Analytic Gradient for Density Functional Theory Based on the Fragment Molecular Orbital Method

Abstract

The equations for the response terms for the fragment molecular orbital (FMO) method interfaced with the density functional theory (DFT) gradient are derived and implemented. Compared to the previous FMO-DFT gradient, which lacks response terms, the FMO-DFT analytic gradient has improved accuracy for a variety of functionals, when compared to numerical gradients. The FMO-DFT gradient agrees with the fully ab initio DFT gradient in which no fragmentation is performed, while reducing the nonlinear scaling associated with standard DFT. Solving for the response terms requires the solution of the coupled perturbed Kohn-Sham (CPKS) equations, where the CPKS equations are solved through a decoupled Z-vector procedure called the self-consistent Z-vector method. FMO-DFT is a nonvariational method and the FMO-DFT gradient is unique compared to standard DFT gradients in that the FMO-DFT gradient requires terms from both DFT and time-dependent density functional theory (TDDFT) theories. Disciplines Chemistry CommentsReprinted ( ABSTRACT: The equations for the response terms for the fragment molecular orbital (FMO) method interfaced with the density functional theory (DFT) gradient are derived and implemented. Compared to the previous FMO−DFT gradient, which lacks response terms, the FMO−DFT analytic gradient has improved accuracy for a variety of functionals, when compared to numerical gradients. The FMO−DFT gradient agrees with the fully ab initio DFT gradient in which no fragmentation is performed, while reducing the nonlinear scaling associated with standard DFT. Solving for the response terms requires the solution of the coupled perturbed Kohn− Sham (CPKS) equations, where the CPKS equations are solved through a decoupled Z-vector procedure called the self-consistent Z-vector method. FMO−DFT is a nonvariational method and the FMO−DFT gradient is unique compared to standard DFT gradients in that the FMO−DFT gradient requires terms from both DFT and time-dependent density functional theory (TDDFT) theories.

Analytic second derivative of the energy for density functional theory based on the three-body fragment molecular orbital method

“…48 The FMO analytic energy gradient has been developed for closed and open-shell Hartree-Fock (HF) [66][67][68][69][70][71] and DFT. 72 The analytic second derivative, which can be used to evaluate Raman activities, 73 has been developed only for HF 46,74 at the two-body level (FMO2). The FMO-RHF Hessian is not fully analytic, because some terms (e.g., contributions from second order responses) that are expected to be small are neglected.…”

Ab initio molecular dynamics of liquid water using embedded-fragment second-order many-body perturbation theory towards its accurate property prediction

“…An application of ab initio MP or CC theory to a bulk liquid has not been possible because of its prohibitive cost. The situation has changed with the advent of the so-called embedded-fragment method 18 , which enables a systematic, fast, and parallel-executable application of an ab initio theory to large molecular clusters, crystals, and even liquids 19 20 21 . The method divides a weakly interacting system into overlapping molecular dimers (“fragments”), which are embedded in the electrostatic environment of the whole system, and then applies well-developed molecular theories and software to these fragments to reconstruct an array of whole-system properties, with a computational cost scalable with respect to both system and computer sizes.…”

“…The effect of three-body interactions was assessed by Komeiji et al 23 in a water cluster of (H 2 O) 32 and that of electron correlation by Mochizuki et al 24 using second-order many-body perturbation (MP2) theory and the 6-31G* basis set on (H 2 O) 64 . Brorsen et al 20 21 repeated FMO-BOMD calculations of liquid water using more rigorous formulas for atomic forces at the HF and DFT level. All of these studies considered only the radial distribution functions and, owing to the smallness of the basis set employed, it is doubtful that the calculated results have much quantitative value.…”

Analytical gradients for projection-based wavefunction-in-DFT embedding

“…[31][32][33][34][35][36][37] The development of analytical nuclear gradients for projection-based embedding will expand its applicability to include geometry optimization, transition state searches, and potentially ab initio molecular dynamics. Analytical nuclear gradients already exist for a number of other embedding methodologies, including the incremental molecular fragmentation method, 38 fragment molecular orbital method, [39][40][41] quantum mechanics/molecular mechanics (QM/MM), [42][43][44] ONIOM, [45][46][47][48] embedded mean-field theory (EMFT), 17,49 frozen density embedding, [50][51][52][53] and subsystem DFT. [54][55][56] However, the projection-based approach provides a number of advantages for WF-in-DFT embedding calculations and leads to a distinct analytical gradient theory, which we derive and numerically demonstrate in several applications.…”