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Letter

An efficient variational principle for the direct optimization of excited states Luning Zhao, and Eric Neuscamman J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00508 • Publication Date (Web): 05 Jul 2016 Downloaded from http://pubs.acs.org on July 10, 2016

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Journal of Chemical Theory and Computation

An efficient variational principle for the direct optimization of excited states Luning Zhao†,‡ and Eric Neuscamman∗,†,‡ †Department of Chemistry, University of California, Berkeley, California, 94720, USA ‡Chemical Science Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA E-mail: [email protected] Abstract We present a variational principle that enables systematically improvable predictions for individual excited states through an efficient Monte Carlo evaluation. We demonstrate its compatibility with different ansatzes and with both real space and Fock space sampling and discuss its potential for use in the solid state. In numerical demonstrations for challenging molecular excitations, the method rivals or surpasses the accuracy of very high level methods using drastically more compact wave function approximations.

Across chemistry and physics, methods for modeling electronically excited states are less robust and less accurate than those for ground states, limiting our ability to design band gaps, harness photochemistry, and interpret the spectroscopic experiments used to characterize matter. The heart of this disparity can be traced to a fundamental disadvantage suffered by excited states, namely the lack of a robust and efficient variational principle analogous to that exploited by ground state methods. While many formal candidates for an excited state variational principle exist, 1–6 their evaluation can be far more expensive than the 1 ACS Paragon Plus Environment


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ground state case due to their reliance on squaring the Hamiltonian in order to measure the energy “distance” between a state and the targeted position in the energy spectrum. Within the world of quantum Monte Calro (QMC), available state-specific approaches include variance minimization 7 and a recent projector method, 8 but the former is unable to specify which state is to be targeted while the latter inherits the combinatorial cost scaling of full configuration interaction QMC (FCIQMC). Inspired by these various approaches, this letter will introduce a Monte Carlo method that directly targets and variationally optimizes an individual excited state while maintaining a cost and optimization method very similar to ground state variational Monte Carlo. To date, the lack of a function that can be efficiently minimized to yield an exact excited state without reference to other states, as the energy E(Ψ) = hΨ|H|Ψi/hΨ|Ψi can be for the ground state, has hindered our ability to achieve direct optimization of excited states, i.e. optimization that does not also require solving for lower states. Instead, existing excited state methods typically require an ansatz to use its variational freedom to satisfy the needs of many eigenstates simultaneously, the difficulty of which has limited our predictive power over the doubly-excited states in light harvesting systems, the spectra of excited state absorption experiments, and the band gaps of transition metal oxides. For example, linear response (LR) methods such as time dependent HF and DFT, 9 CI singles (CIS), 9 equation of motion CC with singles and doubles (EOM-CCSD), 10 and LR density matrix renormalization group (DMRG) 11–13 are limited by the requirement that all excited states of interest must be found in the ground state’s LR space, which for a nonlinear ansatz is typically much less flexible than its full variational space. In many other cases, such as state-averaged complete active space methods, 14,15 some VMC approaches, 16 directly targeted DMRG, 17 and FCIQMC, 18,19 crucial ansatz components (often the one particle basis) are required to be the same for all states. While these methods clearly do not take full advantage of an ansatz’s variational freedom, they have been preferred due to the historical difficulties associated with the direct optimization of individual excited states.

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This report presents a new variational method consisting of two parts: first, a function Ω(Ψ) whose global minimum is an excited eigenstate, and second, a method for evaluating and minimizing Ω whose cost scales polynomially for a wide class of approximate wave functions. We will begin by proving that Ω has the necessary properties to be the basis of an excited state variation principle, after which we will detail our method for minimizing it. During this discussion, we will explain which wave functions are compatible with the approach, as well as its general applicability in molecules and solids. Finally, we will present numerical examples that demonstrate the method’s potential. We employ the function

Ω(Ψ) ≡

ω−E hΨ|(ω − H)|Ψi = 2 hΨ|(ω − H) |Ψi (ω − E)2 + σ 2

(1)

where σ 2 = hΨ|(H − E)2 |Ψi/hΨ|Ψi is the variance and the energy shift ω is assumed to be placed in between distinct eigenvalues of H in order to target the eigenstate whose energy is immediately above it. Assuming real numbers for brevity, we proceed to prove that this eigenstate is the global minimum of Ω as follows. First, we write an exact ansatz as a P linear combination of all eigenstates of H, |Ψe i = i ci |ii, and rewrite Ω in terms of H’s eigenvalues. P 2 c (ω − Ei ) Ω(~c ) = P i 2i 2 i ci (ω − Ei )

(2)

Differentiating with respect to the elements of ~c, we see that ~c is a stationary point (SP) if and only if

0 = ci (ω − Ei ) 1 − (ω − Ei )Ω ∀ i.

