Dynamically weighted multireference perturbation theory: Combining the advantages of multi-state and state-averaged methods

Analytical nuclear gradient and derivative coupling theories for multireference perturbation methods

Electron correlations should be appropriately included in quantum chemistry calculations to accurately describe the energy and wave functions. In multiconfigurational methods, the reference functions are written as linear combinations of multiple electronic configurations to describe static correlations. Using the multiconfigurational reference functions, it is also possible to correct for dynamical correlations using various methods. Geometry optimizations and dynamics simulations are among the most prominent applications of quantum chemistry methods. Such applications become much more straightforward when analytical nuclear gradients are available. Many efficient algorithms for computing analytical nuclear gradients and derivative coupling using multireference perturbation theories (MRPTs) have recently been developed. This work aims to provide a comprehensive and easy-to-follow review of analytical gradient theories and the properties of methods for obtaining analytical gradients and derivative coupling methods using MRPTs. We also briefly review the practical applications of these methods in performing nonadiabatic dynamics simulations.

A Density Functional Valence Bond Study on the Excited States

The accurate description of excited states is crucial for the development of electronic structure theory. In addition to determining excitation energies, strong state interactions arise when electronic states with the same symmetry are degenerate or nearly degenerate, often requiring a multi-state treatment. These strong correlation effects and state interactions can be effectively handled by the Hamiltonian matrix correction-based density functional valence bond (hc-DFVB) method, a multi-reference density functional theory capable of accurately describing electronic state interactions. In this paper, we explore the low-lying excited states of four isoelectronic systems (C2H, CN, CO+, BO) using valence bond methods, including the valence bond self-consistent field (VBSCF) and hc-DFVB methods. Our results show that the hc-DFVB method provides significantly better excitation energies compared to VBSCF. Furthermore, hc-DFVB can reliably predict the correct ordering of excited states, whereas VBSCF shows some ordering inconsistencies. By categorizing the VB structures into groups based on point group symmetry, we can extract the key structural contributions and bonding pictures of each state from the weight distribution of these groups. Additionally, we study the potential energy curves for lithium fluoride (LiF) and a mixed-valence spiro cation, demonstrating the superior performance of hc-DFVB when applied to the study of near-degenerate excited states in the avoided crossing region.

Perturbation theory in the complete degenerate active space (CDAS-PT2)

Methods based on the multireference perturbation theory (MRPT) with the one-electron zeroth-order Hamiltonian are widely used for the description of excited states, for example, due to their relatively low computational cost. However, current methods have a common drawback-use of a model space with low size. In this article, we propose the MRPT method with the model space extended to the complete active space. The one-electron zeroth-order Hamiltonian suitable for this extension is formulated. The proposed method was applied to common models, such as LiF, ethylene, and trans-butadiene. It was shown to have accuracy superior to XMCQDPT2 in most cases, especially in the case of the small active space.

Hybrid Density Functional Valence Bond Method with Multistate Treatment

Recently, a hybrid density functional valence bond (VB) method, λ-DFVB(U), has been proposed and shown to give accuracy that is comparable to that of CASPT2 in calculations of atomization energies, atomic excitation energies, and reaction barriers, while its computational cost is approximately the same as the valence bond self-consistent-field (VBSCF) method. However, the interaction between electronic states is not included in λ-DFVB(U) since the last step of λ-DFVB(U) is not a diagonalization of the Hamiltonian matrix on the electronic state basis. Therefore, λ-DFVB(U) gives the wrong topology of the potential energy surfaces (PESs) near the conical intersection region. In the present paper, we propose a novel hybrid density functional VB method with multistate treatment, named λ-DFVB(MS), in which an effective Hamiltonian matrix is constructed on the basis of the diabatic states obtained by the valence-bond-based compression approach for the diabatization scheme, and the interaction between electronic states can be included through the diagonalization of the effective Hamiltonian matrix. Test calculations show that λ-DFVB(MS) gives the correct topology of the PESs near the conical intersection region. We also show that the VBSCF wave function with selected VB structures can be applied as a reference in λ-DFVB(MS).

