Simple Models for Difficult Electronic ExcitationsGiuseppe 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 challengingtopics in the chemistry of excited states: double excitations, charge-transfer states,and conical intersections. The results are obtained by using the Initial MaximumOverlap Method (IMOM) which is a modified version of the Maximum OverlapMethod (MOM). The new algorithm converges better than the original,especially for these difficult problems. By considering several case studies, weshow that a single-determinant framework provides a simple and accuratealternative for modeling excited states in cases where other low-cost methods,such as CIS and TD-DFT, either perform poorly or fail completely.

1. INTRODUCTION

Understanding electronically excited states is important inmany fields such as photovoltaics, optics, synthetic chemistry,and biology. Quantum chemistry has played a key role inimproving this understanding, but while ground states areusually studied using a small set of well-established methodssuch as Density Functional Theory (DFT),1,2 Møller−PlessetPerturbation Theory, and Coupled-Cluster Theory, the arsenalof approaches to excited states is very large, including Multi-Reference Configuration Interaction (MRCI),3−5 CompleteActive Space Configuration Interaction (CASCI),6−8 CompleteActive Space Self-Consistent Field (CASSCF),9,10 RestrictedActive Space Self-Consistent Field (RASSCF),11 CASPT2,12−14

Multi-Reference Møller−Plesset Perturbation theory(MRMP),15 Symmetry-Adapted Cluster-Configuration Inter-action (SAC−CI),16−18 Equation-Of-Motion Coupled Cluster(EOM-CC),19−21 Linear Response Coupled Cluster (LR-CC),22,23 Configuration Interaction Singles (CIS),24−26 CISwith perturbative treatment of doubles (CIS(D)),25 Time-Dependent DFT (TD-DFT),27−30 Constrained DFT,31 many-body Green’s functions methods (GW),32,33 and others. Formany researchers, the choice is bewildering.The abundance of excited-state methods results partly from

the belief that whereas ground states are often described well bysingle-determinant methods excited states are usually multi-reference in character, especially if the state is doubly excited orin the vicinity of a conical intersection.34,35 However,specification of a multireference wave function is more difficultand often requires a delicate combination of chemical intuition,experience, trial, and error that incurs a substantially highercomputational cost.The single-reference CIS and TD-DFT methods address

some of these concerns but, notwithstanding their success, havesome important weaknesses. Both CIS and TD-DFT (withinthe adiabatic local density approximation36) are incapable ofdescribing doubly excited states37−40 and often fail near conicalintersections between ground and excited states.40 Moreover,

CIS can overestimate excitation energies by as much as 2 eV,36

and TD-DFT performs poorly for Rydberg41−44 and charge-transfer states45−48 unless a long-range corrected functional isused.49,50

These weaknesses arise because neither CIS nor TD-DFTallows the molecular orbitals (MOs) to relax in the excitedstate, and consequently, they struggle in cases where thatrelaxation would be significant. However, several years ago weshowed that relaxed excited single-determinant wave functionscan 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-determinantapproximation can handle challenging cases where other low-cost excited-state methods fail. Section 2 presents a modifiedMOM algorithm that converges more reliably to the desiredexcited state. Then, in Sections 3−5, we present resultsobtained by applying the new algorithm to problems involvingdouble excitations, charge-transfer states, and conical inter-sections.All of the excitation energies reported here are vertical. That

is, the energy of the excited state is calculated at the structure ofthe ground state, and no attempt has been made to correct forzero-point vibration energy. If we use our modified MOM tofind an excited-state solution of the self-consistent field (SCF)equations, we refer to the resulting energy using the unadornedname of the functional (e.g., BLYP). On the other hand, if weuse the conventional time-dependent approach to estimate theexcitation energy from the ground state, we refer to the energyby prefixing “TD” to the name of the functional (e.g., TD-BLYP). As we show, these two approaches often yield strikinglydifferent models of excited-state energetics.

