Adaptive Configuration Interaction for Computing Challenging

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Adaptive configuration interaction for computing challenging electronic excited states with tunable accuracy Jeffrey B. Schriber, and Francesco A. Evangelista J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00725 • Publication Date (Web): 11 Sep 2017 Downloaded from http://pubs.acs.org on September 14, 2017

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Journal of Chemical Theory and Computation

Adaptive configuration interaction for computing challenging electronic excited states with tunable accuracy Jeffrey B. Schriber and Francesco A. Evangelista∗ Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University Atlanta, Georgia, 30322, USA E-mail: [email protected]

Abstract

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We introduce and analyze various approaches to compute excited electronic states using our 4

recently developed adaptive configuration interaction (ACI) method [J.B. Schriber and F.A. Evangelista, J. Chem. Phys. 144, 161106 (2016)]. These ACI methods aim at describing

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multiple electronic states with equal accuracy, including challenging cases like multi-electron, charge transfer, and near-degenerate states. We develop both state-averaged and state-specific

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approaches to compute excited states whose absolute energy error can be tuned by a userspecified energy error threshold, σ. State averaged schemes are found to be more efficient, in

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that they obtain all states simultaneously in one computation, but lose some degree of statewise tunability. State-specific algorithms allow for direct control of the error of each state,

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though states must be computed sequentially. We compare each method using methylene, LiF, and all-trans polyene benchmark data.

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1. INTRODUCTION Perhaps the biggest difficulty in modeling electronic excited states is in quantifying varying de-

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grees of dynamic and static electron correlation equally well among states of different character.

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This challenge is exacerbated, for example, in multielectron excited states, which frequently occur 18

in chemically relevant systems containing transition metals or highly conjugated hydrocarbons. 1–3 Conventional excited-state electronic structure methods, including time-dependent density func-

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tional theory (TD-DFT) 4,5 and the equation-of-motion coupled cluster hierarchy, 6,7 often cannot efficiently compute multielectron excited states due to their considerable multireference character

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in which double or higher excitations dominate. Another example of challenging excited states are avoided crossings and conical intersections, which play a fundamental role in non-radiative pro-

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cesses. 8–10 In these cases, two or more electronic states with potentially different character become degenerate or near-degenerate, a situation that is difficult to describe with single-reference methods

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and state-specific multireference approaches, especially when the states involved are the groundand first-excited state. In this work, we generalize our previously developed adaptive configura-

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tion interaction (ACI), 11 which is very well-suited for treating large numbers of strongly correlated electrons, to the general description of excited states, including multielectron and near-degenerate

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states. In order to describe electronic states in which static correlation plays a dominant role, a mul-

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tireference method is typically required. The most common choice is to build a multideterminantal reference wave function by permuting a chosen number of electrons in a set of selected active orbitals.

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While these complete active space (CAS) wave functions, with (CASSCF) or without (CASCI) orbital optimization, are good references for excited state computations, the number of variational

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parameters in CASCI wave functions grows combinatorially with the number of electrons and orbitals and quickly becomes prohibitive. Therefore, efficient alternatives to CASCI are required to

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describe the excitations of large, chemically-relevant molecular systems. Wave function factorization techniques, 12–18 stochastic approaches, 19–28 symmetry projection

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schemes, 29–35 and various truncated configuration interaction (CI) methods 11,36–56 have risen as the most economical, near-exact representations of compressed full CI (FCI) or CASCI wave functions.

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Moreover, these techniques have allowed accurate descriptions of unprecedentedly large systems with significant multireference character by finding an efficient, sub-combinatorial scaling of the number

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of parameters with respect to the size of the active space. The density matrix renormalization group (DMRG) 12–18 and full CI quantum Monte Carlo (FCIQMC) 20,22 have been particularly successful in

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describing the ground states of large systems by correlating to near exactness active spaces beyond 2 ACS Paragon Plus Environment

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for example, the renormalized basis states are in general not optimal for multiple roots, and can be sequentially optimized for each root in a state-specific scheme. Recently introduced alternatives

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include derivations of Tamm-Dancoff (TDA) or random phase approximations (RPA) from a DMRG matrix product state reference, 65 in addition to time-dependent formalisms. 70 For FCIQMC, state-

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specific algorithms can be achieved by using a Gaussian projector shifted near the energy of the desired root, 66 or by projecting out all previously computed lower roots from the solution of the

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current root. 68,71,72 However, near-degenerate electronic states can pose a significant challenge for state-specific methods, and typically an excited state method that computes states simultaneously

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is required. Alternatively, methods that compute states simultaneously usually incorporate some form of

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state-averaging, in which a common basis is used to describe multiple states. In DMRG, this averaging is realized by renormalizing basis states with respect to multiple roots, though this procedure

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generally requires more basis states to achieve accuracy comparable with ground state DMRG calculations. 63 Coe and Patterson introduced a modified Monte Carlo CI (MCCI), which stochastically

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samples Hilbert space by selecting determinants based on a criterion averaged over roots of interests, and then forms and diagonalizes a matrix within the selected space. 73 Blunt et al. recently proposed

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a modified FCIQMC method in which all states are computed simultaneously, and orthogonalized with a Graham-Schmidt procedure at each iteration. 68 Finally, Rodríguez-Guzmán and co-workers

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presented a method in which ground and excited states are expanded in terms of nonorthogonal variationally-optimized symmetry-projected configurations. 74

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Another method well-suited to accurately describe very large excited states is the variational 2RDM (v-2RDM) method, 75–80 which forgoes wave functions entirely and directly optimizes the one

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and two-particle reduced density matrices subjected to N -representability conditions. The v-2RDM method can treat the lowest energy state of a given symmetry and total spin with high accuracy, 80,81

