α-CASSCF: An Efficient, Empirical Correction for SA-CASSCF To

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α‑CASSCF: An Efficient, Empirical Correction for SA-CASSCF To Closely Approximate MS-CASPT2 Potential Energy Surfaces James W. Snyder, Jr.,†,‡ Robert M. Parrish,†,‡ and Todd J. Martínez*,†,‡ †

Department of Chemistry and The PULSE Institute, Stanford University, Stanford, California 94305, United States SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States

‡

S Supporting Information *

ABSTRACT: Because of its computational efficiency, the state-averaged complete active-space self-consistent field (SA-CASSCF) method is commonly employed in nonadiabatic ab initio molecular dynamics. However, SA-CASSCF does not effectively recover dynamical correlation. As a result, there can be qualitative differences between SA-CASSCF potential energy surfaces (PESs) and more accurate reference surfaces computed using multistate complete active space second-order perturbation theory (MSCASPT2). Here we introduce an empirical correction to SA-CASSCF that scales the splitting between individual states and the state-averaged energy. We call this the αCASSCF method, and we show here that it significantly improves the accuracy of relative energies and PESs compared with MS-CASPT2 for the chromophores of green fluorescent and photoactive yellow proteins. As such, this method may prove to be quite valuable for nonadiabatic dynamics.

T

focus on accurately describing the ground state and usually require an initial estimate of the correlation energy, which can be scaled or extrapolated. These methods can essentially be split into two classes. One class of methods seeks to approximate the exact Schrö dinger equation directly by summing and scaling the energies (or energy differences) computed with two or more theoretical methods and modest basis sets.29−33 The other class attempts to extrapolate the results computed at a particular level of theory in a computationally tractable basis set to the infinite basis set limit.34−38 With high enough levels of theory as input, one expects such approaches to provide very accurate energetics. In this work, we seek to provide such an extrapolative approach but with no explicit computation of the dynamical correlation energy. This would limit the computational expense, making long-time-scale dynamics feasible. Moreover, we demand that the method be applicable to excited states, including regions of near-degeneracy around conical intersections. An alternative approach proposed by Frutos et al. simply scaled the energy of each SA-CASSCF state and gradient by a constant factor.20 This was not an entirely new idea. Frequency scaling is commonly employed to adjust the harmonic frequencies computed at local minima using Hartree−Fock, which are overestimated by a factor of ∼0.89.40−42 Scaling the entire PES is essentially the same concept, except the scaling factor is employed at all geometries (and not just at local minima). Interestingly, the energy scaling parameter employed by Frutos et al. is 0.795, which is exceptionally close to the

he state-averaged complete active-space self-consistent field1−8 (SA-CASSCF) method is one of the most widely used methods in nonadiabatic ab initio molecular dynamics. Because of the computational efficiency of recent implementations,9−11 thousands of sequential energy and response property evaluations can be easily performed for large molecules. Moreover, SA-CASSCF is often at least qualitatively accurate and can correctly describe the topology of the potential energy surface (PES) in the vicinity of a conical intersection,12−15 even in cases where one of the degenerate states is the ground state. Despite these attractive qualities, SACASSCF largely neglects dynamic electron correlation. As a result, the PESs produced by SA-CASSCF and more accurate reference methods, such as multistate complete active-spacesecond order perturbation theory16,17 (MS-CASPT2), are sometimes qualitatively different.18−23 This diminishes the reliability of results obtained with SA-CASSCF, although this problem can be ameliorated to some degree by tuning the active and state spaces to reproduce MS-CASPT2 results. Unfortunately, inexpensive methods that recover dynamic electron correlation, including time-dependent density functional theory24,25 (TD-DFT) and approximate coupled-cluster singles-and-doubles26 (CC2), do not accurately capture the topology of the PES near certain classes of conical intersections.13,27,28 For this reason, there is significant value in developing a computationally efficient method that incorporates dynamical correlation effects into the SA-CASSCF PES without diminishing its ability to accurately describe conical intersections. The use of a few empirical parameters provides such an opportunity. There have been numerous attempts to recover correlation energy using empirically derived parameters.29−39 Most works © XXXX American Chemical Society