(3)

Recalling that ω is assumed to be distinct from any of H’s eigenvalues, we see that ~c cannot be a SP if any two of its elements that correspond to distinct Hamiltonian eigenvalues are



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non-zero, as this would prevent (1 − (ω − Ei )Ω) from vanishing for both of them. In other words, |Ψe i cannot be a SP of Ω unless the nonzero values in ~c all correspond to one (possibly degenerate) eigenvalue of H. As the eigenstates of H are clearly SPs of Ω(~c), we see that |Ψe i is a SP if and only if it is an eigenstate. At one of these SPs, Ω(~c) thus simplifies to 1/(ω − Ei ) and takes on negative values for SPs with Ei > ω. The global minimum (i.e. most negative value) of Ω(~c) therefore corresponds to the SP with Ei immediately above ω, as this maximizes the magnitude of 1/(ω − Ei ) while keeping its sign negative. (Note that this directionality, i.e. that we target the state above ω, is in our view the key difference ˜ (2) ˆ 2 between Ω(Ψ) and the related, non-directional ∆ x = hΨ|(ω − H) |Ψi form considered by Messmer 2 ). As |Ψe i can describe any state in Hilbert space, this value will be less than or equal to that of any approximate ansatz, thus achieving the variational property we desire. Note that this proof requires no assumption about the system’s boundary conditions, and so this variational principle has the potential to be applied to periodic systems as well as to molecules, for example to calculate band gaps by optimizing the wave function of the eigenstate at the bottom of the conduction band in a semiconductor. While formally interesting, the mere existence of a variational function for excited states is not useful without an efficient way to evaluate and minimize it. Indeed, the presence of H 2 makes the straightforward evaluation of Ω drastically more expensive than the ground state function E, which is why studies that have worked implicitly with this function in the past 17,20 have, to the best of our knowledge, always approximated this term (see discussion of harmonic Ritz methods below). As is done in variance evaluation 7 (and also by Booth and Chan, 8 but in a projector context) we avoid explicitly squaring H by resolving identities via complete sums over states, P

− H)|Ψi . m hΨ|(ω − H)|mihm|(ω − H)|Ψi

Ω(Ψ) = P

m hΨ|mihm|(ω

(4)

We may evaluate this sum (up to a controllable statistical uncertainty that obeys the zero


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variance principle) through Monte Carlo integration as P m∈ξ Wm ΩM C (Ψ) = P 2 m∈ξ |Wm |

Wm ≡

hm|(ω − H)|Ψi hm|Ψi

(5)

where the elements of ξ are sampled from |hm|Ψi|2 via a Metropolis walk (note that the normalization constants for numerator and denominator cancel and that we have corrected for the ratio-of-mean bias using Tin’s modified ratio estimator 21 ). Thus any ansatz admitting efficient evaluations for Wm will be compatible with our approach. This includes the wide class of wave functions already used in ground state VMC for molecules and solids, such as Slater-Jastrow (SJ), 22 multi-Slater-Jastrow (MSJ), 23 the Jastrow antisymmetric geminal power (JAGP), 24–29 and in principle even matrix product states. 30,31 Moreover, the method is applicable to both real space, in which case m is a position vector, and Fock space, in which case m is an occupation number vector. Furthermore, although we have not observed statistical pathologies in our initial tests, we are mindful that σ 2 can have infinite uncertainty if estimated naively, 32 and so it may be profitable in future to adopt a sampling scheme 33 that avoids this difficulty. In summary, essentially all the tools for ground state VMC appear to be useful and available for our proposed method. Crucially, this availability extends to wave function optimization methods, as we now demonstrate by presenting a generalization of the ground state linear method (LM). 34,35 Performing a linear expansion of |Ψi with respect to its variational parameters ~u,

|Ψi →

X

ai |Ψi i,

(6)

i

|Ψ0 i ≡ |Ψi

|Ψi i ≡ ∂|Ψi/∂ui ∀ i > 0,

(7)

we minimize Ω with respect to ~a by solving the generalized eigenproblem X

hΨi | (ω − H) − λ (ω − H)2 |Ψj iaj = 0.