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and its ability to compute excited states of the same symmetry based on the equations of motion of the 1-RDM has also been recently reported. 82

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Two of the most commonly used selected CI methods, CIPSI 37 and MRD-CI, 38,83 both have been modified to compute excited states using a single space of selected determinants. 39,60,84,85

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Though the definition of the selection importance criterion is different for each method, both add determinants to the model space based on the maximum value of the selection criterion with respect 4 ACS Paragon Plus Environment

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to all electronic states. This approach is equivalent to adding determinants to the model space that are important to at least one state being studied. While the success of both methods has prompted

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numerous excited state studies, 86–92 [See Ref. 93 and references therein] there has been a lack of comparison of numerical results to determine the most effective way to compute excited states with

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selected CI methods. In this work, we provide a detailed analysis of both simultaneous and sequential ACI methods for

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computing excited state wave functions. Although our analysis is focused on ACI, our comparisons of various excited state methods is applicable to selected CI methods in general. One of the hallmark

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features of ACI is its ability to compute ground state energies to a user-defined accuracy. 11 Similar in spirit to the recently proposed variance-matching variational Monte Carlo method of Robinson and

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Neuscamman, 94 which computes all states to a similar quality as monitored by the wave function’s energy variance, we seek an adaptation of ACI such that any excited state energy can be efficiently

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computed with a specifiable error. As shown in this work, ACI can be generalized to compute excitation energies with state-specific error control and near perfect error cancellation.

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2. THEORY 2.1. Review of ACI.

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We begin with a brief overview of ACI. 11 ACI is an iterative selected CI

method which generates and optimal space of Slater determinants such that the total error in the ground state energy approximately matches a user defined parameter, σ: |E0FCI − E0ACI (σ)| ≈ σ,

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(1)

where E0FCI and E0ACI are the FCI and ACI ground state energies, respectively. The procedure is as follows:

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1. At iteration k, we define a reference space, P (k) = {Φµ }, and associated reference wave function, (k)

ΨP =

X

c(k) µ Φµ ,

(2)

µ∈P (k)

118

(k)

where expansion coefficients cµ are obtained by diagonalizing the Hamiltonian in the P (k) space. The procedure is started with a reference space, P (0) , comprised of all determinants 5 ACS Paragon Plus Environment


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in a small CASCI (< 1000 determinants). 2. All singly and doubly excited determinants are generated from P (k) to form the first order

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interacting space, F (k) . An estimate of the energy contribution to the reference space, ǫ(ΦI ), is computed for each determinant ΦI ∈ F (k) . This energy importance measure is defined as

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(k)

the variational correlation energy of a two-state system involving ΨP and ΦI , and takes the form, 95 1 ǫ(ΦI ) = ∆I − 2

q 2 2 ∆I + 4VI ,

(3)

where the Epstein–Nesbet denominator (∆I ) and coupling (VI ) are defined as (k) ˆ (k) ˆ |ΦI i , ∆I = hΨP | H |ΨP i − hΦI | H (k) ˆ VI = hΨP | H |ΦI i .

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(4) (5)

The generation of all singles and doubles has a computational cost that scales as |F (k) | ≈ |P (k) |NO2 NV2 , where |P (k) | is the number of determinants in the set P (k) , while NO and NV

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are the number of occupied and virtual orbitals, respectively. The quantity VI is computed as VI =

ˆ c(k) µ hΦµ | H |ΦI i ,

X

(6)

µ∈P (k) 130

by an algorithm that generates all single and double excitations out of the determinants in P (k) . To reduce the memory necessary to store the vector VI , we apply screening and store

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(k) ˆ |ΦI i | > τV , where τV is a only those elements for which one of the contributions |cµ hΦµ | H

user-specified screening threshold. Unless otherwise noted, calculations reported in this work 134

use a screening threshold τV = 10−12 Eh . 3. We define the secondary space, Q(k) , as the smallest possible subset of F (k) such that the

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accumulation of error estimates for sampled determinants not in Q(k) is approximately equal to σ, the energy selection parameter. In practice, this truncation of F (k) is performed by

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accumulating the ǫ(ΦI ) estimates in increasing order until σ is reached, X

|ǫ(ΦI )| ≤ σ.

ΦI ∈F (k) \Q(k)


(7)

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The total model space M (k) is then defined as the union of the secondary space and the 140

reference, M (k) = Q(k) ∪ P (k) .

(8)

In our current implementation, this step has a computational cost that scales as |F (k) | ln |F (k) | 142

due to sorting of the error estimates |ǫ(ΦI )|. In principle, sorting can be avoided by using an optimization algorithm that identifies an error threshold η such that the difference between

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σ and the sum of the error estimates |ǫ(ΦI )| for the determinants with |ǫ(ΦI )| < η is zero. This can be achieved by introducing a function f (η),

f (η) =

X

(9)

|ǫ(ΦI )|θ(η − |ǫ(ΦI )|) − σ,

ΦI ∈F (k)

146

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and seeking the largest η such that f (η) ≤ 0. The function f (η) is monotonous (albeit not P continuous) and has one zero if σ ≤ ΦI ∈F (k) |ǫ(ΦI )| (otherwise, all determinants should be included). Therefore, η may be found by using the bisection method with cost proportional to |F (k) |.

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4. The Hamiltonian is formed and diagonalized in the model space, M (k) , yielding a total energy (k)

EM and wave function, (k)

X

|ΨM i =

(k)

(10)

CI |ΦI i .