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

Article

pubs.acs.org/JCTCCite This: J. Chem. Theory Comput. 2018, 14, 1501−1509

© 2018 American Chemical Society 1501 DOI: 10.1021/acs.jctc.7b00994J. Chem. Theory Comput. 2018, 14, 1501−1509

2. METHODThe MOM provides an alternative to the aufbau principle fordetermining which MOs to occupy on each cycle of an SCFcalculation. Rather than choose the lowest energy MOs, wechoose those with the largest projection into the span of theoccupied MOs of the previous SCF cycle. Given the MOoverlap matrix

= †O (C ) SCprevious current(1)

and projections = ∑( )p Oj i ij2 1/2

, we simply choose to occupy

the MOs with the largest pj values; all other aspects of the SCFare unchanged.The aufbau criterion drives the SCF toward the lowest-

energy solution of the SCF equations. If it is replaced by theMOM criterion, and if the SCF algorithm finds stationarypoints and not just energy minima (e.g., Direct Inversion ofIterative Subspace (DIIS)), the SCF can discover higher-energysolutions of those equations, and we have demonstrated thatthese correspond to the excited states of the system.51−53 Thisapproach is not limited to the lowest-energy solutions of eachsymmetry type.Notwithstanding the success of the MOM, cases sometimes

arise wherein the SCF converges to an undesired solution, andthis is especially common in systems with near-degeneracies.Such behavior led us to develop a modified MOM protocol inwhich we choose to occupy the MOs with the largest projectioninto the space spanned by the occupied MOs of the initial guess.Thus, the MO overlap matrix becomes

= †O (C ) SCinitial current(2)

This strategythe Initial Maximum Overlap Method(IMOM)encourages the SCF to find a solution of the SCFequations in the neighborhood of the initial guess. Weimplemented it in the Q-CHEM 4.4.1 package54 and used it togenerate 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 ofattraction of the target state. Often, it is sufficient to perform aground-state calculation and simply promote an electron froman occupied to a virtual orbital. The orbitals are then allowed torelax in the ensuing SCF calculation. However, even with agood guess, it is possible for the MOs in a MOM calculationgradually to drift away from the initial guess. The IMOMprevents this by anchoring the SCF to that guess, and thissimple algorithmic modification significantly enhances itsusefulness over the MOM.Figure 1 illustrates this point by comparing the convergence

of the two approaches for calculating the 22P excited state ofthe boron atom. The initial guess, which is identical for bothcalculations, was obtained from the ground-state MOs bypromoting an electron from the occupied 2px orbital to theunoccupied 3py orbital. The IMOM calculation converges tothe desired 1s22s23p1 configuration, whereas the MOMcalculation collapses onto the ground state.Table 1 contains overlap values between relevant py orbitals

which help explain the differences observed in Figure 1. Initiallythe occupied SCF orbital has an overlap with the guess orbitalof 0.648, but as the MOM SCF proceeds, the occupied orbitaldrifts away from the guess causing the overlap values todecrease monotonically (column 3). After cycle 3, there exists apreviously unoccupied orbital that now has greater overlap with

the guess than any of the occupied orbitals. The IMOM selectsthis 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 inenergy between cycles 3 and 4 in Figure 1. The final overlapvalues in columns 3 and 6 show that the IMOM orbital has alarger overlap (0.526) with the guess orbital than the MOMorbital (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 beobtained using the MOM. This is clear from the fact that theprotocols become equivalent when applied to a convergedsolution of the SCF equations. The key advantage of the newapproach is that it is much easier to generate initial guesses thatlead to the desired target states.All reported calculations use the spin-unrestricted formal-

ism.55,56

3. DOUBLE EXCITATIONSThe term “doubly excited” is commonly used to describe stateswhose configuration interaction (CI) expansions include

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

Table 1. Orbital Overlap Values between Various py Orbitalson Different SCF Cycles of HF/aug-cc-pVTZ Calculationson the Boron Atoma

MOM IMOM

Cycle ′o o 2 o i 2 v i 2 ′o o 2 o i 2 v i 2

1 0.648 0.648 0.320 0.648 0.648 0.3202 0.982 0.518 0.445 0.982 0.518 0.4453 0.988 0.412 0.545 0.011 0.545 0.4124 0.999 0.411 0.544 0.999 0.537 0.4115 1.000 0.410 0.545 0.999 0.530 0.4116 1.000 0.410 0.545 0.995 0.523 0.3917 0.998 0.549 0.3638 0.994 0.483 0.4289 0.997 0.528 0.38210 1.000 0.526 0.38411 1.000 0.526 0.384

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

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doubly substituted configurations with large amplitudes.However, being dependent on the chosen reference config-uration, this definition is ambiguous.If the ground-state wave function Ψ and excited-state wave

function Ψ for an n-electron system are single determinants ofthe MOs ψi and ψj, respectively, the excitation number57

∑ ∑η ψ ψ= − |⟨ | ⟩|ni

n

j

n

i j2

(3)

measures the number of electrons in the excited state whichoccupy the space spanned by the virtual orbitals of the groundstate. Thus, for example, doubly substituted versions of Ψwould have η = 2. Post-excitation orbital relaxation leads tosmall deviations from this ideal value, but η allows us easily todecide whether or not a state is doubly excited.3.1. The H2 Molecule. The 1σg

