Low-cost molecular excited states from a state-averaged resonating

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Quantum Electronic Structure

Low-cost molecular excited states from a stateaveraged resonating Hartree-Fock approach Jacob Nite, and Carlos A. Jimenez-Hoyos J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00579 • Publication Date (Web): 16 Aug 2019 Downloaded from pubs.acs.org on August 19, 2019

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Journal of Chemical Theory and Computation

Low-cost molecular excited states from a state-averaged resonating Hartree–Fock approach Jacob Nite and Carlos A. Jiménez-Hoyos∗ Department of Chemistry, Wesleyan University, Middletown, CT 06459 E-mail: [email protected]

Abstract Quantum chemistry methods that describe excited states on the same footing as the ground state are generally scarce. In previous work, Gill et al. (J. Phys. Chem. A 2008, 112, 13164) and later Sundstrom and Head-Gordon (J. Chem. Phys. 2014, 140, 114103) considered excited states resulting from a non-orthogonal configuration interaction (NOCI) on stationary solutions of the Hartree–Fock equations. We build upon those contributions and present the state-averaged resonating Hartree–Fock (saResHF) method, which differs from NOCI in that spin-projection and orbital relaxation effects are incorporated from the onset. Our results in a set of small molecules (alanine, formaldehyde, acetaldehyde, acetone, formamide, and ethylene) suggest that sa-ResHF excitation energies are a notable improvement over configuration interaction singles (CIS), at a mean-field computational cost. The orbital relaxation in sa-ResHF, carried in the presence of a spin-projection operator, generally results in excitation energies that are closer to the EOM-CCSD and experimental values than the corresponding NOCI ones.

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1

Introduction

The accurate description of electronically excited states has been a long standing quest in quantum chemistry. Time-dependent density functional theory offers a reasonable compromise of computational cost vs accuracy for low-lying excited states. Unfortunately, it is often unreliable as it tends to overstabilize charge-transfer states. Configuration interaction singles (CIS) is often the simplest wavefunction method to treat excited states. But it too is unreliable because it lacks electron correlation as well as orbital relaxation effects. There have been recent efforts to develop alternative, low-cost methods to describe excited states. For instance, Liu et al. developed an orbital-optimized CIS strategy where not only the CIS amplitudes but also the orbitals are relaxed 1 . Shea and Neuscamman considered a CIS expansion with an alternative Lagrangian that targets excited states 2 . Oosterbaan et al. used a non-orthogonal configuration interaction supplemented with singles excitations to describe core excited states 3 . Some of us had previously explored an excited variation after mean-field projection (VAMP) strategy that provides access, at mean-field computational cost, to the low-energy spectrum of molecular systems 4 . While we were able to describe the low-energy excited states of formaldehyde, we think the use of different ansätze for each of the excited states is suboptimal and may impact the quality of the excitation energies obtained. As a motivation for this work, Gill and co-workers 5 noted that higher-lying self-consistent field (SCF) or Hartree–Fock (HF) solutions can provide a reasonably accurate zeroth order description of low-energy excited states. This is even when ignoring the fact that such higherlying SCF solutions have non-zero overlap with the ground state and cannot be considered, in principle, as true excited state wavefunctions. The authors pointed out, nevertheless, that the overlap remained small in the systems they considered. The logical extension to the SCF solutions found by Gill et al. is to employ a nonorthogonal configuration approach (NOCI), in which a CI hamiltonian is diagonalized using a basis of non-orthogonal SCF solutions. While this approach has been outlined by many 2


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others, it has never gained the wide usage of orthogonal approaches 6–8 . This is most likely due to a combination of difficulty in finding appropriate SCF solutions to use in the CI expansions, efficiently optimizing to the states that are saddle points in the potential energy surface, and the spin-contamination present in UHF solutions 9 . In subsequent work, Sundstrom and Head-Gordon 9 expanded on the previous work in their NOCI calculations of the low-energy spectrum of polyene chains. By performing an actual diagonalization of the Hamiltonian among the set of SCF solutions rather than neglecting the non-zero overlap, the states calculated were better representations of true excited states. Using the SCF procedure of Gill et al., they had access to HF determinants that represented single excitations from a reference restricted ground state. In addition, they carried out the spin-projection of all the states. The importance of spin-projection cannot be underestimated: low-energy excited states tend to be dominated by single excitations and such excitations out of a reference restricted HF (RHF) determinant necessarily break spin symmetry and are therefore spin-contaminated. NOCI solutions can often outperform time-dependent HF and CIS because they incorporate orbital relaxation effects, which are completely neglected at the singles excitation level, and because they can formally access doubly-excited determinants: it is possible that one of the low-lying SCF solutions can be mostly characterized as a double-excitation out of the reference HF wavefunction. Our intention with this work is to explore an ingredient that is still missing: the orbital relaxation of each of the determinants participating in the NOCI solutions in the presence of each other and the spin-projection operator. To this end, we present the result of stateaveraged resonating Hartree–Fock (ResHF) solutions. Here, ResHF is our preferred way to call the orbitally optimized multi-determinant Hartree–Fock solutions which Fukutome 10 introduced in the context of accounting for ground state correlations. The rest of the paper is organized as follows. In Sec. 2 we provide details of the formalism we use, including how the variational optimization is carried out. In Sec. 3 we provide some

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of the computational details underlying our work. Section 4 presents our calculated vertical excitation energies and oscillator strengths for alanine, formaldehyde, acetaldehyde, acetone, and formamide as representative examples of small organic molecules. Lastly, in Sec. 5 we present our conclusions from this work.