ΦI ∈M (k) 152

(k)

The procedure ends when EM converges, which occurs when the determinantal makeup of M (k) is identical between iterations [M (k) = M (k+1) ]. This diagonalization is usually the

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most expensive step in the ACI procedure and has a cost between |M (k) | and |M (k) |NO2 NV2 . 5. If the energy is not converged, the updated reference P (k+1) space is generated from the

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most important subset of the M (k) space. Specifically, P (k+1) is formed by storing determi(k)

(k)

nants in M (k) with the largest |CI |2 values, until the accumulation of the |CI |2 values of 158

determinants kept reaches a second user-defined value, X

|Cµ(k) |2 ≤ 1 − γσ,

Φµ ∈P (k+1)


(11)


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where γ is a user-specified constant with units (energy)−1 . Following our previous work, 11 160

we set γ = 1 Eh−1 . This step requires sorting of the determinant coefficients and scales as |M (k) | ln |M (k) |. However, as noted above, sorting may also be avoided in this step. Analo-

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gous to the initiator approximation in FCIQMC, 22 this step greatly increases the efficiency of ACI by only allowing determinants important in the model space to generate candidate

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determinants in F (k) . The aimed selection strategy gives the user nearly perfect control over the error for ground state

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calculations. This control is achieved in step 3 by accurately identifying determinants that need to be explicitly treated in the diagonalization procedure and ignoring those determinants whose cumu-

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lative energy contribution is smaller than the desired error. Another recently proposed selected CI method, heat-bath CI (HCI), 52,62 uses a very similar algorithm to ACI with the important excep-

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tion that its generation of the model space is done without sampling any unselected determinants. Specifically, it selects determinants individually using a threshold ǫ1 , only adding a determinant ΦI

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to the model space for which the inequality, ∀ΦJ ∈ P (k) ,

|HIJ cJ | > ǫ1

(12)

holds. Sorting of the integrals allows the selection of determinants to be done with perfect efficiency, 174

meaning that no determinants are sampled that are not added to the model space. Additionally, HCI continually grows the model space, so that at iteration k + 1 all the model space determinants

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are retained, P (k+1) = M (k) ,

(13)

rather than truncating it as is done with ACI. The resulting method is extremely efficient, but does 178

not provide an inherent means of estimating the total energy error. The predictive control over the error in ACI relies on a fairly complete sampling of unselected determinants, and a potentially

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fruitful development in ACI would be to somehow adopt the heat-bath sampling method while still preserving the predictive power of the current method. In the following section, we generalize the

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ground state ACI method to determinant spaces capable of describing excited state wave functions.


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2.2. Excited State Methods in ACI. 184

In principle, excited state wave functions can be obtained

nearly automatically by solving for the lowest eigenvalues of the Hamiltonian in the ground state determinant space (M0 ) optimized by ACI. However, the determinant set M0 is biased towards

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the ground state and provides no way to control the accuracy of higher eigenvalues. One solution is a state-averaged (SA) approach in which the definition of the selection importance criterion,

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ǫ(ΦI ), is modified to reflect importance with respect to multiple roots, resulting in a single space of determinants capable of describing all roots of interest. Alternatively, an algorithm may be preferred

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wherein determinant selection is done separately for each root, allowing each state to be described by potentially unique determinant spaces. We expand upon both approaches below.

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For all methods considered, we generalize our definition of the energy importance criterion, ǫ(ΦI ), to apply to any arbitrary state n, q 1 2 , ∆I,n − ∆2I,n + 4VI,n 2

(14)

(k) (k) ˆ ˆ |ΦI i , |ΨP,n i − hΦI | H ∆I,n = hΨP,n | H

(15)

ǫ(ΦI , n) = where,

(k) ˆ VI,n = hΨP,n | H |ΦI i ,

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(16)

(k)

and |ΨP,n i refers to the n-th eigenstate of the P (k) space. For converged wave functions, the (k) superscript will be dropped.

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By combining different definitions of selection criteria and model spaces, it is possible to formulate several excited state methods. In this work we consider five excited state ACI methods, which,

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for convenience are summarized in Table 1. We separate these approaches into two categories: methods that use a single model space for all states, and methods that use or combine different

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model spaces. What follows is a detailed description of each excited state method.

2.2.1. State-averaged methods based on a single model space. 202

The goal of a state-

averaged ACI (SA-ACI) procedure is to obtain a single compact space of determinants capable of describing multiple electronic states with controlled accuracy. Algorithmically, the only differ-



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ence in SA-ACI compared to the ground state algorithm is the definition of the energy importance criterion, ǫ(ΦI , n), where we propose two ways to define a determinant’s importance in lowering the

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energy of multiple states. One option is to simply take the average importance of a determinant (|ǫ(ΦI , n)|) among N roots of interest,

ǫ¯(ΦI ) =

N −1 1 X |ǫ(ΦI , n)|. N

(17)

n=0

208

In addition, the reference space is updated by selecting determinants based on an average of the expansion coefficient, −1 1 NX |Cµ,n |2 ≤ 1 − γσ, N (k+1)

X

Φµ ∈P 210

(18)

n=0

where Cµ,n is the CI coefficient from root n corresponding to the reference space determinant Φµ . This averaged procedure, denoted SA-ACIavg , only requires one optimization to obtain multiple

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roots, which are rigorously orthogonal as they are eigenvectors of a common Hamiltonian. However, the near-exact energy control in ground state ACI is exchanged for a control over the average error

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among roots. The loss of state-specific error control may become potentially hazardous if one state has a

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significantly different determinantal makeup with respect to the other computed roots, where the optimized space will become biased towards those roots described by similar determinants. This

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biasing can be remedied either by adding user-specified weights to each state, or by defining the selection importance criterion as a maximum function rather than an average. Specifically, we can

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redefine ǫ¯(ΦI ) as, ǫ¯(ΦI ) =

max

0≤n≤N −1

(19)

|ǫ(ΦI , n)|,

and similarly the maximum of the expansion coefficient among roots is then used to truncate the 222

model space M to update the reference. This procedure, denoted as SA-ACImax , will build a final model space of determinants which are important to at least one root of interest rather than all to

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roots of interest in averaged way.

