Simple models for difficult electronic excitations

Article Cite This: J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

pubs.acs.org/JCTC

Simple Models for Difficult Electronic Excitations Giuseppe M. J. Barca, Andrew T. B. Gilbert, and Peter M. W. Gill* Research School of Chemistry, Australian National University, Acton ACT 2601, Australia ABSTRACT: We present a single-determinant approach to three challenging topics in the chemistry of excited states: double excitations, charge-transfer states, and conical intersections. The results are obtained by using the Initial Maximum Overlap Method (IMOM) which is a modified version of the Maximum Overlap Method (MOM). The new algorithm converges better than the original, especially for these difficult problems. By considering several case studies, we show that a single-determinant framework provides a simple and accurate alternative for modeling excited states in cases where other low-cost methods, such as CIS and TD-DFT, either perform poorly or fail completely.

CIS can overestimate excitation energies by as much as 2 eV,36 and TD-DFT performs poorly for Rydberg41−44 and chargetransfer states45−48 unless a long-range corrected functional is used.49,50 These weaknesses arise because neither CIS nor TD-DFT allows the molecular orbitals (MOs) to relax in the excited state, and consequently, they struggle in cases where that relaxation would be significant. However, several years ago we showed that relaxed excited single-determinant wave functions can be found using the Maximum Overlap Method (MOM)51,52 and that these are useful approximations.53 In this paper, we ask how well the single-determinant approximation can handle challenging cases where other lowcost excited-state methods fail. Section 2 presents a modified MOM algorithm that converges more reliably to the desired excited state. Then, in Sections 3−5, we present results obtained by applying the new algorithm to problems involving double excitations, charge-transfer states, and conical intersections. All of the excitation energies reported here are vertical. That is, the energy of the excited state is calculated at the structure of the ground state, and no attempt has been made to correct for zero-point vibration energy. If we use our modified MOM to find an excited-state solution of the self-consistent field (SCF) equations, we refer to the resulting energy using the unadorned name of the functional (e.g., BLYP). On the other hand, if we use the conventional time-dependent approach to estimate the excitation energy from the ground state, we refer to the energy by prefixing “TD” to the name of the functional (e.g., TDBLYP). As we show, these two approaches often yield strikingly different models of excited-state energetics.

1. INTRODUCTION Understanding electronically excited states is important in many fields such as photovoltaics, optics, synthetic chemistry, and biology. Quantum chemistry has played a key role in improving this understanding, but while ground states are usually studied using a small set of well-established methods such as Density Functional Theory (DFT),1,2 Møller−Plesset Perturbation Theory, and Coupled-Cluster Theory, the arsenal of approaches to excited states is very large, including MultiReference Configuration Interaction (MRCI),3−5 Complete Active Space Configuration Interaction (CASCI),6−8 Complete Active Space Self-Consistent Field (CASSCF),9,10 Restricted Active Space Self-Consistent Field (RASSCF),11 CASPT2,12−14 Multi-Reference Møller−Plesset Perturbation theory (MRMP),15 Symmetry-Adapted Cluster-Configuration Interaction (SAC−CI),16−18 Equation-Of-Motion Coupled Cluster (EOM-CC),19−21 Linear Response Coupled Cluster (LRCC),22,23 Configuration Interaction Singles (CIS),24−26 CIS with perturbative treatment of doubles (CIS(D)),25 TimeDependent DFT (TD-DFT),27−30 Constrained DFT,31 manybody Green’s functions methods (GW),32,33 and others. For many researchers, the choice is bewildering. The abundance of excited-state methods results partly from the belief that whereas ground states are often described well by single-determinant methods excited states are usually multireference in character, especially if the state is doubly excited or in the vicinity of a conical intersection.34,35 However, specification of a multireference wave function is more difficult and often requires a delicate combination of chemical intuition, experience, trial, and error that incurs a substantially higher computational cost. The single-reference CIS and TD-DFT methods address some of these concerns but, notwithstanding their success, have some important weaknesses. Both CIS and TD-DFT (within the adiabatic local density approximation36) are incapable of describing doubly excited states37−40 and often fail near conical intersections between ground and excited states.40 Moreover, © XXXX American Chemical Society

Received: September 24, 2017 Published: February 14, 2018 A

DOI: 10.1021/acs.jctc.7b00994 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

2. METHOD The MOM provides an alternative to the auf bau principle for determining which MOs to occupy on each cycle of an SCF calculation. Rather than choose the lowest energy MOs, we choose those with the largest projection into the span of the occupied MOs of the previous SCF cycle. Given the MO overlap matrix O = (Cprevious)† SCcurrent

and projections pj = ∑i Oij 2

(

(1) 1/2

)

, we simply choose to occupy

the MOs with the largest pj values; all other aspects of the SCF are unchanged. The auf bau criterion drives the SCF toward the lowestenergy solution of the SCF equations. If it is replaced by the MOM criterion, and if the SCF algorithm finds stationary points and not just energy minima (e.g., Direct Inversion of Iterative Subspace (DIIS)), the SCF can discover higher-energy solutions of those equations, and we have demonstrated that these correspond to the excited states of the system.51−53 This approach is not limited to the lowest-energy solutions of each symmetry type. Notwithstanding the success of the MOM, cases sometimes arise wherein the SCF converges to an undesired solution, and this is especially common in systems with near-degeneracies. Such behavior led us to develop a modified MOM protocol in which we choose to occupy the MOs with the largest projection into the space spanned by the occupied MOs of the initial guess. Thus, the MO overlap matrix becomes O = (Cinitial )† SCcurrent

Figure 1. SCF energies for HF/aug-cc-pVTZ calculations on the boron atom using the MOM and IMOM protocols. The IMOM SCF calculation converges to the target state, whereas the MOM calculation drifts away from the initial guess and collapses to the ground state. See text for further details.

