PCM); Ea = 33.5 (D-PCM),

34.0 kcal/mol (C-PCM). The geometry parameters and the energies are close to the gas

phase values obtained with the same DFT model. These results are fairly stable with

respect to the basis set: upon expanding it to 6-311++G(2d,p), Ea and ∆Etc change by

less than 1.5 kcal/mol and 0.7 kcal/mol, respectively. Table 2.6 summarizes solvent

effects on Etc and Ea computed using different approaches.

Next, we considered a new version of the continuum solvation model, SVPE15,

which has recently been implemented in GAMESS(US)9, for the single point calcu-

lations of relative energies ∆Etc and Ea at the gas-phase geometry parameters. At the

PBE0/6-31+G(d,p) level, we obtained a considerable reduction of the activation energy

compared to the PCM results: Ea = 24.6 kcal/mol, while the energy ∆Etc = 2.1 kcal/mol

was almost the same as in the PCM model. According to a comment by Chipman15, the

improvements introduced in SVPE do affect the reaction barrier estimated with the con-

tinuum solvation models, especially for charged substrates. The large solvent effect on

Ea (9.1 kcal/mol reduction relative to the gas-phase value) is consistent with the more

ionic character of the TS (see Table 2.5).

Finally, we combined the SVPE model with the adequate description of the elec-

tronic structure of the solute and carried out the calculations of relative energies with the

CASSCF(12/11)/cc-pVDZ description for the chromophore. The quantities, ∆Etc = 2.6

69

and Ea = 9.9 kcal/mol, can now be directly compared to the experimental free energy

difference of 2.3 and 15.4 kcal/mol39. The reduction of about 15 kcal/mol in Ea when

going from DFT to CASSCF results within the SVPE solvation model is consistent with

the corresponding reduction (13 kcal/mol) in the gas-phase calculations. Moreover, DFT

and CASSCF agree on the magnitude of the reduction of Ea due to solvent (9.1 kcal/mol

and 12.6 kcal/mol, respectively). Thus, the absolute value of Ea is overestimated by DFT

due to the multiconfigurational character of the electronic structure at the TS geometry,

which is correctly captured by CASSCF.

Aqueous solution: Explicit solvent molecules

The relative energies ∆Etc and Ea for the HBDI anion inside a cluster of water molecules

were computed using a QM/MM technique. To build the starting point, the chromophore

molecule at the gas-phase TS geometry was placed in a sphere of 200 water molecules

using the VMD computer program42. After running short molecular dynamics trajecto-

ries at various temperatures, the energy was minimized using molecular mechanics with

the CHARMM force field43 by keeping the solute species frozen. Then the solvent out-

side the first solvation shell was removed with 49 water molecules completely covering

the chromophore remaining in the system. This saddle point structure was re-optimized

for all geometric degrees of freedom by QM/MM using DFT with the PBE0 functional

and the 6-31+G(d,p) basis for the QM-part (chromophore), EFP16 for QM-solvent inter-

actions, and the empirical TIP3P potential for water-water interactions. The steepest

descent pathways taken in both directions from the optimized TS lead to the cis- and

trans-isomers of the trapped chromophore (Fig. 2.7). The energies obtained with this

method, ∆Etc = 5.0 kcal/mol and Ea = 26.0 kcal/mol, are consistent with the DFT/SVPE

calculations (Table 2.6).

70

Figure 2.7: Cis-trans isomerization of the chromophore inside a cluster ofwater molecules. Relative energies of the stationary points are computed withQM(CASSCF(12/11)/cc-pVDZ)/EFP/MM(TIP3P).

71

Tabl

e2.

6:C

is-t

rans

ener

gydi

ffer

ence

∆E t

can

dth

een

ergy

barr

ier

Ea

for

the

cis-

tran

siso

mer

izat

ion

ofth

eH

BD

Iani

onca

lcul

ated

atva

riou

slev

elso

fthe

ory

for

the

chro

mop

hore

mol

ecul

ean

dso

lven

t.

Met

hod

∆E t

c,kc

al/m

olE

a,kc

al/m

ol

Chr

omop

hore

Solv

ent

Gas

phas

eSo

lutio

nSh

ift

Gas

phas

eSo

lutio

nSh

ift

PBE

0/6-

31+G

(d,p

)2.

334

.5D

-PC

M2.

1−

0.2

33.5

−1.

0C

-PC

M2.

3+

0.0

34.0

−0.

5SV

PE2.

1−

0.2

24.6

−9.

9Q

M/M

Ma

5.0

+2.

726

.0−

8.5

CA

SSC

F(12

/11)

/cc-

pVD

Z3.

522

.5SV

PE2.

6−

0.9

9.9

−12

.6Q

M/M

Ma

2.1

−1.

411

.1−

11.4

MR

MP2

/cc-

pVD

Z3.

726

.2

aC

alcu

late

dw

ithQ

M/M

M:E

FPfo

rthe

solv

ent-

QM

part

inte

ract

ion

and

TIP

3Pfo

rthe

wat

er-w

ater

inte

ract

ion.

72

The energies of the cis-, trans- and TS structures were also calculated using

CASSCF(12/11) for in the QM-part. The interactions between the chromophore

molecule and solvent and between the water molecules were handled with EFP and

TIP3P, respectively. In line with the gas-phase and dielectric continuum model results,

the activation energy is lower at the CASSCF level relative to DFT: ∆Etc = 2.1 kcal/mol,

Ea = 11.1 kcal/mol.

The relative energies can be further refined by including the dynamical correlation

effects. In the gas phase, accounting for dynamical correlation via MRMP2 raises the

CASSCF activation barrier by 3.7 kcal/mol. Applying that correction, we estimate the

activation energy for the cis-trans isomerization of the GFP chromophore in aqueous

solution to be 14.8 kcal/mol. The remaining discrepancy with the experimental value of

15.4 kcal/mol39 can be partly attributed to the differences between the potential energy

barrier and the free energy barrier. The present QM/MM model with only 49 explicit

water molecules is not sufficient for describing the statistical state in solution. This water

solvation shell completely covers the chromophore molecule and accounts for principal

environmental effects, but precise free energy values along the isomerization pathway

should be estimated by using more appropriate condensed phase models. As shown,

e.g. in Ref. 108, no considerable changes in conclusions are expected if the dielectric

continuum model is applied on top of the model with explicit water molecules.

2.4 Chapter 2 conclusions

In this work we exploit modern quantum chemical methods for calculations of the

electronic properties of the GFP chromophore and compare the results to the gas-

phase absorption spectrum obtained by photodestruction spectroscopy in the ion-storage

ring3, 28, 29.

73

The experimental action spectrum of the denatured gas phase anionic GFP chro-

mophore features a broad line (2.4–2.8 eV) with a maximum at 2.59 eV and a minor fea-

ture at 2.3 eV. Wavefunction-based and DFT calculations estimate VDE of 2.4–2.5 eV.

Thus, we assign the minor peak as due to the photodetachment transition. Based on

our estimate of VDE, the absorption band at 2.6 eV corresponds to the transition to the

resonance state embedded in an electron detached continuum, and the broad character

of the spectrum is at least partially due to the interaction with the continuum states.

The resonance nature of the ππ∗ state suggest a finite lifetime, and that autoioniza-

tion channel should be considered when modeling the anionic GFP photocycle. The

triplet state is found to be well below the photodetachment threshold (vertical excita-

tion energy 1.86 eV). Thus, the two states are expected to have very different lifetimes,

which makes the suggested3 population trapping in the triplet state even more essential

for explaining slow fragmentation kinetics. The resonance nature of the ππ∗ state in

the anionic GFP might be responsible for very different behavior of the photofragment

yield of the anionic and protonated GFP3, however, more detailed electronic structure

calculations are required in order to suggest a viable mechanism. An important question

is how photoinduced isomerization and other structural changes40, 41 affect the relative

states energies.

All wavefunction-based and TD-DFT methods agree on the nature of the transition

lending the intensity to the resonance state, which is a bright ππ∗ transition (HOMO-

LUMO in a small basis set), however, quantitative agreement is more difficult to achieve.

Most importantly, small basis set calculations discretize the ionization continuum, and

the results of such calculations provide only a crude estimate of the energy of the res-

onance state. In order to account for basis set effects, the stabilization analysis can be

74

used; however, in view of the large size of the GFP chromophore molecule, we were

able to only conduct it with the CIS and BNL methods.

Nevertheless, it is instructive to compare vertical excitation energies of the ππ∗ state

computed with different methods in a moderate basis set and with the experimental band

maximum. While the CIS/aug-cc-pVTZ excitation energy is more than 1 eV off, per-

turbative inclusion of double excitations by SOS-CIS(D) yields a value which is within

0.1 eV from the experimental band maximum. The EOM-CCSD values computed in the

modest 6-31+G* basis are within 0.38 eV from the experimental absorption maximum.

An analysis of the EOM-CCSD wavefunction confirms the dominant one-electron char-

acter of the ππ∗ state, however, perturbative inclusion of triple excitations and a larger

basis set are required for a converged (with respect to the level of theory) EOM-CC

value. Based on previous studies, a proper account of dynamical correlation by includ-

ing triple excitations and increasing basis set can change the vertical excitation energy

by as much as 0.3 eV. Basis set effects evaluated using inexpensive SOS-CIS(D) calcu-

lations affect vertical excitation energies by 0.14 eV upon the transition from cc-pVDZ

to aug-cc-VTZ.

Additional uncertainties arise from the ground state geometry. For example, dif-

ferent choices of the ground-state geometry (optimized with CASSCF, DFT, and MP2)

introduce an uncertainty of 0.1 eV in the vertical excitation energies. A relatively strong

dependence of the excitation energy on the structure suggests additional broadening of

the absorption band due to vibrational excitations of the chromophore.

The best MRMP2 estimate is within 0.07 eV from the experimental band maximum,

however, the inclusion of basis set effects (using the SOS-CIS(D)/aug-cc-pVTZ esti-

mate) increases the difference to 0.21 eV. Overall, the observed variations demonstrate

75

that it is unrealistic to expect an accuracy better than 0.1 eV from computational proto-

cols applicable to a molecule of this size even for non-resonance excited states.

BNL/cc-pVDZ vertical excitation energy of the bright ππ∗ transition is above

MRMP2/cc-pVDZ value by 0.9 eV. Since the self-interaction error is considerably

reduced in BNL, the photodetachment continuum is likely not contaminated by spu-

rious low-lying charge-transfer states ubiquitous in TD-DFT calculations.

Calculations of the cis-trans isomerization pathway require wavefunctions that are

flexible enough to reflect the multiconfigurational character of the transition state.

CASSCF gives a qualitatively correct curve, whereas DFT and SOS-MP2 fail at the

twisted geometries. The gas-phase CASSCF value of the barrier height is 10 kcal/mol

higher than the experimental value, however, the inclusion of solvent effects brings it

down to 9.9–11.1 kcal/mol, which agrees well with 15.4 kcal/mol derived from experi-

mental measurements. Including dynamical correlation correction yields 14.8 kcal/mol,

which is within 0.8 kcal/mol from the experiment. Good agreement between SVPE and

QM/MM calculations further supports the validity of our results. Large solvent effect on

Ea is due to the more ionic character of the TS, in which negative chare is localized on

imidazolin ring, in contrast to the equilibrium structure in which the negative charge is

delocalized between the two rings. Our calculations help resolve previous disagreement

between theory and experiment with respect to the GFP chromophore isomerization in

aqueous solution.

76

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80

Chapter 3: The effect of oxidation

on the electronic structure of the green

fluorescent protein chromophore

3.1 Chapter 3 introduction

The unique properties of green fluorescent protein (GFP) exploited in novel bioimag-

ing techniques have revolutionized many areas in life sciences1–3. The photophysics

of GFP and other wild-type and mutated photoactive proteins have been a subject of

many experimental and theoretical studies4, 5. Yet, intrinsic electronic properties of the

chromophores and their interactions with the protein matrix are not fully understood.

The absorption spectrum of the wild-type GFP has two broad (0.2–0.4 eV) bands at

3.13 eV (396 nm) and 2.60 eV (476 nm) assigned to a neutral and an anionic (deproto-

nated) forms of the chromophore, respectively. The fluorescence spectrum is also broad

and has a maximum at 2.44 eV (508 nm). Detailed mechanistic studies have demon-

strated that while both the neutral and the anionic forms of the GFP chromophore are

responsible for absorption, fluorescence occurs from the anionic form (see, for example,

recent reviews4, 5 on the photophysics of GFP and references therein).

81

The absorption and fluorescence wave lengths and quantum yields can be controlled

by structural modifications, and a number of GFP mutants and other photoactive pro-

teins have been characterized5, 6. The changes in photophysics can be due to structural

modifications of the chromophore itself or in the surrounding protein. Moreover, the

photochemical properties of some chromophores from the GFP family can be affected

by light giving rise to so-called optical highlighting techniques. By using light of vary-

ing intensity and duration, the fluorescence can be turned on and off (photoswitching),

and its spectrum (both wave length and intensities) can be changed (photoconversion).

Both reversible and irreversible photoconversion have been described in the literature5, 7.

Photoconversion phenomena are not yet well understood at the molecular level. The

established mechanisms involve photoinduced isomerizations (e.g., cis-trans transfor-

mation), changes in protein microenvironment, as well as more significant structural

changes such as bond breaking, proton and charge transfer between the protein and

the chromophore5, 7. For example, the intensity of GFP fluorescence is increased by

two orders of magnitude upon irreversible photoconversion by UV or visible light (4.9–

2.6 eV) to the anionic form accompanied by the decarboxylation of an adjacent glu-

tamine residue8, 9. That also leads to a change in the absorption spectrum: increased

absorbance of blue light and diminished absorbance in the UV region. In another FP,

Dronpa, photoconversion between a protonated dark and an anionic bright forms is

reversible and involves cis-trans isomerization10. A different photoconversion type is

exemplified by Kaede FP, which can be irreversibly switched from green to red fluores-

cent form by the cleavage of a peptide bond leading to the extension of the π-system of

the chromophore7.

82

Photoinduced redding of GFP is a photoconversion that has not yet been explained.

It was observed11 that under anaerobic conditions illumination of a GFP mutant (GFP-

mut2) with blue light (2.5–2.7 eV or 460–500 nm) for several seconds results in red

fluorescence at 1.8–2.0 eV (610–680 nm). Similar behavior was observed for wild-type

GFP and other proteins. GFP photoconversion was more efficient when using higher

energy UV light (3.3–3.7 eV or 340–380 nm)11. A more detailed study of Elowitz et. al

reported that illumination of GFP by blue light converts GFP to a stable (under anaer-

obic conditions) form that absorbs green light at 2.36 eV (525 nm) and emits green,

yellow and red light (emission maxima at 2.21 eV, 2.10 eV and 2.07 eV)12. Based on

the measured timescale of brightening (0.7 sec), this study suggested that the photoac-

tivation proceeds in two steps: blue light (2.54 eV, 488 nm) stimulates a fast transition

to an excited intermediate which then decays slowly to the red-emitting GFP state (the

second step can proceed in the dark).