2

Formalism

2.1

Excited VAMP

In Ref. 4 , we proposed the use of the excited variation after mean-field projection (VAMP) strategy to obtain the low-energy spectrum of molecular systems. We now proceed to a brief description of it, disregarding the symmetry projection to simplify the presentation. In its simplest form, a single Slater determinant is used for each state. Let us assume that the ground state determinant |Φ0 i has already been found; we equate |Ψ0 i ≡ |Φ0 i, where the notation implies that the |Ψ0 i state is chosen as the ground state determinant |Φ0 i. The excited VAMP strategy proposes an ansätz for the first excited state |Ψ1 i as |Ψ1 i = 1 − Sˆ1 |Φ1 i,

(1)

where |Φ1 i is a single Slater determinant and Sˆ1 is a projector |Φ0 i hΦ0 | Sˆ1 = hΦ0 |Φ0 i

(2)

that guarantees orthogonality with respect to the ground state. While the ansätz used for the first excited state is not strictly a Slater determinant, all the flexibility in the ansätz is determined by the orbitals that define |Φ1 i. In our previous work, we opted for a variational optimization of the proposed ansätz. Naturally, the procedure described above for the first excited state can be generalized to higher ones. Let us assume that the ground state and q − 1 excited states are already 4


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available. An ansätz for the q-th excited state is written as |Ψq i = 1 − Sˆq |Φq i,

(3)

where |Φq i is a single Slater determinant and Sˆq is the projector

Sˆq =

q−1 X

|Φr i (A−1 )rs hΦs |,

(4)

r,s=0

Ars = hΦr |Φs i.

(5)

In this case, all the variational flexibility of the ansätz lies in the determinant |Φq i. While we successfully used the excited VAMP strategy in Ref. 4 , the different ansätze used for different states (due to the use of the projection operator in Eq. 3) can lead to an imbalanced description.

2.2

Non-Orthogonal Configuration Interaction (NOCI)

An alternative strategy to obtain the low energy spectrum of a molecular system based on non-orthogonal determinants is NOCI. For our purposes, NOCI is simply the diagonalization of the Hamiltonian matrix among a set of available, non-orthogonal in general, Slater determinants. That is, given a manifold {|Φr i; r = 0, . . . , q} of Slater determinants, NOCI states will result from the solution to the generalized eigenvalue problem

H f = Af

(6)

ˆ s i, Hrs = hΦr |H|Φ

(7)

Ars = hΦr |Φs i,

(8)

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where f is the matrix of eigenvectors and is the (diagonal) matrix of eigenvalues. The p-th excited state is then expanded as

|Ψp i =

q X

fkp |Φk i.

(9)

k=0

NOCI can provide a reasonable description of the low-energy spectrum of a molecule if each of the determinants approximately describes one of the low-lying states. Here, we emphasize the importance of Gill and co-worker’s contribution 5 realizing that higher-lying SCF solutions can provide an approximate description of an excited state. In retrospect, this is not too surprising given that a CIS state dominated by a single natural transition can often be approximated as a single Slater determinant. Some of us have used a similar ansätz in a different context, to account for ground state correlations 11 . In that case, the ground state is expanded as

0

|Ψ i =

q X

fk0 |Φk i.

(10)

k=0

where the coefficients fk0 and determinants |Φk i are variationally optimized to make the ground state energy as low as possible. This ansätz was proposed by Fukutome 10 as the resonating Hartree–Fock (ResHF) method. Note that the nature of the determinants that are optimal to describe ground-state correlations will generally be very different than those determinants that can provide a reasonable description of low-energy excited states.

2.3

State-Averaged Resonating Hartree–Fock (sa-ResHF)

In this work, we use the same ansätz as that used in NOCI, but we include the optimization of the underlying Slater determinant manifold {|Φr i; r = 0, . . . , q}. To this purpose, we follow the state-averaged (SA) strategy commonly used in multi-configurational self-consistent field

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(MC-SCF) approaches and introduce the SA Lagrangian

L=

p X

wj j

(11)

j=0

where wj is the weight of state j in L (with the condition wj > 0) and j is the energy of the j-th state. Note that p ≤ q, that is only the first p states are included in the Lagrangian, with the rest being discarded. In order to carry out the minimization of L with respect to the determinants {|Φr i} we follow a similar strategy as that used in Ref. 11 , where a Thouless matrix Z r is used to parametrize each determinant in the expansion. At convergence, the orbital gradient vanishes, i.e., X ∂ L = wj r ∂Zai j

P

s

∗ ˆ − j |Φs i fsj h(Φr )ai |H frj P ∗ = 0 ∀ r, a, i. t s st ftj fsj hΦ |Φ i

(12)

Here, |(Φr )ai i denotes a single excitation of determinant |Φr i. The equal weight for all states of interest in the sa-ResHF Lagrangian (Eq. 11) helps to avoid selective optimization of one state: there is no risk of a variational collapse of the expansion to the ResHF ground state. In addition, each state is fundamentally described by the same ansätz, which should lead to a more balanced approach than the one used in excited VAMP.