2.2.2. Methods based on state-specific model spaces. 226

Perhaps the most attractive feature

of ground state ACI is that the error is predicted by σ to remarkable accuracy. SA-ACI methods 10 ACS Paragon Plus Environment

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achieve this in an averaged way because all states are computed with the same set of determinants. 228

To better preserve error prediction in higher roots, we propose a second class of ACI methods which form separate model spaces Mn with corresponding wave functions ΨMn for each root n of

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interest, a procedure we define as unconstrained ACI. Unfortunately, the model space wave functions, ΨMn , are in general not mutually orthogonal when optimized separately. In fact, the degree of

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nonorthogonality between states can be viewed as a measure of the inaccuracy of individually optimized wave functions, since all states become orthogonal in the FCI limit (σ → 0). We propose

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three methods which attempt to preserve state-wise energy control while maintaining orthogonality among roots.

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Two of the following methods begin with an unconstrained ACI calculation, yielding separate determinant spaces for each root. In order to correctly converge each state to the right solution,

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a reasonable initial guess is required. To achieve correct convergence to each root, we start each computation with three or four warmup iterations of SA-ACI, select the desired root, and then

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optimize it individually. To avoid the possibility of root flipping, we do all optimizations within a root-following scheme

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State Combined ACI. The N non-orthogonal states are first optimized separately with unconstrained ACI to produce a set of converged model spaces {Mn }, with which we can define a final

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model space MSC as the union of all individually selected spaces,

MSC = M0 ∪ M1 ∪ · · · ∪ MN −1 .

(20)

Finally, the matrix representation of the Hamiltonian in the MSC space can be diagonalized to 246

yield orthogonal eigenstates for the N roots. This state-combined procedure (SC-ACI) can also be viewed as a more rigorous way to remove any determinant space biasing in SA-ACIavg . While the

248

energy errors derived from the individually converged model subspaces {Mn } are well controlled by σ, the final energies computed from the final model space MSC may not be. Since this final model

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space will have more determinants than any individually optimized space (unless all spaces Mn are identical), then the energies of the final model space will be lower than their individually optimized

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counterparts. This energy lowering effect should not be considered entirely a good consequence, as the lowering may be inconsistent among roots, resulting in poorer error cancellation. Nonetheless, 11 ACS Paragon Plus Environment


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σ remains an approximate upper bound to the expected error for each state. Multistate ACI. Another approach to overcome the nonorthogonality problem in uncon-

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strained ACI is to simply recouple the nonorthogonal solutions to yield a multistate ACI (MS-ACI). Specifically, we form the multi-state Hamiltonian in the basis of individually optimized states, and

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then solve the resulting generalized eigenvalue problem for N roots,

HC = SCE,

(21)

where the Hamiltonian (H) and overlap (S) matrices are defined as ˆ |ΨMn i (H)mn = hΨMm | H

260

m, n = 0, . . . , N − 1,

(22)

and (S)mn = hΨMm |ΨMn i

m, n = 0, . . . , N − 1,

(23)

while E = diag(E0 , . . . , EN −1 ) is a diagonal matrix. Though simple, this approach is vulnerable to 262

numerical instability when the eigenvalues of the overlap matrix, S, are nearly zero, corresponding to near linear-dependencies in the basis of individually optimized states. Near-degenerate electronic

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states may also pose a significant challenge for the initial computation of nonorthogonal wave functions, however, and issues in correctly identifying the root to be optimized and root-flipping could

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preclude the use of an ACI method. Additionally, nonorthogonal wave functions create difficulties for certain properties and computations that require multiple wave functions, such as transition

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dipole moments. Orthogonality-constrained ACI. The final excited state method, orthogonality constrained

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ACI (OC-ACI), sequentially optimizes determinant spaces Mn in the orthogonal complement of the space spanned by the previous roots. This strategy has been successfully applied in the context

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of density functional theory to obtain both valence 96 and core-excited states. 97 In this procedure, the selection and screening for P (k) and M (k) spaces is identical to the unconstrained ACI, but the

274

computation of the energies and wave functions for these spaces differs. Specifically, in the optimization of the n-th root, we define the projector onto the space com-


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Table 1 Comparison of excited state ACI methods.

Method

Energy importance criterion

SA-ACIavg

1 X ǫ(ΦI , n) N +1

Model space

N

Single average space (Mavg )

n=0

SA-ACImax SC-ACI MS-ACI OC-ACI

276

Single average space (Mmax )

max ǫ(ΦI , n) 0≤n≤N        ǫ(ΦI , n)      

n Mn )

Union of individual model spaces (MSC =

S

Union of individual model spaces (MMS =

S

n Mn )

Different spaces per state (M0 ⊆ M1 ⊆ M2 . . .)