Recently, photoinduced redding of GFP has been observed under aerobic condi-

tions13. In the presence of different oxidizing agents (e.g., K3Fe(CN)6, benzoquinone,

etc), irradiation with 2.54 eV (488 nm) light results in the bleaching of the green fluo-

rescence and the appearance of the red signal with excitation and emission maxima at

2.16 eV (575 nm) and 2.04 eV (607 nm), respectively. Measurements at different light

intensities determined that the redding is a one-photon process. Based on kinetic data,

which showed that the green fluorescence disappears faster than the red fluorescence

builds up, a two-step mechanism involving oxidation of GFP was suggested13. By test-

ing a variety of electron acceptors, it was found that the redding was achieved using

compounds with the standard reduction potential E◦ in the range of +0.42 to −0.32 V.

The yield of the redox reaction between GFP and cytochrome c was found to be 1.7

suggesting that a two-electron oxidation of GFP is responsible for the redding (that is,

83

two reduced cytochrome c molecules are produced per one converted GFP molecule).

The stability of the red form under the aerobic conditions points out that the mecha-

nisms of anaerobic and oxidative redding are, most likely, different13. Moreover, the

excitation spectra of the two forms are different: the maxima are 2.36 and 2.16 eV (525

and 575 nm) for anaerobic and oxidative redding forms of GFP, respectively. Bogdanov

et al.13 hypothesized that anaerobic redding proceeds by photoreduction producing a

stable radical (which will be sensitive to the presence of oxygen). Their findings suggest

that oxidative redding produces more stable species. Other GFP variants, i.e., AcGFP1,

TagGFP, zFP506, amFP486, and ppuGFP2 also undergo oxidative redding, however, the

blue and cyan mutants (that have notably different chromophores) of GFP (EBFP and

ECFP) do not13.

Theoretical modeling noticeably contributes to the studies of the GFP and related

model systems14–24 (for a recent review, see Ref. 25). In this chapter we charac-

terize the electronic structure of two different oxidized states of the deprotonated 4’-

hydroxybenzylidene-2,3-dimethylimidazolinone (HBDI) anion (Fig. 3.1). We consider

singly and doubly oxidized forms and discuss excitation energies and oscillator strengths

of the electronic transitions in these species. To avoid confusion, it is important to clar-

ify the terminology. Here we refer to deprotonated HBDI as the HBDI anion, or simply

anion. The (neutral) species derived by removing one electron from the anion is called

the HBDI radical, and doubly oxidized deprotonated HBDI is called the HBDI cation.

We also present the relevant ionization energies and estimate standard reduction poten-

tials corresponding to these oxidation steps. The calculations are performed for the

isolated gas-phase chromophores, however, possible effects of the solvent on reduction

potentials are also discussed.

84

(a)

(b)

Figure 3.1: Structures of deprotonated HBDI (HBDI anion, top) and the cation(bottom). The structure of the radical is similar to that of the cation, with theunpaired electron being on the carbon that hosts positive charge in the cation.

The protein environment is crucial for the function of the chromophore. The rigid

protein matrix restricts the chromophore range of motions and impedes twisting motion

in the excited state, which strongly affects the S1–S0 oscillator strength. For example,

the chromophore in solution does not fluoresce, and the fluorescence is restored only in

glass matrices4. The photoinduced conversion of the neutral to the anionic form also

85

involves the protein. It is facilitated by a hydrogen bond network26–28 and can be cou-

pled to the decarboxylation of the glutamate 222 residue8, 9. The electrostatic field of the

protein may induce spectral shifts, although gas-phase studies suggested that the effect

of the protein environment on the excited states energies may be relatively small29, 30.

Despite the role of the protein, an important step towards understanding of the function

of GFP is characterization of the intrinsic properties of the isolated chromophore. By

focusing on the electronic properties of the chromophore, we can quantify changes in

absorption and fluorescence due to specific structural changes and to test different mech-

anistic ideas. Our calculations show that the absorption of the singly oxidized species

(radical) is blue shifted with respect to the HBDI anion. However, the doubly oxidized

system (cation), which has a stable closed-shell electronic structure, features red-shifted

absorption at 2.02 eV. The proposed mechanism involves two-step oxidation proceeding

via electronically excited states.

3.2 Computational details

Equilibrium geometries of closed-shell ground-state species (i.e., the HBDI anion and

cation) were optimized by RI-MP2/cc-pVTZ. The ground state structure of the doublet

radical was computed using ωB97X/6-311(+,+)G(2df,2pd). The harmonic vibrational

frequencies and IR intensities of the anionic and cationic forms of deprotonated HBDI

were computed with RI-MP2/cc-pVTZ.

Excitation energies of the cation were computed using SOS-CIS(D) and EOM-EE-

CCSD. SOS-CIS(D) calculations employed the cc-pVTZ and cc-pVDZ bases31. EOM-

CCSD calculations used the 6-311G* basis set32, 33. For the anion, SOS-CIS(D) exci-

tation energies are in excellent agreement with more accurate MR-PT. The oscillator

strengths for SOS-CIS(D) were computed using the underlying CIS wave functions,

86

whereas for EOMEE-CCSD, the transition density matrices computed using EOMEE

wavefunctions were employed.

Electronic structure calculations for the open-shell radical species are more chal-

lenging due to spin contamination of the doublet reference. This effect can be mitigated

by using better reference orbitals, e.g., those optimized for the MP2 wave functions, i.e.,

O2 method. We employed O2 orbitals in the SOS-CIS(D) calculations of the doublet

states. Moreover, excited states in doublet radicals can also develop a notable doubly

excited character causing poor performance of single-reference methods using the dou-

blet reference34–37. This can be avoided by employing the spin-flip (SF) methods38–40

utilizing a high-spin quartet reference, i.e., (π2)1(π1)

1(π∗)1. The SF approach mitigates

spin contamination and provides more balanced description of the ground and relevant

excited states. Our EOMSF-CCSD calculations (which employed B3LYP orbitals to

control spin-contamination of the quartet reference) showed that the wave functions of

the excited states are of singly-excited character (with respect to the doublet reference)

thus validating the SOS-CIS(D) results. The 〈S2〉 values for the three lowest states of

the doublet are 0.78. 0.79, and 0.81. The EOMSF-CCSD and SOS-CIS(D) calculations

are in quantitative agreement. Finally, MR-PT calculations on the doublet states are also

in agreement with EOMSF-CCSD and SOS-CIS(D).

MR-PT calculations employed a modified version of second-order multiconfigura-

tional quasidegenerate perturbation theory (MCQDPT2)41. Originally, the correspond-

ing programs were implemented in GAMESS (US)42 and PC GAMESS43. Recently,

serious bugs in the MCQDPT2 algorithm were noticed44 and resolved in the new vari-

ant called XMCQDPT2 as implemented in PC GAMESS/Firefly44.

The XMCQDPT2 calculations used the complete active space self-consistent field

(CASSCF) state-averaged (SA) electron density over several lowest-lying states. As

87

CASSCF does not account for dynamical electron correlation45, the CASSCF state

ordering is often different from the correct order predicted by higher-level calculations.

This is the case for the doublet radical: the fourth lowest-lying state in the XMCQDPT2

calculation appears the seventh in CASSCF. Since it is one of the states of interest (i.e.,

the D2 state whose orbital character is described below), we had to use an extensive

state-averaging procedure including eight states in the consideration. The active space

for SA-CASSCF was chosen as 12 electrons distributed over 11 orbitals, which is a

truncated representation of the complete set of 14 π-orbitals forming the conjugated

system of the chromophore. The MCQDPT2 based approaches have been shown to

yield accurate results for a variety of chromophores41, 46, 47. The underlying CASSCF

wave functions are free of spin contamination.

The energy differences between different oxidized states were computed as follows.

The vertical energy difference between the anion and the neutral radical, i.e., the first

VDE, was computed by EOMIP-CCSD/6-311G* (2.33 eV) and corrected for basis set

increase from 6-311G* to aug-cc-pVTZ using a Koopmans estimate (0.21 eV). Thus,

our best estimate of VDE is 2.54 eV.

The vertical energy gap between the cation and the anion is computed as the dif-

ference between the RI-MP2/cc-pVTZ total energies of the respective species. The dif-

ferences between the doublet and the cation were computed from the relative energies

of the cation and the anion and the anion’s VDE. Finally, the adiabatic values were

obtained from vertical gaps by using relaxation energies of the neutral and the cation,

which were computed by RI-MP2/cc-pVTZ (cation) and ωB97X/6-311(+,+)G(2df,2pd)

(radical) using the geometries described above.

EOM and SOS-CIS(D) calculations were performed using Q-Chem48. XMCQDPT2

calculations were carried out with PC GAMESS/Firefly43.

88

Table 3.1: Vertical excitation energies (eV) and oscillator strengths of the ππ∗ statesof the doublet HBDI radical. The values of 〈S2〉 are also shown.

Method D1 〈S2〉 D2 〈S2〉

SOS-CIS(D)/cc-pVDZa 1.64 (0.02) 1.04 3.53 (1.35) 0.90SOS-CIS(D)/cc-pVTZa 1.52 (0.02) 1.01 3.37 (1.34) 0.89EOM-SF-CCSD/6-311G*b 1.93 (0.01) 0.79 4.14 (1.04) 0.81XMCQDTP2/6-311G*c 1.80 (0.01) 3.27 (0.66)

a Using O2 orbitals. Oscillator strength are computed at the CIS level. b Using B3LYPorbitals. c Based on SA(8)-CASSCF(12/11).

3.3 Results and discussion

Relevant MOs and electronic states of the gas-phase HBDI anion are shown in Figs. 3.2

and 3.3. Gas-phase photodestruction spectra show the absorption maximum of the

anionic GFP chromophore at 2.59 eV29, 30, which is remarkably close to the absorption

of wild-type GFP at 2.61 eV. Most of the electronic structure studies agree on the char-

acter of the bright state14–17, 19, 21–23, 49, which is of a π→ π∗ type (Fig. 3.2), although

there is a spread in reported excitation energies. Our recent theoretical study revealed

that the bright state is, in fact, a resonance state embedded in a photodetachment con-

tinuum. These calculations determined that the vertical detachment energy (VDE) of

the HBDI anion is 2.4–2.5 eV, which is below the bright state. A recent experimental

study50, which also employed electronic action spectroscopy, reported the vertical exci-

tation energy of 2.60 eV and determined VDE to lie within 2.48–2.8 eV, in excellent

agreement with our computational predictions.

89

Figure 3.2: Relevant molecular orbitals of deprotonated HBDI (HF/6-311G*). Inthe ground state of the anion, both π1 and π2 are doubly occupied, and the brightstate is derived by the π1→ π∗ excitation. Oxidized forms are derived by removingthe electrons from π1.

As evident from Fig. 3.3, the electronic excitation using 2.6 eV (and higher) pro-

duces an excited state that can photodetach an electron producing a doublet neutral radi-

cal. Thus, our first step is to characterize electronic transitions of this form. Three lowest

electronic states of the doublet and the corresponding excitation energies and oscillator

strengths are shown in Fig. 3.4 and summarized in Table 3.1. The ground state of the

doublet is derived by the ionization from the anion’s HOMO (π1 from Fig. 3.2). The

two excited states are of a mixed character and include almost equal contributions from

the π1→ π∗ and π2→ π∗ excitations. Consistently with the allylic character of the three

relevant MOs, both excitations have large transition dipole moments, and their plus and

minus combinations give rise to the bright and dark states at 4.14 eV ( fl = 1.04) and

90

Figure 3.3: Energy levels and electronic states of the HBDI anion and radical.The bright absorbing π1π∗ state at 2.62 eV is a resonance state embedded in aphotodetachment continuum beginning at 2.54 eV. The respective MOs are shownin Fig. 3.2.

Figure 3.4: Electronic states of the doublet HBDI radical. The respective MOs areshown in Fig. 3.2.

1.93 eV ( fl = 0). The SOS-CIS(D) and MR-PT calculations agree with EOM-SF-CCSD

on the character of the excited states and yield slightly lower excitation energies. Over-

all, the bright state of the doublet is strongly blue-shifted relative to the anionic form.

Therefore, the doublet radical is an unlikely candidate for the red GFP form. The energy

spacing between the two excited doublet states is 2.21 eV and the oscillator strength is

non-negligible (0.01). Thus, red-shifted electronic transitions between D2 and D1 are

possible.

Doubly ionized HBDI anion (i.e., the cation) is a well-behaved closed-shell system

derived by removing two electrons from π1. Its electronic states are summarized in

91

Table 3.2: Vertical excitation energies (eV) and oscillator strengths of the excitedstates of the HBDI cation at its equilibrium geometry.

Method 1A′ ( fl) 1A′′

SOS-CIS(D)/cc-pVDZa 2.10 (1.20) 2.14SOS-CIS(D)/cc-pVTZa 2.02 (1.17) 2.09EOM-CCSD/6-311G* 2.53 (0.78) 2.10XMCQDPT2/6-311G*b 2.27 (0.84)

a Oscillator strength are computed at the CIS level. b Based on SA(2)-CASSCF(12/11).

Figure 3.5: Electronic states of the doubly ionized HBDI anion. The respectiveMOs are shown in Fig. 3.2.

Fig. 3.5 and in Table 3.2. The lowest singlet excited state (1A′) is at 2.02 eV (SOS-

CIS(D)/cc-pVTZ level) and has a large oscillator strength. It is red-shifted relative to

the S0 → S1 transition in the anion by 0.6 eV, which is similar in magnitude to the

observed experimental shift in absorption. The EOM-EE-CCSD excitation energy is

higher, but the shift relative to the EOM-EE value of the S1 excitation energy in the

anion (at 3.1 eV at the EOM-EE/6-311G* level) is very close to the SOS-CIS(D) one.

Thus, this species is a possible candidate for the red fluorescent form of GFP produced

in oxidative redding experiments13.

The red shift is consistent with the changes in the character of the electronic transi-

tion, which is a π2→ π1 excitation, as opposed to the π1→ π∗ transition in the anion.

92

Figure 3.6: Energy diagram of the relevant electronic states of deprotonated HBDI.

Thus, in terms of the anionic orbitals, the S1 state of the cation is HOMO−1→HOMO

excitation. Because of a relatively dense spectrum of the occupied orbitals (which

becomes even denser in the positively charged system), the gap between HOMO− 1

and HOMO is smaller than the gap between the HOMO and the LUMO. Thus, double

ionization is expected to result in a red shift in the lowest electronic transition.

Interestingly, the π2→ π∗ state of the cation is much higher in energy (approximately

at 3.16 eV, fl = 0.0261). The closest electronic state is a dark state, 1A′′ derived from

the HOMO−3→ HOMO transition (in terms of the anionic orbitals).