Table 1. Orbital Overlap Values between Various py Orbitals on Different SCF Cycles of HF/aug-cc-pVTZ Calculations on the Boron Atoma MOM

(2)

This strategythe Initial Maximum Overlap Method (IMOM)encourages the SCF to find a solution of the SCF equations in the neighborhood of the initial guess. We implemented it in the Q-CHEM 4.4.1 package54 and used it to generate all the results reported in this paper. To use either the IMOM or the MOM, the SCF calculation must begin with orbitals that lie within the respective basins of attraction of the target state. Often, it is sufficient to perform a ground-state calculation and simply promote an electron from an occupied to a virtual orbital. The orbitals are then allowed to relax in the ensuing SCF calculation. However, even with a good guess, it is possible for the MOs in a MOM calculation gradually to drift away from the initial guess. The IMOM prevents this by anchoring the SCF to that guess, and this simple algorithmic modification significantly enhances its usefulness over the MOM. Figure 1 illustrates this point by comparing the convergence of the two approaches for calculating the 22P excited state of the boron atom. The initial guess, which is identical for both calculations, was obtained from the ground-state MOs by promoting an electron from the occupied 2px orbital to the unoccupied 3py orbital. The IMOM calculation converges to the desired 1s22s23p1 configuration, whereas the MOM calculation collapses onto the ground state. Table 1 contains overlap values between relevant py orbitals which help explain the differences observed in Figure 1. Initially the occupied SCF orbital has an overlap with the guess orbital of 0.648, but as the MOM SCF proceeds, the occupied orbital drifts away from the guess causing the overlap values to decrease monotonically (column 3). After cycle 3, there exists a previously unoccupied orbital that now has greater overlap with

2

Cycle

o o′

1 2 3 4 5 6 7 8 9 10 11

0.648 0.982 0.988 0.999 1.000 1.000

oi

2

0.648 0.518 0.412 0.411 0.410 0.410

IMOM

vi

2

0.320 0.445 0.545 0.544 0.545 0.545

o o′

2

0.648 0.982 0.011 0.999 0.999 0.995 0.998 0.994 0.997 1.000 1.000

oi

2

0.648 0.518 0.545 0.537 0.530 0.523 0.549 0.483 0.528 0.526 0.526

vi

2

0.320 0.445 0.412 0.411 0.411 0.391 0.363 0.428 0.382 0.384 0.384

a Orbitals are indicated by o = the occupied orbital; o′ = the occupied orbital from the previous SCF cycle; i = the initial guess orbital; and v = the lowest unoccupied virtual py orbital. The bold row highlights the IMOM occupancy “flip”.

the guess than any of the occupied orbitals. The IMOM selects this orbital causing a sudden change in the occupied orbitals. This occupancy “flip” gives rise to the low overlap value (0.011) between successive orbitals seen in column 5 and the jump in energy between cycles 3 and 4 in Figure 1. The final overlap values in columns 3 and 6 show that the IMOM orbital has a larger overlap (0.526) with the guess orbital than the MOM orbital (0.410), as desired. It is worth stressing that the set of solutions that can be obtained using the IMOM is identical to that which can be obtained using the MOM. This is clear from the fact that the protocols become equivalent when applied to a converged solution of the SCF equations. The key advantage of the new approach is that it is much easier to generate initial guesses that lead to the desired target states. All reported calculations use the spin-unrestricted formalism.55,56

3. DOUBLE EXCITATIONS The term “doubly excited” is commonly used to describe states whose configuration interaction (CI) expansions include B


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cluster configuration interaction (SAC−CI)80 reduce the errors to 0.5 eV, but only multireference perturbative treatments (e.g., MRMP and CASPT2) were able to achieve errors of 0.1−0.29 eV for benzene,66,81 0.27−0.54 eV for naphthalene,66 0.15 eV for anthracene, and 0.25 eV for naphthacene.71,82,83 For consistency with the other molecules, we classify the states of benzene using D2h symmetry, and its first singlet excited state, which is 1 1E2g in D6h, becomes a degenerate Ag + B1g pair in D2h. The lowest totally symmetric, π → π*, singly and doubly excited singlet states are interesting. In benzene, the single excitation is much lower in energy than the double, but the ordering reverses in anthracene66,71 and larger molecules.84 To discover whether standard DFT methods reproduce this reordering, we used the IMOM to compute the BLYP/6311G* energies of these states in benzene, naphthalene, anthracene, and pleiadene (Figure 2) at ground-state BLYP/

doubly substituted configurations with large amplitudes. However, being dependent on the chosen reference configuration, this definition is ambiguous. If the ground-state wave function Ψ and excited-state wave function Ψ̂ for an n-electron system are single determinants of the MOs ψi and ψ̂ j, respectively, the excitation number57 n

η=n−

n

∑ ∑ |⟨ψi|ψĵ ⟩|2 i

(3)

j

measures the number of electrons in the excited state which occupy the space spanned by the virtual orbitals of the ground state. Thus, for example, doubly substituted versions of Ψ would have η = 2. Post-excitation orbital relaxation leads to small deviations from this ideal value, but η allows us easily to decide whether or not a state is doubly excited. 3.1. The H2 Molecule. The 1σ2g ground state of the H2 molecule has been a benchmark for quantum chemical methods since the dawn of quantum mechanics.58 Accurate energies of the lowest doubly excited 1σ2u state, an autoionizing resonance, were first obtained by Bottcher and Docken59 and later by others.60−63 We performed SCF calculations on both states with R = 1.4 bohr using a modified aug-mcc-pV8Z basis64 to which additional diffuse s, p, and d shells were added and from which the g and higher shells were removed. The HF and CI excitation energies (Table 2) are similar because excitation preserves the electron pair, and so the correlation energies of the states are comparable. Table 2. Total Energies (E, in hartree), Excitation Energies (ΔE, in eV), Self-Interaction Energies (ESIE, in eV), and η Values for the 1σ2g → 1σ2u Excitation in H2 %Fock E(1σ2u) E(1σ2g ) ΔE ESIE(1σ2u) ESIE(1σ2g ) η

BLYP

B3LYP

HF-LYP

HF

Full CI

0 −0.16250 −1.17031 27.42 −1.781 0.007 2

20 −0.16016 −1.18071 27.77 −1.219 0.217 2

100 −0.09950 −1.17198 29.18 0 0 2

100 −0.08076 −1.13363 28.65 0 0 2

100 −0.11755 −1.17428 28.75 0 0 

Figure 2. Polycyclic hydrocarbons with low-lying doubly excited states.

