Extended screened exchange functional derived from transcorrelated density functional theory

We propose a new formulation of the correlation energy functional derived from the transcorrelated method in use in density functional theory (TC-DFT). An effective Hamiltonian, H_TC, is introduced by a similarity transformation of a many-body Hamiltonian, H, with respect to a complex function F: H_TC=1FHF. It is proved that an expectation value of H_TC for a normalized single Slater determinant, Dⁿ, corresponds to the total energy: E[n] = ⟨Ψⁿ|H|Ψⁿ⟩/⟨Ψⁿ|Ψⁿ⟩ = ⟨Dⁿ|H_TC|Dⁿ⟩ under the two assumptions: (1) The electron density nr associated with a trial wave function Ψⁿ = DⁿF is v-representable and (2) Ψⁿ and Dⁿ give rise to the same electron density nr. This formulation, therefore, provides an alternative expression of the total energy that is useful for the development of novel correlation energy functionals. By substituting a specific function for F, we successfully derived a model correlation energy functional, which resembles the functional form of the screened exchange method. The proposed functional, named the extended screened exchange (ESX) functional, is described within two-body integrals and is parametrized for a numerically exact correlation energy of the homogeneous electron gas. The ESX functional does not contain any ingredients of (semi-)local functionals and thus is totally free from self-interactions. The computational cost for solving the self-consistent-field equation is comparable to that of the Hartree-Fock method. We apply the ESX functional to electronic structure calculations for a solid silicon, H^- ion, and small atoms. The results demonstrate that the TC-DFT formulation is promising for the systematic improvement of the correlation energy functional.