93

Fig. 3.6 shows an overall energy diagram for the forms of HBDI. Adiabatically, the

doubly oxidized form is 9.98 eV above the ground state of the anion. The respective

value of VDE (corresponding to removing two electrons) is 10.37 eV. The adiabatic IE

from the ground state of the doublet radical is 7.59 eV (computed using the anions’s

VDE value of 2.54 eV and 0.15 eV relaxation energy of the neutral radical). Another

relevant value is the energy gap between the excited states (D1 and D2) of the doublet

radical and the cation. Using the same values of vertical detachment and relaxation

energies, and 1.52 eV and 3.37 eV for the vertical D0 → D1,2 excitation energies, we

estimate the ionization energy of the electronically excited doublet radical as 6.07 eV

and 4.22 eV, respectively. Finally, the energy gap between the excited singlet state of

the anion and the two excited states of the radical are 1.29 eV and 3.14 eV, respectively.

To evaluate whether two-electron oxidation of the anion is possible using the elec-

tron acceptors employed in Ref. 13, it is necessary to estimate standard reduction poten-

tials corresponding to the detachment and ionization transitions described above. Rig-

orous calculations of the oxidation/reduction potentials require free energy calculations,

however, a simpler approach based on the correlation between detachment/ionization

energies and the standard reduction potentials can be employed51:

Eo = (−2.59±0.26)+(0.56±0.03) ·VIE, (3.1)

where E◦ corresponds to the following half-reaction:

A++ e = A (3.2)

and VIE is the ionization energy of A. This equation was derived by comparing experi-

mental values of the reversible oxidation potentials (in DMF or ACN solvent) for a set of

94

Table 3.3: Standard oxidation potentials corresponding to different one-electronoxidation transitions computed using gas-phase detachment and ionization ener-gies (see text).

Transition IE, eV E◦, V

D0→ cation 7.59 1.66±0.49D1→ cation 6.07 0.81±0.44D2→ cation 4.22 −0.23±0.39S1→D1 1.29 −1.87±0.30S1→D2 3.14 −0.83±0.35

14 organic molecules with their gas-phase ionization energies. Applying this equation,

we arrive at the values of E◦ summarized in Table 3.3. Eq. (3.1) was derived using IEs

in the range of 5.6–9.0 eV and its application to lower values (such as S1→ D1,2 transi-

tions from Table 3.3) assumes that the trend can be extrapolated. Thus, estimations for

the lower IEs are of a semiquantitative value only.

Oxidation of the ground-state doublet radical (D0) corresponds to E◦ = 1.66 V,

which is much larger than E◦ of the strongest oxidizing agent employed in Ref. 13

(K3Fe(CN)6, E◦ = 0.42 V). However, the oxidation of the electronically excited neutral

radical corresponds to E◦ = 0.81 and E◦ = −0.12 V for the D1 and D2 states, respec-

tively. Thus, electronically excited doublet radical (in the D2 state) can be oxidized even

by the weakest oxidizing agent from Ref. 13 (E◦ =−0.32 V), whereas the oxidation of

D1 is less likely. Finally, the electronically excited anion can be easily oxidized produc-

ing excited states of the doublet (E◦ = −1.87 and E◦ = −0.83 V for D1 and D2 states,

respectively). Note that the S1→D0 transition is spontaneous in the gas phase and does

not require the presence of oxidizing agents due to the autoionizing nature of S1. By

95

comparing electronic configurations of the doublet states (see Fig. 3.4) and the S1 state

(Fig. 3.3), we note that these transitions include significant one-electron character.

Thus, a possible mechanism of the oxidative redding emerging from the present

study is

The first step involves photoexcitation, and blue light is sufficient to generate this tran-

sition. The second and third steps are one-electron oxidation steps with E◦ =−0.88 V,

which is within the reach of the oxidizing agents employed in Ref. 13. The two-electron

oxidation is consistent with the stoichiometric results from Ref. 13. Moreover, the

closed-shell character of the cation is consistent with the chemically stable nature of

the red form of GFP. Moreover, the closed-shell character of the cation is consistent

with the relatively chemically stable nature of the red form of GFP. Although the chem-

ical structure of the cation suggests high reactivity with respect to nucleophilic agents

such as water, the protein may be sufficiently effective in shielding this fragile species.

Finally, the absorption of the cationic form is red-shifted by 0.6 eV, and the resulting

value of 2.02 eV is in a good agreement with the experimental excitation energy of

2.12 eV. The structural differences between the chromophores of the FPs that undergo

oxidative redding (AcGFP1, TagGFP, zFP506, amFP486, and ppuGFP2), and those that

do not (EBFP and ECFP) provide additional support to the proposed mechanism. The

chromophores in the former group contain a phenolic moiety that can be easily con-

verted to the anionic deprotonated form and thus oxidized, whereas in EBFP and ECFP

the imidazolin moiety is connected to nitrogen-containing aromatic fragments that are

expected to be basic rather than acidic.

96

Whereas the present model is consistent with the available experimental information,

it cannot rule out possible involvement of the protein via proton transfer or other chem-

ical changes (such as covalent bond breaking leading to the extension of the π-system

of the chromophore7, 24). Thus, it would be desirable to observe oxidative redding in

the isolated chromophore, e.g., in solution. However, it is essential for the proposed

mechanism that the excited states involved in oxidation, S1 of the anion and D1,2 of

the radical, are relatively long-lived. Thus, it may be problematic to observe oxida-

tive redding of isolated HBDI in solution, where the internal conversion is markedly

faster than in the protein (which is responsible for the loss of fluorescence), except for

low-temperature glass matrices (in which fluorescence is restored). Finally, the protein

environment may change the predicted energetics by stabilizing different states differ-

ently. However, the observed strong red shift in the doubly oxidized species originates

in strong changes in the electronic structure of the chromophore, and it is unlikely that

the energies of these transitions will be affected by the protein much stronger than the

energies of the transitions in the anion. Our preliminary QM/MM calculations, which

treated the chromophore and the nearest amino acid side chains as the quantum subsys-

tem and the rest of the model protein system as the collection of effective fragments, do

not show dramatic structural changes in the chromophore binding site for the anionic

and cationic species. We also note that the present model does not address the nature of

the anaerobic redding of GFP, which will be discussed elsewhere.

Ionization-induced structural changes

As discussed in Chapter 2, the equilibrium structure the HBDI anion can be explained

by considering two resonance structures shown in Fig. 3.1 (upper panel). Consequently,

the negative charge is delocalized between the phenolic and imidazolin oxygens and the

97

Table 3.4: Selected geometric parameters (bond lengths in A, angles in degrees) ofthe HBDI anion, and its singly and doubly oxidized forms.

Species CP-CB CI-CB CP-CB-CI OP-C OI-C

Aniona 1.393 1.385 134.8 1.248 1.236Cationa 1.386 1.389 127.8 1.226 1.197Radicalb 1.413 1.369 129.6 1.228 1.209

a RI-MP2/cc-pVTZ b ωB97X/6-311(+,+)G(2df,2pd)

CI–CB and CP–CB bond orders are scrambled. The analysis of the structural param-

eters and the wave functions demonstrated that the phenolic form dominates, i.e., the

CP–CB bond is longer than CI–CB. Double ionization eliminates the phenolic struc-

ture entirely producing the quinoid structure shown in Fig. 3.1 (bottom). The positive

charge located on the imidazolin carbon is stabilized by electron-donating methyl group

(the NBO analysis52 suggests considerable delocalization of the hole over imidazolin

moiety). The structure of the radical can be described as the quinoid structure with the

unpaired electron on the imidazolin’s methyl-substituted carbon.

Table 3.4 summarizes important geometric parameters of the anion, neutral, and

the cation. Consistently with the above analysis, both CO bonds become shorter in the

cation, and the CP–CB bond becomes shorter than CI–CB. However, the bond alternation

remains small due to conjugation. Moreover, because of the two resonance structures

present in the anion, the overall structural differences between the anion and the cation

are less than could be expected by considering only the dominant phenolate structure.

The ionization-induced structural changes strongly affect vibrational frequencies,

which may be exploited in the experimental characterization of the red form of GFP,

e.g., by using IR or Raman spectroscopy as in Refs. 9, 53 (a detailed summary of the

vibrational spectroscopy studies of GFP can be found in Ref. 54). Fig. 3.7 shows the

98

Figure 3.7: Infrared spectra of the anionic and cationic forms of deprotonatedHBDI computed with RI-MP2/cc-pVTZ.

computed IR spectra of anionic and cationic deprotonated HBDI. The anion spectrum is

dominated by a band at 1691 cm−1, which is an asymmetric CO stretch strongly coupled

with an asymmetric bridge vibration. In the cation, this mode changes its character to an

almost pure bridge vibration, loses intensity and is blue-shifted. Instead, those modes

gain intensity that involve the imidazolin moiety hosting a positive charge. The two

most intense bands are predominantly CH3 vibrations at 1534 and 1435 cm−1, followed

by lower frequency bands involving skeletal imidazolin vibrations.

3.4 Chapter 3 conclusions

We characterized changes in the electronic structure of the deprotonated HBDI anion

due to one- and two-electron oxidation processes. One-electron oxidation produces

a doublet radical with strongly blue-shifted absorption, whereas the absorption in the

cation is red-shifted by almost 0.6 eV with respect to the anion. From the electronic

structure point of view, the cation has a closed-shell singlet ground state, and is expected

to be relatively stable chemically.

99

Our results suggest that the doubly oxidized species (the deprotonated HBDI cation)

may be responsible for the oxidative redding of GFP, which was recently reported13. The

proposed mechanism involves two-step oxidation via electronically excited states and is

consistent with the available experimental information13, i.e., it is a one-photon process

including two-electron oxidation resulting in chemically stable species. Moreover, all

proteins that undergo oxidative redding contain a phenolic moiety and are expected to

have similar oxidative properties, whereas the two proteins that do not exhibit redding

(EBFP and ECFP) have nitrogen-containing aromatic fragments that are less efficient

electron donors. This suggests that the oxidative redding is an intrinsic property of

the chromophore, although chemical changes involving the protein cannot be ruled out.

Moreover, the irreversible nature of the oxidative redding13 suggests that the protein is

likely to change in response to the changes in the chromophore’s oxidation state.

The closed-shell character of the cation is consistent with the relatively chemically

stable nature of the red form of GFP. Although the chemical structure of the cation

suggests high reactivity with respect to nucleophilic agents such as water, the protein

may be sufficiently effective in shielding this fragile species.

Ionization-induced structural changes of the chromophore result in significant

changes in vibrational spectra (the frequencies as well as the character of the modes

and IR intensities) which may exploited in the experimental validation of the proposed

mechanism.

We would like to emphasize that simple molecular orbital considerations predict red-

shifted absorption for any doubly-oxidized species. Thus, while the predicted values of

the absorption maximum may change once the protein environment is taken into account

(or at a higher level of theory), the overall trend remains unchanged. This mechanism of

100

redding is distinctly different from previously characterized ones in which redding was

achieved by extending the π-system of the chromophore7.

101

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106

Chapter 4: Electronically excited states

of uracil

4.1 Chapter 4 introduction

This chapter presents calculations of the electronic structure and the excited states of

uracil using high-level MRCI and EOM-CC methods with the purpose of obtaining reli-

able and accurate theoretical estimates of vertical excitation energies. We investigate

the dependence of the excitation energies on the basis set, the possible doubly-excited

character of the states, as well as dynamical correlation effects using several EOM-CC

methods with triple excitations1–5.

Our best estimates of the vertical nπ∗ and ππ∗ excitations are 5.0 eV and 5.3 eV,

respectively, with estimated error bars of less than 0.1 eV. We attribute the 0.2 eV

difference between the computed vertical excitations and the maximum of the exper-

imental absorption to a strong vibronic interaction between the lowest A′′ and A′ states

leading to intensity borrowing by the forbidden transition. Experimental observations

of Kong et al.6 suggest the strong vibronic coupling between the states. The EOM-

CCSD/aug-cc-pVTZ excitation energy of the nπ∗ state is only 0.23 eV higher than our

best estimate, which is well within a conservative estimate of the EOM-CCSD error

bars7. Thus, no evidence of a significant overestimation of the excitation energies in

107

uracil by EOM-CCSD has been observed. We also investigate solvent effects, and pro-

vide accurate estimates for vertical excitation energies of triplet and other singlet states.

4.2 Computational details

Density functional theory (DFT) with the B3LYP8 functional and the 6-

311G(2df,2pd)9, 10 basis set are employed for ground-state structure optimization.

Pople’s 6-31G11 and 6-311G9 basis sets with polarization and diffuse functions are

used in the study. Their performance is compared with Dunning’s cc-pVDZ and aug-

cc-pVTZ12 bases, as well as the NASA Ames atomic natural orbital (ANO)13, 14 basis

set contracted as double-ζ and augmented with diffuse functions taken from the aug-

cc-pVDZ12 basis set. In many calculations, mixed basis sets are employed: a higher

quality basis is used on heavy atoms. The mixed basis sets are designated with a for-

ward slash; for example, 6-311+G(d)/6-31G(d,p) means that 6-311+G(d) is used on the

carbon, nitrogen, and oxygen atoms, and 6-31G(d,p) is used on the hydrogens. All basis

sets include the pure angular momentum spherical harmonics (5 d-functions).

The lowest eight core orbitals are frozen in all the calculations. Where specified, the

highest eight virtual orbitals are frozen as well.

Active spaces are denoted as (n,m), which stands for n electrons in m orbitals.

The EOM-CC(2,3) and EOM-CCSDt calculations are performed with two active

spaces. The first one contains 10 electrons in 8 orbitals: five highest occupied (two

a′ in-plane lone pairs and three a′′ π orbitals) and three lowest virtual orbitals (one a′

orbital and two a′′ π∗ orbitals). We denote this active space as (10,8). The second active

space extends (10,8) by including two more lowest virtual orbitals, both of which are a′.

This active space is denoted (10,10).

108

Two different MRCI expansions are used. In both, the eight core and the highest

eight virtual orbitals are frozen. The first expansion, denoted MRCI1, is based on a

(14,10) active space, which consists of two a′ in-plane lone pair orbitals, five occupied

a′′ π orbitals, and three virtual a′′ π∗ orbitals. The MOs are obtained from a state-

averaged CASSCF(14,10) calculation with five states included in the average. Single

excitations outside the active space are allowed from the active and all σ orbitals. The

expansion consists of 16–32 million CSFs, depending on the basis set.

The second expansion (MRCI2) uses a smaller (12,9) active space, which includes

only one a′ lone pair orbital and the same eight a′′ orbitals as in (14,10). A previous

study15 showed that (12,9) and (14,10) produce very similar energies of the first two

excited states, which are the most important for the photophysical properties of uracil.

MRCI2 is used to refine those energies. The MOs are obtained from state-averaged

CASSCF(12,9) with five states included in the average. Single and double excitations

are allowed from the active space and only single excitations from the σ orbitals. The

Davidson correction is also computed for these MRCI2 energies. This MRCI2 expan-

sion consists of about 100–330 million CSFs. Because of the size of the expansion, only

the first two excited states are calculated at the MRCI2 level. Moreover, since the active

space has only one a′ orbital, only one A′′ state can be computed.