6-311G structures. Initial guesses for the singly and doubly excited states were obtained from the ground-state MOs by promoting, respectively, one or two electrons from an occupied π orbital (usually the HOMO) to the lowest unoccupied π* orbital. Table 3 compares the resulting excitation energies with CASSCF, MRMP, and experimental values, and Figure 3 reveals the energy reordering as the system size increases. In all cases, the BLYP calculations give the correct ordering and pleasingly accurate excitation energies, with a mean absolute deviation from the experimental values of only 0.15 eV. The worst result, an error of 0.41 eV for the singly excited 3 1Ag state of anthracene, can be compared with CASSCF and MRMP errors of 1.24 and 0.32 eV, respectively. The 1 1E2g state in benzene has been reported to be a doubly excited state arising from the (HOMO) 2 → (LUMO) 2 transition. This assignment is based on large amplitudes of doubly substituted configurations that appear in the CASSCF81 and MRMP wave functions.66 However, this state can be accurately modeled with a single determinant, and its η value (1.0058) strongly suggests that, in fact, it is only singly excited. We infer from this that the important doubles in the CASSCF wave function serve largely to describe correlation and

Because the self-interaction errors65 1 ESIE = E J + E X (4) 2 (where EJ and EX are Coulomb and exchange energies) are much larger in the excited state than in the ground state, the DFT results improve as the percentage of Fock exchange increases. This highlights the need for exchange functionals that are accurate for excited states. Because the electrons are excited from a gerade to an ungerade MO, the overlap integrals in eq 3 vanish, and η is predicted by all levels of theory to be exactly 2. 3.2. Polycyclic Hydrocarbons. It is difficult to model π → π* excited states of benzene and polyacenes accurately. While semi-empirical methods, such as the Pariser−Parr−Pople (PPP) method,73−75 can give good results, ab initio methods often struggle to obtain comparable performance. Early CI studies gave excitation energy errors exceeding 1 eV for some valence states of benzene.76−78 Multireference configuration interaction (MRCI)79 and symmetry-adapted C


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Journal of Chemical Theory and Computation Table 3. Excitation Energies of the Lowest Totally Symmetric Singly and Doubly Excited States of Aromatic Systems Present Work

Previous Work

Experimental

Molecule

State

η

BLYP

CASSCF

MRMP

Work

Benzene

2 5 2 4 2 3 2 3

1

1.0058 2.0034 1.0191 2.0019 2.0013 1.0141 2.0118 1.0818

7.70 10.21 5.66 6.77 4.62 4.92 2.46 3.43

8.0166  5.8666 6.7569 5.4266 6.5771  

7.7366  5.6566 6.7669 5.0366 5.2871  

7.8067  5.5268  4.7170 5.3370 2.4672 3.6172

Naphthalene Anthracene Pleiadene

Ag 1 Ag 1 Ag 1 Ag 1 Ag 1 Ag 1 A1 1 A1

Figure 4. Ethylene + tetrafluoroethylene complex.

Figure 3. Lowest valence totally symmetric singly and doubly excited states for C6H6, C10H8, C14H10, and C18H12 using BLYP/6- 311G*.

relaxation and should not be interpreted as indicating that the state is doubly excited. Figure 5. Variation of the excitation energy ΔE of the first CT state of C2H4 + C2F4 with the distance R.

4. CHARGE-TRANSFER STATES A typical electronic excitation creates an electron−hole pair as the electron moves from one MO to another. If the electron and hole are separated by a significant distance R, the result is termed a charge-transfer (CT) state. Because of their charges, the electron and hole attract, causing the energy of the system to rise as 1/R as the donor and acceptor separate. CIS is able to reproduce the 1/R dependence of the excitation energy for CT states, despite giving large (0.5−2 eV) excitation energy errors.36 In contrast, TD-DFT calculations fail to capture the 1/R behavior, not because of flaws in TD-DFT itself but because of the adiabatic approximation universally adopted in its implementations.47,48 Because excited-state DFT energies obtained using the IMOM do not make this approximation, one anticipates that they will avoid the undesirable features of TD-DFT. To test this, we have compared the performance of DFT with CIS and TD-DFT for the lowest CT states of two supramolecular systems: ethylene + tetrafluoroethylene and bacteriochlorin + zincbacteriochlorin. 4.1. Ethylene + Tetrafluoroethylene. CT states of the C2H4 + C2F4 complex (Figure 4) have been studied previously by Dreuw et al. using TD-DFT and CIS.47 To capture the correct 1/R behavior, they proposed a hybrid approach which combines TD-DFT and CIS and which yielded reasonable estimates for the CT excitation energies. However, because of its reliance on CIS, their approach is not very accurate. Figure 5 compares the excitation energies of the first CT state predicted by EOM-CCSD, CIS, B3LYP, M08-HX, and

TD-B3LYP as the distance R is varied. The 6-31G* basis set was used for all calculations. As anticipated, TD-B3LYP is qualitatively wrong, while EOM-CCSD, CIS, B3LYP, and M08HX all reproduce the correct 1/R behavior. The failure of TDB3LYP can be traced to an incompletely modeled interaction between the electron and the hole. Although CIS captures the correct decay behavior, it predicts an excitation energy at R = 4.6 which is 0.65 eV greater than the EOM-CCSD reference (Table 4). The B3LYP error at this point is also large (−0.48 eV), but the M08-HX functional85 reduces this to 0.12 eV. 4.2. Bacteriochlorin + Zn-Bacteriochlorin. Bacteriochlorins (7,8,17,18-tetrahydroporphyrins) are the chromophoric moiety of bacteriochlorophylls (BChl) which are found in purple bacteria, green bacteria, and heliobacteria.86 Their photochemistry has aroused broad scientific interest from the development of artificial light-harvesting antennae for photoTable 4. Excitation Energies (in eV) of the First CT Transfer States of C2H4 + C2F4 and BC + Zn-BC Complexes

D

Complex

R/Å

M08HX

B3LYP

CIS

TDB3LYP

EOMCCSD

C2H4 + C2F4 BC + ZnBC BC + ZnBC

4.60 11.20 12.04

10.84 3.28 3.41

10.48 3.33 3.49

11.61 3.69 3.78

6.79 1.82 1.84

10.96  


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Journal of Chemical Theory and Computation active devices87−90 to photodynamic therapy for cancer destruction.91 The key step in the process involves the absorption of light and a transfer of the singlet excitation energy via protein−BChl complexes to the photosynthetic reaction center.88 It is obvious, therefore, that the theoretical study of such excitations requires a detailed understanding of charge-transfer (CT) states. In nature, the zinc-bacteriochlorin (Zn-BC) is linked to the bacteriochlorin (BC) through a phenylene bridge. However, the phenylene group has only a minor influence on the CT states, and in our study, we follow the approach of Dreuw et al.,48 adopting the model shown in Figure 6. This allows the

5. CONICAL INTERSECTIONS A conical intersection (ConInt) is a subset of the nuclear coordinate space where the adiabatic potential energy surfaces

Figure 6. Bacteriochlorin + zinc-bacteriochlorin complex. Figure 8. Energy difference ΔE between the D0 and D1 states of equilateral H3 as the interatomic distance R varies. Green triangles show the MECI predicted by Mielke et al.95

distance between the chromophores to be varied from that determined by the bridge, which is 12.04 Å. (Note that our definition of R, which measures the distance between the centers of nuclear charge of each monomer, differs from theirs.) B3LYP/6-31G* structures of the complex were optimized for several R between 11.2 and 16.2 Å. Excitation energies of the first CT state were calculated using M08-HX, B3LYP, CIS, and TD-B3LYP with the 6-31G* basis, and these are shown in Figure 7. As we saw for the C2H4 + C2F4 complex, the DFT and CIS methods predict the correct 1/R dependence but TDB3LYP fails.