ˆ M,n ) plementary to the first n − 1 roots (Q

ˆ M,n = 1 − Q

n−1 X

|ΨM,m i hΨM,m | ,

(24)

m=0

where |ΨM,m i is the optimized wave function for root m computed from the Mm model space. Thus, 278

this projector makes the n-th root orthogonal to all n−1 roots in the intersection of their subspaces. At each iteration, the P (k) and M (k) space wave functions are computed by projecting out the converged wave functions of the previous roots, ˆQ ˆ M,n |Ψ (k) i = EP,n |Ψ (k) i ˆ M,n H Q P ,n P ,n

(25)

ˆ M,n H ˆQ ˆ M,n |Ψ (k) i = EM,n |Ψ (k) i , Q M ,n M ,n

(26)

where the lowest eigenstate of this projected Hamiltonian yields the desired excited state wave 280

function. Note that this procedure is equivalent to the Gram-Schmidt orthogonalization of an excited state wave function with respect to lower roots, with the requirement that the Gram-

282

Schmidt orthogonalized remains variationally optimal. In practice, Eqs. (25)-(26) are implemented by projecting out eigenvectors for the first n − 1 roots from the trial vectors in the Davidson–Liu

284

algorithm. 98,99 The excited state ACI methods are all developed in the freely available open-source software,

286

Forte, 100 written as a plugin to Psi4. 101,102



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3. RESULTS AND DISCUSSION

288

3.1. Methylene.

We compare all ACI excited state methods using benchmark data on the

three lowest singlet A1 states of methylene. These states provide a rigorous test for the various ACI 290

excited state methods due to their different determinantal character. 103–108 The 1 1 A1 state is closedshell with a single dominant configuration, while the 2 1 A1 state has considerable double excitation

292

character and the 3 1 A1 is a multi-configurational open-shell singlet biradical. The geometries used were taken from Ref. 105 and are of C2v symmetry. The cc-pVDZ basis was used, 109 augmented

294

with additional s functions for both the carbon and hydrogen atoms, as outlined in Ref. 105, and no orbitals were frozen, resulting in a FCI space equivalent to a CAS(8e,27o). Table 2 Methylene vertical excitation energies (eV) obtained with truncated CI, EOM-CCSD, CC3, and ACI methods. All computations use restricted Hartree–Fock orbitals and the modified cc-pVDZ basis described in Ref. 105 with no frozen orbitals, resulting in a CAS(8e,27o).

All values shown are errors with respect to FCI, for which the excitation energy is instead shown. Number of variational Excitation Energy Error (eV) 1 Method parameters (Np ) 2 A1 3 1 A1 CISD 3,607 2.639 1.990 CISDT 83,689 1.113 0.035 CISDTQ 1,020,759 0.046 0.032 EOM-CCSD CC3

3,607 83,689

1.456 0.471

−0.005 −0.004

0.056 0.021 0.028 −0.016 −0.007

0.220 0.095 0.063 0.026 0.008

54,575 112,563 70,253 23,563/24,401/47,049 23,563/24,094/47,875

0.003 0.002 0.003 < 0.001 < 0.001

0.027 0.015 0.013 0.008 < 0.001

77,145,700

4.655

6.514

σ = 0.25 eV SA-ACIavg SA-ACImax SC-ACI MS-ACI OC-ACI

6279 12684 8871 1513/2525/6250 1513/2059/6282 σ = 0.054 eV

SA-ACIavg SA-ACImax SC-ACI MS-ACI OC-ACI FCI


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296


In the limit σ → 0, all ACI methods are equivalent to FCI. However, the rates at which each method converges to FCI as σ is reduced will likely be different, as each method uses a

298

different selection algorithm. Table 2 shows the error in excitation energy for the two excited states studied computed with conventional CI truncated to double (SD), triple (SDT), and quadruple

300

(SDTQ) excitations, equation of motion coupled cluster with singles and doubles (EOM-CCSD), 110 approximate coupled cluster with triples (CC3), 111 and excited-state ACI methods. For CI and CC

302

methods, the computed 2 1 A1 excitation energy converges to the correct result much faster than for the 2 1 A1 vertical excitation energy. This result is well understood considering that the 3 1 A1

304

state, though a multireference open-shell singlet, is most significantly composed of determinants with excitation rank up to triples. For the 2 1 A1 state, however, higher excitations are required,

306

and the inclusion of triples in CC methods is required even to achieve sub-eV accuracy. For the ACI methods, this trend does not hold—the 2 1 A1 state is more easily described than is

308

the 3 1 A1 state, and only slightly. Regardless of excitation rank, the 2 1 A1 state is largely dominated by a single determinant, so the ACI methods converge very quickly to the FCI result for this typically

310

challenging case. With a threshold of σ = 0.054 eV, all ACI methods produce excitation energy errors on the order of 1 meV or better for the 2 1 A1 state. The 3 1 A1 excitation energy is more

312

challenging since it is multideterminantal, and for the OC-ACI and MS-ACI methods it requires additional determinants to achieve the same level of accuracy. Among the ACI methods, the OC-

314

ACI shows the best performance for methylene as it gives ∼1-8 meV accuracy while requiring much fewer determinants than the other methods. Even for the modest σ values shown, we see that ACI,

316

particularly OC-ACI, can compute excited state energies more accurately and efficiently than the CI and CC methods in our example.

318

To further understand the varying efficiencies of each ACI method, we can analyze the absolute error of each state with respect to the required number of determinants, shown in Figure 2. Note

320

that the efficiencies of SA-ACIavg , SA-ACImax , and SC-ACI are very similar for all three states, meaning that they produce comparable absolute errors with a similar number of determinants.

322

In contrast, OC-ACI and MS-ACI show an improved efficiency for the 1 1 A1 and 2 1 A1 states. These lowest two states can be treated with fewer determinants than the 3 1 A1 state to achieve

324

a similar absolute error, but this property cannot be exploited by SA-ACI and SC-ACI as they employ a single set of determinants for all states. In general, the efficiencies of ACI methods which 15 ACS Paragon Plus Environment



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Table 3 Comparison of LiF avoided crossing curves. We report the average error for the lowest two Σ+ states, the non paralellism error (NPE)a , the standard deviation (σ∆E ) with respect to FCI, and the average number of variational parameters (Np ) required for each computation. For OC-ACI computations, the two numbers represent the average number of determinants required for ground and excited states, respectively. All values are reported in mEh , and computed from seven bond distances from 8.5 a.u. to 14.5 a.u. incremented by 1 a.u. in order to enable comparison with PMC-SD and SA-MCCI. We do not show data for MS-ACI and SC-ACI as they diverge near the avoided crossing. All ACI computations used restricted Hartree–Fock orbitals and the custom basis set defined in Ref. 112, with the two 1s-like molecular orbitals frozen on each atom resulting in a CAS(8,27).