Received: April 18, 2017 Accepted: May 17, 2017 Published: May 17, 2017 2432

DOI: 10.1021/acs.jpclett.7b00940 J. Phys. Chem. Lett. 2017, 8, 2432−2437

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The Journal of Physical Chemistry Letters canonical frequency scaling factor of 0.89 (once the appropriate square root is taken). Although the energy scaling method of Frutos et al. was highly successful for the treatment of retinal protonated Schiff base (RPSB), there are a few potential problems. For example, scaling the entire PES by a constant factor assumes that the relative energy of all points is approximately correct within a constant factor. This is clearly not the case for the green fluorescent protein chromophore, for example.19 Moreover, the absolute energies obtained by this scaling procedure are problematic. For example, a typical absolute energy might be on the order of a thousand hartrees, implying that the scaling correction changes the absolute energy by almost 100 hartree (almost 3000 eV). Nonetheless, the efficacy of this method for RPSB leads one to wonder whether there is a similar energy and gradient scaling scheme that works for a broad array of molecules and leads to more reasonable absolute energies. We introduce such an energy and gradient scaling theory and show that it is surprisingly accurate for a variety of molecules and multiple excited states. Instead of scaling the entire energy (and consequently its gradient), we consider the state-averaged energy and the state-specific energy splittings separately. We then leave the state-averaged energy from SA-CASSCF untouched and scale only the energy splittings by a constant factor α. We call this the α-CASSCF method. Because it is a scaling scheme, α-CASSCF exhibits the same efficiency as SACASSCF. However, unlike SA-CASSCF, it produces a potential energy surface with low nonparallelity error (NPE) relative to the more accurate MS-CASPT2 reference. Therefore, we believe this method will prove useful for nonadiabatic dynamics and photochemistry. The energy of the Θth state at the SA-CASSCF level of theory can be written in terms of two components, the stateaveraged energy and the splitting from this average CAS EΘCAS = ESA + ΔEΘ

Figure 1. Transformation between the SA-CASSCF potential energy surface and the α-CASSCF surface is displayed for a fictitious system. At a given point, the excitation energy is scaled by α. However, the relative energy may increase or decrease between different structures, and by extension, barriers may be raised or lowered. The conical intersection seam space does not change.

positions as well as the relative energy between various critical points are different. Barrier heights may also vary. One feature that does remain unchanged when comparing the SA-CASSCF and α-CASSCF potential energy surfaces is the location of the seam space for conical intersections. However, the relative energy of points along the seam may change if more than two states are included in the state-averaged space. Moreover, the topology in the vicinity of any conical intersection43−45 (e.g., “sloped” versus “peaked”) may change. This feature of αCASSCF enables it to approximately reproduce the PESs computed with MS-CASPT2. However, the α-CASSCF PES also depends on the choice of active space (number of electrons and orbitals) and the number of states included in the stateaveraging. Therefore, care must be taken to ensure that these are selected appropriately. This can be readily accomplished using a simple fitting procedure, as discussed below. Although the α-CASSCF method might be considered partially semiempirical (involving one parameter that must be determined), it is substantially more efficient than parameterfree methods like multireference configuration interaction and MS-CASPT2. Because the dominant computational cost in αCASSCF is just the underlying SA-CASSCF calculation, it inherits the cost and scaling behavior of SA-CASSCF. With modern algorithms,9−11 the computational scaling (with respect to molecular size) of SA-CASSCF is as low as O(N2) without approximations. Linear scaling methods (for fixed active space) can be further envisioned by orbital localization.46−48 Statespecific gradients and nonadiabatic coupling vectors are easily computed with existing SA-CASSCF response property machinery. In the case of the gradient

(1)

We posit that SA-CASSCF systematically overestimates the splitting by an approximately constant multiplicative factor, which we will call α. As such, the α-CASSCF energy can be expressed in the following way CAS CAS EΘαCAS = ESA + αΔEΘ = αEΘCAS + (1 − α)ESA

(2)

Here α is an empirical parameter that can be used to tune the splitting to approximately match a more accurate reference or experimental observable. Note that the empirical parameter α will be chosen to be system-specific but not geometry-specific. At a given geometry, the excitation energy at the α-CASSCF level of theory between states Θ and Φ is equal to the excitation energy at the SA-CASSCF level of theory scaled by α. EΘαCAS − EΦαCAS = α(EΘCAS − EΦCAS)

(3)

This relative energy relationship, however, does not hold between molecular structures because the state-averaged energy depends on the geometry.