Figure 9. Protonated Schiff Base of Retinal (PSBR).

(PESs) of two electronic states of a molecule are degenerate. ConInts frequently play a key role in the reactions, spectroscopy, and dynamics of molecules, especially those of biochemical interest.92,93 Due to the degeneracy of the PESs, excited-state calculations involving ConInts are challenging. In particular, linear response-based methods, e.g., CIS and TD-DFT, fail when the HOMO−LUMO gap is small or zero. We test the ability of single-determinant methods to model PESs in the vicinity of ConInts by considering the H3 and retinal molecules. 5.1. The H3 Molecule. The study of ConInts in H3 has a long history, both theoretical and experimental, and we encourage the interested reader to study the survey94 by Halász et al. for further details. There are four ConInts involving the three lowest-energy electronic states, but we focus on the ConInt with D3h symmetry as it is characterized by a single interatomic distance R. For most values of R, the doublet ground state D0 has 2E′ symmetry. However, for very small R, the 2A1′ state is lower in energy, thus creating the ConInt. Mielke et al. reported95 accurate PESs for the H + H2 reaction at a highly correlated level with the aug-cc-pVDZ, aug-ccpVTZ, and aug-cc-pVQZ basis sets. They found the minimal energy ConInt (MECI) at R = 0.495 Å, and we use this value to

Figure 7. Variation of the excitation energy ΔE of the first CT state of BC + Zn-BC with the distance R. The black-dotted line shows the natural separation R = 12.04 Å.

At R = 12.04 Å, the CIS excitation energy of 3.78 eV (Table 4) is close to the value (3.79 eV) obtained by Dreuw et al. using their hybrid approach.48 Higher levels of theory are prohibitive for the BC/Zn-BC system, and no experimental results are available. However, our results for the C2H4 + C2F4 complex suggest that CIS probably overestimates the excitation energy and that the M08-HX results are probably the most accurate. E


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Figure 10. Variation of the S0 → S1 excitation energy ΔE with the torsional angle ϕ in PSBR.

have provided evidence that the S1−S0 ConInt in retinal is responsible for the ultrafast photoisomerization of the molecule, and this was later corroborated by experiments by Polli et al.104 Unfortunately, Levine et al.40 report that ConInts involving a closed-shell singlet ground state cannot be found by either TD-DFT or CIS because “matrix elements connecting the initial state and the response states are excluded from the formulation”. The reaction path for the photoisomerization of retinal undoubtedly involves complicated motions of all of the nuclei. However, it is dominated by the torsion rotation about the C15−C16 double bond,101,102,105 and the MECI is expected to lie near ϕ = 90° (Figure 9). The equilibrium geometry was found at B3LYP/6-31G*, and frozen-geometry scans for 0 ≤ ϕ ≤ 180° were then performed. Using the 6-31G* basis, the ground (S0) and excited (S1) energies were computed using the IMOM at the BLYP and B3LYP levels, and the resulting PESs are shown in Figure 10. Both BLYP and B3LYP predict a ConInt near ϕ = 90°, lying approximately 2 eV above the equilibrium structure. This is slightly lower than CASSCF-based estimates of around 2.3 eV in the work of Molnar et al.101 and of Andruniów et al.105

assess the performance of various DFT, TD-DFT, and CIS methods. The energies of the two states were computed using the conventional CIS method, four IMOM-based methods (HF, HF-LYP, BLYP, and B3LYP), and three TD-DFT methods (TD-HF-LYP, TD-BLYP, and TD-B3LYP). In all cases, the aug-cc-pVDZ basis set96 was used. The energy differences are plotted in Figure 8, and on such plots, any ConInts appear as cusps on the horizontal axis. Using the IMOM, we were able to find the ConInt at all four single-determinant levels considered. CIS is also able to model the ConInt, and for the narrow domain of R shown in Figure 8, the CIS and HF energies are very similar. This is no coincidence for if the CIS state intersected the ground state at a different value of R the CIS solution would have a lower energy than the ground state, leading to a contradiction. None of the TD-DFT models yields a ConInt. Both TDBLYP and TD-B3LYP show discontinuities at the ConInt because the ground-state reference changes from 2A1′ to 2E′ at this point. These different references give different excitation energies for the D1 state, leading to the discontinuity. The TDHFLYP PES is continuous and has a cusp that coincides with that obtained using HF-LYP. However, it does not correspond to a ConInt as the solutions are not degenerate at this point. HF-LYP, the most accurate of the methods considered, predicts a MECI at R = 0.483 Å, which is only 0.012 Å below the Mielke value (R = 0.495 Å), and HF and CIS predict a MECI at 0.016 Å above Mielke’s. Both BLYP and B3LYP predict a MECI at a bond length that is almost 0.1 Å shorter than Mielke’s. 5.2. Retinal. The photoisomerization of the 11-cis retinal chromophore to its all-trans form in the rhodopsin protein is the primary process involved in vision.97 Many attempts have been made to explain this process from an electronic structure point of view, including pioneering ab initio calculations by Du and Davidson98 in 1990 on the excited states of the protonated Schiff base of retinal (PSBR). Early theoretical studies of photoisomerization of protonated Schiff base cations were reported by Bonačić-Koutecký et al.99 They showed that the isomerization of the formaldiminium cation (CH2NH+2 ) occurs through a ConInt between the S1 and S0 states at an N−C bond twist-angle of 90°. Since then, many theoretical studies100−103

6. CONCLUDING REMARKS We have examined single-determinant approximations for excited states involving double excitations, charge-transfer, and conical intersections. These determinants correspond to higher-energy solutions of the SCF equations and are widely believed to be difficult to obtain. However, the new IMOM protocol provides a straightforward and reliable method for obtaining these solutions, and we have shown that they may be preferable to other low-cost excited-state methods. For double excitations, which cannot be described by CIS or TD-DFT, IMOM-based HF or DFT calculations are among the few low-cost options available. Moreover, we find that the single-determinant energies obtained in this way are remarkably accurate and can rival far more expensive methods such as CASSCF and MRMP. It is especially pleasing to discover how accurately the 1σ2u resonance state in H2 is modeled by HF theory. F