2.4

Symmetry-Projection

Symmetry-projection can be incorporated in a way that results in only minor modifications of the formalism introduced above. As described in detail in Ref. 12 , spin-projection to a state with spin j can be obtained using the operator 2j + 1 j Pˆmk = 8π 2

Z

s∗ ˆ dΩDmk (Ω)R(Ω),

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(13)


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s where Ω = (α, β, γ) stands for the set of Euler angles, Dmk is an element of Wigner’s D-

ˆ matrix, and R(Ω) is a spin-rotation operator. The evaluation of symmetry projected matrix elements such as overlaps can be carried out as

Ars =

X

j s fk fk0 hΦr |Pˆkk 0 |Φ i,

(14)

kk0

where fk are linear coefficients. In this work, we restrict ourselves to the use of UHF determinants (which preserve ms as a good quantum number). This has two consequences: the double sum over k and k 0 elements in Eq. 14 collapses to a single term, and the threedimensional integration from Eq. 13 reduces to a one-dimensional integration, i.e., 2j + 1 Ars = 2

π

Z

dβ sin(β) dsmm (β) hΦr | exp(−iβ Sˆy )|Φs i,

(15)

0

where we have assumed that determinants |Φr i and |Φs i have both ms = m. The integration can be carried out numerically using a small number of grid points. Note that, since exp(−iβ Sˆy )|Φs i is itself a Slater determinant, Eq. 15 reduces to a weighted sum of overlaps over non-orthogonal determinants. The orbital optimization of the sa-ResHF Lagrangian including spin-projection is not much different from the one shown in Eq. 12: each matrix element is substituted by the projected version in an analogous form to Eq. 15. We stress out that the orbital optimization is carried out in the presence of the projection operator, i.e., it is the symmetry-projected state-averaged energy that is optimized in the orbital optimization step.

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Computational Details

The sa-ResHF (including spin projection) calculations have been performed with an in-house code, supplied with one- and two-electron integrals evaluated with the Gaussian 16 suite of programs. 13 In all cases, the (ground state) geometries were optimized with RB3LYP and

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a cc-pVDZ basis set using Gaussian 16. In the case of alanine, we used the lowest energy conformer as described in the literature. 14 For all our excited state calculations we use the aug-cc-pVDZ basis set. While a larger basis set may be desired to reach better convergence of Rydberg excitations, our current code requires all two-electron integrals to be available in memory. We do not think, however, that the basis set limitation will change the conclusions of this work. We used a 6-point quadrature grid to carry out the spin projection in all cases. Following Gilbert et al., 5 the initial guess for sa-ResHF calculations consisted of the RHF ground state along with other higher energy SCF solutions. In previous works, such higher energy SCF solutions have been obtained using the maximum overlap method (MOM). In our case, we obtained those solutions in a different way: having carried out a CIS calculation, we prepared initial guesses of excited determinants by using the natural transition orbitals with largest weight for each low-lying CIS solution. We fed such SCF initial guesses (necessarily of UHF type) to a HF solver using a Newton-Raphson (NR) procedure. It is known that NR tends to converge to the closest stationary point, which may or may not be a local minimum. In most instances, we were able to reach stationary SCF solutions with energy higher than that of the starting reference HF without difficulty. While our current in-house code does not support symmetry, the sa-ResHF optimizations were performed taking advantage of the determinant symmetry in an ad-hoc way. All determinants sharing identical symmetry were optimized simultaneously, but independently from other. Once each symmetry block of determinants were optimized, the entire Hamiltonian was diagonalized to produce the state averaged energies. The overlap between symmetricallydistinct determinants was checked to be zero prior to and after optimization. The sa-ResHF calculations were compared against EOM-CCSD calculations using the aug-cc-pVDZ basis set calculated using Gaussian 16 (unless otherwise noted).

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4

Results and Discussion

We discuss our results for the alanine, formaldehyde, acetaldehyde, acetone, formamide, and ethylene molecules. In what follows, we will use the acronym NOCI to refer to calculations that include spin-projection and the solution to the generalized eigenvalue problem, but use determinants obtained from higher energy SCF solutions. In contrast, the acronym saResHF will be used for calculations that include spin-projection and where the solution to the generalized eigenvalue problem accounts for orbital relaxation effects. Note that NOCI results lack orbital relaxation due to both spin-projection and the presence of other states.