2 ground state of the H2molecule has been a benchmark for quantum chemical methodssince the dawn of quantum mechanics.58 Accurate energies ofthe lowest doubly excited 1σu

2 state, an autoionizing resonance,were first obtained by Bottcher and Docken59 and later byothers.60−63

We performed SCF calculations on both states with R = 1.4bohr using a modified aug-mcc-pV8Z basis64 to whichadditional diffuse s, p, and d shells were added and fromwhich 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 thecorrelation energies of the states are comparable.

Because the self-interaction errors65

= +E E E12SIE J X (4)

(where EJ and EX are Coulomb and exchange energies) aremuch larger in the excited state than in the ground state, theDFT results improve as the percentage of Fock exchangeincreases. This highlights the need for exchange functionals thatare 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 ispredicted 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. Whilesemi-empirical methods, such as the Pariser−Parr−Pople(PPP) method,73−75 can give good results, ab initio methodsoften struggle to obtain comparable performance.Early CI studies gave excitation energy errors exceeding 1 eV

for some valence states of benzene.76−78 Multireferenceconfiguration interaction (MRCI)79 and symmetry-adapted

cluster configuration interaction (SAC−CI)80 reduce the errorsto 0.5 eV, but only multireference perturbative treatments (e.g.,MRMP and CASPT2) were able to achieve errors of 0.1−0.29eV for benzene,66,81 0.27−0.54 eV for naphthalene,66 0.15 eVfor anthracene, and 0.25 eV for naphthacene.71,82,83

For consistency with the other molecules, we classify thestates of benzene using D2h symmetry, and its first singletexcited 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 singleexcitation is much lower in energy than the double, but theordering reverses in anthracene66,71 and larger molecules.84 Todiscover whether standard DFT methods reproduce thisreordering, we used the IMOM to compute the BLYP/6-311G* energies of these states in benzene, naphthalene,anthracene, and pleiadene (Figure 2) at ground-state BLYP/

6-311G structures. Initial guesses for the singly and doublyexcited states were obtained from the ground-state MOs bypromoting, respectively, one or two electrons from an occupiedπ 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 3reveals the energy reordering as the system size increases. In allcases, the BLYP calculations give the correct ordering andpleasingly accurate excitation energies, with a mean absolutedeviation from the experimental values of only 0.15 eV. Theworst result, an error of 0.41 eV for the singly excited 3 1Agstate of anthracene, can be compared with CASSCF andMRMP 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 ofdoubly substituted configurations that appear in the CASSCF81

and MRMP wave functions.66 However, this state can beaccurately 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 CASSCFwave function serve largely to describe correlation and

Table 2. Total Energies (E, in hartree), Excitation Energies(ΔE, in eV), Self-Interaction Energies (ESIE, in eV), and ηValues for the 1σg

2 → 1σu2 Excitation in H2

BLYP B3LYP HF-LYP HF Full CI

%Fock 0 20 100 100 100E(1σu

2) −0.16250 −0.16016 −0.09950 −0.08076 −0.11755E(1σg

2) −1.17031 −1.18071 −1.17198 −1.13363 −1.17428ΔE 27.42 27.77 29.18 28.65 28.75ESIE(1σu

2) −1.781 −1.219 0 0 0ESIE(1σg

2) 0.007 0.217 0 0 0η 2 2 2 2

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

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relaxation and should not be interpreted as indicating that thestate is doubly excited.

4. CHARGE-TRANSFER STATESA typical electronic excitation creates an electron−hole pair asthe electron moves from one MO to another. If the electronand hole are separated by a significant distance R, the result istermed a charge-transfer (CT) state. Because of their charges,the electron and hole attract, causing the energy of the systemto 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 failto capture the 1/R behavior, not because of flaws in TD-DFTitself but because of the adiabatic approximation universallyadopted in its implementations.47,48 Because excited-state DFTenergies obtained using the IMOM do not make thisapproximation, one anticipates that they will avoid theundesirable features of TD-DFT. To test this, we havecompared the performance of DFT with CIS and TD-DFTfor the lowest CT states of two supramolecular systems:ethylene + tetrafluoroethylene and bacteriochlorin + zinc-bacteriochlorin.4.1. Ethylene + Tetrafluoroethylene. CT states of the