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(15) Hirao, K. Multireference Møller-Plesset method. Chem. Phys. Lett. 1992, 190, 374. (16) Nakatsuji, H.; Hirao, K. Cluster expansion of the wavefunction. Symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J. Chem. Phys. 1978, 68, 2053. (17) Nakatsuji, H. Cluster expansion of the wavefunction. Excited states. Chem. Phys. Lett. 1978, 59, 362. (18) Nakatsuji, H. Cluster expansion of the wavefunction. Electron correlations in ground and excited states by SAC (symmetry-adaptedcluster) and SAC CI theories. Chem. Phys. Lett. 1979, 67, 329. (19) Rowe, D. J. Equations-of-Motion Method and the Extended Shell Model. Rev. Mod. Phys. 1968, 40, 153. (20) Mukherjee, D.; Mukherjee, P. K. A response-function approach to the direct calculation of the transition-energy in a multiple-cluster expansion formalism. Chem. Phys. 1979, 39, 325−335. (21) Sekino, H.; Bartlett, R. J. A linear response, coupled-cluster theory for excitation energy. Int. J. Quantum Chem. 1984, 26, 255. (22) Dalgaard, E.; Monkhorst, H. J. Some aspects of the timedependent coupled-cluster approach to dynamic response functions. Phys. Rev. A: At., Mol., Opt. Phys. 1983, 28, 1217. (23) Koch, H.; Jørgensen, P. Coupled cluster response functions. J. Chem. Phys. 1990, 93, 3333. (24) Del Bene, J. E.; Ditchfield, R.; Pople, J. A. Self-Consistent Molecular Orbital Methods. X. Molecular Orbital Studies of Excited States with Minimal and Extended Basis Sets. J. Chem. Phys. 1971, 55, 2236. (25) Foresman, J. B.; Head-Gordon, M.; Pople, J. A.; Frisch, M. J. Toward a systematic molecular orbital theory for excited states. J. Phys. Chem. 1992, 96, 135−149. (26) Head-Gordon, M.; Rico, R. J.; Oumi, M.; Lee, T. J. A doubles correction to electronic excited states from configuration interaction in the space of single substitutions. Chem. Phys. Lett. 1994, 219, 21−29. (27) Runge, E.; Gross, E. K. U. Density-functional theory for Timedependent systems. Phys. Rev. Lett. 1984, 52, 997. (28) Gross, E. K. U.; Kohn, W. Local density-functional theory of frequency-dependent linear response. Phys. Rev. Lett. 1985, 55, 2850. (29) Gross, E. K. U.; Kohn, W. Time-dependent density-functional theory. Adv. Quantum Chem. 1990, 21, 255. (30) Casida, M. E. In Recent Advances in Density Functional Methods; Chong, D. P., Ed.; World Scientific: Singapore, 1995; p 155. (31) Kaduk, B.; Kowalczyk, T.; Van Voorhis, T. Constrained density functional theory. Chem. Rev. 2012, 112, 321. (32) Hedin, L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys. Rev. 1965, 139, A796. (33) Strinati, G.; Mattausch, H.; Hanke, W. Dynamical correlation effects on the quasiparticle Bloch states of a covalent crystal. Phys. Rev. Lett. 1980, 45, 290. (34) González, L.; Escudero, D.; Serrano-Andrés, L. Progress and challenges in the calculation of electronic excited states. ChemPhysChem 2012, 13, 28. (35) Serrano-Andrés, L.; Serrano-Pérez, J. J. Handbook of Computational Chemistry; Springer, 2012; pp 485−557. (36) Dreuw, A.; Head-Gordon, M. Single-reference ab initio methods for the calculation of excited states of large molecules. Chem. Rev. 2005, 105, 4009−4037. (37) Tozer, D. J.; Handy, N. C. On the determination of excitation energies using density functional theory. Phys. Chem. Chem. Phys. 2000, 2, 2117−2121. (38) Maitra, N. T.; Zhang, F.; Cave, R. J.; Burke, K. Double excitations within time-dependent density functional theory linear response. J. Chem. Phys. 2004, 120, 5932. (39) Cave, R. J.; Zhang, F.; Maitra, N. T.; Burke, K. A dressed TDDFT treatment of the 21Ag states of butadiene and hexatriene. Chem. Phys. Lett. 2004, 389, 39. (40) Levine, B. G.; Ko, C.; Quenneville, J.; Martinez, T. J. Conical intersections and double excitations in time-dependent density functional theory. Mol. Phys. 2006, 104, 1039−1051.

Charge-transfer states are also modeled well by single determinants, and the correct 1/R behavior is predicted even for functionals whose potentials are asymptotically incorrect. Finally, conical intersections, which are particularly challenging for both CIS and TD-DFT, are satisfactorily treated by IMOM-based DFT.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Giuseppe M. J. Barca: 0000-0001-5109-4279 Peter M. W. Gill: 0000-0003-1042-6331 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS P.M.W.G. thanks Prof. Nimrod Moiseyev for an interesting discussion about resonances, the National Computational Infrastructure (NCI) for supercomputer time, and the Australian Research Council for funding (Grants DP140104071 and DP160100246).