4.1

Alanine

We begin by considering the case of singlet excitations in alanine, which we use for illustrative purposes. We initialize the sa-ResHF calculation with the higher-energy SCF solutions, obtained as described in Sec. 3, used in NOCI. We present in Table 1 the overlap between each of the converged HF solutions after spin-projection. (The raw overlaps are in agreement with those obtained using the MOM method and presented in Ref. 5 .) Table 1 presents the overlap and Hamiltonian matrices obtained before and after the SA orbital relaxation is carried out. While most overlaps are below ≈ 0.1 (in absolute value) before the optimization, the overlap between determinants 3 1A and 5 1A is around 0.35. The magnitude of this overlap is minimally affected by the presence of the spin projection operator. The presence of such large off-diagonal overlap matrix elements makes imperative the solution to the generalized eigenvalue problem in order to obtain true Hamiltonian eigenvalues in the subspace of interest. We also show in Table 1 the calculated overlaps after the orbital optimization step has been completed as a proxy for how the orbital optimization modified the original determinants obtained as HF solutions. While this is only a proxy, we do observe that the resulting overlap matrix is different, in this case becoming more diagonally dominant. The excitation energies, obtained as solutions to the sa-ResHF generalized

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Table 1: Overlap and Hamiltonian (in a.u.) matrix elements for alanine using 6 lowest “singlet” (ms = 0) spin-projected HF determinants before and after orbital relaxation. State

1 1A

2 1A

3 1A

4 1A

5 1A

6 1A

1.000 −0.001 −0.002 −0.129 0.091

1.000 −0.044 −0.359 −0.070

1.000 0.010 −0.080

1.000 −0.111

1.000

−570.794 0.627 1.379 73.899 −51.889

−570.755 24.952 205.130 39.813

−570.725 −5.849 45.483

−570.674 63.307

−570.726

1.000 0.051 0.027 −0.012

1.000 0.014 0.004

1.000 0.011

1.000

−570.765 −29.153 −15.231 7.061

−570.760 −8.190 −2.546

−570.753 −6.554

−570.745

NOCI overlap 1 1A 2 1A 3 1A 4 1A 5 1A 6 1A

1.000 0.001 0.017 0.006 0.042 0.082

NOCI Hamiltonian 1 1A 2 1A 3 1A 4 1A 5 1A 6 1A

−570.961 −0.539 −9.938 −3.336 −23.764 −47.030

sa-ResHF overlap 1 1A 2 1A 3 1A 4 1A 5 1A 6 1A

1.000 0.005 0.001 0.006 0.066 0.031

1.000 −5. × 10−4 2. × 10−5 −0.001 0.034

sa-ResHF Hamiltonian 1 1A 2 1A 3 1A 4 1A 5 1A 6 1A

−571.015 −3.129 −0.470 −3.247 −37.916 −17.882

−570.816 0.260 −0.014 0.467 −19.364

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eigenvalue problem, are shown in Table 2. Table 2: Vertical singlet excitation energies (in eV) for alanine obtained from solving the spin-projected ResHF eigenvalue problem before and after the SA orbital optimization step.

state

a

HFa

NOCI sa-ResHF EOM-CCSDb

1 1A −321.925346c − − − 1 2A 4.65 4.54 5.51 5.88 3 1A 5.62 5.62 6.60 6.32 1 4A 6.47 6.38 7.40 7.24 1 5A 6.49 6.50 7.52 7.48 6 1A 7.97 8.26 7.54 7.61 Excitation energies obtained from the various stationary HF solutions (without spin projection). b Excitation energies from Ref. 15 c Ground state reference energy (in a.u.).

The excitation energies of alanine using the sa-ResHF are in good agreement with previous work. The EOM-CCSD calculations in Ref. 15 agree well with the sa-ResHF energies: the major difference is observed for 2 1A where our result is 0.45 eV lower. Other excitation energies deviate from the EOM-CCSD results by approximately 0.1 eV. In contrast, the original calculations by Gill et al. (lacking spin-projection and diagonalization) deviate by as much as 1 eV compared to the EOM-CCSD results. We emphasize that the computational cost of the sa-ResHF presented in this work has the same scaling as the one from the original NOCI calculations: there is a global ngrid pre-factor (associated with the spin-projection) and a quadratic dependence on the number of determinants used (as opposed to linear), but the mean-field scaling is unchanged.

4.2

Carbonyls

Tables 3 through 5 show the calculated vertical excitation energies resulting from sa-ResHF calculations in formaldehyde, acetaldehyde, and acetone. For these molecules, we considered both singlet and triplet excitations. In general, we observe that the predicted sa-ResHF excitation energies for singlet states are in better agreement with EOM-CCSD values (which 12


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in turn are in good agreement with available experimental values) than the corresponding NOCI excitation energies. For both formaldehyde and acetaldehyde we see particularly good agreement with EOM-CCSD values. In those cases, sa-ResHF excitation energies are in better agreement with EOM-CCSD than those predicted with CIS, which tends to underestimate the excitation energies. The sa-ResHF calculations of the triplet states also agree well with experiment, and provide a substantial improvement over NOCI as well as CIS excitation energies. In addition to calculating the triplet states using UHF “triplet” (ms = 1) determinants, we also calculated triplet excitations using UHF “singlet” (ms = 0) determinants, projected to a triplet state in the spin-projection step. The latter excitation energies were consistently higher in energy than those calculated with the triplet determinants, although the magnitude of the difference varied from a few tenths of an eV to over an eV. In general, we expect that NOCI results using “singlet” determinants will be relatively poor: the stable HF solutions are most likely optimized targeting an average of singlet and triplet excitation energies and may in general be better for the singlet states. However, sa-ResHF calculations further relax the orbitals in the presence of both the spin-projection operator and the full manifold of determinants. We speculate that for some states with large multi-reference character the “singlet” determinants may work better, but otherwise the “triplet” determinants may prove superior. The sa-ResHF excitation energies are generally superior to the corresponding NOCI results, suggesting that the orbital optimization provides a significant improvement. As an example, the NOCI and sa-ResHF excitation energies for the 1A1 states in formaldehyde are significantly different. This highlights the importance of the orbital relaxation effect. The oscillators strengths for formaldehyde from sa-ResHF, CIS, and EOM-CCSD calculations as well as the experimental ones (when available) are shown in Table 3. The oscillator strength values are mildly correlated between the four sources. While there are large discrepancies between sa-ResHF, CIS, and EOM-CCSD, all three methods qualitatively match the experimental values, with the 2 1A1 , 1 1B2 , and 2 1B2 having small oscillator strengths.