Several iterative and non-iterative EOM-CC methods, including EOM-CCSD,

active-space EOM-CCSDt, and CR-EOM-CCSD(T) are fully interfaced with the

molecular mechanics module of NWChem16, which enables excited-state simula-

tions of molecules in realistic environments within the quantum-mechanical/molecular-

mechanical (QM/MM) framework. Excited-state studies of biologically relevant species

have indicated need for balanced inclusion of correlation effects, as well as effects of

109

a fluctuating environment17–19. In the present work, a similar approach is employed to

compute the excitation energies of uracil in the aqueous solution.

Solvent effects are evaluated by the QM/MM approach20 as implemented in

NWChem. The uracil molecule at the equilibrium ground-state geometry is embedded

in a 30 A-wide cubic box containing 887 water molecules. The QM region consists of

the uracil molecule, and the rest of the system is treated at the MM level using the SPC/E

water model21. To ensure proper solvent structure around uracil, the solvent part of the

system is first optimized, equilibrated over the course of 50 ps of molecular dynamics

simulations, and then re-optimized again. During this simulation, the electrostatic field

of uracil is represented by a set of fixed effective charges obtained from an electrostatic

potential (ESP) fitting using B3LYP calculations with the 6-31G(d) basis set. Molecu-

lar dynamics equilibration is performed at the constant temperature of 298.15 K with a

15 A cutoff. The resulting configuration is then used to calculate the vertical excitation

energies using the B3LYP and EOM-CCSDt levels of theory with the 6-31G(d) basis

set. The (10,10) active space is used in the QM part.

EOM-CCSD and EOM-CC(2,3) calculations are performed with Q-Chem22, CR-

EOM-CCSD(T), EOM-CCSDt, and some EOM-CCSD computations are carried out

with NWChem16, 23. The COLUMBUS24, 25 suite of programs is used for MRCI calcu-

lations.

4.3 Results and discussion

4.3.1 Structure and electronic states of uracil

In the ground electronic state, uracil is a planar molecule. Fig. 4.1 defines atomic labels

in its diketo tautomer. Table 4.1 gives equilibrium geometrical parameters optimized

110

Figure 4.1: Structure of uracil defining atom labels referred to in Table 4.1.

by DFT and compares them to experimental values26 obtained by averaging 32 uracil

residue structures found in a crystallographic database. The standard deviation of the

experimental data is about 0.01 A for distances and about 1.0◦ for angles. The differ-

ences between the calculated and experimental parameters do not exceed 0.03 A for

bond lengths and 2.5◦ for angles. The equilibrium structure also agrees well with the

CC2/aug-cc-pVQZ optimized geometry reported by Fleig et al.27: the maximum differ-

ence between the bond lengths is 0.015 A.

Frontier MOs calculated with RHF/aug-cc-pVDZ are depicted in Fig. 4.2. 26 a′ and

27 a′ are occupied in-plane lone pairs on oxygen atoms. They lie below 28 a′′ and 29 a′′,

two occupied π orbitals. Two virtual π∗ orbitals, 34 a′′ and 35 a′′, contain contributions

from orbitals on all the heavy atoms. There are two low-lying diffuse orbitals: the lowest

unoccupied MO 30 a′ and a higher orbital 38 a′.

The ππ∗ states correspond to transitions from the π orbitals 28 a′′ and 29 a′′ to the π∗

orbitals 34 a′′ and 35 a′′ (Table 4.2). Transitions originating from the lone pair orbitals

26 a′ and 27 a′ form the nπ∗ states. The transition from HOMO to the lowest diffuse

orbital gives rise to a low-lying Rydberg state.

111

Figure 4.2: Frontier molecular orbitals of uracil at the ground state equilibriumgeometry. Orbital energies are calculated with RHF/aug-cc-pVDZ.

112

Table 4.1: Equilibrium geometry of uracil. The calculated structure is optimizedwith B3LYP/6-311G(2df,2pd). Experimental values are obtained by averagingdimensions found in crystal structuresa. Atomic labels are defined in Fig. 4.1, dis-tances in A, angles in degrees.

Bond Calc.b Exp. Angle Calc.b Exp.

N1–C2 1.3908 1.379 C6–N1–C2 123.57 121.3C2–N3 1.3801 1.373 N1–C2–N3 112.92 114.8N3–C4 1.4092 1.383 C2–N3–C4 128.15 127.0C4–C5 1.4559 1.440 N3–C4–C5 113.40 114.7C5–C6 1.3426 1.338 C4–C5–C6 119.97 119.2C6–N1 1.3706 1.380 C5–C6–N1 121.99 122.8C2–O2 1.2084 1.218 O2–C2–N3 124.43 122.0C4–O4 1.2110 1.227 O4–C4–C5 126.18 125.4

N1–H1 1.0060 H1–N1–C2 115.03N3–H3 1.0112 H3–N3–C4 116.20C5–H5 1.0769 H5–C5–C6 121.92C6–H6 1.0808 H6–C6–N1 115.42

a Ref. 26. b Nuclear repulsion energy Enuc = 357.159959 hartree.

Table 4.2 lists leading electronic configurations as well as EOM singles amplitudes

obtained in an EOM-CCSD/aug-cc-pVDZ calculation. The total norm of singly excited

amplitudes R21 is about 0.9 for all the excited states, which suggests that they do not

carry a significant doubly excited character. This conclusion is also supported by small

weight of triple excitations in the EOM-CCSD(2,3) wave function, as discussed below.

113

Table 4.2: Leading electronic configurations in the EOM-CCSD/aug-cc-pVDZwave function. r is the weight of a configuration, R2

1 is the norm of the EOM singlesamplitudes.

State Configurations r R21

X1A′ · · ·(26 a′)2(27 a′)2(28 a′′)2(29 a′′)2

11A′′ (nπ∗) · · ·(26 a′)2(27 a′)(28 a′′)2(29 a′′)2(34 a′′) 0.22 0.91· · ·(26 a′)2(27 a′)(28 a′′)2(29 a′′)2(35 a′′) 0.33

11A′ (ππ∗) · · ·(26 a′)2(27 a′)2(28 a′′)2(29 a′′)(34 a′′) 0.28 0.91· · ·(26 a′)2(27 a′)2(28 a′′)2(29 a′′)(35 a′′) 0.44

21A′′ (Ry) · · ·(26 a′)2(27 a′)2(28 a′′)2(29 a′′)(30 a′′) 0.68 0.93· · ·(26 a′)2(27 a′)2(28 a′′)2(29 a′′)(38 a′′) 0.08

21A′ (ππ∗) · · ·(26 a′)2(27 a′)2(28 a′′)(29 a′′)2(34 a′′) 0.34 0.89· · ·(26 a′)2(27 a′)2(28 a′′)(29 a′′)2(35 a′′) 0.31

13A′ (ππ∗) · · ·(26 a′)2(27 a′)2(28 a′′)2(29 a′′)(34 a′′) 0.30 0.95· · ·(26 a′)2(27 a′)2(28 a′′)2(29 a′′)(35 a′′) 0.46

13A′′ (nπ∗) · · ·(26 a′)2(27 a′)(28 a′′)2(29 a′′)2(34 a′′) 0.21 0.92· · ·(26 a′)2(27 a′)(28 a′′)2(29 a′′)2(35 a′′) 0.33

4.3.2 Effect of the one-electron basis set

The upper panel of Fig. 4.3 shows changes in the excitation energies of singlet and

triplet states as the basis set on heavy atoms is expanded with diffuse and polarization

functions. Changes in the oscillator strength are shown in the lower panel.

States within 5.5 eV from the ground state do not exhibit a significant dependence

as the basis set is augmented: the excitation energy changes by less than 0.05 eV

(Table 4.3). The change is appreciable for higher states (11A′ and upward). Among

the studied states, the low-lying Rydberg 21A′′ state is affected the most. At least one

114

Figure 4.3: Excitation energies (upper panel) and oscillator strengths (lower panel)of the lowest excited states of uracil calculated with EOM-CCSD and various one-electron basis sets.

115

Tabl

e4.

3:Ve

rtic

alE

OM

-CC

SDex

cita

tion

ener

gies

ofur

acil

(eV

).G

roun

dst

ate

CC

SDen

ergi

esar

esh

own

inha

rtre

e,os

cilla

tor

stre

ngth

sar

egi

ven

inpa

rent

hese

s.T

hesi

ngle

tCC

SDre

fere

nce

wav

efu

nctio

nw

asus

edfo

rbo

thsi

ngle

tand

trip

letE

OM

calc

ulat

ions

.

Bas

isse

tG

roun

dst

ate

11 A′′

11 A′

21 A′′

31 A′′

21 A′

13 A′

13 A′′

(nπ∗ )

(ππ∗ )

(Ryd

berg

)(π

π∗ )

(nπ∗ )

6-31

1G(d

)/−

413.

814

317

5.28

5.84

6.71

6.97

3.92

4.99

6-31

G(1

2.35×

10−

5 )(0

.220

3)(8.5

32×

10−

6 )(0

.055

73)

6-31

1G(d

)/−

413.

846

864

5.26

5.84

6.69

6.96

3.92

4.97

6-31

G(d

,p)

(12.

27×

10−

5 )(0

.221

9)(8.2

69×

10−

6 )(0

.054

18)

6-31

1+G

(d)/

−41

3.86

472

95.

225.

706.

166.

596.

803.

894.

956-

31G

(d,p

)(0.9

84×

10−

5 )(0

.236

7)(0

.002

429)

(1.2

54×

10−

6 )(0

.058

51)

6-31

1(2+

)G(d

)/−

413.

866

067

5.22

5.70

6.01

6.58

6.79

3.89

4.95

6-31

G(d

,p)

(0.7

94×

10−

5 )(0

.235

4)(0

.002

514)

(0.6

95×

10−

6 )(0

.055

42)

6-31

1(2+

)G(d

f)/−

414.

008

904

5.21

5.71

6.12

6.58

6.82

3.89

4.94

6-31

G(d

,p)

(0.9

84×

10−

5 )(0

.238

5)(0

.002

666)

(0.5

98×

10−

6 )(0

.056

77)

aug-

AN

O-D

Z−

413.

953

957

5.25

5.60

6.05

6.76

3.86

4.98

(0.0

00)

(0.2

110)

(0.0

0277

3)(0

.046

66)

aug-

cc-p

VD

Z−

413.

822

857

5.22

5.58

6.00

6.57

6.72

3.87

4.95

(0.0

26×

10−

5 )(0

.208

9)(0

.002

660)

(0.7

54×

10−

6 )(0

.045

69)

aug-

cc-p

VT

Z−

414.

131

547

5.23

5.59

116

set of diffuse functions is required to describe it correctly, and with a second set of dif-

fuse functions its excitation energy drops by 0.15 eV. Fleig et al.27 report similar effects

of improving the basis set. They also show that the CC2 excitation energies for the

four lowest valence singlet states of uracil change by 0.01 eV by extending the basis

set from aug-cc-pVTZ to aug-cc-pVQZ, whereas the energy of the lowest Rydberg state

changes by 0.05 eV. Thus, we consider our aug-cc-pVTZ energies of the valence states

converged within approximately 0.01 eV with respect to the one-electron basis set.

The oscillator strength of the allowed ππ∗ transitions changes very slightly as the

basis set is expanded. In the case of the dark nπ∗ transitions, the oscillator strength

drops by an order of magnitude upon addition of a set of diffuse functions on heavy

atoms.

We also note the excellent performance of the augmented ANO double-zeta and

aug-cc-pVDZ bases, which produce excitation energies for the two lowest states within

0.02 eV from aug-cc-pVTZ (see Table 4.3), whereas the CC2 energies from Ref. 27

change by 0.08–0.11 eV between the aug-cc-pVDZ and aug-cc-pVTZ bases.

Overall, both polarization and diffuse functions are required to obtain the correct

order and converged energy of the excited states. The basis set effects can account for as

much as 0.25 eV for the ππ∗ state, while the nπ∗ state is less sensitive. We will use the

aug-ANO-DZ and aug-cc-pVTZ basis sets in the best estimations of excitation energies.

As summarized in Table 4.3, EOM-CCSD calculations with the 6-311(2+)G(df)/6-

31G(d,p), aug-ANO-DZ, and aug-cc-pVTZ bases give excitation energies of 5.21–

5.25 eV for the lowest nπ∗ state and 5.59–5.71 eV for the ππ∗ transition.

117

4.3.3 Effects due to including higher excitations

Methods with explicit active-space triples, EOM-CC(2,3) and EOM-CCSDt, are used to

estimate the effect of non-dynamical correlation. Results obtained with the 6-31G(d)

basis set are presented in Table 4.4. Due to the high computational cost, only the

energies of the first two transitions can be calculated with EOM-CCSDt(I). Including

active-space triples affects the EOM-CCSD vertical excitation energies by no more than

0.15 eV for the lowest two singlet states.

The EOM-CC(2,3) expansion is similar to that of EOM-CCSDt(I), only different

reference wave functions are used for the two methods: CCSD and CCSDt, respectively.

Unlike EOM-CCSDT, EOM-CC(2,3) is not size-intensive3. This can be the cause of the

opposite direction of the excitation energy shift produced by the two methods: EOM-

CC(2,3) transition energies are blue-shifted, while EOM-CCSDt(I) energies are red-

shifted with respect to EOM-CCSD. The resulting large discrepancy of about 0.2 eV

(5.42 eV versus 5.20 eV and 5.96 eV versus 5.80 eV for the nπ∗ and ππ∗ transitions,

respectively) is attributed to the same reason. However, both methods agree on the

energy difference between the excited states. The transition energy between the first

two excited singlet states (11A′′→ 11A′) is in the 0.54–0.61 eV range for all the explicit

triples methods, slightly down from the 0.63 eV of EOM-CCSD. Thus, both models

with active-space triples confirm that there is no significant doubly excited character in

the lowest valence states of uracil, in agreement with the small basis CC3 results from

Ref. 27.

Table 4.5 shows contributions from the singles, doubles, and triples parts of the

excitation operator to the EOM-CC(2,3)/6-31G(d) wave function in the (10,10) active

space. The excitation operator R is described by Eqns. (1.10) and (1.12). The norm of

the doubly excited with respect to the reference part R2 is 0.16 and 0.13 for the nπ∗ and

118

Table 4.4: Effect of triple excitations on vertical excitation energies. Shift in exci-tation energies due to triple excitations is shown in parentheses. All energies are ineV.

