Simple models for difficult electronic excitations - Journal of Chemical

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Simple models for difficult electronic excitations Giuseppe M. J. Barca, Andrew T.B. Gilbert, and Peter M.W. Gill J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00994 • Publication Date (Web): 14 Feb 2018 Downloaded from http://pubs.acs.org on February 15, 2018

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Journal of Chemical Theory and Computation

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, ACT 2601, Australia E-mail: [email protected]

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 modelling excited states in cases where other lowcost methods, such as CIS and TD-DFT, either perform poorly or fail completely.

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 Multi-Reference 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 ClusterConfiguration Interaction (SAC-CI), 16–18 EquationOf-Motion Coupled Cluster (EOM-CC), 19–21 Linear Response Coupled Cluster (LR-CC), 22,23 Configuration Interaction Singles (CIS), 24–26 CIS with 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. 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, ex-

cited states are usually multi-reference in character, especially if the state is doubly-excited or in the vicinity of a conical intersection. 34,35 However, specification of a multi-reference wavefunction 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 approximation 36 ) are incapable of describing doublyexcited states 37–40 and often fail near conical intersections 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 Rydberg 41–44 and charge-transfer states, 45–48 unless a long-range corrected functional is used. 49,50 These weaknesses arise because neither CIS nor TDDFT 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 singledeterminant wavefunctions 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 low-cost excited-state methods fail. Section 2 presents a modified MOM algorithm that converges more reliably to the desired excited state. Then, in Sections 3 to 5, we present results obtained by applying the new algorithm to problems involving double excitations, chargetransfer states and conical intersections. All of the excitation energies reported here are vertical, i.e. 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 will 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 will refer to the energy by prefixing “TD” to the name of the functional (e.g. TD-BLYP). As we will show, these

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Table 1: Orbital overlap values between various py orbitals on different SCF cycles of HF/aug-cc-pVTZ calculations on the boron atom. Orbitals are indicated by o, the occupied orbital, o0 , the occupied orbital from the previous SCF cycle, i, the initial guess orbital, and v, the lowest unoccupied virtual py orbital. MOM Cycle 1 2 3 4 5 6 7 8 9 10 11

0

2

|ho|o i| 0.648 0.982 0.988 0.999 1.000 1.000

2

|ho|ii| 0.648 0.518 0.412 0.411 0.410 0.410

IMOM |hv|ii| 0.320 0.445 0.545 0.544 0.545 0.545

two approaches often yield strikingly different models of excited-state energetics. -��

2 2P

-�� -�� -�� -�� -��

1 2P �

�

�

�

��

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.

2

2

Method

The MOM provides an alternative to the aufbau 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 (1) P and projections pj = ( i Oij )1/2 , we simply choose to occupy the MOs with the largest pj values; all other aspects of the SCF are unchanged. The aufbau criterion drives the SCF towards the

0

2

|ho|o i| 0.648 0.982 0.011 0.999 0.999 0.995 0.998 0.994 0.997 1.000 1.000

2

|ho|ii| 0.648 0.518 0.545 0.537 0.530 0.523 0.549 0.483 0.528 0.526 0.526

2

|hv|ii| 0.320 0.445 0.412 0.411 0.411 0.391 0.363 0.428 0.382 0.384 0.384

lowest-energy 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

(2)

This strategy – the Initial Maximum Overlap Method (IMOM) – encourages the SCF to find a solution of the SCF equations in the neighbourhood of the initial guess. We implemented it in the Q-Chem 4.4.1 package 54 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 2 2P excited state of the boron atom. The initial guess,


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Table 2: Total energies (E, in hartree), excitation energies (∆E, in eV), self-interaction energies (ESIE , in eV) and η values for the 1σg2 → 1σu2 excitation in H2 . %Fock E(1σu2 ) E(1σg2 ) ∆E ESIE (1σu2 ) ESIE (1σg2 ) η

BLYP 0 –0.16250 –1.17031 27.42 –1.781 0.007 2

B3LYP 20 –0.16016 –1.18071 27.77 –1.219 0.217 2

which is identical for both calculations, was obtained from the ground-state MOs by promoting an electron from the occupied 2 px orbital to the unoccupied 3 py orbital. The IMOM calculation converges to the desired 1 s2 2 s2 3 p1 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 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 doubly-substituted configurations with large amplitudes. However, being dependent on the chosen reference configuration, this definition is ambiguous. If the ground-state wavefunction Ψ and excited-state ˆ for an n-electron system are single dewavefunction Ψ terminants of the MOs ψi and ψˆj , respectively, the ex-