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Table 3: Vertical singlet and triplet excitation energies (in eV) for formaldehyde obtained from NOCI and sa-ResHF calculations. Calculated oscillator strengths are shown in parentheses. state

transition NOCI

sa-ResHF

CIS

EOM-CCSD

exptla

0.00 7.74 (0.039) 10.46 (0.214) 3.60 (0.000) 8.30 (0.000) 9.27 (0.008) 6.77 (0.014) 7.79 (0.026)

0.00 9.54 (0.117) 9.85 (0.173) 4.61 (0.000) 10.14 (0.000) 10.03 (0.000) 8.51 (0.032) 9.39 (0.050)

0.00 8.04 (0.060) 9.85 (0.152) 4.08 (0.000) 8.60 (0.000) 9.49 (0.001) 7.03 (0.022) 7.98 (0.043)

0.00 8.14 (0.017b )

5.07 9.20 3.79 8.70 8.14 9.03

6.16 7.91 3.61 8.61 6.86 7.79

5.86 7.96 3.50

5.07 9.20 3.79 10.05 8.70 8.14 9.03

6.16 7.91 3.61 8.63 8.61 6.86 7.79

5.86 7.96 3.50

singlet states 1 1A1 2 1A1 3 1A1 1 1A2 2 1A2 1 1B1 1 1B2 2 1B2

n→3pb2 π→π ∗ n→π ∗ n→3pb1 σ→π ∗ n→3sa1 n→3pa1

0.00 6.92 10.96 2.71 7.56 8.22 5.97 7.03

4.07 8.37 9.0b 7.11 (0.028b ) 7.97 (0.032b )

triplet states (using ms = 1 determinants) 1 3A1 2 3A1 1 3A2 1 3B1 1 3B2 2 3B2

π→π ∗ n→3pb2 n→π ∗ σ→π ∗ n→3sa1 n→3pa1

4.44 6.94 2.54 7.36 5.99 6.97

5.74 8.15 3.84 8.62 7.17 8.10

6.83 7.79

triplet states (using ms = 0 determinants) 1 3A1 2 3A1 1 3A2 2 3A2 1 3B1 1 3B2 2 3B2

π→π ∗ n→3pb2 n→π ∗ n→3pb1 σ→π ∗ n→3sa1 n→3pa1

5.50 7.04 2.59 7.81 7.59 6.09 7.10

5.94 8.46 3.97 9.15 8.98 7.49 8.47

MAEc MBEc

c

6.83 7.79

1.02 0.36 0.93 −0.91 0.12 0.77 a Experimental energies from Ref. 16 unless otherwise noted. b Ref. 17 The mean absolute error and mean bias error are calculated with respect to the EOM-CCSD energies.

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Table 4: Same as Table 3, but for acetaldehyde. state

NOCI

sa-ResHF

CIS

EOM-CCSD

exptla

singlet states 1 1A0 2 1A0 3 1A0 4 1A0 5 1A0 1 1A00 2 1A00 3 1A00

0.00 5.84 6.55 6.92 10.88 3.07 6.76 8.43

0.00 0.00 6.81 (0.026) 8.43 (0.017) 7.53 (0.063) 9.19 (0.110) 7.90 (0.013) 9.40 (0.031) 8.71 (0.007) 9.90 (0.330) 4.12 (8. × 10−6 ) 5.01 (1. × 10−4 ) 7.81 (0.034) 9.44 (0.013) 9.58 (0.006) 10.08 (0.000)

0.00 6.81 (0.022) 7.52 (0.074) 7.89 (0.032) 9.54 (0.115) 4.41 (1. × 10−4 ) 7.79 (0.011) 9.42 (0.000)

6.82 7.46 7.75 4.28

triplet states (using ms = 1 determinants) 1 3A0 2 3A0 3 3A0 4 3A0 1 3A00 2 3A00 3 3A00

4.66 5.80 6.52 6.88 2.93 6.83 7.61

6.13 7.15 7.89 8.23 4.39 8.18 9.05

5.36 8.11 8.86 9.24 4.24 8.88 9.36

6.26 6.67 7.37 7.77 3.98 8.68 7.76

5.99 6.81 7.44 7.8 3.97

6.26 6.67 7.37 7.77 3.98 8.68 7.76

5.99 6.81 7.44 7.8 3.97


6.92 7.84 5.76 6.65 2.99 5.93 7.03

6.34 7.44 8.20 8.57 4.58 8.50 9.41

MAEb 0.97 b MBE −0.84

0.48 0.29 b

5.36 8.11 8.86 9.24 4.24 8.88 9.36

1.10 0.98 a Experimental energies from Ref. 16 With respect to the EOM-CCSD energies.