Method 11A′′ 11A′ 31A′′ 21A′

(nπ∗) (ππ∗) (nπ∗) (ππ∗)

6-31G(d) basis set

EOM-CCSD 5.30 5.93 6.73 7.05

(10,8) active spaceEOM-CC(2,3) 5.42 (0.12) 5.96 (0.03) 6.93 (0.20) 6.92 (−0.13)EOM-CCSDt(III) 5.30 (0.00) 5.91 (−0.02) 6.73 (0.00) 7.04 (−0.01)EOM-CCSDt(II) 5.31 (0.01) 5.89 (−0.04) 6.76 (0.03) 7.02 (−0.03)EOM-CCSDt(I) 5.20 (−0.10) 5.80 (−0.13)

(10,10) active spaceEOM-CC(2,3) 5.44 (0.14) 5.98 (0.05) 6.95 (0.22) 6.93 (−0.12)EOM-CCSDt(III) 5.30 (0.00) 5.91 (−0.02) 6.73 (0.00) 7.04 (−0.01)EOM-CCSDt(II) 5.31 (0.01) 5.89 (−0.04) 6.76 (0.03) 7.02 (−0.03)

CR-EOM-CCSD(T) 5.19 (−0.11) 5.65 (−0.28) 6.77 (−0.28)

aug-cc-pVTZ basis set

EOM-CCSD 5.23 5.59CR-EOM-CCSD(T) 5.00 (−0.23) 5.25 (−0.34)

ππ∗ states, respectively; the norm of R3 does not exceed 0.01 for both. Thus, there is no

evidence of a multi-configurational reference or a doubly excited character, for which

explicit higher excitations are required. Therefore, the perturbative correction can be

applied to recover the effect of all triple excitations.

119

Table 4.5: Norms of the amplitudes Rm,k from Eq. (1.15) in the EOM-CC(2,3)/6-31G(d) wave function of uracil. The (10,10) active space is used. Total norm is notunity in virtue of biorthogonal properties of EOM-CC.

State R20 R2

1 R22 R2

3

X1A′ 0.99 0.00 0.00 0.0111A′′ 0.00 0.83 0.16 0.0111A′ 0.00 0.86 0.13 0.0121A′′ 0.00 0.86 0.13 0.0121A′ 0.00 0.78 0.21 0.02

In the EOM-CCSDt(III)→ (II)→ (I) series of calculations (Table 4.4), the active-

space constraint for the excitations given by Eqns. (1.14)–(1.19) gradually relaxes, giv-

ing rise to a partial account for dynamical correlation. The increasing magnitude of the

correction in the series indicates a considerable effect of dynamical correlation. The

reduced subset of triples is not capable of fully accounting for dynamical correlation,

and we employ CR-EOM-CCSD(T) for that purpose, which indeed produces a larger

correction of 0.11–0.28 eV. A comparable correction (0.2 eV) for the lowest ππ∗ state

was obtained at the CC3 level in a similar quality basis set27. Overall, the vertical

excitation energies calculated with CR-EOM-CCSD(T)/6-31G(d) for the nπ∗ and ππ∗

transitions are 5.19 eV and 5.65 eV, respectively.

Since dynamical correlation converges slowly with the basis set, it is important to

evaluate the effect of triple excitations in a large basis set. In the aug-cc-pVTZ basis, the

CR-EOM-CCSD(T) excitation energies of the two lowest states are below the respective

EOM-CCSD values by 0.23–0.34 eV, which is appreciably larger than the 6-31G(d)

values.

120

The observed tendency of triples to lower EOM-CCSD excitation energies is con-

sistent with recent studies27, 28.

4.3.4 MRCI results

The results of MRCI calculations are presented in Table 4.6. Due to its high com-

putational requirements, MRCI2 was only used for the first two singlet excited states.

As reported previously15, the MRCI2 expansion, which includes more correlation than

MRCI1, does not change the excitation energies dramatically. That indicates that the

σ–π correlation plays a more important role than double excitations from the active

space.

The MRCI expansions used in these calculations cannot describe Rydberg states

since there are no Rydberg orbitals included in the active space. So even when the basis

set has diffuse functions, the Rydberg states will be too high in energy, and that is why

the second A′′ state in the MRCI results is always nπ∗, even though in the coupled cluster

calculations Rydberg states appear below the second nπ∗.

With the 6-31G(d,p) basis set, the MRCI2 excitations are 5.16 eV and 5.89 eV for

the first two singlet states compared to 5.30 eV and 5.93 eV at the EOM-CCSD/6-

31G(d) level. Thus the MRCI2 energies are blue-shifted by 0.14 eV and 0.04 eV for

the two states compared to EOM-CCSD. With the 6-311+G(d)/6-31G(d,p) basis set, the

MRCI2 energies become 4.87 eV and 5.70 eV. The corresponding excitation energies

using EOM-CCSD and the same basis set [6-311+G(d)/6-31G(d,p)] are 5.22 eV and

5.70 eV. The two methods agree well on the ππ∗ energy, but the disagreement for nπ∗

has now increased to 0.35 eV. This is mainly because the MRCI2 energy of the lowest

singlet shows a surprisingly large effect on the basis set. With the aug-ANO-DZ basis

set the energy of ππ∗ is 5.65 eV, which is very similar to the EOM-CCSD value of

121

Tabl

e4.

6:M

RC

Iexc

itatio

nen

ergi

es(e

V)o

fthe

four

low

ests

ingl

etex

cite

dst

ates

ofur

acil.

Osc

illat

orst

reng

thsa

regi

ven

inpa

rent

hese

s.

Bas

isse

tC

SFs

11 A′′

11 A′

31 A′′

21 A′

(nπ∗ )

(ππ∗ )

(nπ∗ )

(ππ∗ )

MR

CI1

6-31

G(d

,p)

16,6

27,7

105.

09(1

4.90×

10−

5 )5.

79(0

.186

)6.

62(0

.000

)6.

57(0

.047

8)6-

311+

G(d

)/6-

31G

(d,p

)28

,159

,230

5.02

(3.2

00×

10−

5 )5.

65(0

.221

)6.

87(5.1

00×

10−

5 )6.

44(0

.035

7)au

g-A

NO

-DZ

32,4

83,5

505.

01(0.5

00×

10−

5 )5.

53(0

.180

)6.

49(1.6

00×

10−

5 )6.

35(0

.038

4)

MR

CI2

6-31

G(d

,p)

112,

036,

212

5.16

5.89

6-31

1+G

(d)/

6-31

G(d

,p)

254,

019,

060

4.87

5.70

aug-

AN

O-D

Z32

0,01

9,06

05.

65

MR

CI2

+D

avid

son

corr

ectio

n6-

31G

(d,p

)5.

095.

606-

311+

G(d

)/6-

31G

(d,p

)4.

765.

21au

g-A

NO

-DZ

5.32

122

5.60 eV. Unfortunately, we could not obtain the energy of the nπ∗ state at that level due

to technical difficulties, so we could not explore further the dependence of the nπ∗ state

on the basis set. The basis set dependence of the nπ∗ state at the CC level and at the

MRCI1 level is much smaller. We do not understand this MRCI2 dependence well at

present, but it may be an effect of the smaller active space used in MRCI2.

The Davidson correction lowers the excitation energy of the ππ∗ state by 0.29–

0.49 eV, indicating the importance of higher than double excitations in the wave func-

tion. The magnitude of the effect is consistent with the EOM-CCSD and CR-EOM-

CCSD(T) results (Table 4.4). The nπ∗ state is not affected as much by the Davidson

correction, which lowers the energy by about 0.1 eV.

4.3.5 Solvent effects

TDDFT calculations of the isolated uracil molecule with the B3LYP functional and the

6-31G(d) basis set give excitation energies of 4.73 eV and 5.37 eV for the nπ∗ and ππ∗

states, respectively (Table 4.7). Account for effects of surrounding water molecules via

QM/MM increases the energies by 0.65 eV and 0.02 eV, respectively.

QM/MM calculations that employ EOM-CCSDt(II) with the 6-31G(d) basis and the

(10,10) active space yield first two excitation energies of 5.75 eV and 5.96 eV, which

are 0.44 eV and 0.07 eV above the respective gas-phase excitation energies.

Solvent effects are in a qualitative agreement with previous TDDFT/PBE0 and

TDDFT/B3LYP calculations, which predict a larger positive shift for the nπ∗ state and

a much smaller shift for the ππ∗ state29–31. In our calculations, the excitation energy

of the nπ∗ state increases by 0.5 eV, and the excitation energy of the ππ∗ state remains

almost the same. Taking into account previous calculations, we estimate the effect on

the ππ∗ state as ±0.1 eV.

123

Table 4.7: Vertical excitation energies (eV) of uracil in the gas phase and watersolution calculated with QM/MM. Energy shifts due to the solvent are shown inparentheses.

Method Environment 11A′′ 11A′

(nπ∗) (ππ∗)

TDDFT/B3LYP/6-31G(d) Gas phase 4.73 5.37Water 5.38 (0.65) 5.39 (0.02)

EOM-CCSDt(II)/6-31G(d)a Gas phase 5.31 5.89Water 5.75 (0.44) 5.96 (0.07)

TD-DFT/PBE0/6-311+G(2d,2p)b Gas phase 4.80 5.26Water 5.27 (0.47) 5.14 (−0.12)

a (10,10) active space. b Ref. 29.

Our calculations do not account for solvent-specific interactions and possible tau-

tomerization. Previous cluster studies of DNA bases with several water molecules32–36

do not suggest a large effect on the excitation energies due to solvent-specific inter-

actions. For example, Yoshikawa and Matsika36, who investigated excited states of

uracil–water isomers, report shifts between−0.03 eV and +0.2 eV for the nπ∗ state and

between −0.1 eV and +0.1 eV for the ππ∗ state.

There is no explicit QM treatment of the hydrogen bond interaction between uracil

and surrounding water molecules in our calculations. Since excitation is localized on

the uracil moiety, we expect the average solvent electric field to have a larger effect on

the excitation energies than hydrogen bonding. That is captured by the QM/MM calcu-

lations through the classical description of the solvent. To quantify the errors due to the

classical as opposed to the fully QM description of the first solvation shell, we perform

an additional TDDFT QM/MM calculation with one of the waters coordinating with an

oxygen atom included into the QM part. Resulting excitation energies of 5.38 eV and

124

5.42 eV for the nπ∗ and ππ∗ states show that the errors due to the classical description

are less than 0.01 eV and 0.03 eV for the two states, respectively.

4.3.6 Best theoretical estimate of vertical excitation energies in

uracil

EOM-CCSD with large augmented basis sets gives vertical excitation energies of about

5.25 eV and 5.60 eV for the nπ∗ and ππ∗ states, respectively (Table 4.8). Including

triple excitations non-iteratively via CR-EOM-CCSD(T) lowers both energies by 0.2–

0.3 eV to 5.0 eV and 5.3 eV. That is our best estimate of the excitation energies. The ππ∗

transition energy computed with MRCI with the Davidson correction is 5.3 eV, which

is in an excellent agreement with the EOM value. We estimate the error bars for the

theoretical values to be ±0.05 eV.

TDDFT/PBE0 agrees with the EOM methods on the ππ∗ energy, but underestimates

the energy of the nπ∗ transition by 0.2 eV. TDDFT/B3LYP underestimates the exci-

tation energies of both states even more. The CASPT2/ANO values37 are below our

best estimates by 0.5 eV (nπ∗) and 0.3 eV (ππ∗). More recent CASPT2 calculations38

employing a larger active space and a smaller basis are closer to the EOM-CC esti-

mates, however, in view of strong basis set dependence, this agreement is accidental.

Empirically-corrected CASPT2/TZVP values28 for the two singlet states are below our

estimates by 0.1 eV and 0.02 eV. However, a relatively modest basis set was used, and

it is not clear how well empirically corrected CASPT2 would perform in a larger basis.

The CC2/TZVP excitation energies28 differ from our estimates by −0.09 eV and

+0.22 eV for the nπ∗ and ππ∗ states, respectively. Consistently with our findings, larger

basis CC2 calculations27 yield considerably different values, which are −0.2 eV and

125

Table 4.8: Best estimates of the vertical excitation energies (eV) of the lowest singletand triplet states compared with previously published results.

Method 11A′′ 11A′ 13A′ 13A′′

(nπ∗) (ππ∗) (ππ∗) (nπ∗)

EOM-CCSD/aug-cc-pVTZ 5.23 5.59CR-EOM-CCSD(T)/aug-cc-pVTZ 5.00 5.25

EOM-CCSD/aug-ANO-DZ 5.25 5.60 3.86a 4.98a

CR-EOM-CCSD(T)/aug-ANO-DZ 5.06 5.30 5.17MRCISD+Q(12,9)/aug-ANO-DZ 5.32

CASPT2(2,10)/Roos ANOb 4.54 5.00CASPT2(14,10)/6-31G(d,p)c 4.93 5.18 3.80 4.71CASPT2/TZVPd 4.90 5.23TDDFT/PBE0/6-311+G(2d,2p)e 4.80 5.26TDDFT/B3LYP/aug-cc-pVTZf 4.64 5.11 3.38 4.26TDDFT/B3LYP/aug-cc-pVTZg 4.62 5.07TDDFT/B3LYP/6-311++G(d,p)h 4.67 5.17CIS(2’)/cc-pVDZi 5.11 5.89RI-CC2/aug-cc-pVQZj 4.80 5.35 3.95 4.61CC2/TZVPd 4.91 5.52EOM-CCSD/TZVPd 5.11 5.70

a The singlet CCSD reference wave function was used. b Ref. 37. c Ref. 38. d Ref. 28.e Ref. 29. f Ref. 39. g Ref. 30. h Ref. 40. i Ref. 41. j Ref. 27.

+0.05 eV from our estimates. These differences are well within the CC2 error bars (see

Sec. 3.2).

The present data, as well as previous studies,27, 28 demonstrate that a large basis

set and an accurate account of dynamical correlation are both essential for converged

results.

126

Figure 4.4: Absorption spectrum of uracil with theoretical estimations ofelectronic transitions calculated with EOM-CCSD/aug-ANO-DZ and CR-EOM-CCSD(T)/aug-ANO-DZ.

Fig. 4.4 shows a UV absorption spectrum42 of uracil vapor obtained at 228◦C and

the EOM-CCSD/aug-ANO-DZ and CR-EOM-CCSD(T)/aug-ANO-DZ vertical excita-

tion energies depicted as a stick spectrum using the EOM-CCSD/aug-ANO-DZ inten-

sities. The experimental spectrum shows a broad feature peaking at 242 nm (5.08 eV),

which is 0.17 eV below the CR-EOM-CCSD(T)/aug-cc-pVTZ vertical excitation energy

for the bright ππ∗ state. As elaborated in the benchmark study of Schreiber et al., there

are numerous reasons complicating the interpretation of the observed absorption max-

ima28. We attribute the difference to strong vibronic coupling of the nπ∗ and ππ∗ states.

Intensity borrowing by the lower-lying dark state can affect the overall shape of the

absorption, shifting it toward lower energies. Strong vibronic interaction is consistent

with the broad featureless shape of the spectrum42, as well as with experimental obser-

vations of Kong and coworkers6, however, more detailed investigations are necessary

for a definite conclusion.

127

4.4 Chapter 4 conclusions

A series of calculations of energies and oscillator strengths of transitions to electroni-

cally excited states of uracil show that using high-quality basis sets (i.e., aug-cc-pVTZ

or aug-ANO-DZ) with diffuse functions is essential for converged results.