∂EΘαCAS ∂E CAS ∂E CAS = α Θ + (1 − α) SA ∂x ∂x ∂x

EΘαCAS(R1⃗ ) − EΘαCAS(R⃗ 2) = α[EΘCAS(R1⃗ ) − EΘCAS(R⃗ 2)] CAS ⃗ CAS ⃗ + (1 − α)[ESA (R1) − ESA (R 2)]

(4)

(5)

requiring only the state-specific and state-averaged SA-CASSCF gradients for evaluation. Both of these are straightforward to compute, as demonstrated in previous work.9−11 The nonadiabatic coupling vectors between states Θ and Φ at the αCASSCF and SA-CASSCF levels of theory are identical

As a result, the α-CASSCF potential energy surface can be qualitatively different from the SA-CASSCF potential energy surface, beyond just a rigid shift or compression factor. This is demonstrated schematically in Figure 1. As one can see, 2433

DOI: 10.1021/acs.jpclett.7b00940 J. Phys. Chem. Lett. 2017, 8, 2432−2437

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The Journal of Physical Chemistry Letters ∂ Θ Φ ∂x

αCAS

=

∂ Θ Φ ∂x

CAS

(RMS) error of the relative energy of each state at each critical point; degenerate states were counted only once. The use of the α scaling parameter substantially reduces the RMS error with respect to the MS-CASPT2 relative energies from 2.06 to 0.23 eV, improving the results at every state and point. This suggests that the NPE between α-CASSCF and MS-CASPT2 is quite low, and we would expect the dynamics of the system treated at the two levels of theory to be qualitatively similar. However, α-CASSCF is substantially faster than MS-CASPT2. Even for a 27-atom system treated with a relatively small active space, α-CASSCF is on the order of 100 times faster than MSCASPT2 for combined energy and gradient evaluations when employing the fastest codes for each method.9,10,50−53 Furthermore, α-CASSCF exhibits weaker computational scaling than MS-CASPT2. As such, α-CASSCF can be employed to treat much larger systems at longer time scales relative to MSCASPT2. Our results demonstrate the advantage of our SA-CASSCF scaling scheme relative to that proposed by Frutos et al., which is also compared in Figure 2.20 The optimal scaling factor for the method of Frutos et al. was determined in the same manner as α and is equal to 0.75, resulting in an RMS error of 0.44 eV relative to MS-CASPT2. As one can see, α-CASSCF is superior at most points, with the difference being particularly salient at the minimum energy conical intersection (MECI). Scaling the total energy by a constant can only ever reduce the relative energy of two geometries and is therefore guaranteed to preserve the relative energetic ordering of geometries. However, MS-CASPT2 exaggerates the energy difference between Twist-I/S1 and the MECI and inverts the relative energy between the MECI and Twist-P/S1 relative to SACASSCF. Our α-CASSCF method faithfully reproduces this behavior. Further tests will be required to make a determination as to the universality of this result. For some of the molecules we have treated (e.g., butadiene and p-coumaric acid) the two scaling methods exhibit similar performance. Our method is clearly superior for the treatment of HBDI, the chromophore of many fluorescent proteins, and enables the high accuracy study of this important class of molecules. The accuracy of α-CASSCF extends beyond its ability to accurately capture the relative energy of a few critical points. We computed the minimum energy pathway between planar trans-p-coumaric acid, the photoactive yellow protein (PYP) chromophore, and 90°-twisted p-coumaric acid at the SA-5CAS(6,5)/6-31G* level of theory with nudged elastic band (NEB).54 Using these points, we computed the PES at the MS5-CAS(6,5)-PT2/6-31G* and (α = 0.8)-5-CAS(6,5)/6-31G* levels of theory. The results are displayed in Figure 3. As before, the parameter α was selected to minimize the RMS error of the relative energy for a set of critical points, in this case each electronic state of the planar and twisted S1 minima. We did not optimize the PESs at each level of theory in this case, as it was cost prohibitive to do so. On average, it required 56.6 h to do a single energy evaluation, let alone a gradient evaluation, on the 20-atom PYP chromophore at the MS-CASPT2 level of theory. In contrast, the combined calculation of the energy and gradient at the SA-CASSCF and α-CASSCF levels of theory required ∼30 s on a single GPU in mixed precision. As can be seen in Figure 3, the α-CASSCF method significantly reduces the error with respect to MS-CASPT2. This demonstrates that a few important control points can be used to determine α, substantially reducing the computational effort required to parametrize the method.