15



Page 16 of 26

Table 5: Same as Table 3, but for acetone. State

NOCI

sa-ResHF

CIS

EOM-CCSD

exptla

0.00 5.93 (0.002) 2.56 (0.000) 5.87 (0.000) 4.95 (0.043) 5.98 (0.013) 6.56 (0.057)

0.00 9.24 (0.029) 5.25 (0.000) 9.20 (0.000) 8.25 (0.040) 9.32 (0.003) 9.88 (0.061)

0.00 7.46 (2. × 10−4 ) 4.56 (0.000) 7.43 (0.000) 6.41 (0.032) 7.60 (0.007) 8.04 (0.042)

7.41 4.38 7.36 6.35 7.45 8.09

singlet states 1 1A1 2 1A1 1 1A2 2 1A2 1 1B2 2 1B2 3 1B2

0.00 6.50 3.16 6.34 5.48 6.52 7.05

triplet states (using ms = 1 determinants) 1 3A1 2 3A1 1 3A2 1 3B2 2 3B2 3 3B2

4.67 6.33 3.04 5.48 6.52 7.07

4.56 6.18 2.94 5.28 6.30 6.93

5.52 9.11 4.51 8.09 9.20 9.74

6.30 7.37 4.16 6.33 7.53 7.99

5.88 4.16

triplet states (using ms = 0 determinants) 2 3A1 b 1 3A2 2 3A2 1 3B2 2 3B2 3 3B2

b

6.51 3.11 6.47 5.59 6.64 7.18

6.48 3.13 6.49 5.57 6.62 7.21

9.11 4.51 9.11 8.09 9.20 9.74

7.37 4.16 7.41 6.33 7.53 7.99

4.16

MAEc 1.02 1.25 1.47 c MBE −1.02 −1.25 1.34 a Experimental energies from Ref. 18 Even though we have not reported a lower energy 1 3A1 state, we have labeled this as 2 3A1 given the character of the transition. c With respect to the EOM-CCSD energies.

16


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While all three computation methods predict a very bright 3 1A1 state, it is known that the π → π ∗ transition in formaldehyde can have significant mixing with Rydberg excitations. 17 . The basis set employed in this study is not well suited to handle states with significant Rydberg mixing, although we still find good agreement between the sa-ResHF calculated excitation energy of 10.43 eV and the 10.1 eV value predicted using a time-dependent HF approach. 17 The sa-ResHF excitation energies for acetaldehyde have good agreement with EOMCCSD values for the singlet states with the largest error being around 0.80 eV for the 5 1A0 state (see Table 4). Other excitation energies show smaller deviations from the EOM-CCSD values. Similar trends are observed for the triplet states, with the sa-ResHF excitation energies being slightly worse than for the singlet states. There is qualitative agreement between CIS and sa-ResHF oscillator strengths, with the notable exception of the 5 1A0 state: CIS predicts it to be bright as opposed to sa-ResHF. This is simply a reflection of the different character between the CIS and sa-ResHF states. The sa-ResHF excitation energies for acetone show the largest deviation from EOMCCSD and experimental values for all carbonyl systems tested, with all excitations having an average negative deviation of approximately 1.5 eV (see Table 5). The sa-ResHF excitation energy errors are consistent between all experimental excited states. This can be due to a significant difference in the dynamical correlation contributions to the ground state and all the low-lying excited states and/or to the fact that low-lying excited states have significant multi-reference character.

4.3

Formamide

We show in Table 6 the calculated vertical excitation energies in formamide. Formamide has two main singlet valence transitions, the n → π ∗ and the π → π ∗ . The sa-ResHF excitation energy for the n → π ∗ transition (5.54 eV) agrees well with the EOM-CCSD value (5.75 eV) as well as the experimental value reported in the literature (5.65 eV) (see Table 6). This can 17



Page 18 of 26

be compared to the fairly poor CIS excitation energy of 6.55 eV. This state has also been well described by previous CASPT2 calculations and coupled cluster calculations coming within a few tenths of an eV. 19 Table 6: Same as Table 3, but for formamide. state

NOCI

sa-ResHF

CIS

0.00 (0.000) 6.91 (0.002) 7.50 (0.074) 7.95 (0.368) 5.54 (0.001) 6.26 (0.022) 7.13 (0.001) 7.72 (0.001)

0.00 8.55 (0.087) 8.92 (0.189) 9.43 (0.140) 6.55 (0.001) 7.67 (0.023) 8.59 (1. × 10−4 ) 9.20 (0.010)

EOM-CCSD

exptla

singlet states 1 1A0 2 1A0 3 1A0 4 1A0 1 1A00 2 1A00 3 1A00 4 1A00

0.00 5.08 6.50 7.57 4.27 5.70 5.94 6.22

0.00 7.75 6.85 7.42 5.75 6.75 7.60 8.21

(0.266) (0.003) (0.091) (0.001) (0.023) (0.001) (0.005)

7.32 (0.37) 5.65

triplet states (using ms = 1 determinants) 1 3A0 2 3A0 3 3A0 1 3A00 2 3A00 3 3A00

