Analytic derivatives for the XYG3 type of doubly hybrid density functionals: Theory, implementation, and assessment

Abstract

We present a theoretical development of the equations required to perform an analytic geometry optimization of a molecular system using the XYG3 type of doubly hybrid (xDH) functionals. In contrast to the well-established B2PLYP type of DH functionals, the energy expressions in the xDH functionals are constructed by using density and orbital information from another standard Kohn-Sham (KS) functional (e.g., B3LYP) for doing the self-consistent field calculations. Thus, the xDH functionals are nonvariational in both the hybrid density functional part and the second-order perturbation part, each of which requires formally to solve a coupled-perturbed KS equation. An implementation is reported here which combines the two parts by defining a total Lagrangian such that only a single set of the Z-vector equations need to be solved. The computational cost with our implementation is of the same order as those for the conventional Møller-Plesset theory to the second order (MP2) and B2PLYP. Systematic test calculations are provided for covalently bonded molecules as well as compounds involving the intramolecular nonbonded interactions for the main group elements. Satisfactory performance of the xDH functionals demonstrates that the extra computer time on top of the conventional KS procedure is well-invested, in particular, when the standard KS functionals and MP2 as well, are problematic.

Doubly hybrid density functionals that correctly describe both density and energy for atoms

“…In particular, in the xDH functionals, a lowerrung DFA at the Jacob's ladder is adopted for its reliability and efficiency to generate orbitals and densities, while a top-rung DH functional is adopted for final energy evaluation for its improved accuracy due to the introduction of the nonlocal correlation effects. The original densities from self-consistent-field (SCF) calculations can be further updated to the xDH relaxed densities for better accuracies by the external potential perturbation (45), which was realized through a modified Z-vector method (45,46) (see SI Materials and Methods for more details). Hence, the xDHs ascend and descend the ladder at will for energy and density, having a better chance that both energy and density can be well described.…”

“…We note that both B3LYP and PBE0 are on their list of the best methods, in the Science paper of Medvedev et al (26), yielding the best densities, while the xDH functionals use these orbitals and densities as the input, which is advantageous by construction. In fact, the B3LYP and PBE0 densities can be updated for an even better accuracy by using a modified Z-vector method (45,46). The Z vector is obtained from the xDH energy gradient calculations to take into account both the perturbation effects from the DFA part and the MP2-like part, coupling the SCF functional and the DH energy functional together.…”

A comparison of geometric parameters from PBE-based doubly hybrid density functionals PBE0-DH, PBE0-2, and xDH-PBE0

“…The other comprises the total difference density matrix P PT2+ , which is derived via the coupled perturbed Kohn-Sham (CP-KS) equation that combines the effects from both E n,DFA xc and E P T 2 c in the final energy expression of the xDH functional. 67 The later part is actually accomplished by defining a total Lagrangian such that only a single set of the Z-vector equations 73 need to be solved. μνκτ = S μνκτ + NS μνκτ is the element of the two-particle density matrix.…”

“…for 67 We would like to point out that analytic gradients for other well-established theory 71 The program is implemented in the NorthWest computational Chemistry (NWChem) software package version 6.1.1. 82 For benchmarking purpose, DFA energies are computed using the NWChem "Xfine" grid.…”

Perturbative treatment of anharmonic vibrational effects on bond distances: An extended langevin dynamics method

“…Various electronic structure methods, which solve the time‐independent electronic Schrödinger equation, have been developed with implementation of analytical gradient algorithms . These have now been routinely applied to geometry optimizations.…”