C2H4 + C2F4 complex (Figure 4) have been studied previouslyby Dreuw et al. using TD-DFT and CIS.47 To capture thecorrect 1/R behavior, they proposed a hybrid approach whichcombines TD-DFT and CIS and which yielded reasonableestimates for the CT excitation energies. However, because ofits 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 setwas used for all calculations. As anticipated, TD-B3LYP isqualitatively wrong, while EOM-CCSD, CIS, B3LYP, and M08-HX all reproduce the correct 1/R behavior. The failure of TD-B3LYP can be traced to an incompletely modeled interactionbetween 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 theEOM-CCSD reference (Table 4). The B3LYP error at thispoint is also large (−0.48 eV), but the M08-HX functional85

reduces this to 0.12 eV.4.2. Bacteriochlorin + Zn-Bacteriochlorin. Bacterio-

chlorins (7,8,17,18-tetrahydroporphyrins) are the chromophor-ic moiety of bacteriochlorophylls (BChl) which are found inpurple bacteria, green bacteria, and heliobacteria.86 Theirphotochemistry has aroused broad scientific interest from thedevelopment of artificial light-harvesting antennae for photo-

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 1Ag 1.0058 7.70 8.0166 7.7366 7.8067

5 1Ag 2.0034 10.21 Naphthalene 2 1Ag 1.0191 5.66 5.8666 5.6566 5.5268

4 1Ag 2.0019 6.77 6.7569 6.7669 Anthracene 2 1Ag 2.0013 4.62 5.4266 5.0366 4.7170

3 1Ag 1.0141 4.92 6.5771 5.2871 5.3370

Pleiadene 2 1A1 2.0118 2.46 2.4672

3 1A1 1.0818 3.43 3.6172

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

Figure 4. Ethylene + tetrafluoroethylene complex.

Figure 5. Variation of the excitation energy ΔE of the first CT state ofC2H4 + C2F4 with the distance R.

Table 4. Excitation Energies (in eV) of the First CT TransferStates of C2H4 + C2F4 and BC + Zn-BC Complexes

Complex R/ÅM08-HX B3LYP CIS

TD-B3LYP

EOM-CCSD

C2H4 + C2F4 4.60 10.84 10.48 11.61 6.79 10.96BC + ZnBC 11.20 3.28 3.33 3.69 1.82 BC + ZnBC 12.04 3.41 3.49 3.78 1.84

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active devices87−90 to photodynamic therapy for cancerdestruction.91 The key step in the process involves theabsorption of light and a transfer of the singlet excitationenergy via protein−BChl complexes to the photosyntheticreaction center.88 It is obvious, therefore, that the theoreticalstudy of such excitations requires a detailed understanding ofcharge-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 CTstates, and in our study, we follow the approach of Dreuw etal.,48 adopting the model shown in Figure 6. This allows the

distance between the chromophores to be varied from thatdetermined by the bridge, which is 12.04 Å. (Note that ourdefinition of R, which measures the distance between thecenters 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 thefirst CT state were calculated using M08-HX, B3LYP, CIS, andTD-B3LYP with the 6-31G* basis, and these are shown inFigure 7. As we saw for the C2H4 + C2F4 complex, the DFT andCIS methods predict the correct 1/R dependence but TD-B3LYP fails.

At R = 12.04 Å, the CIS excitation energy of 3.78 eV (Table4) is close to the value (3.79 eV) obtained by Dreuw et al. usingtheir hybrid approach.48 Higher levels of theory are prohibitivefor the BC/Zn-BC system, and no experimental results areavailable. However, our results for the C2H4 + C2F4 complexsuggest that CIS probably overestimates the excitation energyand that the M08-HX results are probably the most accurate.

5. CONICAL INTERSECTIONSA conical intersection (ConInt) is a subset of the nuclearcoordinate space where the adiabatic potential energy surfaces

(PESs) of two electronic states of a molecule are degenerate.ConInts frequently play a key role in the reactions, spectros-copy, and dynamics of molecules, especially those ofbiochemical interest.92,93

Due to the degeneracy of the PESs, excited-state calculationsinvolving ConInts are challenging. In particular, linearresponse-based methods, e.g., CIS and TD-DFT, fail whenthe HOMO−LUMO gap is small or zero. We test the ability ofsingle-determinant methods to model PESs in the vicinity ofConInts by considering the H3 and retinal molecules.