HF-LYP 100 –0.09950 –1.17198 29.18 0 0 2

HF 100 –0.08076 –1.13363 28.65 0 0 2

Full CI 100 –0.11755 –1.17428 28.75 0 0 —

citation number 57 η =n−

n X n 2 X hψi |ψˆj i 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, doublysubstituted versions of Ψ would have η = 2. Postexcitation 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 1σg2 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 1σu2 state, an auto-ionizing resonance, were first obtained by Bottcher and Docken 59 and later by others. 60–63 We performed SCF calculations on both states with R = 1.4 bohr using a modified aug-mcc-pV8Z basis 64 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. Because the self-interaction errors 65 ESIE = 21 EJ + EX

(4)

(where EJ and EX are Coulomb and exchange energies) are much larger in the excited state than in the ground state, 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 (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



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Table 3: Excitation energies of the lowest totally symmetric singly- and doubly-excited states of aromatic systems.

State 2 1 Ag 5 1 Ag

Present work η BLYP 1.0058 7.70 2.0034 10.21

Naphthalene

2 1 Ag 4 1 Ag

1.0191 2.0019

5.66 6.77

5.86 66 6.75 69

5.65 66 6.76 69

5.52 68 —

Anthracene

2 1 Ag 3 1 Ag

2.0013 1.0141

4.62 4.92

5.42 66 6.57 71

5.03 66 5.28 71

4.71 70 5.33 70

Pleiadene

2 1 A1 3 1 A1

2.0118 1.0818

2.46 3.43

— —

— —

2.46 72 3.61 72

Molecule Benzene

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 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 1 E 2g 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 anthracene 66,71 and larger molecules. 84 To discover whether standard DFT methods reproduce this reordering, 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 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.

Previous work CASSCF MRMP 8.01 66 7.73 66 — —

Experimental work 7.80 67 —

Benzene (C6 H6 )

Anthracene (C14 H10 )

Naphthalene (C10 H8 )

Pleiadene (C18 H12 )

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

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 1 Ag state of anthracene, can be compared with CASSCF and MRMP errors of 1.24 eV and 0.32 eV, respectively. The 1 1 E 2g 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 CASSCF 81 and MRMP wavefunctions. 66 However, this state can be accurately modelled 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 wavefunction serve largely to describe correlation and relaxation, and should not be interpreted as indicating that the state is doubly excited.


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, M08HX 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 EOMCCSD, CIS, B3LYP and M08-HX all reproduce the correct 1/R behavior. The failure of TD-B3LYP can be traced to an incompletely modelled interaction between the electron and the hole.

!" #$

Single excitation Double excitation "!""##$$ %" #$

'" #$

&" #$

&" #$

'" #" &" #"

() *)

("% *"+

("+ *,

(", *"&

2

Figure 3: The lowest valence totally symmetric singly and doubly excited states for C6 H6 , C10 H8 , C14 H10 and C18 H12 using BLYP/6-311G*.

4

∆E / eV

1.5

Charge-Transfer States

0

F

F

F

5

6

7

8

9

10

R/Å

Figure 5: Variation of the excitation energy ∆E of the first CT state of C2 H4 + C2 F4 with the distance R. 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 functional 85 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 photoactive devices 87–90 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 centre. 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 distance between the chromophores to be varied from that determined by the bridge, which is 12.04 ˚ A. (Note that our definition of R, which measures the distance between the centres of nuclear charge of each monomer, differs from theirs.)

Ethylene + Tetrafluoroethylene

F

1

0.5

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, TDDFT 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

EOM−CCSD CIS B3LYP M08−HX TD−B3LYP

R

Figure 4: The ethylene + tetrafluoroethylene complex. CT states of the C2 H4 + C2 F4 complex (Figure 4) have been studied previously by Dreuw et al. using TDDFT and CIS. 47 To capture the correct 1/R behavior, they proposed a hybrid approach which combines TDDFT and CIS and which yielded reasonable estimates