j


(8)


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Assuming we were already near the minimum, in which case all ~a elements except a0 will be small, then we may use ~a to update ~u through a reverse Taylor expansion exactly as in the ground state LM. In practice, we may shift the eigenproblem 34,35 to ensure this assumption is valid even when far from the minimum. This linearize, diagonalize, and update procedure may be iterated to convergence in the same manner as Newton’s method, allowing us to optimize the ansatz’s linear and nonlinear parameters variationally via the minimization of Ω. As the matrix elements for the eigenproblem can be evaluated through the same stochastic identity resolution as described above, we arrive at a full-fledged and efficient method for the evaluation and minimization of Ω for any ansatz that can be efficiently used with the ground state LM in either molecules or solids. The precise cost scaling will of course depend on the choice of Ψ, with examples including Ns Ne3 for real space SJ and JAGP and Ns Ne4 for Hilbert space JAGP, where Ns and Ne are the number of samples and electrons, respectively, both of which will grow linearly with system size. Note the similarity of this eigenvalue equation to the harmonic Davidson equation that arises in applications 17,20,36,37 of the harmonic Ritz principle 38,39 for targeting interior eigenvalues of a matrix. In fact, some of these approaches 17,20 appear to have been minimizing an approximation to Ω with respect to linear parameters, in which P H 2 P was approximated by P HP HP , where P is the projector into the subspace corresponding to the linear parameters in question. Except for its controllable statistical uncertainty, the present approach makes no approximation when evaluating Ω and can optimize both linear and nonlinear parameters. As our methodology is expected to be used with an approximate ansatz in practice, it is important to consider the consequences of a non-zero value for the variance σ 2 in Eq. (1). To guide this discussion, we plot in Figure 1 the value of Ω against the shift ω for the JAGP applied to the first two excited states of a regular, 1.5Å-edge-length H6 hexagon. We first notice that instead of diverging at ω = E as would occur for an exact ansatz, Ω instead has a finite minimum near ω = E − σ, which is in fact the analytic solution for the minimum of Eq. (1) when E and σ are held fixed. A direct consequence of this downward shift of


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Ω’s minimums is that the value of ω at which the character of the global minimum switches states is shifted downwards as well, and no longer occurs at the energy of the lower state. The practical consequence is that the range of ω values that target a given state (i.e. those for which that state is the global minimum of Ω) gets shifted downward due to the non-zero variance of the wave function. Happily, the variance is readily evaluated in VMC, and so the magnitude of these shifts may be readily estimated and accounted for when selecting an ω with which to target a state. A second consequence of using an approximate rather than exact ansatz is that the optimized wave function, and thus also its energy, may now depend on the precise choice of ω. In our tests so far, we have observed this energy dependence to be quite small (for an example, see Figure S1), and so the precise choice of ω does not meaningfully affect our results. This observation can be explained by considering that as a wave function becomes more accurate, the point in wave function variable space that minimizes Ω must align with the corresponding stationary point for the energy, and as the energy’s first derivatives are zero at such a point, small changes to the wave function induced by adjusting ω will have only a very small effect on the energy value. However, in cases where the wave function approximation is poor, this mechanism will likely break down and a strong dependence of the energy on ω may arise. We therefore propose that in the future, ω should be chosen to minimize Ω for the state in question, thus removing it as a free parameter and preventing the user for adjusting it to select a desired result. We note that optimizing ω automatically alongside the wave function would require only small changes to the optimization algorithm and would have the added benefit of making the method more black-box. Other consequences of using an approximate ansatz include the possibility of symmetry breaking, as occurs in ground state variational methods, and also the loss of orthogonality between the approximate eigenstates. If desired, these issues can be addressed by performing a configuration interaction between the optimized wave functions, which is trivial to perform within VMC for any ansatzes that may be efficiently optimized via Ω. Note that we only