Active-space EOM-CC calculations with explicit triple excitations demonstrate that

the excited states do not exhibit a doubly excited character, and therefore EOM-CCSD

provides a good zero-order wave function for including remaining dynamical correlation

via perturbative treatment of triple excitations. The latter affects excitation energies by

as much as 0.3 eV. Thus, the EOM-CCSD errors for the excitation energies of uracil do

not exceed the conventional estimates of the EOM-EE-CCSD error bars.

Our best estimate of excitation energies calculated with CR-EOM-CCSD(T)/aug-cc-

pVTZ places the nπ∗ singlet excited state at E(11A′′

)=5.00 eV, and the ππ∗ singlet state

at E(11A′

)=5.25 eV above the ground state. An excellent agreement between high-level

MRCI calculations, which place the ππ∗ state at 5.32 eV, and EOM-CC further supports

our conclusions. A solvent (water) affects the excitation energies of the two states by

+0.5 eV and less than 0.1 eV, respectively.

We conclude that the maximum of the broad UV spectrum of uracil does not corre-

spond to the position of the vertical ππ∗ excitation, possibly due to strong vibronic cou-

pling with the lower-lying dark nπ∗ state. Our results indicate that previously reported

CASPT2 and TDDFT calculations underestimate the excitation energies.

128

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133

Chapter 5: Direct location of the

minimum point on intersection seams

of potential energy surfaces with

equation-of-motion coupled-cluster

methods

5.1 Chapter 5 introduction

The dynamics of chemical reactions is often interpreted in terms of nuclear motions on

a single potential energy surface (PES), which is based on the (non-relativistic) Born-

Oppenheimer (BO), or adiabatic, approximation1. The BO approximation separates

slow and fast degrees of freedom: the nuclei move in the mean field created by the

electrons, which adjust their positions instantaneously. This is valid for many chem-

ical reactions, however there is a plethora of processes for which the single-surface

approximation breaks down2. Examples of such nonadiabatic processes, which involve

transitions between different electronic states, include organic photochemistry3 with a

134

Figure 5.1: The potential energy surface slices for the ground and the first excitedstates of a substituted ethylene. There is an energy barrier that prevents themolecule from rotating freely around the double bond. The top of the barrier onthe ground state PES corresponds to the minimum on the excited state PES.

significant share of biologically important chemical reactions4, as well as spin-forbidden

processes5.

Consider, for instance, the isomerization of retinal, an important step in the chem-

istry of vision6. A prototype reaction7, the cis-trans photoisomerization of substituted

ethylene, is sketched in Fig. 5.1. Upon electronic excitation, the system slides downhill

towards the minimum on the excited state PES, which corresponds to a local maximum

on the lower PES. At that point, the two surfaces approach each other, and the probabil-

ity of a transition back to the ground state increases. Once on the ground state PES, the

system relaxes to either the cis- or trans-product.

A set of points of confluence of two adiabatic PESs forming a continuous hyperline

is called an intersection seam8. A nonadiabatic transition from one electronic state to

another is more likely to happen near the crossing and is enabled by electronic couplings

135

between the states. The nature of the coupling can be different: electronic states of the

same multiplicity and symmetry can be coupled through the derivative coupling, while

spin-orbit interaction facilitates the transition between states of different multiplicities.

While the nonadiabatic transition can occur at any nuclear configuration, its probability

is higher if the energy gap between the states is small.

A full description of a nonadiabatic process requires a calculation of nuclear dynam-

ics that brings the system from the Franck-Condon region towards the point where the

two states are nearly degenerate, and the probability of the nonadiabatic transition is

larger (see, for example, Ref. 9, 10 and references therein). To describe the transition,

both electronic and nuclear degrees of freedom should be considered.

Alternatively, one can gain considerable mechanistic insight and even quantitative

information about the probability of nonadiabatic transitions from the geometry and

energy of the minimum energy point on the intersection seam (minimum energy cross-

ing point, MECP, MSX)11–19. The MECP can be thought of as a characteristic point

of the process, just like the transition state in a thermal chemical reaction. The energy

of the MECP can help assess the possibility of the process, and with the use of the

nonadiabatic transition state theory20, the rate of the nonadiabatic process can be esti-

mated. The geometry of MECP suggests which modes are most efficient in facilitating

the transition21.

Several methods of finding the MECP have been proposed over the past

years11–13, 15–19. The crossing optimization procedure is very similar to that of finding

the minimum on a PES, with the difference that constraints must be enforced to ensure

the degeneracy of the states of interest. The algorithm of locating the MECP on a coni-

cal intersection seam is the same as for crossings of different symmetry or spin potential

surfaces, only the constraints (and the dimensionality of the seam) are different.

136

Many methods employ the Lagrange multiplier technique to constrain the optimiza-

tion. Some approaches require the calculation of a nonadiabatic coupling vector as well,

which is rather demanding and have been implemented only for a handful of wavefunc-

tions. For example, Ragazos et al.12 and Manaa et al.15 introduced analytical for-

malisms for the calculation of the nonadiabatic coupling vector at the multi-reference

self-consistent field (MCSCF) and the multi-reference configuration interaction (MRCI)

levels of theory and used it for seam minimization. Levine et al.19 avoid the laborious

evaluation of the coupling vector by using a tuning parameter. This implementation

works with the time-dependent density functional theory (TD-DFT) for excited states.

This chapter reports an implementation of the projected gradient method16 within

the equation-of-motion coupled-cluster (EOM-CC) family of methods. The algorithm

does not require the use of the Lagrange multiplier method and enables the constraints in

the minimization procedure by projecting the energy gradient. The method was imple-

mented within the Q-Chem quantum chemistry package22.

5.2 Theory

5.2.1 Adiabatic approximation

The full non-relativistic Hamiltonian of a molecular system H depends explicitly on

both nuclear and electronic coordinates, R and r, respectively:

H(r,R) = Tn(R)+Te(r)+Vee(r)+Vne(r,R)+Vnn(R) (5.1)

= Tn(R)+He(r;R),

137

where Te and Tn are electronic and nuclear kinetic energy operators, respectively, Vee,

Vne, and Vnn are potential energy operators for electron-electron, nuclei-electron, and

nuclei-nuclei interactions. The electronic Hamiltonian He(r;R) is defined as the full

Hamiltonian minus the nuclear kinetic energy operator and thus depends explicitly on

the electronic coordinates and only parametrically on the nuclear coordinates.

Eigenfunctions of the full Hamiltonian satisfy the Schrodinger equation:

H(r,R)Ψi(r,R) = εiΨi(r,R). (5.2)

The solutions of the so-called electronic Schrodinger equation are defined as follows:

He(r;R)φi(r;R) = Eiφi(r;R), (5.3)

and may be used as a basis to represent the full wavefunction Ψ(r,R):

Ψ(r,R) = ∑j

φ j(r;R)χ j(R), (5.4)

where χ j(R) are expansion coefficients called nuclear wave functions.

Substituting representation (5.4) into the full Schrodinger equation (5.2) and multi-

plying by φk from the left, we arrive at:

∑k′〈φk|He +Tn |φk′ 〉r χk′ = ε∑

k′′〈φk|φk′′〉r χk′′ , (5.5)

138

where the subscript r denotes integration over the electronic coordinates. After expand-

ing Tn and using 5.3:

(ε−Ek−Tn−T′′kk

)χk = ∑

k′ 6=k

(T′kk′+T′′kk′

)χk′, (5.6)

where:

Tn =−∑A

12MA

∂2

∂R2A, (5.7)

T′i j =−∑A

1MA

⟨φi

∣∣∣∣ ∂

∂RA

∣∣∣∣φ j

⟩r

∂

∂RA, (5.8)

T′′i j =−∑A

12MA

⟨φi

∣∣∣∣ ∂2

∂R2A

∣∣∣∣φ j

⟩r. (5.9)

A solution to the coupled set of equations (5.6) yields χ j(R), and therefore the solu-

tion to the full problem (5.2). If the off-diagonal terms on the right hand side of (5.6)

vanish, the overall wavefunction becomes just a single product of the electronic and

nuclear parts, and the nuclear motions on different potential surfaces become uncou-

pled. This is the essence of the Born-Oppenheimer approximation: complete adiabatic

separation of the nuclear and electronic motions (see, for example, Ref. 1).

The approximation therefore requires that the coupling terms in (5.6) are small and

can be neglected. Thus, it is important to understand when they may become signifi-

cant. Using perturbation theory, one can rewrite the electronic part of the nonadiabatic

coupling terms in (5.8) as:

⟨φi

∣∣∣∣ ∂

∂R

∣∣∣∣φ j

⟩=

⟨φi

∣∣∣∂He∂R

∣∣∣φ j

⟩E j−Ei

, (5.10)

139

which suggests that the coupling between two electronic states becomes large when they

are nearly degenerate. In such situations, the adiabatic approximation breaks down, and

transitions between different electronic states become possible.

5.2.2 Crossing location and minimization

Although the set of equations (5.6) is infinite, the energy denominator in (5.10) suggests

that the coupling between only two electronic states needs to be taken into account in the

regions of proximity of their PESs, and the couplings with other states can be neglected.

The description of conical intersections and the algorithms for locating MECPs are

based on a first order perturbative treatment of the electronic Schrodinger equation in

the vicinity of the intersection8, 23. Recently, second-order treatment that accounts for

the curvature of the seam, in addition to the first-order linear terms, has been reported24.

At the intersection point R, the electronic Hamiltonian has the following matrix

form in the so-called “crude adiabatic basis” {φi(r;R0)}, eigenstates of the electronic

Hamiltonian at a nearby point R0:

He(R) =

H11 H12

H12 H22

, (5.11)

where Hi j(R) =⟨φi(r;R0) |He(R)|φ j(r;R0)

⟩r.

After finding the eigenvalues of the Hamiltonian (5.11), one can readily write the

conditions on the matrix elements that ensure that the states are degenerate:

H11−H22 = 0,

H12 = 0.(5.12)

140

The above result is the essence of the Neumann–Wigner theorem, or the noncrossing

rule25, which states that two conditions have to be imposed on the Hamiltonian in order

for the states to be degenerate and, therefore, limits the dimensionality of the intersection

seam to N− 2, where N is the number of internal degrees of freedom. If the coupling

term is zero due to the symmetry restrictions (e.g., between the states of different spatial

symmetry or different multiplicity), the dimensionality of the seam is N−1.

By applying degenerate perturbation theory (see, for example, Ref. 23) to the eigen-

problem with Hamiltonian (5.11), conditions (5.12) become:

∂

∂R [E1(R)−E2(R)]∣∣∣R=R0

δR = 0,⟨φ1(r;R)

∣∣∣ ∂

∂R

∣∣∣φ2(r;R)⟩

r

∣∣∣R=R0

δR = 0.(5.13)

By defining the gradient difference vector:

g(R0) =∂

∂RE1(R)|R=R0−

∂

∂RE2(R)|R=R0, (5.14)

and the derivative coupling vector:

h(R0) =

⟨φ1(r;R)

∣∣∣∣ ∂

∂R

∣∣∣∣φ2(r;R)

⟩r

∣∣∣∣R=R0

, (5.15)

we arrive at: g(R0) ·δR = 0,

h(R0) ·δR = 0.(5.16)

Non-zero g and h vectors define a so-called g-h, or branching, plane26, 27. The degener-

acy of the states is preserved in the subspace complement to the g-h plane and is lifted

141

only along the plane. Thus, MECP can be found by following the gradient on one of the

PES under constraint (5.16).

In this work, we implemented a reduced version of the projected gradient method16

for locating the minimum energy point on an intersection. The method requires two

vectors, the gradient difference g as defined in (5.14), the derivative coupling h as in

(5.15), and the projection of the gradient on the higher PES onto the subspace orthogonal

to the branching plane:

G2 = P∂

∂RE2, (5.17)

where P is the projector that performs the required operation. The gradient difference

vector is multiplied by the energy difference in order to yield zero on the seam:

G1 =E2−E1√|g|2

g. (5.18)

The total gradient used in the optimization procedure is defined as:

G = G1 +G2. (5.19)

Following G1, one arrives to the seam, while G2 is a gradient along the seam towards

MECP.

The advantage of this technique is that it can be readily used with any existing opti-

mization algorithm. The disadvantage is that it requires both gradients and nonadiabatic

couplings.

Analytical calculations of the h vector have been implemented for the MCSCF and

MRCI methods12, 15. For EOM-CC, however, only the mathematical procedure has been

developed28, and no implementation has been reported so far. In this work we consider

the situations when the nonadiabatic derivative coupling vector is zero by symmetry, e.g.

142

the spatial parts of the wavefunctions of the states carry different irreducible representa-

tions or the states are of different multiplicities. In this case, only analytic gradients are

required29.

5.3 Computational details

All calculations were performed with the Q-Chem22 package. Various versions of the

EOM-CC method were employed for excited states calculations. The 6-31G basis set30

was used for N+3 and NO2, and 6-31G* basis31 — for the para-benzyne diradical. To

validate the code, MECPs were located by generating dense numerical grids for N+3 and

NO2. Then the algorithm was tested by setting the calculations with different starting

geometries and employing different tolerance criteria.

5.4 Benchmarks and examples

5.4.1 Cyclic trinitrogen cation

The cyclic trinitrogen cation is a closed-shell system with a manifold of low-lying near-

degenerate excited states32, 33.

The equilibrium ground state geometry of N+3 is an equilateral triangle (D3h),

whereas low-lying excited states undergo Jahn-Teller distortions to C2v. The C2v point

group symmetry will be used to label the excited states.

Both HOMO and LUMO in the ground state orbital picture of the cation are doubly

degenerate. Therefore, four near-degenerate electronic states are derived upon the exci-

tations from the highest occupied lone pair orbitals to the lowest unoccupied π∗ orbitals.

Among those, the 21A2 and 11B1 adiabatic states have PESs that form two intersection

143

seams. One of the seams lies along the nuclear space coordinate that preserves the D3h

point group symmetry (this is the Jahn-Teller seam), and the other one is found in the

subspace where the molecule is distorted from the equilateral triangular geometry to

C2v. There are 3 symmetry-identical MECP of the C2v type.

This interesting feature — 4 very closely located conical intersections — was char-

acterized by Dillon and Yarkony33 and makes the intersection to appear glancing rather

than conical32.

As in the recent study32, we employed EOMEE-CCSD to describe the excited states

of N+3 . As long as the molecule preserves the C2v symmetry, the wavefunctions of

the electronic states of interest belong to different irreps, A2 and B1, and therefore the

derivative coupling matrix element vanishes. The intersection seam between the states

is two-dimensional.

To validate the algorithm, the 21A2/11B1 crossing seams were first scanned in order

to find the minimum energy points. The locations of these points, as well as the points

of triple degeneracy, are listed in Table 5.1.

To assess the performance of the algorithm, we attempted various geometries as the

starting points for the optimization procedure. Usually the algorithm locates the mini-

mum energy points on the 21A2/11B1 seams in less than 20 iterations (Table 5.2). Just

like the local minimum on a global PES found with an optimization algorithm depends

on the starting point, the seam located with the projected gradient method depends on

the choice of the initial molecular geometry.