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REFERENCES

(1) Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. 1964, 136, B864. (2) Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133. (3) Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular ElectronicStructure Theory; John Wiley and Sons: New York, 2000. (4) Buenker, R. J.; Peyerimhoff, S. D. CI method for the study of general molecular potentials. Theor. Chim. Acta 1968, 12, 183. (5) Buenker, R. J.; Peyerimhoff, S. D. Critically heterosymmetric biradicaloid geometries of of protonated Schiff bases. Theor. Chim. Acta 1974, 35, 33. (6) Potts, D. M.; Taylor, C. M.; Chaudhuri, R. K.; Freed, K. F. The improved virtual orbital-complete active space configuration interaction method, a packageable efficient ab initio many-body method for describing electronically excited states. J. Chem. Phys. 2001, 114, 2592−2600. (7) Abrams, M. L.; Sherrill, C. D. Natural orbitals as substitutes for optimized orbitals in complete active space wavefunctions. Chem. Phys. Lett. 2004, 395, 227−232. (8) Slavíc̆ek, P.; Martínez, T. J. Ab initio floating occupation molecular orbital-complete active space configuration interaction: An efficient approximation to CASSCF. J. Chem. Phys. 2010, 132, 234102. (9) Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M. A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48, 157. (10) Roos, B. O. The complete active space SCF method in a fockmatrix-based super-CI formulation. Int. J. Quantum Chem. 1980, 18, 175. (11) Malmqvist, P. A.; Rendell, A.; Roos, B. O. The restricted active space self-consistent-field method, implemented with a split graph unitary group approach. J. Phys. Chem. 1990, 94, 5477−5482. (12) Andersson, K.; Malmqvist, P. A.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. Second-order perturbation theory with a CASSCF reference function. J. Phys. Chem. 1990, 94, 5483. (13) Andersson, K.; Malmqvist, P. A.; Roos, B. O. Second-order perturbation theory with a complete active space self-consistent field reference function. J. Chem. Phys. 1992, 96, 1218−1226. (14) Roos, B. O.; Andersson, K.; Fülscher, M. P.; Malmqvist, P. A.; Serrano-Andrés, L.; Pierloot, K.; Merchán, M. Multiconfigurational perturbation theory: applications in electronic spectroscopy. Adv. Chem. Phys. 2007, 93, 219. G


Article

Journal of Chemical Theory and Computation (41) Tozer, D. J.; Handy, N. C. Improving virtual Kohn-Sham orbitals and eigenvalues: application to excitation energies and static polarizabilities. J. Chem. Phys. 1998, 109, 10180. (42) Casida, M. E.; Jamorski, C.; Casida, K. C.; Salahub, D. R. Molecular excitation energies to high-lying bound states from timedependent density-functional response theory: characterization and correction of the time-dependent local density approximation ionization threshold. J. Chem. Phys. 1998, 108, 4439. (43) Casida, M. E.; Salahub, D. R. Asymptotic correction approach to improving approximate exchange-correlation potentials: Time-dependent density functional theory calculations of molecular excitation spectra. J. Chem. Phys. 2000, 113, 8918−8935. (44) Tozer, D. J.; Handy, N. C. The importance of the asymptotic exchange-correlation potential in density functional theory. Mol. Phys. 2003, 101, 2669−2675. (45) Tozer, D. J.; Amos, R. D.; Handy, N. C.; Roos, B. O.; SerranoAndrés, L. Does density functional theory contribute to the understanding of excited states of unsaturated organic compounds? Mol. Phys. 1999, 97, 859. (46) Sobolewski, A.; Domcke, W. Ab initio study of the excited-state coupled electron-proton-transfer process in the 2-aminopyridine dimer. Chem. Phys. 2003, 294, 73. (47) Dreuw, A.; Weisman, J. L.; Head-Gordon, M. Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange. J. Chem. Phys. 2003, 119, 2943. (48) Dreuw, A.; Head-Gordon, M. Failure of time-dependent density functional theory for long-range charge-transfer excited states:? the zincbacteriochlorin-bacteriochlorin and bacteriochlorophyll-spheroidene complexes. J. Am. Chem. Soc. 2004, 126, 4007. (49) Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. A long-range-corrected time-dependent density functional theory. J. Chem. Phys. 2004, 120, 8425−8433. (50) Yanai, T.; Tew, D. P.; Handy, N. C. A new hybrid exchangecorrelation functional using the Coulomb-attenuating method (CAMB3LYP). Chem. Phys. Lett. 2004, 393, 51−57. (51) Gilbert, A. T. B.; Besley, N. A.; Gill, P. M. W. Self-consistent field calculations of excited states using the Maximum Overlap Method (MOM). J. Phys. Chem. A 2008, 112, 13164. (52) Besley, N. A.; Gilbert, A. T. B.; Gill, P. M. W. Calculation of core excited states using the Maximum Overlap Method (MOM). J. Chem. Phys. 2009, 130, 124308. (53) Barca, G. M. J.; Gilbert, A. T. B.; Gill, P. M. W. Communication: Hartree-Fock description of excited states of H2. J. Chem. Phys. 2014, 141, 111104. (54) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; Ghosh, D.; Goldey, M.; Horn, P. R.; Jacobson, L. D.; Kaliman, I.; Khaliullin, R. Z.; Kús, T.; Landau, A.; Liu, J.; Proynov, E. I.; Rhee, Y. M.; Richard, R. M.; Rohrdanz, M. A.; Steele, R. P.; Sundstrom, E. J.; Woodcock, H. L., III; Zimmerman, P. M.; Zuev, D.; Albrecht, B.; Alguire, E.; Austin, B.; Beran, G. J. O.; Bernard, Y. A.; Berquist, E.; Brandhorst, K.; Bravaya, K. B.; Brown, S. T.; Casanova, D.; Chang, C.-M.; Chen, Y.; Chien, S. H.; Closser, K. D.; Crittenden, D. L.; Diedenhofen, M.; DiStasio, R. A., Jr.; Do, H.; Dutoi, A. D.; Edgar, R. G.; Fatehi, S.; Fusti-Molnar, L.; Ghysels, A.; Golubeva-Zadorozhnaya, A.; Gomes, J.; Hanson-Heine, M. W. D.; Harbach, P. H. P.; Hauser, A. W.; Hohenstein, E. G.; Holden, Z. C.; Jagau, T.-C.; Ji, H.; Kaduk, B.; Khistyaev, K.; Kim, J.; Kim, J.; King, R. A.; Klunzinger, P.; Kosenkov, D.; Kowalczyk, T.; Krauter, C. M.; Lao, K. U.; Laurent, A.; Lawler, K. V.; Levchenko, S. V.; Lin, C. Y.; Liu, F.; Livshits, E.; Lochan, R. C.; Luenser, A.; Manohar, P.; Manzer, S. F.; Mao, S.-P.; Mardirossian, N.; Marenich, A. V.; Maurer, S. A.; Mayhall, N. J.; Oana, C. M.; Olivares-Amaya, R.; O’Neill, D. P.; Parkhill, J. A.; Perrine, T. M.; Peverati, R.; Pieniazek, P. A.; Prociuk, A.; Rehn, D. R.; Rosta, E.; Russ, N. J.; Sergueev, N.; Sharada, S. M.; Sharma, S.; Small, D. W.; Sodt, A.; Stein, T.; Stück, D.; Su, Y.-C.; Thom, A. J. W.; Tsuchimochi, T.; Vogt, L.; Vydrov, O.; Wang, T.; Watson, M. A.; Wenzel, J.; White, A.; Williams, C. F.; Vanovschi, V.; Yeganeh, S.; Yost, S. R.; You, Z.-Q.; Zhang, I. Y.; Zhang, X.; Zhao, Y.; Brooks, B. R.; Chan, G. K. L.; Chipman, D. M.; Cramer,