(6)

This is the case because the wave-function parameters are determined entirely at the SA-CASSCF level of theory, and the α parameter does not affect how the wave function changes with respect to a perturbation to the nuclear coordinates. Of course, this means that the interstate coupling vector, often called h, is scaled by α because the energy difference is scaled by α. (See eq 3.) Similarly, the gradient difference vector, often called g, is scaled by α, which is clear based on eqs 3 and 5. This scaling preserves the Berry phase.49 Before discussing our results, we introduce some terminology that we will employ throughout this paper. We will frequently use the notation SA-Ns-CAS(m,n) to describe a SA-CASSCF wave function, where Ns is the number of states being averaged, m is the number of active electrons, and n is the number of active orbitals. Similarly, the α-CASSCF wave function will be denoted (α=a)-Ns-CAS(m,n), where a is the numerical value of the parameter α, and the MS-CASPT2 wave function will be denoted MS-Ns-CAS(m,n)-PT2. In all cases, the weights of each of the Ns states are equal and all Ns states have the same spin multiplicity. More details are provided in the Supporting Information. Critical point searches for anionic 4-hydroxybenzylidene-1,2dimethylimidazoline (HBDI), the green fluorescent protein chromophore, were performed at the SA-2-CAS(4,3)/6-31G*, (α = 0.73)-2-CAS(4,3)/6-31G*, and MS-2-CAS(4,3)-PT2/631G* levels of theory. The results are shown in Figure 2. The parameter α was selected to minimize the root-mean-squared

Figure 2. Critical points on the potential energy surface of HBDI computed with MS-CASPT2, SA-CASSCF, α-CASSCF, and the method of Frutos et al. The critical points include the Franck− Condon point (FC), the planar S1 minimum (Planar), the phenyl twisted S1 minimum (Twist-P), the imidazole twisted S1 minimum (Twist-I), and the minimum energy conical intersection (MECI). For this system, the optimal value of α was determined to be 0.73, and the RMS error versus MS-CASPT2 is 0.23, 0.44, and 2.06 eV for αCASSCF, the method of Frutos et al., and SA-CASSCF. A scaling parameter of 0.75 was used to compute the relative energies using the method of Frutos et al. to minimize the RMS error relative to MSCASPT2. Despite this, α-CASSCF is a better estimate of MS-CASPT2 at most points, especially the MECI. This result indicates that the αCASSCF potential energy surface is close to parallel to the MSCASPT2 surface. 2434