5


Page 6 of 14

H

H

Complex C2 H4 + C2 F4

˚ R/A N 4.60

BC + ZnBC

11.20

H

N

BC + ZnBC

H

M08-HX N 10.84

12.04

B3LYP 10.48

CIS 11.61

TD-B3LYP 6.79

EOM-CCSD 10.96

3.33

3.69

1.82

—

3.49

3.78

1.84

—

H 3.28 N

3.41

H H H H

H H

H H

H

H

H N

N

N H

R

H

N H

N Zn

N

N

H

5

H

N

H H

H H

H H

H H

H

H + zinc-bacteriochlorin comFigure 6: Bacteriochlorin plex N N Zn N N structures of the complex were optiB3LYP/6-31G* mized for several R between 11.2 and 16.2 ˚ A. Excitation H H energies of the first CT state were calculated using M08H HX,HB3LYP, CIS and TD-B3LYP with the 6-31G* basis, and these are shown in Figure 7. As we saw for the C2 H4 +C2 F4 complex, the DFT and CIS methods predict the correct 1/R dependence but TD-B3LYP fails. At R = 12.04 ˚ A, 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 C2 H4 + C2 F4 complex suggest that CIS probably overestimates the excitation energy and that the M08-HX results are probably the most accurate.

0.6

B3LYP M08−HX CIS TD−B3LYP

0.4 ∆E / eV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

H transfer states of the C2 H4 + C2 F4 and BC + ZnBC complexes. Table 4: Excitation energies (in eV) of the first CT

0.2

0 11

12

13

14 R/Å

15

16

Figure 7: Variation of the excitation energy ∆E of the first CT state of BC + Zn-BC with the distance R. The red line shows the natural separation R = 12.04 ˚ A.

Conical Intersections

A conical intersection (ConInt) is a subset of the nuclear coordinate 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, 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 TDDFT, 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 survey 94 by Halász et al. for further details. There are four ConInts involving the three lowest-energy electronic states but we will focus on the ConInt with D3h symmetry as it is characterised by a single interatomic distance R. For most values of R, the doublet ground state D0 has 2 E 0 symmetry. However, for very small R, the 2 A01 state is lower in energy, thus creating the ConInt. Mielke et al. reported 95 accurate PESs for the H+H2 reaction at a highly correlated level with the aug-cc-pVDZ, aug-cc-pVTZ and aug-cc-pVQZ basis sets. They found the minimal energy ConInt (MECI) at R = 0.495 ˚ A and we use this value to assess the performance of various DFT, TDDFT 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 TDB3LYP). In all cases, the aug-cc-pVDZ basis set 96 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.


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0.1

0.1

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0

0.5

0

0.51

trafast 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”. 0.485

0.49

15

14 7 0.5

0.3

0.3

0.2

0.2

1

0.39

0.4

0.41

0

3 0.41

0.42

0.43

Figure 8: Energy difference ∆E 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

None of the TD-DFT models yields a ConInt. Both TD-BLYP and TD-B3LYP show discontinuities at the ConInt because the ground-state reference changes from 2 0 A1 to 2 E 0 at this point. These different references give different excitation energies for the D1 state, leading to the discontinuity. The TD-HFLYP 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 ˚ A which is only 0.012 ˚ A below the Mielke value (R = 0.495 ˚ A) and HF and CIS predict a MECI at 0.016 ˚ A above Mielke’s. Both BLYP and B3LYP predict a MECI at a bond length that is almost 0.1 ˚ A shorter than Mielke’s.

5.2

6

2

0.1

0.1

13

10

0.4

0.4

0

12

8

0.5

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 Davidson 98 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 Bonaˇcić-Kouteck´ y et al. 99 They showed that the isomerization of the formaldiminium cation (CH2 NH2+ ) occurs through a ConInt between the S1 and S0 states at an N−C bond twist-angle of 90◦ . Since then, many theoretical studies 100–103 have provided evidence that the S1 − S0 ConInt in retinal is responsible for the ul-

11

5 4

16

φ

17 18 19

20

NH 2

9

Figure 9: Protonated Schiff Base of Retinal (PSBR). 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/631G* 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

6

Concluding Remarks

We have examined single-determinant approximations for excited states involving double excitations, chargetransfer 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 1σu2 resonance state in H2 is modelled by HF theory. Charge-transfer states are also modelled well by single determinants and the correct 1/R behavior is predicted even for functionals whose potentials are asymptotically incorrect.


Journal of Chemical Theory and Computation 3 S1 B3LYP S1 BLYP S0 B3LYP S0 BLYP 2 ∆E / eV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 14

1

0 0

30

60

90 φ / degrees

120

150

180

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

Finally, conical intersections, which are particularly challenging for both CIS and TD-DFT, are satisfactorily treated by IMOM-based DFT.

7

Acknowledgement

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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