Another interesting feature of the manifold of the electronic states in N+3 is addi-

tional accidental degeneracies. The 11A2 and 21A2 are degenerate at a point in the

internal coordinate space which happens to correspond to the molecule having the D3h

point group symmetry. Since 21A2 and 11B1 are rigorously degenerate in D3h, the point

144

Table 5.1: Geometries and total energies of the intersection seam minima andpoints of triple degeneracy in N+

3 calculated by EOM-CCSD/6-31G. All bondlengths are given in angstroms, all angles are given in degrees, all energies aregiven in a.u.

R(NN) α(NNN) E(11A2

)E(21A2

)E(11B1

)E(21B1

)D3h seam minimum1.4556 60.00 -162.836070 -162.822635 -162.822635 -162.816033

C2v seam minimum1.4476 60.78 -162.837652 -162.821903 -162.821903 -162.815864

Points of triple degeneracy1.6671 60.00 -162.795662 -162.795661 -162.799368 -162.7956631.6041 60.00 -162.812016 -162.807857 -162.807860 -162.807858

becomes a point of a triple 11A2/21A2/11B1 intersection. The 11B1 and 21B1 elec-

tronic states have a point of degeneracy as well, which is therefore the point of a triple

21A2/11B1/21B1 intersection.

The 11A2/21A2/11B1 and 21A2/11B1/21B1 triple intersections were located at D3h

geometries with the N–N distance R(NN) = 1.6671 A and R(NN) = 1.6041 A, respec-

tively. Scans of the PESs around the points of intersections are shown in Fig. 5.2.

Note that the triple intersections are found at the D3h symmetry and in fact are cross-

ings of PESs belonging to different irreducible representations, even though it is not

immediately clear from the C2v notation. Therefore, the problem with complex energy

values34 does not arise.

145

Table 5.2: Results of energy minimization of the 21A2/11B1 intersection seam inN+

3 calculated by EOM-CCSD/6-31G. All bond lengths are given in angstroms, allangles are given in degrees.

Starting geometry Iterations Final geometryR(NN) α(NNN) R(NN) α(NNN)

Tolerancea ∆g = 3×10−4 (Q-Chem default)1.2000 60.00 6 1.4553 60.001.4200 60.00 4 1.4555 60.001.4600 70.00 16 1.4556 59.991.5400 50.00 5 1.4557 60.001.6000 90.00 8 1.4475 60.79

Tolerance ∆g = 3×10−5

1.2000 60.00 7 1.4556 60.001.4200 60.00 5 1.4556 60.001.4600 70.00 16 1.4556 59.991.5400 50.00 6 1.4556 60.001.6000 90.00 9 1.4476 60.78

Tolerance ∆g = 1×10−5

1.2000 60.00 7 1.4556 60.00b

1.4200 60.00 5 1.4556 60.001.4600 70.00 17 1.4556 60.001.5400 50.00 6 1.4556 60.001.6000 90.00 9 1.4476 60.78c

aTolerance on the maximum gradient component. The optimization procedure con-verges if this and one of the two criteria — energy change ∆E = 10−6 a.u. and max-imum displacement ∆q = 1.2× 10−3 A — are satisfied. bNuclear repulsion energyEnr = 53.441794 a.u., total energy Etot =−162.822635 a.u. cNuclear repulsion energyEnr = 53.528406 a.u., total energy Etot =−162.821900 a.u.

5.4.2 Nitrogen dioxide

The PESs of the ground and first excited states of the NO2 molecule, X2A1 and A2B2,

form an intersection35. Crossing points have been located for different N–O bond dis-

tances with the O–N–O angle between 107◦ and 108◦36. 146

Figure 5.2: D3h-constrained scans of adiabatic PESs of N+3 around the points of

intersections calculated by EOM-CCSD/6-31G.

When the molecule is constrained to the C2v point group symmetry, the spacial parts

of the wavefunctions of the states belong to different irreps, A1 and B2, and the derivative

coupling matrix element between them vanishes.

If the symmetry is lowered further to the Cs point group symmetry, and the N–O

bond lengths are allowed to vary independently, both the ground state and first excited

state wavefunctions become of the A′ symmetry. The derivative coupling matrix element

between the two states is no longer zero, and the intersection becomes truly conical.

The calculations were performed using the EOMIP-CCSD method. The closed-shell

CCSD wavefunction for NO−2 was used as the reference for the EOM-IP calculations.

To locate and minimize the seam numerically, the two surfaces were first scanned.

The N–O distance was varied from 1.20 A to 1.40 A, and at each point the O–N–O

angle was varied from 90◦ to 130◦, as shown in Fig. 5.3. These results are summarized

in Table 5.3. The minimum of the seam was located at R(NO) = 1.3045 A, α(ONO) =

106.75◦.

147

Figure 5.3: The seam of the X2A1/A2B2 intersection in nitrogen dioxide. At everyN–O distance, the PESs of both states were scanned to find the O–N–O angle atwhich the states are degenerate.

Table 5.3: Points of the C2v constrained seam of the X2A1/A2B2 intersection innitrogen dioxide. All bond lengths are given in angstroms, all angles are given indegrees, all energies are given in a.u.

R(NO) α(ONO) E(X2A1

)1.2000 106.027 -204.2236931.2200 106.125 -204.2339341.2400 106.241 -204.2414191.2600 106.376 -204.2465031.2800 106.531 -204.2494951.3000 106.706 -204.2506731.3045 106.748 -204.2507131.3200 106.901 -204.2502791.3400 107.117 -204.2485281.3600 107.353 -204.2456081.3800 107.611 -204.2416861.4000 107.889 -204.236910

148

Table 5.4: Results of C2v-constrained energy minimization of the X2A1/A2B2 inter-section seam in NO2 calculated by EOMIP-CCSD/6-31G. All bond lengths aregiven in angstroms, all angles are given in degrees.

Starting geometry Iterations Final geometryR(NO) α(ONO) R(NO) α(ONO)

Tolerancea ∆g = 3×10−4 (Q-Chem default)1.2000 100.00 13 1.3047 106.761.3000 90.00 14 1.3045 106.781.3000 120.00 15 1.3047 106.75

Tolerance ∆g = 3×10−5

1.2000 100.00 15 1.3046 106.741.3000 90.00 18 1.3046 106.751.3000 120.00 17 1.3046 106.75

Tolerance ∆g = 1×10−5

1.2000 100.00 17 1.3046 106.75b

1.3000 90.00 19 1.3046 106.751.3000 120.00 18 1.3046 106.75

aTolerance on the maximum gradient component. The optimization procedure con-verges if this and one of the two criteria — energy change ∆E = 10−6 a.u. and max-imum displacement ∆q = 1.2× 10−3 A — are satisfied. bNuclear repulsion energyEnr = 61.603681 a.u., total energy Etot =−204.250712 a.u.

Various starting geometries were used in order to test the performance of the algo-

rithm. Table 5.4 contains the initial points attempted and the results obtained.

5.4.3 para-Benzyne diradical

The para-benzyne diradical has the 1Ag ground state and a low-lying 3B1u triplet excited

state37–42. In both states, the equilibrium structure is of the D2h point group symmetry.

149

Figure 5.4: The singlet and the Ms=0 triplet target states of the para-benzyne dirad-ical derived from the high-spin triplet reference by spin-flip.

Figure 5.5: Geometrical parameters used to describe the para-benzyne diradical inTable 5.5.

The ground singlet state and the excited triplet state are derived from distributions

of two electrons over the lower b1u and the higher ag frontier molecular orbitals. These

states can be accurately described by the SF method43 as spin-flipping excitations form

the high-spin triplet reference state, as shown in Fig. 5.4.

The equilibrium geometries of the para-benzyne diradical were calculated at the

EOMSF-CCSD/6-31G* level. The geometries agree with the SF-DFT ones from

Ref. 40, as shown in Table 5.5.

Since the adiabatic excitation energy of the triplet state is only about 0.03 eV, one

could anticipate a low-energy 1Ag/3B1u intersection. The ground state equilibrium

geometry found in40 was used as the starting point for locating the minimum of the

crossing seam. The convergence was achieved in about 50 iterations. The progress of

the intersection optimization is shown in Fig. 5.6.

150

Tabl

e5.

5:E

quili

briu

mge

omet

ries

and

the

ME

CP

ofth

e1 A

gan

d3 B

1uel

ectr

onic

stat

esca

lcul

ated

byE

OM

SF-C

CSD

/6-

31G

*.G

eom

etri

calp

aram

eter

sar

ede

fined

inFi

g.5.

5.A

llbo

ndle

ngth

sar

egi

ven

inan

gstr

oms,

alla

ngle

sar

egi

ven

inde

gree

s.

Met

hod

R(C

C1)

R(C

C2)

R(C

H)

α(C

CC

1)α(C

CC

2)α(C

CH)

1 Ag

grou

ndst

ate

SF-D

FT(5

0,50

)/6-

31G

*a1.

419

1.35

81.

077

124.

811

7.6

123.

6C

ASS

CF(

8,8)

/6-3

1G*b

1.41

11.

383

124.

011

8.0

EO

MSF

-CC

SD/6

-31G

*c1.

424

1.37

41.

087

124.

411

7.8

123.

0

3 B1u

exci

ted

stat

eSF

-DFT

(50,

50)/

6-31

G*

1.39

81.

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152

Figure 5.7: Relative energies of the 1Ag and 3B1u states of para-benzyne diradical attheir equilibrium structures and MECP. Both at the singlet and triplet equilibriumgeometry, the singlet is below the triplet. The two states intersect 0.65 eV above theground state minimum.

The MECP was located 0.65 eV above the ground state equilibrium energy as shown

in the energy diagram in Fig. 5.7. The nuclear configuration at this point is significantly

distorted compared to the ground state geometry. Note that although this distortion

increases the splitting between the frontier MOs due to increased through-space interac-

tion between the radical centers, it stabilizes the ag orbital, due to the favorable through-

bond interactions45.

5.5 Discussion: Performance of the algorithm and

details of implementation

Our implementation of the projected gradient algorithm employs the minimization tech-

nique used in Q-Chem: the Newton-Raphson method in conjunction with the eigenvalue

following algorithm46. An approximate guess Hessian matrix used for the optimization

153

is updated using the BFGS formula. We modified the EOM-CCSD/CCSD code29 to

allow simultaneous analytical calculation of the gradients for two states. The two com-

ponents of the gradient (5.19) used in the minimization procedure are evaluated as given

by (5.17) and (5.18).

We found that the convergence of the algorithm strongly depends on the relative

absolute values of the two gradient components. If one of the components has a much

greater absolute value than the other, after the step is taken, the smaller gradient changes

its value significantly at the new point. Often, this leads to surges and oscillations of the

energy during the minimization that spoils the convergence. For example, the optimiza-

tion of the 1Ag/3B1u intersection in the para-benzyne diradical fails if the G1 and G2

vectors are taken with the same weights, as in (5.19). We employed a modified version

of the algorithm when the two components of the total gradient can be scaled with coef-

ficients specified by the user. When the G2 vector is scaled such that its absolute value

is approximately the same as that of the G1 vector, the procedure converges as shown in

Fig. 5.6.

In this context, the Lagrange multiplier algorithm (e.g., Ref. 14) seems to be more

robust, at it adjusts the coefficients automatically during the optimization procedure,

thus insuring proper treatment of the problem. This, however, should not be considered a

serious drawback of the projected gradient algorithm as with user-controlled coefficients

comparable performance is achieved.

The scaling problem is related to the quality of the Hessian. In the straightforward

implementation the approximate Hessian for the upper PES is used, which is not the

Hessian of the function which is being optimized. The issue could be rectified by

improving Hessian update procedure. Alternatively, scaling coefficients can be used

in order to improve convergence. By default, the components of the gradient are not

154

scaled. However, the tests showed that scaling the G2 vector to one-tenth of its original

absolute value significantly improves the convergence.

Yarkony47 and Chachiyo and Rodriguez17 addressed this problem by modifying the

algorithms as follows. Rather than optimizing the energy difference, they minimize the

average energy between the states, which demonstrated better performance.

5.6 Chapter 5 conclusions

We implemented a reduced variant of the projected gradient method16 within the

equation-of-motion coupled-cluster (EOM-CC) family of methods in the Q-Chem quan-

tum chemistry package22. The algorithm employs analytic gradients29 and assumes zero

derivative couplings. Thus, it is applicable for minimizing the crossing seams between

states of different symmetry and/or multiplicity. The method was applied to locate the

MECPs between: (i) the excited states of N+3 described by EOMEE-CCSD; (ii) the two

lowest states of NO2 described by EOMIP-CCSD; and (iii) the singlet and triplet states

of the para-benzyne diradical described by EOMSF-CCSD.

The benchmarks demonstrated that the convergence depends on the quality of the

initial guess, as well as local steepness/curvature of the seam, e.g., the convergence is

poor when the magnitude of the two gradient components is very different. Scaling of

the two components of the gradient was shown to improve the convergence.

155

5.7 Chapter 5 references

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[16] M.J. Bearpark, M.A. Robb, and H.B. Schlegel. A direct method for the locationof the lowest energy point on a potential surface crossing. Chem. Phys. Lett.,223(3):269–274, 1994.

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[23] M. Dallos, H. Lischka, R. Shepard, D.R. Yarkony, and P.G. Szalay. Analytic eval-uation of nonadiabatic coupling terms at the MR-CI level. II. Minima on the cross-ing seam: Formaldehyde and the photodimerization of ethylene. J. Chem. Phys.,120:7330–7339, 2004.

[24] F. Sicilia, L. Blancafort, M.J. Bearpark, and M.A. Robb. Quadratic description ofconical intersections: Characterization of critical points on the extended seam. J.Phys. Chem. A, 111:2182–2192, 2007.

[25] J. von Neumann and E. Wigner. On the behavior of eigenvalues in adiabatic pro-cesses. Z. Phys., 30:467–470, 1929.

[26] G.J. Atchity, S.S. Xantheas, and K. Ruedenberg. Potential energy surfaces nearintersections. J. Chem. Phys., 95:1862–1876, 1991.

[27] D.R. Yarkony. Conical intersections: Diabolical and often misunderstood. Acc.Chem. Res., 31:511–518, 1998.

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[29] S.V. Levchenko, T. Wang, and A.I. Krylov. Analytic gradients for the spin-conserving and spin-flipping equation-of-motion coupled-cluster models with sin-gle and double substitutions. J. Chem. Phys., 122:224106–224116, 2005.

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[38] T.D. Crawford, E. Kraka, J.F. Stanton, and D. Cremer. Problematic p-benzyne:Orbital instabilities, biradical character, and broken symmetry. J. Chem. Phys.,114:10638–10650, 2001.

[39] L.V. Slipchenko and A.I. Krylov. Singlet-triplet gaps in diradicals by the spin-flipapproach: A benchmark study. J. Chem. Phys., 117:4694–4708, 2002.

