Analytical energy gradients for second-order multireference perturbation theory using density fitting

Abstract

We present algorithms for computing analytical energy gradients for multi-configuration self-consistent field methods and partially internally contracted complete active space second-order perturbation theory (CASPT2) using density fitting (DF). Our implementation is applicable to both single-state and multi-state CASPT2 analytical gradients. The accuracy of the new methods is demonstrated for structures and excitation energies of valence and Rydberg states of pyrrole, as well as for structures and adiabatic singlet-triplet energy splittings for the hydro-, the O,O(')-formato-, and the N,N(')-diiminato-copper-dioxygen complexes. It is shown that the effects of density fitting on optimized structures and relative energies are negligible. For cases in which the total cost is dominated by the integral evaluations and transformations, the DF-CASPT2 gradient calculations are found to be faster than the corresponding conventional calculations by typically a factor of three to five using triple-ζ basis sets, and by about a factor of ten using quadruple-ζ basis sets.

Nuclear Energy Gradients for Internally Contracted Complete Active Space Second-Order Perturbation Theory: Multistate Extensions

“…The details of the Z-CASSCF equations and the contraction to the gradient integrals using density fitting are found in Ref. 21 (see also Ref. 27).…”

“…Let us use a short-hand notation Ax = b for the Z-vector equation, where x is a column vector consisting of Z xy and z I,N and b is that consisting of Y xy and y I,N . 20,21 The matrix A on the left-hand side is the second derivatives of the SA-CASSCF energy with respect to the orbital rotations and variations of the CI coefficients. Since the SA-CASSCF solutions are minima in the parameter space, this matrix is guaranteed to be positive definite.…”

“…The equations for the nuclear gradients of internally contracted multireference theories are extremely complex, in part because the configuration interaction (CI) coefficients in the CASSCF reference wave functions are geometry dependent. The nuclear gradient implementation was, therefore, limited to uncontracted [15][16][17][18][19] or partially contracted 14,20,21 analogues until very recently, when MacLeod and Shiozaki realized it for a fully internally contracted variant using an automatic code generation approach. 22 The use of full internal contraction in our program allows us to evaluate nuclear gradients far more efficiently than the existing implementations based on uncontracted and partially contracted multireference perturbation theories.…”

Analytical Derivative Coupling for Multistate CASPT2 Theory

“…40,45,52 After solving the Z-vector equation for Z and z, we compute the effective density matrices (γ eff and Γ eff ) and the Lagrange multipliers V, whose explicit forms are given in Refs. 40 and 45.…”

Communication: Automatic code generation enables nuclear gradient computations for fully internally contracted multireference theory

“…The use of partially internally contracted or uncontracted basis functions 10 (referred to as WK-CASPT2 in the following) for first-order wave functions greatly simplifies the equations, making it tractable to manually implement nuclear gradients for such variants. [11][12][13][14][15] This is because, in these methods, parts (or all) of the first-order wave function are expanded in terms of excited Slater determinants,…”