Linearized Pair-Density Functional Theory

Multiconfiguration pair-density functional theory (MC-PDFT) is a post-SCF multireference method that has been successful at computing ground- and excited-state energies. However, MC-PDFT is a single-state method in which the final MC-PDFT energies do not come from diagonalization of a model-space Hamiltonian matrix, and this can lead to inaccurate topologies of potential energy surfaces near locally avoided crossings and conical intersections. Therefore, in order to perform physically correct ab initio molecular dynamics with electronically excited states or to treat Jahn-Teller instabilities, it is necessary to develop a PDFT method that recovers the correct topology throughout the entire nuclear configuration space. Here we construct an effective Hamiltonian operator, called the linearized PDFT (L-PDFT) Hamiltonian, by expanding the MC-PDFT energy expression to first order in a Taylor series of the wave function density. Diagonalization of the L-PDFT Hamiltonian gives the correct potential energy surface topology near conical intersections and locally avoided crossings for a variety of challenging cases including phenol, methylamine, and the spiro cation. Furthermore, L-PDFT outperforms MC-PDFT and previous multistate PDFT methods for predicting vertical excitations from a variety of representative organic chromophores.

Assessment of State-Averaged Driven Similarity Renormalization Group on Vertical Excitation Energies: Optimal Flow Parameters and Applications to Nucleobases

We present a comprehensive excited-state benchmark for the state-averaged (SA) driven similarity renormalization group (DSRG) [Li, C.; Evangelista, F. A. J. Chem. Phys. 2018, 148, 124106]. Following the QUEST database [Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; Loos, P.-F. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2021, 11, e1517], 280 vertical transition energies of 35 medium-sized molecules are computed using the SA-DSRG derived second- and third-order perturbation theories (PT2/PT3) along with a nonperturbative approach [sq-LDSRG(2)]. Comparing to the theoretical best estimates, the optimal flow parameter is found to be 0.35 and 2.0 Eh-2 for SA-DSRG-PT2 and SA-DSRG-PT3, respectively. For SA-sq-LDSRG(2), a flow parameter of 1.5 Eh-2 provides converged equations without compromising the accuracy. We then assess the accuracy of the SA-DSRG hierarchy using these parameters. The SA-DSRG-PT2 scheme outperforms the level-shifted CASPT2 by 0.10 eV in mean absolute error (MAE), yet this accuracy is slightly inferior than that of CASPT2 with the ionization-potential-electron-affinity shift. Both SA-DSRG-PT3 and SA-sq-LDSRG(2) yield a MAE of 0.10 eV, which is comparable to that of CASPT3 (0.09 eV). Finally, we compute vertical excitation energies of several low-lying singlet states of nucleobases. The SA-sq-LDSRG(2) approach provides highly accurate results for π → π* excitations, while n → π* transitions are better described by SA-DSRG-PT3.

Analytic First-Order Derivatives of (X)MS, XDW, and RMS Variants of the CASPT2 and RASPT2 Methods

Crossings between states involve complex electronic structures, making the accurate characterization of the crossing point difficult. In this study, the analytic derivatives of three complete active space second-order perturbation theory (CASPT2) variants as well as an extension of the restricted active space (RASPT2) are developed. These variants are applied to locating minimum energy conical intersections. Our results demonstrate that the three CASPT2 variants predict qualitatively similar results, but a recently developed variant, the rotated multistate CASPT2 (RMS-CASPT2), is least sensitive to the number of states considered in the calculation. We demonstrate that CASPT2 and the reference self-consistent field calculations predict qualitatively different energetics and bond lengths.

Analytical Gradient Theory for Spin-Free State-Averaged Second-Order Driven Similarity Renormalization Group Perturbation Theory (SA-DSRG-MRPT2) and Its Applications for Conical Intersection Optimizations