[40] Y. Shao, M. Head-Gordon, and A.I. Krylov. The spin-flip approach within time-dependent density functional theory: Theory and applications to diradicals. J.Chem. Phys., 118:4807–4818, 2003.

[41] H.H. Wenk, M. Winkler, and W. Sander. One century of aryne chemistry. Angew.Chem. Int. Ed. Engl., 42:502–528, 2003.

[42] V. Vanovschi, A.I. Krylov, and P.G. Wenthold. Structure, vibrational frequencies,ionization energies, and photoelectron spectrum of the para-benzyne radical anion.Theor. Chim. Acta, 120:45–58, 2008.

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[44] A.E. Clark and E.R. Davidson. Model studies of hydrogen atom addition andabstraction processes involving ortho-, meta-, and para-benzynes. J. Am. Chem.Soc., 123:10691–10698, 2001.

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159

Chapter 6: Future work

As exemplified by the calculations here as well as other works1–3, it is possible to pro-

duce theoretical estimates of real life experimental data on large systems using high

quality modeling techniques. The accuracy of such estimates depends directly on the

amount of detail included in the model. Typical biochemical reactions involve a rel-

atively small center, where chemical bonds are actually broken and created, and sur-

rounding solvent or protein environment. While the environment does not participate in

the reaction directly, it influences virtually every aspect of the process.

Hybrid quantum mechanical/molecular mechanical (QM/MM) techniques are

designed to give a handle on such complex systems. Their principle is to employ ab

initio theory for the reaction center and describe the surroundings using less sophisti-

cated and usually much cheaper methods.

The QM/MM approach has been used successfully for single point calculations and

simple PES explorations that can yield basic energetics of chemical reactions. How-

ever, modeling vibration spectra, free energy, reaction dynamics, and many other studies

require extensive PES sampling. For example, to calculate vibrational states one needs

multidimensional PES scans, and ab initio molecular dynamics (AIMD) and Monte

Carlo simulations call for random points on PES.

160

Figure 6.1: Large biological systems such as this active center of GFP require treat-ment of the QM core and the MM environment using hybrid QM/MM methods.

Using CC and EOM methods for the quantum subsystem in QM/MM for the purpose

of sampling PES has a prohibitive cost. Because we desire to keep the attractive proper-

ties of the methods, rather than substituting them with cheaper methods, we propose to

increase their efficiency to the extent where direct AIMD calculations on multiple PES

become feasible.

The most challenging problem of CC and EOM theories is their computational com-

plexity. CCSD and EOMEE-CCSD scale as N6, where N is the size of the molecule.

That means that a dimer requires roughly 64 times more computation effort than the

original system. Our approach is to attack the scaling by performing CC and EOM

calculations using reduced-rank (compressed) tensors.

161

Local and reduced-rank methods. The physical principle underlying reduced scal-

ing electronic structure methods is the locality of electronic correlation. Previous

attempts to use it have proven successful resulting in near linear-scaling local CCD4

and CCSD5 implementations. However, the approach has two serious shortcomings.

One is that the equations in the entire hierarchy of CC methods have to be rederived and

reprogrammed. The other is the doubt that the same principle of locality can be applied

to both ground and excited states equally.

Tensor decomposition techniques can be used to overcome the problems of the local

methods. They put a solid mathematical base under the same physical idea, and with the

new tensor library the original CC and EOM equations need not be reimplemented.

The reduced-rank tensor representations are based on tensor decompositions

in which d-dimensional tensors are factorized into a sum of vectors or lower-

dimensionality tensors6–11. The two main strategies are canonical product (CP)12 and

Tucker (or HOSVD)13 decompositions. The TT (Tree-Tucker, Three-dimensional Ten-

sor) decomposition14, 15 combines the qualities of CP and HOSVD and resolves some

of their shortcomings.

Canonical product decomposition. Following a recent proposal supported by

encouraging preliminary results12, we first consider the CP or canonical decomposition

(CANDECOMP) format6. In this format, a four-dimensional tensor A(w,x,y,z) (i.e.,

two-electron integrals or T2 amplitudes) is expanded as:

A =R

∑r=1

awr ⊗ax

r⊗ayr⊗az

r, (6.1)

where one-dimensional vectors aur are the representing vectors and the summation length

R is called rank of the representation.

162

The trivial CP decomposition is obtained as follows: the first representing vector

codes the index of the first dimension as a unit vector, the second one does the same

for the second dimension, and the last representing vector contains the values corre-

sponding to such multi-index. Obviously, the rank of such decomposition is n3 (for the

two-electron integrals), and the storage requirements (R = n3 sets of four n-dimensional

vectors need to be stored) are increased relative to the full four-index tensor (n4 ele-

ments). However, if the rank can be significantly reduced, then the storage requirements

(4 ·n ·R) of the decomposed tensor can become much smaller and linear scaling can, in

principle, be achieved.

Recently, an efficient (and easily parallelizable) rank-reduction algorithm (based on

the accelerated gradient method) have been developed by Hackbusch and coworkers12

and applied to two-electron integrals and MP2 amplitudes. They demonstrated that

ranks of these basic quantities can be significantly reduced by this algorithm giving rise

to n1.2−n2.5 depending on the thresholds used. The scaling of their algorithm depends

on the ratio of the initial and final ranks. Thus, the cost of the rank reduction step

can be decreased if one starts with a representation which is more compact than the

trivial decomposition. The authors12 suggested that using RI or Cholesky decomposed

integrals as a starting point of the rank reduction procedure can considerably increase

the efficiency of this most time-consuming step.

Once the integrals are represented in the CP format, the contractions with any two-

index tensors (MO coefficients C or T1 amplitudes) can be trivially performed at the

163

reduced cost. For example, the integral transformation (from AO to MO basis) assumes

the following form:

〈ab|cd〉 = ∑µνλσ

CµaCνbCλcCσd ∑r

ξµr ⊗ξ

νr ⊗ξ

λr ⊗ξ

σr

= ∑r

(∑µ

Cµaξµr

)⊗(

∑ν

Cνbξνr

)⊗

(∑λ

Cλcξλr

)⊗

(∑σ

Cσdξσr

)= ∑

rva

r ⊗ vbr ⊗ vc

r⊗ vdr (6.2)

Thus, the complexity of the transformation is reduced to O(n ·Nvirt ·R), to be compared

with the O(n4 ·Nvirt +n3 ·N2virt +n2 ·N3

virt +n ·N4virt) scaling of the conventional algorithm.

Most importantly, the rank does not change upon this transformation.

The complexity of contractions between two four-index CP tensors is also reduced,

however, such contractions do change the rank, i.e., the rank of the product obtained

by contracting two tensors of ranks R′ and R′′ equals R′ ·R′′, as demonstrated by the

following example:

∑ab

tabi j 〈kl||ab〉 = ∑

ab

(S

∑s=1

tsi ⊗ ts

j⊗ tsa⊗ ts

b

)(R

∑r=1

vrk⊗ vr

l ⊗ vra⊗ vr

b

)

=RS

∑q=1

tqi [i]t

qj [ j]v

rk[k]v

rl [l]

(∑a

tsa[a]v

ra[a]∑

btsb[b]v

rb[b]

)

=RS

∑q=1

tqi ⊗ tq

j ⊗ vrk⊗ vr

l (6.3)

Thus, the rank reduction procedure should be applied after each such tensor operation.

A minor complication arises due to the terms involving energy denominators, which

can be efficiently dealt with by the Laplace transform based on approximating 1/x by

exponentials sums16–18, as done in the Laplace MP2 method16–21.

164

In sum, both the storage requirement and the cost of contractions between tensors

represented in the low-rank CP format is dramatically reduced compared to the original

tensors. However, the contractions (except when one of the tensors is a two-index tensor)

lead to a rank increase. Thus, to preserve the low-rank representation, a rank reduction

procedure should be performed after each contraction.

Rank reduction. The rank-reduction procedure amounts to the minimization problem

of the difference between an original tensor A and its lower-rank approximation, A:

||A− A|| ≤ ε (6.4)

where ε determines the accuracy. The accelerated gradient (AG) algorithm22 has been

successfully employed in Ref. 12. The most time-consuming step in the AG algorithm is

the calculation of the gradient. The complexity of this step is O(d · R ·n · (R+R)) where

d is the dimension of the tensor, and R and R are the ranks of A and A, respectively.

Thus, the cost of this step depends critically on the initial rank. The authors of

Ref. 12, who employed the trivial decomposition as the initial step for rank reduction,

have suggested that a better starting point can be provided by using RI or Cholesky

decomposed integrals.

To further reduce the cost of the rank reduction algorithm, the sliced factorization

procedure has been developed12, in which the initial reduction is performed for smaller

blocks of data. This algorithm can be trivially parallelized.

We plan to begin by implementing the parallelized AG procedure including sliced

reduction following the algorithm described in Ref. 12. We then will explore the follow-

ing approaches for improving the starting point for the rank reduction of the integrals:

165

(i) RI decomposed integrals; (ii) Cholesky decomposed integrals; (iii) using localized

MOs; (iv) using frozen natural orbitals (FNO)23.

HOSVD decomposition. There are multiple ways to define SVD for more than two

dimensions. In this study we use the Tucker decomposition as high-order SVD

A =R

∑t1=1

R

∑t2=1

R

∑t3=1

R

∑t4=1

g(t1, t2, t3, t4)a(w, t1)a(x, t2)a(y, t3)a(z, t4) (6.5)

This form assumes a multidimensional core which is expanded by matrices a(x, t).

R is the rank of Tucker decomposition. Because the core has the same dimensionality

as the original tensor, the number of parameters in this decomposition does not have a

reduced scaling. In the equation above, even after the reduction of the rank, the number

of parameters grows as O(n4). Therefore, an improvement is required to use Tucker

decomposition for reduced-scaling methods.

TT decomposition. In the TT decomposition, a target tensor A(w,x,y,z) is represented

as

A = ∑α1α2α3α4

awα1

axα1α2

ayα2α3az

α3(6.6)

TT decomposition involves three-index tensors at most, thereby enabling reduced-

scaling representation of higher-order tensors with appropriate rank reduction. In addi-

tion, the decomposition algorithm is based on partial SVD of unfolding matrices, which

is better positioned to solve the rank reduction problem than the iterative algorithms

used in CP decomposition.

* * *

166

The suite of CC and EOM methods, including calculations of energies, properties

and gradients, is implemented in Q-Chem 4 using the new library for tensor algebra (see

Appendix). The design of the library allows us to reuse the CC and EOM expressions,

but introduce new types of tensors that are based on the decompositions.

This project has three milestones:

1. Enable the tensor library to store and use tensors in a readily available decompo-

sition format, for example RI. While this step does not reduce the scaling of the

method, it improves the performance by reducing the amount of the data set.

2. Introduce decomposition algorithms and corresponding tensor formats. Use the

infrastructure created in Step 1 to use these new compressed formats.

3. Optimize specific tensor operations to take advantage of the compressed formats.

If rank reduction is achieved successfully by the decompositions, this step will

lead to a reduced-scaling implementation of the methods.

167

6.1 Chapter 6 references

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Appendix: Tensor algebra library

Efficient object-oriented implementation of CC and EOM-CC methods. Pro-

grammable expressions in wave function methods are dominated by tensor contractions

(generalized matrix multiplications). For example, the CCD correlation energy is given

by

ECCD =14 ∑

i jabtabi j 〈i j||ab〉

where tabi j is the two-body cluster operator, and 〈i j||ab〉 is the electron repulsion integrals

in the molecular orbital basis. Both are four-index antisymmetric tensors.

The challenge of implementing many-body methods lies in the complexity of the

expressions, which consist of numerous tensor products. For example, the CCSD cluster

amplitude equations involve more than thirty contractions between the amplitudes (T1

and T2) and the integrals. All of them have to be programmed correctly and efficiently.

Object-oriented computer languages such as C++ allow the programmer to manage

the complexity and code such equations in a compact and readable form that facilitates

validation and debugging. To take full advantage of this philosophy, we developed a

set of expression templates that use overloaded C++ operators. They allow one to code

programmable equations using an intuitive syntax.

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For example, the second-order Møller-Plesset perturbation theory (MP2) electronic

correlation energy is computed as

EMP2 = ∑ia

tai fia +

14 ∑

i jabtabi j 〈i j||ab〉, where

tai =

fia

εi− εaand tab

i j =〈i j||ab〉

εi + ε j− εa− εb

The C++ function that computes the energy using the tensor algebra library may look

similar to this.

double mp2_energy(btensor<2> &t1, btensor<4> &t2,

btensor<2> &f_ov, btensor<4> &i_oovv) {

letter i, j, a, b;

return dot_product(t1(i|a), f_ov(i|a)) +

0.25 * dot_product(t2(i|j|a|b), i_oovv(i|j|a|b));

}

The expression templates are designed so that most common programming mistakes

are detected at compilation time rather than runtime thus improving the robustness of

the code. When expressions are executed, they are translated into a series of block

tensor operations. Since the conversion takes place at runtime, the library can take

full advantage of its computational setup (details unknown at compilation time such as

memory and disk availability, type and number of processors, etc.)

The level of abstraction offered by the expression templates allows us to introduce

changes in the low-level library code without modifying the expressions. Once coded

using this interface, all methods immediately take advantage of improvements in the

188

Figure A.1: Components of the tensor algebra library.

library. One example of such improvement is transition to a new computational plat-

form. Once the tensor algebra library features the ability to use certain hardware or par-

allelism, all codes automatically take advantage of that. Other examples are new types

of symmetry, different representations of tensors such as algebraic decompositions, or

exploiting sparsity in local correlation approaches.

Library for tensor algebra in electronic structure calculations. We implemented

the principles presented here in the form of a C++ tensor algebra library. It supports

virtual memory (utilizing disk for storage when there is not enough RAM), symmetries

found in electronic structure calculations and parallel computation on capable hardware.

The library is now a part of Q-Chem.

The tensor library has a layered structure (Fig. A.1):

• The interface is a tool that enables the entry of programmable expressions by pro-

viding functions and overloaded arithmetic operators for tensor operations such as

addition, scaling, contraction, dot and direct products. It also serves a secondary

189

purpose of preliminary optimization of the expression, which is converted to a

sequence of block tensor operations.

• Block tensor operations are a set of routines that produce the symmetry and

numerical result of basic and general block tensor linear algebra. Operations such

as addition of multiple block tensors, general contraction of two block tensors,

products of two block tensors are used as building blocks to compute complex

expressions and implemented as individual classes. These routines first calculate

the symmetry of the result, and then compute the numerical data accounting for

symmetry and avoiding redundancies. Each unique block of the resulting block

tensor is computed by invoking simple tensor operations on the blocks of the

arguments. In parallel environments the blocks of the result can be calculated

simultaneously using shared input data. Concurrency is implemented using MPI,

OpenMP, native threads and other similar techniques.

• Simple tensor operations work on individual dense blocks of block tensors and

perform tensor linear algebra on multidimensional arrays. Depending on the

nature of the operation, it can compute the result directly or invoke lower-level

numerical libraries.

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