Method CISDT CISDTQ

Average Error (mEh ) X 1 1 Σ+ 2 1 Σ+ 14.075 15.584 1.295 1.273

NPE (mEh ) X 2 1 Σ+ 9.825 10.231 1.275 1.378

σ∆E 4.400 0.593

Np 76,365 978,423

1 Σ+

SA-ACIavg (50) SA-ACIavg (10) SA-ACIavg (1) SA-ACIavg (0.5)

42.853 10.243 0.891 0.448

48.873 10.052 0.830 0.410

13.896 1.262 0.089 0.055

21.110 1.314 0.109 0.071

7.368 0.437 0.047 0.029

488 3,341 57,798 95,711

OC-ACI(50) OC-ACI(10) OC-ACI(1) OC-ACI(0.5)

46.185 10.721 0.963 0.471

46.150 9.685 0.871 0.442

2.845 1.043 0.102 0.038

11.205 1.593 0.077 0.041

2.234 0.721 0.055 0.020

246/261 1576/1929 37,986/38,675 65,144/67,980

– 1.571

– 1.143

– 1.000

– 2.000

0.586 0.497

2734 5 × 106

b SA-MCCI c PMC-SD a

NPE is computed as the minimum error along the curve subtracted from the maximum error along the curve. Values taken from Ref. 73, Np refers to configuration state functions. c Values computed from data in Ref. 71, Np refers to the approximate number of walkers in projector Monte Carlo simulation. b

near-degenerate electronic states. 112–114 Near equilibrium, the ground state wave function is qualita332

tively single-determinantal with ionic character, and the first excited state is an open-shell biradical singlet dominated by low-weighted pairs of covalent configurations, requiring a multireference treat-

334

ment. At the avoided crossing point, the ionic and covalent character swaps between these two states, thus requiring a balanced treatment of strong and weak correlation to achieve good accuracy

336

for both states. 115,116 All LiF computations used the Li (9s 5p)/[4s 2p]; F (9s 6p 1d)/[4s 3p 1d] basis defined in Ref.

338

112 and restricted Hartree–Fock orbitals with the 1s-like molecular orbitals frozen on each atom, resulting in the explicit treatment of 8 electrons in 27 orbitals. To ensure proper convergence, we use

340

a γ value of 0.1 mEh−1 . The FCI solution requires 7.7 × 107 determinants and predicts the avoided 17 ACS Paragon Plus Environment



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360


SA-ACIavg methods. As shown in the previous example, the SA-ACImax method has an efficiency that is comparable or smaller than that of the SA-ACIavg scheme, therefore, we do not report results

362

for it. SA-ACImax results are not reported since this In Figure 3, we report absolute energy errors in

364

mEh for CISDT, CISDTQ, OC-ACI, and SA-ACIavg for σ = 10 and 1 mEh . CISDT and CISDTQ fail to describe the multireference character of the covalent state with the same accuracy as the ionic

366

state, while both states are described to similar accuracy in the ACI methods. For σ = 10 mEh , the OC-ACI(10) and SA-ACIavg (10) produce errors near 10 mEh throughout the curve, with deviations

368

up to about 2 mEh near the avoided crossing point. Decreasing σ to 1 mEh gives nearly identical results for the two ACI methods, which both consistently yield an error near the preselected value

370

without noticeable deviations at this scale. In Table 3 we report the number of variational parameters in each computation in addition to

372

the average error of each state, the non-parallelism error (NPE) and the standard deviation (σ∆E ) of each curve with respect to FCI. The OC-ACI(10) and SA-MCCI produce an avoided crossing

374

curve of similarly good quality using a modest number of parameters. To achieve a comparably accurate PES, PMC-SD requires each state to be computed to within 1 mEh , while all ACI methods

376

can produce a given σ∆E value by computing each state with an order of magnitude less accuracy. The systematic improvability allows the ACI methods to compute the avoided crossing with NPEs

378

on the order of 40 µEh . The OC-ACI and SA-ACIavg give reasonably similar energy errors for all chosen σ values, but

380

with OC-ACI requiring fewer determinants without exception. Furthermore, the OC-ACI shows consistently better NPEs for all σ values shown. This result is somewhat surprising in that the SA-

382

ACIavg is very well suited for this particular problem. While the determinantal character for each state is different and swaps at the avoided crossing point, the total determinantal makeup of both

384

states in combination remains fairly unchanged throughout dissociation. Despite the determinantal basis for the SA-ACIavg Hamiltonian remaining fairly constant, it still shows greater NPEs with

386

respect to OC-ACI because it lacks the direct state-specific error control of OC-ACI.

3.3. 388

Extended polyenes.

The trans-polyene series (Cn Hn+2 ) is a set of nearly one-dimensional

conjugated carbon chains, which contain numerous low-lying π-π ∗ excited states often with double 19 ACS Paragon Plus Environment


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excitation character. As the chain length increases, the energies of π bonding and anti bonding or390

bital pairs converge, which further enhances the strong correlation effects present in these low-lying excited states. 117–119

392

In the following sections, we first compare all ACI methods with available FCI benchmarks on octatetraene to establish the capacity of each method to describe these low-lying transitions. Next,

394

we apply the OC-ACI to a series of extended polyenes.