C. J.; Goddard, W. A., III; Gordon, M. S.; Hehre, W. J.; Klamt, A.; Schaefer, H. F., III; Schmidt, M. W.; Sherrill, C. D.; Truhlar, D. G.; Warshel, A.; Xu, X.; Aspuru-Guzik, A.; Baer, R.; Bell, A. T.; Besley, N. A.; Chai, J.-D.; Dreuw, A.; Dunietz, B. D.; Furlani, T. R.; Gwaltney, S. R.; Hsu, C.-P.; Jung, Y.; Kong, J.; Lambrecht, D. S.; Liang, W.; Ochsenfeld, C.; Rassolov, V. A.; Slipchenko, L. V.; Subotnik, J. E.; Van Voorhis, T.; Herbert, J. M.; Krylov, A. I.; Gill, P. M. W.; Head-Gordon, M.; Neuscamman, E.; Yang, J. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys. 2015, 113, 184−215. (55) Pople, J. A.; Nesbet, R. K. Self-consistent orbitals for radicals. J. Chem. Phys. 1954, 22, 571. (56) Pople, J. A.; Gill, P. M. W.; Handy, N. C. Spin-unrestricted character of Kohn-Sham orbitals for open-shell systems. Int. J. Quantum Chem. 1995, 56, 303−305. (57) Barca, G. M. J.; Gilbert, A. T. B.; Gill, P. M. W. Excitation number: characterizing multiply excited states. J. Chem. Theory Comput. 2018, 14, 9−13. (58) James, H. M.; Coolidge, A. S. The ground state of the hydrogen molecule. J. Chem. Phys. 1933, 1, 825−835. (59) Bottcher, C.; Docken, K. Autoionizing states of the hydrogen molecule. J. Phys. B: At. Mol. Phys. 1974, 7, L5. (60) Moiseyev, N.; Corcoran, C. Autoionizing states of H2 and H2− using the complex-scaling method. Phys. Rev. A: At., Mol., Opt. Phys. 1979, 20, 814−817. (61) Collins, L. A.; Schneider, B. I. Linear algebraic approach to electronic excitation of atoms and molecules by electron impact. Phys. Rev. A: At., Mol., Opt. Phys. 1983, 27, 101. (62) Hazi, A. U.; Derkits, C.; Bardsley, J. N. Theoretical study of the lowest 1Σg+ doubly excited state of H2. Phys. Rev. A: At., Mol., Opt. Phys. 1983, 27, 1751. (63) Guberman, S. L. The doubly excited autoionizing states of H2. J. Chem. Phys. 1983, 78, 1404. (64) Mielke, S. L.; Schwenke, D. W.; Peterson, K. A. Benchmark calculations of the complete configuration-interaction limit of BornOppenheimer diagonal corrections to the saddle points of isotopomers of the H+H2 reaction. J. Chem. Phys. 2005, 122, 224313. (65) Johnson, B. G.; Gonzales, C. A.; Gill, P. M. W.; Pople, J. A. A density functional study of the simplest hydrogen abstraction reaction. Effect of self-interaction correction. Chem. Phys. Lett. 1994, 221, 100− 108. (66) Hashimoto, T.; Nakano, N.; Hirao, K. Theoretical study of the valence π→π* excited states of polyacenes: benzene and naphthalene. J. Chem. Phys. 1996, 104, 6244. (67) Nakashima, N.; Inoue, H.; Sumitani, M.; Yoshihara, K. Laser flash photolysis of benzene. III. Sn←S1 absorption of gaseous benzene. J. Chem. Phys. 1980, 73, 5976. (68) Dick, B.; Hohlneicher, G. Two-photon spectroscopy of the lowlying singlet states of naphthalene and acenaphthene. Chem. Phys. Lett. 1981, 84, 471. (69) Rubio, M.; Merchán, M.; Ortí, E.; Roos, B. O. A theoretical study of the electronic spectrum of naphthalene. Chem. Phys. 1994, 179, 395−409. (70) Dick, B.; Hohlneicher, G. Two-photon spectroscopy of dipoleforbidden transitions: the low-lying singlet states of anthracene. Chem. Phys. Lett. 1981, 83, 615. (71) Kawashima, Y.; Hashimoto, T.; Nakano, H.; Hirao, K. Theoretical study of the valence π→π* excited states of polyacenes: anthracene and naphthacene. Theor. Chem. Acc. 1999, 102, 49−64. (72) Kolc, J.; Downing, J. W.; Manzara, A. P.; Michl, J. π,πbiradicaloid hydrocarbons. The pleiadene family. II. A doubly excited state of pleiadene. J. Am. Chem. Soc. 1976, 98, 930−937. (73) Pariser, R.; Parr, R. G. A semi-empirical theory of the electronic spectra and electronic structure of complex unsaturated molecules. I. J. Chem. Phys. 1953, 21, 466. (74) Pariser, R.; Parr, R. G. A semi-empirical theory of the electronic spectra and electronic structure of complex unsaturated molecules. II. J. Chem. Phys. 1953, 21, 767. H