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spaces for provitamin D3.55 Second, select α to reproduce an experimental observable, such as the absorption spectrum. Ideally, the results would then be validated by comparison with experiments for which the parameter α was not optimized. We have developed a new method called α-CASSCF, which empirically corrects SA-CASSCF to more closely approximate MS-CASPT2 potential energy surfaces. α-CASSCF maintains all of the advantages of SA-CASSCF, including its efficiency and ability to capture the topology of the PES near conical intersections. The use of the α scaling parameter substantially improves the results for HBDI, p-coumaric acid, butadiene, and cyclohexadiene (results for latter two are provided in the Supporting Information). This indicates that α-CASSCF works for a wide variety of systems. Although α-CASSCF is not a panacea (see results for uracil in Supporting Information) and surely not a replacement for MS-CASPT2 or other methods that calculate dynamic correlation explicitly, one can expect it will almost invariably be more accurate than SA-CASSCF. Because α-CASSCF is essentially a scaling scheme for SACASSCF, it still exhibits some of the deficiencies of SACASSCF, such as state inversion and energy discontinuity problems. In most cases, these problems can be resolved with a judicious selection of the active and state spaces18,23 as well as dynamic weighting schemes.56,57 Nonetheless, α-CASSCF seems to mimic dynamic electron correlation effects at a small fraction of the cost of MS-CASPT2 and other highaccuracy reference methods without explicit computation of the correlation energy. As a result, we expect that α-CASSCF will prove to be very useful for the purposes of nonadiabatic ab initio molecular dynamics of large molecules for long time scales.

Figure 3. Minimum energy potential energy surface connecting planar trans-p-coumaric acid and 90°-twisted p-coumaric acid is displayed. Points along the pathway were determined using nudged elastic band at the SA-5-CAS(6,5)/6-31G* level of theory. Single point energy evaluations were then performed with MS-CASPT2 and α-CASSCF. For this case, α was optimized to be 0.80. The potential energy surfaces determined with MS-CASPT2 and α-CASSCF exhibit low nonparallelity error, despite only optimizing α for the relative energy at the reactant and product structures.

Although one does not expect that α is a universal parameter, we have found that optimization often leads to values near 0.8. This is strikingly close to the average optimal scaling parameter of 0.89 employed in Hartree−Fock frequency scaling,40−42 which corresponds to α = 0.79. There are a variety of ways to determine the optimal α at varying computational expense. Because the objective of the α-CASSCF method is to minimize the NPE between SA-CASSCF and a higher accuracy reference, the best approach to determine α would be to scan the entire potential energy surface with a high accuracy method and select α to minimize the NPE. Of course, this is cost prohibitive for most interesting problems. In this work, we selected α to minimize the RMS error between a few critical points and demonstrated that this improves the results across the entire PES. We believe that this method is generally the most robust, yet computationally tractable, α optimization strategy. In cases where even this strategy is intractable one might select α to minimize the RMS error between states at a single point, such as the Franck−Condon point. In fact, doing this for HBDI would result in α = 0.71 (compared with α = 0.73 from the more involved procedure). This would have resulted in an RMS error of 0.28 eV for the full set of critical points, which is only modestly worse than the 0.23 eV RMS error obtained for α = 0.73. The reason for this close level of agreement is due to the fact that the most striking error for SA-CASSCF is the blue shift at the Franck−Condon point. For the largest systems treatable with α-CASSCF, even one reference MS-CASPT2 calculation might be impossible. In this case, we recommend two possible approaches. First, optimize α for a representative subsystem. For example, provitamin D3 is too large to treat with MSCASPT2, but it has a cyclohexadiene chromophore. Because cyclohexadiene is quite small, α can be optimized for cyclohexadiene with the critical point method and used for provitamin D3 computations. In fact, this strategy was successfully used in prior work to select the active and state

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b00940.

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Computational details, molecular geometries for comparisons presented in the text, and further benchmarks of αCASSCF for butadiene, cyclohexadiene, uracil, and provitamin D3 molecules. (PDF) Coordinates for all molecules used in benchmarks and associated input/output files. (ZIP)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Todd J. Martínez: 0000-0002-4798-8947 Notes

The authors declare the following competing financial interest(s): T.J.M. is a cofounder of PetaChem, LLC.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation (Grant No. ACI-1450179) and used the XStream computational resource supported by the NSF MRI Program (Grant No. ACI-1429830). J.W.S. is grateful to the National Science Foundation for an NSF Graduate Fellowship. 2435

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DOI: 10.1021/acs.jpclett.7b00940 J. Phys. Chem. Lett. 2017, 8, 2432−2437