3.3.1. Comparison of Methods. 396

To investigate the performance of each ACI method on

large numbers of dense states, we studied the eight lowest singlets of octatetraene, the all-trans polyene with eight carbon atoms and four double bonds. The geometry was obtained from a DFT

398

optimization at the B3LYP/cc-pVDZ level of theory using C2h symmetry. 120 The ACI and CASCI computations employed a CAS(8,16) π-double active space in which all eight π electrons were

400

correlated in sixteen π bonding and anti bonding orbitals, all within the cc-pVDZ basis. To achieve fast convergence in ACI computations, we use Pipek-Mezey split-localized orbitals 121,122 in which

402

the active π orbitals occupied in the initial RHF reference wave function are localized separately from the active π orbitals unoccupied in the reference. Additionally, the ACI computations were

404

performed without symmetry for better localization, though we report the results using C2h labels. The CAS(8,16) is within the scope of CASCI calculations, allowing direct comparison of absolute

406

and excitation energies computed with ACI. Table 4 shows the absolute and excitation energy errors with respect to CASCI of the lowest 8 states of octatetraene for all ACI methods computed with

408

σ = 10 and 1 mEh . Similarly with the LiF example, the unconstrained ACI computation required for MS-ACI and SC-ACI suffered from incurable variational collapse for σ = 10 mEh . In this case,

410

the 2 1 Bu state collapses into the 1 1 Bu state, so MS-ACI is not feasible, but we do report SC-ACI values determined from the seven roots the unconstrained ACI did converge. For σ = 1 mEh , all

412

correct roots were found in the unconstrained ACI. In Table 4, we see that OC-ACI and MS-ACI consistently give the lowest errors in excitation

414

energy, 0.24 and 0.20 mEh respectively, compared to SC-ACI (0.37 mEh ), SA-ACIavg (0.77 mEh ) , and SA-ACImax (0.56 mEh ) for σ = 1 mEh . However, SC-ACI consistently gives the lowest absolute

416

error for each state, and for many states the SA-ACI methods give lower absolute errors than do the state-specific ACI methods. Most notably, the ground state absolute energy errors for SC-ACI 20 ACS Paragon Plus Environment

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Table 4 Comparison of excited state ACI methods for the lowest eight singlet electronic states of octatetraene (C8 H10 ) computed with σ = 1 and 10 mEh . All computations use split-localized orbitals, with an active space including all sixteen π orbitals defined in the cc-pVDZ basis. We report both the excitation energy errors (∆Eex ) and absolute energy errors (∆Eabs ) with respect to FCI in mEh .

X 1 Ag

3 1 Bu

5 1 Ag

Average of |∆E|

9.41 12.13 4.27 −1.85

∆Eex (σ = 10 mEh ) 6.75 9.40 12.06 6.12 12.69 12.52 3.45 4.49 4.32 −0.51 −0.62 −2.15

8.99 11.76 3.76 −2.20

7.43 9.93 4.10 −0.22

9.43 11.16 4.19 1.44

0.85 0.60 0.40 −0.22 −0.30

0.71 0.57 0.37 −0.20 −0.24

∆Eex (σ = 1 mEh ) 0.68 0.73 0.91 0.50 0.54 0.56 0.35 0.37 0.38 −0.18 −0.18 −0.22 −0.20 −0.23 −0.26

0.75 0.56 0.36 −0.23 −0.27

0.77 0.59 0.38 −0.14 −0.19

0.77 0.56 0.37 0.20 0.24

15.30 16.63 8.61 10.13

12.75 15.81 7.92 10.81

∆Eabs (σ = 10 mEh ) 10.09 12.74 9.79 16.36 7.10 8.14 12.15 12.04

15.39 16.20 7.97 10.51

12.33 15.45 7.41 10.46

10.76 13.61 7.76 12.44

11.59 13.44 7.32 11.40

1.12 0.81 0.59 0.92 0.88

0.98 0.78 0.56 0.95 0.94

∆Eabs (σ = 1 mEh ) 0.95 0.99 0.70 0.75 0.54 0.56 0.96 0.96 0.99 0.96

1.17 0.77 0.57 0.93 0.93

1.02 0.77 0.55 0.91 0.91

1.04 0.79 0.57 1.00 1.00

0.94 0.70 0.52 0.97 0.97

1 1 Bu

SA-ACIavg SA-ACImax SC-ACI OC-ACI

11.97 12.96 4.95 −2.53

SA-ACIavg SA-ACImax SC-ACI MS-ACI OC-ACI

SA-ACIavg 3.34 SA-ACImax 3.67 SC-ACI 3.65 OC-ACI 12.66

SA-ACIavg SA-ACImax SC-ACI MS-ACI OC-ACI

418

4 1 Ag

2 1 Ag

0.26 0.21 0.19 1.14 1.18

State 2 1 Bu 3 1 Ag

(0.19 mEh ), SA-ACIavg (0.26 mEh ), and SA-ACImax (0.21 mEh ) are significantly lower than the corresponding error of OC-ACI (1.18 mEh ) and MS-ACI (1.14 mEh ), which are much closer to

420

the value predicted by σ. It is this over-stabilization of the ground state in SC-ACI and SA-ACI methods which causes the large errors in the excitation energies. In fact, OC-ACI and MS-ACI

422

both give an average absolute energy error of 0.97 mEh , close to σ = 1.0 mEh , with little deviation from each state, resulting in very accurate excitation energies.