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observed symmetry breaking in the case of CH2 ’s 5th and 6th excited states, for which we report the energies after a 2 state re-diagonalization. Finally, note that all the effects of using an approximate ansatz may be systematically eliminated by increasing the ansatz’s flexibility, ensuring the same systematically improvable accuracy that makes the ground state variational principle so powerful. Before presenting results, we should also point out an important difference between optimizing E and Ω. Using Ω, the quality of the wave function depends strongly on both the value of its energy and its variance, as they are both important for shrinking the magnitude of the denominator in Eq. (1). Ω minimization thus has as much in common with variance minimization 7 as it does with pure energy minimization. Therefore, just as it is biased (in some cases 40 by 0.5 eV) to take an energy difference between states when one is optimized for σ 2 and the other for E, it would be biased to take such a difference between states when one was optimized for E and the other for Ω. For this reason, we report results soley for wave functions optimized via Ω, even when energy minimization is possible due to a state being the ground state or the lowest in its symmetry. As a demonstration in Fock space, we applied the method to optimize the Hilbert space JAGP 29 for singlet excited states in an H6 ring (Figure 1), CH2 (Figure 2), and C2 (Figure 3). See the supplemental information for full computational details. In CH2 , the two doubly excited states are absent in CIS due to HF’s limited LR space and are treated poorly by EOM-CCSD. While CCSD’s LR space contains doubles, it lacks the triples necessary to describe the orbital relaxations that should accompany the excitation. Although JAGP’s LR space also lacks triples, which becomes clear when one considers that its Jastrow factor can be written as a constrained CC doubles operator, 41,42 it agrees much better with full CI (FCI), 43 because the variational minimization of Ω explores regions of parameter space beyond the LR regime. The excited states of H6 are even more challenging, each having 12 or more normalized FCI coefficients above 0.1 as compared to 8 or fewer for CH2 . Nonetheless, the same pattern emerges: large errors in EOM-CCSD are reduced by an order of magnitude


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in variationally optimized JAGP. We should stress that multi-reference perturbation theory is preferred over EOM-CCSD for double excitations, and that we compare to EOM-CCSD not because it is a benchmark, but because it is among the most reliable polynomial cost methods in wide use. C2 provides further evidence of JAGP’s superiority to EOM-CCSD for double excitations while also revealing the limits of the ansatz’s flexibility. While JAGP delivers 0.1 eV accuracy versus FCI for excited states 1, 2, 4, and 5, it shows an error almost as large as EOM-CCSD for state 3, a complicated excitation involving four different electrons in a mixture of double excitations. Moreover, JAGP’s accuracy (and that of EOM-CCSD for the single excitations), is more dependent on error cancellation in this case, as seen in the total energy data provided in the supplemental material. This raises the important point that, just like selecting a ground state ansatz to be balanced at, say, both equilibrium and stretched geometries, it is important in the present approach to select an ansatz that is not obviously unbalanced for the different excited states involved. To show the method’s systematic improvability and compatibility with a real space Monte Carlo walk, we have also treated C2 with a MSJ ansatz consisting of short configuration state function (CSF) expansions and spline-based 1- and 2-body Jastrow factors (Figure 4). For each state, we selected CSFs with coefficients above a given threshold from a complete active space (CAS) wave function, leading to fewer than 10 (65) CSFs per state for a threshold of 0.1 (0.01). Under variational optimization (with the random walk now in real space), the worst-case MSJ excitation energy error is found to drop from 0.3 to 0.1 eV upon lowering the threshold, as expected for a systematically improvable method. As a benchmark we use Davidson-corrected multi-reference CI (MRCI+Q) in a triple-zeta basis, which for 44 excited state 5 (the 1 Σ+ and g state) is within 0.03 eV of the recent quadruple-zeta DMRG

FCIQMC 18 benchmarks. Significantly, our MSJ result for this state (2.57eV) is within 0.1 eV of these benchmarks (2.47eV) and cc-pVTZ auxiliary field results (2.65eV), 45 despite containing fewer than 100 variational parameters, compared to more than 4,000 in EOM-CCSD,