4.76 5.06 6.06 5.60 6.21 4.17

6.37 7.21 7.79 5.81 6.53 7.52

5.73 8.46 9.05 5.90 7.31 8.45

5.82 6.76 7.34 5.41 6.54 7.53


6.49 5.77 6.31 4.23 5.54 7.07 6.25

6.63 7.40 7.99 5.99 7.23 8.19 8.79

MAEb MBEb

1.14 -1.12

0.51 1.10 0.26 1.08 a Experimental energies from Ref. 20 With respect to the EOM-CCSD energies.

b

5.73 8.46 9.05 5.90 7.31 8.45 9.07

5.82 6.76 7.34 5.41 6.54 7.53 8.14

The π → π ∗ singlet state has proven more challenging to accurately describe theoretically. 20,21 The state is known to have significant amount of mixing with the Rydberg states n → s and n → 3pπ . 20–22 The sa-ResHF excitation energy is 7.95 eV, which is in good 18


Page 19 of 26 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


agreement with the EOM-CCSD (7.42 eV) experimental value (7.32 eV). 20 This error compares favorably to other theoretical methods used to study these transitions. Schreiber et al. employed CASPT2 (7.44 eV) and various coupled cluster methods (8.27 eV using CC3) and found significant deviations in the calculation of the π → π ∗ state and very accurate energies with both CASPT2 and the CC methods for the n → π ∗ . 19 This state was also studied by Gilbert et al. using the maximum overlap method (MOM) with the B3LYP functional giving energies for the n → π∗ and the π → π ∗ states of 5.22 eV and 6.29 eV respectively. Oscillator strengths were also found to be in qualitative agreement with other methods and experiment. The sa-ResHF value for the π → π ∗ state matches that of the experimental value within the precision of the experimental measurements (0.37). 20 Previous CASPT2, CC, and CASSCF/MRCI calculations also provide a similar oscillator strength as well as confirming the near zero oscillator strength of the n → π ∗ transition. 19,21,22 . The experimental methods confirm the experimental observation of the two valence transitions having a dark state and a bright state.

4.4

Ethylene

Table 7 shows the calculated vertical excitation energies in ethylene. In this case, the saResHF method underestimates energies for both singlet and triplet states. The fact that all excitation energies are underestimated probably reflects an imbalanced description between the ground and the excited states, with the ground state being more poorly described. The degree of error varies between 0.12 eV and 1.12 eV with respect to the EOM-CCSD energies, the largest variance of any system tested. While the sources of error in sa-ResHF calculations in ethylene are the same as in other molecules, we observe that the high symmetry of the molecule plays a role in the quality of the results. Examining the symmetries of each state in 7, nearly all states calculated belong to independent symmetry groups, with the exception of the pairs of the 1B1g , 3B1g , and 1Ag states. Since all determinants retained their symmetry during SCF optimization, nearly two-thirds of the excited states calculations 19



contained only a single determinant. This made many excited state calculations equivalent to single-determinant symmetry-projected HF calculations: states of different symmetry do not interact. If the multi-reference character or the dynamical correlation is significantly different between the ground state and states with other symmetry, there is little chance for partial cancellation of errors due to the state-averaged approach.

5

Conclusions

Building upon the NOCI strategy used by Sundstrom and Head-Gordon, we have presented the sa-ResHF approach for the calculation of excited states in molecular systems. In saResHF, we introduce orbital optimization of a state-averaged Lagrangian in the presence of both spin-projection and the Slater determinant manifold. Our results confirm that in a number of cases the orbital optimization makes a large difference in the predicted excitation energies. Generally, the resulting sa-ResHF excitation energies are in better agreement with EOM-CCSD calculations than those of NOCI, though there are some exceptions. While in this work we have not provided an exhaustive assessment of the merits of the sa-ResHF method, our results do indicate that it is a promising approach to describe the low-energy spectrum of a molecular system at a low computational cost. For the sa-ResHF strategy the major remaining source of error can be attributed to the lack of dynamic correlation. In that regard, these results should be viewed as an alternative to SA complete active space SCF (CAS-SCF) calculations. Both lack dynamic correlation in their standard forms, but can provide good reference wavefunctions for post-SCF treatments. We plan to develop approaches that account for the missing dynamic correlation in the near future.

Acknowledgement This work was supported by a generous start-up package from Wesleyan University. JN and 20


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Table 7: Same as Table 3, but for ethylene. state

NOCI

sa-ResHF

CIS

0.00 7.85 (0.000) 7.76 (0.000) 6.68 (0.000) 8.10 (0.000) 6.73 (0.000) 7.61 (0.443) 6.05 (0.116)

0.00 8.85 (0.000) 9.06 (0.000) 7.81 (0.000) 9.24 (0.000) 7.99 (0.000) 7.82 (0.515) 7.21 (0.092)

EOM-CCSD Exptla

singlet states 1 1Ag 2 1Ag 1 1Au 1 1B1g 2 1B1g 1 1B2g 1 1B1u 1 1B3u

0.00 8.46 6.72 6.04 7.74 8.53 6.66 8.09

0.00 9.01 9.27 8.04 8.62 8.09 8.10 7.36

(0.000) (0.000) (0.000) (0.000) (0.000) (0.373) (0.080)

