Note on a rigorous derivation of self-consistent double-hybrid functional theory via generalized Kohn-Sham theory and cumulant approximation

Abstract

In this short note, we present a rigorous derivation of the one-body double-hybrid density functional (OBDHF) theory, a self-consistent double-hybrid density functional framework that unifies the generalized Kohn-Sham (GKS) formalism with one-body M{\o}ller-Plesset second-order perturbation (OBMP2) theory. Conventional double-hybrid density functionals suffer from a fundamental theoretical inconsistency arising from the non-self-consistent treatment of the perturbative MP2 correlation, in which the orbitals entering the correlation energy expression are not variationally optimized with respect to the full double-hybrid energy functional. To address this deficiency, we construct a model energy functional as a linear combination of semilocal density functional approximation XC, a fraction $\alpha_x$ of exact Hartree-Fock (HF) exchange, and a fraction $\alpha_c$ of OBMP2 correlation. By virtue of the one-body operator structure of OBMP2, the perturbative correlation contribution is embedded directly and self-consistently into the GKS effective Hamiltonian, without recourse to the optimized effective potential (OEP) construction or perturbative orbital relaxation corrections. Through functional differentiation of the total OBDHF energy with respect to the orbitals, we derive the OBDHF effective Hamiltonian and the associated self-consistent field equations in a rigorous and transparent manner. This formulation provides a theoretically well-founded and practically tractable pathway to fully self-consistent double-hybrid density functional theory calculations within the GKS framework, resolving the self-consistency problem inherent in conventional double-hybrid functionals.