Second-order multireference-driven similarity renormalization group perturbation theory (DSRG-MRPT2) provides an efficient means of correcting the dynamical correlation with the multiconfiguration reference function. The state-averaged DSRG-MRPT2 (SA-DSRG-MRPT2) method is the simplest means of treating the excited states with DSRG-MRPT2. In this method, the Hamiltonian dressed with dynamical correlation is diagonalized in the CASCI state subspace (SA-DSRG-MRPT2c) or the configuration subspace (SA-DSRG-MRPT2). This work develops analytical gradient theory for spin-free SA-DSRG-MRPT2(c) with the density-fitting approximation. We check the accuracy of the analytical gradients against the numerical gradients. We present applications for optimizing minimum energy conical intersections (MECI) of ethylene and retinal model chromophores (PSB3 and RPSB6). We investigate the dependence of the optimized geometries and energies on the flow parameters and reference relaxations. The smoothness of the SA-DSRG-MRPT2(c) potential energy surfaces near the reference (complete active space self-consistent field) MECI is comparable to the XMCQDPT2 one. These results render SA-DSRG-MRPT2(c) theory a promising approach for studies of conical intersections.

Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics

We report a spin-free formulation of the multireference (MR) driven similarity renormalization group (DSRG) based on the ensemble normal ordering of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)]. This ensemble averages over all microstates of a given total spin quantum number, and therefore, it is invariant with respect to SU(2) transformations. As such, all equations may be reformulated in terms of spin-free quantities and they closely resemble those of spin-adapted closed-shell coupled cluster (CC) theory. The current implementation is used to assess the accuracy of various truncated MR-DSRG methods (perturbation theory up to third order and iterative methods with single and double excitations) in computing the constants of 33 first-row diatomic molecules. The accuracy trends for these first-row diatomics are consistent with our previous benchmark on a small subset of closed-shell diatomic molecules. We then present the first MR-DSRG application on transition-metal complexes by computing the spin splittings of the [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ molecules. A focal point analysis (FPA) shows that third-order perturbative corrections are essential to achieve reasonably converged energetics. The FPA based on the linearized MR-DSRG theory with one- and two-body operators and up to a quintuple-ζ basis set predicts the spin splittings of [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ to be -35.7 and -17.1 kcal mol^-1, respectively, showing good agreement with the results of local CC theory with singles, doubles, and perturbative triples.

Toward an Accurate Ab Initio Description of Low-Lying Singlet Excited States of Polyenes

The low-lying excited states of carotenoids play a crucial role in many important biophysical processes such as photosynthesis. Most of these excited states are strongly correlated, which makes them both challenging for a qualitative ab initio description and an engaging model system for trying out emerging multireference methods. Among these methods, driven similarity renormalization group (DSRG) and its perturbative version (DSRG-MRPT2) are especially attractive in terms of both accuracy and moderate numerical complexity. In this paper, we applied density matrix renormalization group (DMRG) followed by DSRG-MRPT2 for the calculation of vertical and adiabatic excitation energies into the 2A_g^-, 1B_u^-, and 1B_u⁺ electronic states of polyenes containing from 8 to 13 conjugating double bonds acting as a model for natural carotenoids. It was shown that the DSRG flow parameter should be adjusted to ensure both the energy convergence with respect to it and the agreement with the experimental data. With the increased flow parameter, the proposed combination of methods provides a reasonable agreement with the experiment. The deviations of the adiabatic excitation energies are less than 1000 cm^-1 for the 2A_g^- and less than 3000 cm^-1 for the excited states of the B_u symmetry, which in terms of accuracy significantly outperforms the N-electron valence state perturbation theory. At the same time, DSRG-MRPT2 is shown to be robust with respect to variation of quality of the DMRG reference wave function such as the orbital optimization or the number of electronic states in the averaging.

On the role of symmetry in XDW-CASPT2

Herewith, we propose two new exponents for the recently introduced XDW-CASPT2 method [S. Battaglia and R. Lindh, J. Chem. Theory Comput. 16, 1555-1567 (2020)], which fix one of the largest issues hindering this approach. By using the first-order effective Hamiltonian coupling elements, the weighting scheme implicitly takes into account the symmetry of the states, thereby averaging Fock operators only if the zeroth-order wave functions interact with each other. The use of Hamiltonian couplings also provides a physically sounder approach to quantitate the relative weights; however, it introduces new difficulties when these rapidly die off to zero. The improved XDW-CASPT2 method is critically tested on several systems of photochemical relevance, and it is shown that it succeeds in its original intent of maintaining MS-CASPT2 accuracy for the evaluation of transition energies and at the same time providing smooth potential energy surfaces around near-degenerate points akin to XMS-CASPT2.