8.26 7.80

7.65 7.11

triplet states (using ms = 1 determinants) 1 3Ag 1 3Au 1 3B1g 2 3B1g 1 3B2g 1 3B1u 1 3B3u

3.38 5.92 6.62 8.28 7.28 6.66 7.74

7.36 7.82 6.71 8.28 6.74 3.38 6.00

8.17 9.04 7.72 8.50 7.86 3.73 6.99

8.66 9.25 7.98 8.25 8.01 4.58 7.23

8.15

8.66 9.25 7.98 8.25 8.01 4.58 7.23

8.15

4.30 6.98

triplet states (using ms = 0 determinants) 1 3Ag 1 3Au 1 3B1g 2 3B1g 1 3B2g 1 3B1u 1 3B3u

5.97 6.66 4.32 6.70 7.77 7.39 8.26

MAEb MBEb

1.05 1.07 0.30 -1.01 −1.05 −0.17 a Experimental energies from Ref. 23 b With respect to the EOM-CCSD energies.

7.43 7.89 6.78 8.37 6.82 3.80 6.08

8.17 9.04 7.72 8.50 7.86 3.73 6.99

21


4.30 6.98


CAJH acknowledge Michael Frisch from Gaussian, Inc. for useful discussions.

Supporting Information Available The following files are available free of charge. • SupportingInfo.pdf: HF determinant energies and overlap and Hamiltonian matrix elements for all systems

References (1) Liu, X.; Ou, Q.; Alguire, E.; Subotnik, J. E. Communication: An Inexpensive, Variational, Almost Black-Box, Almost Size-Consistent Correction to Configuration Interaction Singles for Valence Excited States. J. Chem. Phys. 2013, 138, 221105. (2) Shea, J. A. R.; Neuscamman, E. Communication: A Mean Field Platform for Excited State Quantum Chemistry. J. Chem. Phys. 2018, 149, 081101. (3) Oosterbaan, K. J.; White, A. F.; Head-Gordon, M. Non-Orthogonal Configuration Interaction with Single Substitutions for Core-Excited States: An Extension to Doublet Radicals. J. Chem. Theory Comput. 2019, 15, 2966–2973. (4) Jiménez-Hoyos, C. A.; Rodr´ıguez-Guzmán, R.; Scuseria, G. E. Excited Electronic States from a Variational Approach Based on Symmetry-Projected Hartree–Fock Configurations. J. Chem. Phys. 2013, 139, 224110. (5) 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, 13164–13171. (6) Jackels, C. F.; Davidson, E. R. The Two Lowest Energy 2A0 States of NO2. J. Chem. Phys. 1976, 64, 2908–2917. 22


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23



¨ Foresman, J. B.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Fox, D. J. Gaussian 16 Revision B.01. Gaussian Inc. Wallingford CT 2016. (14) Blanco, S.; Lesarri, A.; López, J. C.; Alonso, J. L. The Gas-Phase Structure of Alanine. J. Am. Chem. Soc. 2004, 126, 11675–11683. (15) Osted, A.; Kongsted, J.; Christiansen, O. Theoretical Study of the Electronic GasPhase Spectrum of Glycine, Alanine, and Related Amines and Carboxylic Acids. J. Phys. Chem. A 2005, 109, 1430–1440. (16) Robin, M. B. Higher Excited States of Polyatomic Molecules; Academic Press: New York, 1985; Vol. 3. (17) Yeager, D. L.; McKoy, V. Equations of Motion Method: Excitation Energies and Intensities in Formaldehyde. J. Chem. Phys. 1974, 60, 2714–2716. (18) Merchán, M.; Roos, B. O.; McDiarmid, R.; Xing, X. A Combined Theoretical and Experimental Determination of the Electronic Spectrum of Acetone. J. Chem. Phys. 1996, 104, 1791–1804. (19) Schreiber, M.; Silva-Junior, M. R.; Sauer, S. P. A.; Thiel, W. Benchmarks for Electronically Excited States: CASPT2, CC2, CCSD, and CC3. J. Chem. Phys. 2008, 128, 134110. (20) Basch, H.; Robin, M. B.; Kuebler, N. A. Electronic Spectra of Isoelectronic Amides, Acids, and Acyl Fluorides. J. Chem. Phys. 1968, 49, 5007–5018. (21) Hirst, J. D.; Hirst, D. M.; Brooks, C. L. Ab Initio Calculations of the Excited States of Formamide. J. Phys. Chem. 1996, 100, 13487–13491. (22) Antol, I.; Eckert-Maksic, M.; Lischka, H. Ab Initio MR-CISD Study of Gas-Phase Basicity of Formamide in the First Excited Singlet State. J. Phys. Chem. A 2004, 108, 10317–10325. 24


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(23) Palmer, M. H.; Beveridge, A. J.; Walker, I. C.; Abuain, T. The Electronic States of Ethylene up to 10 eV Studied by Electron Impact Spectroscopy and Ab Initio Configuration Interaction and Iterative Natural Orbital Calculations. Chem. Phys. 1986, 102, 63–75.

25



Graphical TOC Entry

Excitation Energy

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

NOCI

sa-ResHF

EOM-CCSD

26

CIS


Page 26 of 26

Low-cost molecular excited states from a state-averaged resonating

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