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The Journal of Chemical Physics

Algorithm for analytic nuclear energy gradients of state averaged DMRG-CASSCF theory with newly derived coupled-perturbed equations

Tsubasa Iino, Toru Shiozaki, Takeshi Yanai

Published Dec 16, 2022• 5 citations

Abstract

We present an algorithm for evaluating analytic nuclear energy gradients of the state-averaged density matrix renormalization group complete-active-space self-consistent field (SA-DMRG-CASSCF) theory based on the newly derived coupled-perturbed (CP) DMRG-CASSCF equations. The Lagrangian for the conventional SA-CASSCF analytic gradient theory is extended to the SA-DMRG-CASSCF variant that can fully consider a whole set of constraints on the parameters of multi-root canonical matrix product states formed at all the DMRG block configurations. An efficient algorithm to solve the CP-DMRG-CASSCF equations for determining the multipliers was developed. The complexity of the resultant analytic gradient algorithm is overall the same as that of the unperturbed SA-DMRG-CASSCF algorithm. In addition, a reduced-scaling approach was developed to directly compute the SA reduced density matrices (SA-RDMs) and their perturbed ones without calculating separate state-specific RDMs. As part of our implementation scheme, we neglect the term associated with the constraint on the active orbitals in terms of the active-active rotation in the Lagrangian. Thus, errors from the true analytic gradients may be caused in this scheme. The proposed gradient algorithm was tested with the spin-adapted implementation by checking how accurately the computed analytic energy gradients reproduce numerical gradients of the SA-DMRG-CASSCF energies using a common number of renormalized bases. The illustrative applications show that the errors are sufficiently small when using a typical number of the renormalized bases, which is required to attain adequate accuracy in DMRG's total energies.

DOI: 10.1063/5.0130636

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We present an algorithm for evaluating analytic nuclear energy gradients of the state-averaged density matrix renormalization group complete-active-space self-consistent field (SA-DMRG-CASSCF) theory based on the newly derived coupled-perturbed (CP) DMRG-CASSCF equations. The Lagrangian for the conventional SA-CASSCF analytic gradient theory is extended to the SA-DMRG-CASSCF variant that can fully consider a whole set of constraints on the parameters of multi-root canonical matrix product states formed at all the DMRG block configurations. An efficient algorithm to solve the CP-DMRG-CASSCF equations for determining the multipliers was developed. The complexity of the resultant analytic gradient algorithm is overall the same as that of the unperturbed SA-DMRG-CASSCF algorithm. In addition, a reduced-scaling approach was developed to directly compute the SA reduced density matrices (SA-RDMs) and their perturbed ones without calculating separate state-specific RDMs. As part of our implementation scheme, we neglect the term associated with the constraint on the active orbitals in terms of the active-active rotation in the Lagrangian. Thus, errors from the true analytic gradients may be caused in this scheme. The proposed gradient algorithm was tested with the spin-adapted implementation by checking how accurately the computed analytic energy gradients reproduce numerical gradients of the SA-DMRG-CASSCF energies using a common number of renormalized bases. The illustrative applications show that the errors are sufficiently small when using a typical number of the renormalized bases, which is required to attain adequate accuracy in DMRG's total energies.

We present an approximate scheme for analytical gradients and nonadiabatic couplings for calculating state-average density matrix renormalization group self-consistent-field wave function. Our formalism follows closely the state-average complete active space self-consistent-field (SA-CASSCF) ansatz, which employs a Lagrangian, and the corresponding Lagrange multipliers are obtained from a solution of the coupled-perturbed CASSCF (CP-CASSCF) equations. We introduce a definition of the matrix product state (MPS) Lagrange multipliers based on a single-site tensor in a mixed-canonical form of the MPS, such that a sweep procedure is avoided in the solution of the CP-CASSCF equations. We apply our implementation to the optimization of a conical intersection in 1,2-dioxetanone, where we are able to fully reproduce the SA-CASSCF result up to arbitrary accuracy.