Compressed-State Multistate Pair-Density Functional Theory

Multiconfiguration pair-density functional theory (MC-PDFT) is a multireference method that can be used to calculate excited states. However, MC-PDFT potential energy surfaces have the wrong topology at conical intersections because the last step of MC-PDFT is not a diagonalization of a model-space Hamiltonian matrix, as done in, for example, multistate second-order perturbation theory (MS-CASPT2). We have previously proposed methods that solve this problem by diagonalizing a model-space effective Hamiltonian matrix, where the diagonal elements are MC-PDFT energies for intermediate states, and the off-diagonal elements are evaluated by wave function theory. One previous method is called variational multistate PDFT (VMS-PDFT), whose intermediate states maximize the trace of the effective Hamiltonian, namely, the sum of the MC-PDFT energies of the model-space states; the VMS-PDFT is very robust but is more computationally expensive than another method, extended multistate PDFT (XMS-PDFT), in which the transformation to intermediate states is accomplished without needing any density functional evaluations. However, although VMS-PDFT was accurate in all cases tested, XMS-PDFT was accurate in only some of them. In the present paper, we propose a new method, called compressed-state multistate PDFT (CMS-PDFT), that is as efficient as XMS-PDFT and as accurate as VMS-PDFT. The new method maximizes the trace of the classical Coulomb energy of the intermediate states such that the electron densities of the intermediate states are compressed. We show that CMS-PDFT performs robustly even where XMS-PDFT fails.

Reduced Density Matrix Cumulants: The Combinatorics of Size-Consistency and Generalized Normal Ordering

Reduced density matrix cumulants play key roles in the theory of both reduced density matrices and multiconfigurational normal ordering. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical with equating the coupled cluster and configuration interaction ansätze. This is shown to be a general mechanism to convert between a multiplicatively separable quantity and an additively separable quantity, as defined by a set of axioms. It is shown that both the cumulants of probability theory and the reduced density matrices are entirely combinatorial constructions, where the differences can be associated with changes in the notion of "multiplicative separability" for expectation values of random variables compared to reduced density matrices. We compare our generating function to that of previous works and criticize previous claims of probabilistic significance of the reduced density matrix cumulants. Finally, we present a simple proof of the generalized normal ordering formalism to explore the role of reduced density matrix cumulants therein. While the formalism can be used without cumulants, the combinatorial structure of expressing RDMs in terms of cumulants is the same combinatorial structure on cumulants that allows for a simple extended generalized Wick's theorem.

Extended Dynamically Weighted CASPT2: The Best of Two Worlds

We introduce a new variant of the complete active space second-order perturbation theory (CASPT2) method that performs similarly to multistate CASPT2 (MS-CASPT2) in regions of the potential energy surface where the electronic states are energetically well separated and is akin to extended MS-CASPT2 (XMS-CASPT2) in case the underlying zeroth-order references are near-degenerate. Our approach follows a recipe analogous to that of XMS-CASPT2 to ensure approximate invariance under unitary transformations of the model states and a dynamic weighting scheme to smoothly interpolate the Fock operator between state-specific and state-average regimes. The resulting extended dynamically weighted CASPT2 (XDW-CASPT2) methodology possesses the most desirable features of both MS-CASPT2 and XMS-CASPT2, that is, the ability to provide accurate transition energies and correctly describe avoided crossings and conical intersections. The reliability of XDW-CASPT2 is assessed on a number of molecular systems. First, we consider the dissociation of lithium fluoride, highlighting the distinctive characteristics of the new approach. Second, the invariance of the theory is investigated by studying the conical intersection of the distorted allene molecule. Finally, the relative accuracy in the calculation of vertical excitation energies is benchmarked on a set of 26 organic compounds. We found that XDW-CASPT2, albeit being only approximately invariant, produces smooth potential energy surfaces around conical intersections and avoided crossings, performing equally well to the strictly invariant XMS-CASPT2 method. The accuracy of vertical transition energies is almost identical to MS-CASPT2, with a mean absolute deviation of 0.01–0.02 eV, in contrast to 0.12 eV for XMS-CASPT2.