424

As mentioned with the previous methylene example, the state-specific ACI methods perform better for relative energies as they allow each state to be described by a different number of de-

426

terminants. Figure 4 shows the absolute error with respect to CASCI of each of the eight states 21 ACS Paragon Plus Environment



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Table 5 Vertical excitation energies (eV) for the polyene series (C2n H2n+2 ) computed with various methods. All ACI, CASCI, and DRMG methods use a CAS(2n,2n), and the NOCI calculation was performed with 10 nonorthogonal determinants as described in Ref 129.

State 2 1 A− g 1 1 Bu− 3 1 A− g

CASCI 4.97 6.13 6.82

OC-ACI(10) 4.96 6.12 6.86

DMRG PAO-CASCI 64 6.33 7.49 7.95

DMRG CASSCF 64 4.69 5.88 6.60

3.97 4.94 5.77

3.97 4.95 5.82

5.40 6.30 7.01

3.76 4.74 5.59

4.10

6 (12,12)

2 1 A− g 1 1 Bu− 3 1 A− g

3.19 3.98 5.12

3.42 4.18 4.81

3.43 4.18 4.85

4.90 5.60 6.28

3.25 4.03 4.78

3.46

8 (16,16)

2 1 A− g 1 1 Bu− 3 1 A− g

2.50 3.10 3.99

3.12 3.71 4.31

4.60 5.15 5.71

2.93 3.57 4.20

2.94

10(20,20)

2 1 A− g 1 1 Bu− 3 1 A− g

2.04 2.51 3.11

12(24,24)

2 1 A− g 1 1 Bu− 3 1 A− g

3.04 3.45 3.89

4.42 4.85 5.31

2.73 3.25 3.78

n CAS 4 (8,8)

446

NOCI 129 5.09

CASCIMRMP 118 4.26 5.30 7.20

1.70 2.05 2.45

n = 12, we use a prescreening threshold value of τV = 10−8 in building the external space F (k) which introduces an error negligible with respect to the chosen value of σ = 10 mEh . For these polyenes,

448

we report the excited states with C2h symmetry labels in addition to the particle/hole symmetry label in order to distinguish states characterized by ionic (+) or covalent (−) valence structures,

450

as is done conventionally (see Ref. 64 for more details). Moreover, this labeling distinguishes the 1 1 Bu− state, which we compute, from the 1 1 Bu+ state, which we do not compute, despite being

452

the second excited state in experimental results. We do not compute the 1 1 Bu+ state because it is largely described by π − σ excitations outside of our selected active space.

454

1 − 1 − Table 1 shows vertical excitation energies for the 2 1 A− g , 1 Bu , and 3 Ag states as computed

with CASCI, OC-ACI(10), DMRG with a projected atomic orbital basis (DMRG-PAO-CASCI), 64 456

DMRG with orbital opitmization (DMRG-CASSCF), 64 non-orthogonal CI (NOCI), 129 and the multireferece Møller–Plesset method with a CASCI reference (CASCI-MRMP). 118 The DMRG meth-

458

ods used the same active space as the OC-ACI, and CASCI-MRMP used a subset of the π valence




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4. CONCLUSIONS

478

In this study, we have detailed five methods for computing challenging excited states using the ACI formalism. These excited state methods iteratively optimize either a single space of determinants or

480

separate determinant spaces in describing multiple electronic states. We rigorously compared each method using challenging excited-state benchmarks characterized by strong correlation and double

482

excitation character, including methylene excited states, the LiF avoided crossing, and many excited states of large polyenes. In these comparisons, we find that the OC-ACI method is generally the most

484

effective and widely applicable because it shows the best error prediction and tunability while still maintaining orthogonality among solutions. We attribute its success to its flexibility; by allowing

486

separate determinant spaces to describe each electronic state, the method can use the appropriate determinants for each state such that their preselected errors are the same. We demonstrate the

488

ability of OC-ACI to compute large polyenes, with up to a CAS(24,24), with remarkable error cancellation, in addition to computing excited states with accuracy competitive with DMRG.

490

In its current form, OC-ACI is well-suited to compute excitation energies of small molecules to high accuracy. Despite its remarkable error cancellation, computations in larger basis sets including

492

extrapolations to the complete basis limit are required for experimental predictability. For small systems, OC-ACI could be a very valuable tool in the benchmarking of excited states, where single-

494

reference methods like CC3 currently represent the highest applied level of theory. In application to larger systems (>100 orbitals), OC-ACI and SA-ACI are most useful in generating reference wave

496

functions which then need to be connected to theories of dynamical correlation to enable comparison to experiment. We are actively working on these extensions.

498

While this article was in revision, we noticed a similar study posted to the arXiv 130 in which the aforementioned HCI is adapted for excited states. This work employs an algorithm similar to

500

SA-ACI in addition to a semi-stochastic multireference perturbation theory, all which allow the authors to accurately compute excitation energies using large basis sets.

502

ACKNOWLEDGEMENTS This work was supported by Emory University startup funds and by a Research Fellowship of the

504

Alfred P. Sloan Foundation. 25 ACS Paragon Plus Environment


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ASSOCIATED CONTENT

506

Supporting information.

We provide Cartesian coordinates of methylene and all polyenes (text

file) in xyz format. For all reported excitation energies in the article, we also provide the absolute 508

energies for the corresponding states. Finally, for the extended polyene computations with SO-ACI, we report RHF energies, absolute SO-ACI energies, and the required number of determinants for

510

each state. This information is available free of charge via the Internet at http://pubs.acs.org.

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(2) Daniel, C. Photochemistry and photophysics of transition metal complexes: quantum chemistry. Coord. Chem. Rev. 2015, 282-283, 19–32.

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(3) Dreuw, A.; Head-Gordon, M. Single-reference ab initio methods for the calculation of excited states of large molecules. Chem. Rev. 2005, 105, 4009–4037.

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