This paper describes an interface between the density matrix renormalization group (DMRG) method and the complete active-space self-consistent field (CASSCF) method and its analytical gradient, as well as an extension to the second-order perturbation theory (CASPT2) method. This interfacing allows large active-space multi-reference computations to be easily performed. The interface and its extension are both implemented in terms of reduced density matrices (RDMs) which can be efficiently computed via the DMRG sweep algorithm. We also present benchmark results showing that, in practice, the DMRG-CASSCF calculations scale with active-space size in a polynomial manner in the case of quasi-1D systems. Geometry optimization of a binuclear iron-sulfur cluster using the DMRG-CASSCF analytical gradient is demonstrated, indicating that the inclusion of the valence p-orbitals of sulfur and double-shell d-orbitals of iron lead to non-negligible changes in the geometry compared to the results of small active-space calculations. With the exception of the selection of M values, many computational settings in these practical DMRG calculations have been tuned and black-boxed in our interface, and so the resulting DMRG-CASSCF and DMRG-CASPT2 calculations are now available to novice users as a common tool to compute strongly correlated electronic wavefunctions.

Accurate prediction of nuclear magnetic resonance (NMR) shielding constants for strongly correlated systems remains challenging. In this context, we present a density matrix renormalization group self-consistent-field (DMRG-SCF) implementation for NMR shielding calculations with large active spaces. Within a gauge-including atomic orbital DMRG-SCF (GIAO-DMRG-SCF) framework, we formulate and solve the coupled-perturbed DMRG-SCF (CP-DMRG-SCF) equations using local-site effective Hamiltonians. This approach enables the direct evaluation of the wave function's response to an external magnetic field. In the limit of small active spaces, the computed NMR shieldings are in agreement with conventional complete active space self-consistent-field (CASSCF) results, while the present approach naturally extends to larger active spaces inaccessible to standard multiconfigurational methods. Applications to a series of molecular systems, particularly strongly correlated transition metal complexes, demonstrate that the proposed implementation can be both reliable and accurate.

We derive analytic energy gradients of the driven similarity renormalization group (DSRG) multireference second-order perturbation theory (MRPT2) using the method of Lagrange multipliers. In the Lagrangian, we impose constraints for a complete-active-space self-consistent-field reference wave function and the semicanonical orthonormal molecular orbitals. Solving the associated Lagrange multipliers is found to share the same asymptotic scaling of a single DSRG-MRPT2 energy computation. A pilot implementation of the DSRG-MRPT2 analytic gradients is used to optimize the geometry of the singlet and triplet states of p-benzyne. The equilibrium bond lengths and angles are similar to those computed via other MRPT2s and Mukherjee's multireference coupled cluster theory. An approximate DSRG-MRPT2 method that neglects the contributions of three-body density cumulant is found to introduce negligible errors in the geometry of p-benzyne, lending itself to a promising low-cost approach for molecular geometry optimizations using large active spaces.

We present the density matrix renormalization group self-consistent field (DMRG-SCF) approach that is analogous to the complete active space self-consisted field (CASSCF) method but instead of using for the description of the active space the full configuration interaction (FCI) method, the DMRG-SCF uses the density matrix renormalization group (DMRG) method. The DMRG-SCF approach, similarly to CASSCF, properly describes the multiconfigurational character of the wave function but avoids the exponential scaling of the FCI method and replaces it with a polynomial scaling. Hence, calculations for a larger number of orbitals and electrons in the active space are possible since the DMRG method provides an efficient tool to automatically select from the full Hilbert space the many-body contracted basis states that are the most important for the description of the wave function.

In a recent work, we introduced the foundations of an orthogonally constrained complete active space self-consistent field (OC-CASSCF) framework that produces state-specific molecular orbitals for mutually orthogonal multiconfigurational electronic states. In the present study, we extend this approach by incorporating a Newton-Raphson orbital-optimization scheme, for which we derive analytical expressions of the orbital gradient and Hessian. Furthermore, we outline a practical route toward the evaluation of analytical nuclear gradients, enabling geometry optimizations within the OC-CASSCF formalism. Benchmark calculations on the three lowest singlet states of LiH and H

_2

O molecules demonstrate a systematic improvement as compared to conventional state-averaged CASSCF, even when using modestly sized active spaces.