5.1. The H3 Molecule. The study of ConInts in H3 has along history, both theoretical and experimental, and weencourage the interested reader to study the survey94 byHalasz et al. for further details. There are four ConIntsinvolving the three lowest-energy electronic states, but we focuson the ConInt with D3h symmetry as it is characterized by asingle interatomic distance R. For most values of R, the doubletground 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 reactionat a highly correlated level with the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets. They found the minimalenergy ConInt (MECI) at R = 0.495 Å, and we use this value to

Figure 6. Bacteriochlorin + zinc-bacteriochlorin complex.

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

Figure 8. Energy difference ΔE between the D0 and D1 states ofequilateral H3 as the interatomic distance R varies. Green trianglesshow the MECI predicted by Mielke et al.95

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

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assess the performance of various DFT, TD-DFT, and CISmethods.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, theaug-cc-pVDZ basis set96 was used. The energy differences areplotted in Figure 8, and on such plots, any ConInts appear ascusps 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 modelthe ConInt, and for the narrow domain of R shown in Figure 8,the CIS and HF energies are very similar. This is nocoincidence for if the CIS state intersected the ground stateat a different value of R the CIS solution would have a lowerenergy than the ground state, leading to a contradiction.None of the TD-DFT models yields a ConInt. Both TD-

BLYP and TD-B3LYP show discontinuities at the ConIntbecause the ground-state reference changes from 2A1′ to 2E′ atthis point. These different references give different excitationenergies for the D1 state, leading to the discontinuity. The TD-HFLYP PES is continuous and has a cusp that coincides withthat obtained using HF-LYP. However, it does not correspondto 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 belowthe Mielke value (R = 0.495 Å), and HF and CIS predict aMECI at 0.016 Å above Mielke’s. Both BLYP and B3LYPpredict a MECI at a bond length that is almost 0.1 Å shorterthan Mielke’s.5.2. Retinal. The photoisomerization of the 11-cis retinal

chromophore to its all-trans form in the rhodopsin protein isthe primary process involved in vision.97 Many attempts havebeen made to explain this process from an electronic structurepoint of view, including pioneering ab initio calculations by Duand Davidson98 in 1990 on the excited states of the protonatedSchiff base of retinal (PSBR). Early theoretical studies ofphotoisomerization of protonated Schiff base cations werereported by Bonacic-Koutecky et al.99 They showed that theisomerization of the formaldiminium cation (CH2NH2

+) occursthrough a ConInt between the S1 and S0 states at an N−C bondtwist-angle of 90°. Since then, many theoretical studies100−103

have provided evidence that the S1−S0 ConInt in retinal isresponsible for the ultrafast photoisomerization of themolecule, and this was later corroborated by experiments byPolli et al.104 Unfortunately, Levine et al.40 report that ConIntsinvolving a closed-shell singlet ground state cannot be found byeither TD-DFT or CIS because “matrix elements connectingthe initial state and the response states are excluded from theformulation”.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 theC15−C16 double bond,101,102,105 and the MECI is expected tolie 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 andB3LYP levels, and the resulting PESs are shown in Figure 10.Both BLYP and B3LYP predict a ConInt near ϕ = 90°, lyingapproximately 2 eV above the equilibrium structure. This isslightly lower than CASSCF-based estimates of around 2.3 eVin the work of Molnar et al.101 and of Andruniow et al.105

6. CONCLUDING REMARKS

We have examined single-determinant approximations forexcited states involving double excitations, charge-transfer,and conical intersections. These determinants correspond tohigher-energy solutions of the SCF equations and are widelybelieved to be difficult to obtain. However, the new IMOMprotocol provides a straightforward and reliable method forobtaining these solutions, and we have shown that they may bepreferable 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 thefew low-cost options available. Moreover, we find that thesingle-determinant energies obtained in this way are remarkablyaccurate and can rival far more expensive methods such asCASSCF and MRMP. It is especially pleasing to discover howaccurately the 1σu

2 resonance state in H2 is modeled by HFtheory.

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

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Charge-transfer states are also modeled well by singledeterminants, and the correct 1/R behavior is predicted evenfor functionals whose potentials are asymptotically incorrect.Finally, conical intersections, which are particularly challeng-

ing for both CIS and TD-DFT, are satisfactorily treated byIMOM-based DFT.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: peter.gill@anu.edu.au.

ORCIDGiuseppe M. J. Barca: 0000-0001-5109-4279Peter M. W. Gill: 0000-0003-1042-6331NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSP.M.W.G. thanks Prof. Nimrod Moiseyev for an interestingdiscussion about resonances, the National ComputationalInfrastructure (NCI) for supercomputer time, and theAustralian Research Council for funding (GrantsDP140104071 and DP160100246).

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