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millions in DMRG, and 2,000 in the FCIQMC trial function. This success, along with MSJ’s high accuracy for C2 ’s other excited states, demonstrates the advantage of optimizing an ansatz directly and variationally for an individual excited state. We have presented a Monte Carlo method for the efficient, variational optimization of a function Ω whose global minimum can be tuned to target individual excited states and which may be used at polynomial cost with a wide range of approximate wave functions. In demonstrations on three molecules with low-lying doubly excited states, the method’s ability to explore an ansatz’s full variational freedom allows for drastic improvements in accuracy compared to linear response theories such as EOM-CCSD, which is among the most reliable polynomial-cost methods for excited states in chemistry (of course, when affordable, exponentially scaling multi-reference methods are preferred). Further, we have shown that for the notoriously difficult double excitations of the carbon dimer, variational optimization allows a very modest multi-Slater Jastrow expansion to achieve accuracies on par with the much more cumbersome DMRG and FCIQMC benchmarks. Given the importance of double excitations in light harvesting and excited state absorption experiments, the method’s compatibility with periodic boundary conditions and thus the solid state, its systematically improvable nature, and its strong similarities to the ground state variational principle, we look forward to its further development and application.

Acknowledgement The authors acknowledge funding from the Office of Science, Office of Basic Energy Sciences, the US Department of Energy, Contract No. DE-AC02-05CH11231. Calculations were performed using the Berkeley Research Computing Savio cluster.

Supporting Information Available The file SuppInfo.pdf provides computational details, total energies, a discussoin of CH2 ’s 10 ACS Paragon Plus Environment

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symmetry breaking, and a plot showing an example of the insensitivity of the energy to the precise choice of ω.

This material is available free of charge via the Internet at http:

//pubs.acs.org/.

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EOM−CCSD MSJ 0.1 MSJ 0.01 Journal of Chemical MRCI+Q 3 CASSCF

Theory and Computation

2 1

Graphical TOC Entry 0

*

−1 Excitation

Error vs MRCI+Q (eV)

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Excitation Energy (eV)

4

2

*

*

Energy Errors in the Carbon Dimer

EOM-CCSD New Method

0

*

*

CIS

−2 1

2 3 4 Excited State Number

*

CASSCF

5


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Excite

−3.5 −4.0

σ

−4.5 −3.3

2 d State

Ω (1/Hartree)

−3.0

Excite

d State

1

−2.5

σ

−3.2 −3.1 −3.0 ω or E (Hartree)

−2.9

Figure 1: Ω vs ω for the first two excited states of an H6 ring in the 6-31G basis, where ΨJAGP is optimized to minimize Ω at each ω value. At bottom right, solid vertical lines show FCI energies for these two states, while filled and slashed bars show deviations from FCI for JAGP and EOM-CCSD, respectively (note JAGP’s deviation for state 2 is too small to be visible). The length-σ arrows give graphical confirmation that the ω value that minimizes Ω is roughly E − σ, as expected from our analysis of Eq. (1).

16 14


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12

CIS EOM−CCSD JAGP FCI

10 8 6 4 2

Error vs FCI (eV)

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0 0.8

*

*

*

*

0.4 0.0

−0.4

1 /1 B1 2 /2A1 3 3/ B2 4 4/ A2 5 5/ B1 6 6/ A2 ExcitedExcited State Number / Symmetry State Number

Figure 2: Singlet excitations for CH2 in a STO-3G basis. Lines mark ω values. Asterisks mark doubly excited states. Note that the 5th and 6th states’ energies are reported after the symmetry-restoring 2x2 re-diagonalization.



5


4

CIS EOM−CCSD JAGP FCI

3 2 1 0

*

*

*

*

*

*

Error vs FCI (eV)

−1 2 0 −2 −4

1 1/ Π" 2 2/ Π" 3 3/ Σ$% 4 4/ Δ$ 5 5/ Δ$ ExcitedExcited State Number / Symmetry State Number

Figure 3: Singlet excitations for C2 in a 6-31G basis. Asterisks mark doubly excited states.

5


4 3

CIS EOM−CCSD MSJ 0.1 MSJ 0.01 MRCI+Q CASSCF

2 1 0

*

*

*

*

*

*

−1

Error vs MRCI+Q (eV)

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

2 0 −2

1 1/ Π" 2 2/ Π" 3 3/ Δ$ 4 4/ Δ$ 5 5/ Σ$% ExcitedExcited State Number / Symmetry State Number

Figure 4: Singlet excitations for C2 in a cc-pVTZ basis with MSJ in real space. Asterisks mark doubly excited states.


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An Efficient Variational Principle for the Direct ... - ACS Publications

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