Article

Journal of Chemical Theory and Computation (75) Pople, J. A. Electron interaction in unsatured hydrocarbons. Trans. Faraday Soc. 1953, 49, 1375. (76) Buenker, R. J.; Whitten, J. L.; Petke, J. D. Ab initio SCF MO and CI calculations on the electronic spectrum of benzene. J. Chem. Phys. 1968, 49, 2261. (77) Peyerimhoff, S. D.; Buenker, R. J. Comparison of the molecular structure and spectra of benzene and borazine. Theor. Chim. Acta 1970, 19, 1. (78) Hay, P. J.; Shavitt, I. Ab initio configuration interaction studies of the π-electron states of benzene. J. Chem. Phys. 1974, 60, 2865. (79) Palmer, M. H.; Walker, I. C. The electronic states of benzene and the azines. I. The parent compound benzene. Correlation of vacuum UV and electron scattering data with ab initio CI studies. Chem. Phys. 1989, 133, 113. (80) Kitao, O.; Nakatsuji, H. Cluster expansion of the wave function. Valence and Rydberg excitations and ionizations of benzene. J. Chem. Phys. 1987, 87, 1169. (81) Lorentzon, J.; Malmqvist, P. A.; Fülscher, M.; Roos, B. O. A CASPT2 study of the valence and lowest Rydberg electronic states of benzene and phenol. Theor. Chim. Acta 1995, 91, 91−108. (82) Bendikov, M.; Duong, H. M.; Starkey, K.; Houk, K. N.; Carter, E. A.; Wudl, F. Oligoacenes: Theoretical Prediction of Open-Shell Singlet Diradical Ground States. J. Am. Chem. Soc. 2004, 126, 7416− 7417 PMID: 15198569. (83) Ghosh, S.; Cramer, C. J.; Truhlar, D. G.; Gagliardi, L. Generalized-active-space pair-density functional theory: an efficient method to study large, strongly correlated, conjugated systems. Chem. Sci. 2017, 8, 2741−2750. (84) Kolc, J.; Michl, J. π,π-biradicaloid hydrocarbons. Pleiadene family. I. Photochemical preparation from cyclobutene precursors. J. Am. Chem. Soc. 1973, 95, 7391−7401. (85) Zhao, Y.; Truhlar, D. G. Exploring the limit of accuracy of the global hybrid meta density functional for main-group: thermochemistry, kinetics, and noncovalent interactions. J. Chem. Theory Comput. 2008, 4, 1849−1868. (86) Saga, Y.; Shibata, Y.; Tamiaki, H. Spectral properties of single light-harvesting complexes in bacterial photosynthesis. J. Photochem. Photobiol., C 2010, 11, 15−24. (87) Kataoka, Y.; Shibata, Y.; Tamiaki, H. Supramolecular energy transfer from photoexcited chlorosomal zinc porphyrin self-aggregates to a chlorin or bacteriochlorin monomer as models of main lightharvesting antenna systems in green photosynthetic bacteria. Bioorg. Med. Chem. Lett. 2012, 22, 5218−5221. (88) Miyatake, T.; Tamiaki, H.; Holzwarth, A. R.; Schaffner, K. Artificial light-harvesting antennae: singlet excitation energy transfer from zinc chlorin aggregate to bacteriochlorin in homogeneous hexane solution. Photochem. Photobiol. 1999, 69, 448−456. (89) Stromberg, J. R.; Marton, A.; Kee, H. K.; Kirmaier, C.; Diers, J. R.; Muthiah, C.; Taniguchi, M.; Lindsey, J. S.; Bocian, D. F.; Meyer, G. J.; Holten, D. Examination of tethered porphyrin, chlorin, and bacteriochlorin molecules in mesoporous metal-oxide solar cells. J. Phys. Chem. C 2007, 111, 15464−15478. (90) Reddy, K. R.; Jiang, J.; Krayer, M.; Harris, M. A.; Springer, J. W.; Yang, E.; Jiao, J.; Niedzwiedzki, D. M.; Pandithavidana, D.; ParkesLoach, P. S.; Kirmaier, C.; Loach, P. A.; Bocian, D. F.; Holten, D.; Lindsey, J. S. Palette of lipophilic bioconjugatable bacteriochlorins for construction of biohybrid light-harvesting architectures. Chem. Sci. 2013, 4, 2036. (91) Joshi, P.; Saenz, C.; Missert, J. R.; Pandey, R. K. In Handbook of Photomedicine; Hamblin, M. R., Huang, Y.-Y., Eds.; Taylor and Francis: Boca Raton, FL, 2014; pp 227−259. (92) Yarkony, D. R. Diabolical conical intersections. Rev. Mod. Phys. 1996, 68, 985. (93) Domcke, W.; Yarkony, D. R. Role of conical intersections in molecular spectroscopy and photoinduced chemical dynamics. Annu. Rev. Phys. Chem. 2012, 63, 325−352. (94) Halász, G.; Vibók, A.; Mebel, A. M.; Baer, M. A survey of ab initio conical intersections for the H+H2 system. J. Chem. Phys. 2003, 118, 3052.

(95) Mielke, S. L.; Garrett, B. C.; Peterson, K. A. A hierarchical family of global analytic Born-Oppenheimer potential energy surfaces for the H+H2 reaction ranging in quality from double-zeta to the complete basis set limit. J. Chem. Phys. 2002, 116, 4142. (96) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007. (97) Yoshizawa, T.; Wald, G. Pre-lumirhodopsin and the bleaching of visual pigments. Nature 1963, 197, 1279−1286. (98) Du, P.; Davidson, E. R. Ab initio study on the excitation energies of the protonated schiff base of 11-cis-retinal. J. Phys. Chem. 1990, 94, 7013−7020. (99) Bonačić-Koutecký, V.; Köhler, J.; Michl, J. Prediction of structural and environmental effects on the S1-S0 energy gap and jump probability in double-bond cis-trans photoisomerization. Chem. Phys. Lett. 1984, 104, 440−443. (100) González-Luque, R.; Garavelli, M.; Bernardi, F.; Merchán, M.; Robb, M. A.; Olivucci, M. Computational evidence in favor of a twostate, two-mode model of the retinal chromophore photoisomerization. Proc. Natl. Acad. Sci. U. S. A. 2000, 97, 9379−9384. (101) Molnar, F.; Ben-Nun, M.; Martinez, T. J.; Schulten, K. Characterization of a conical intersection between the ground and first excited state for a retinal analog. J. Mol. Struct.: THEOCHEM 2000, 506, 169−178. (102) Coto, P. B.; Sinicropi, A.; De Vico, L.; Ferré, N.; Olivucci, M. Characterization of the conical intersection of the visual pigment rhodopsin at the CASPT2//CASSCF/AMBER level of theory. Mol. Phys. 2006, 104, 983−991. (103) Levine, B. G.; Martinez, T. J. Isomerization through conical intersections. Annu. Rev. Phys. Chem. 2007, 58, 613−634. (104) Polli, D.; Altoé, P.; Weingart, O.; Spillane, K. M.; Manzoni, C.; Brida, D.; Tomasello, G.; Orlandi, G.; Kukura, P.; Mathies, R. A.; Garavelli, M.; Cerullo, G. Conical intersection dynamics of the primary photoisomerization event in vision. Nature 2010, 467, 440−443. (105) Andruniów, T.; Ferré, N.; Olivucci, M. Structure, initial excited-state relaxation, and energy storage of rhodopsin resolved at the multiconfigurational perturbation theory level. Proc. Natl. Acad. Sci. U. S. A. 2004, 101, 17908.

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Simple models for difficult electronic excitations

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