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, $HTC$⁠, is introduced by a similarity transformation of a many-body Hamiltonian, $H$⁠, with respect to a complex function F: $HTC=1FHF$⁠. It is proved that an expectation value of $HTC$ for a normalized single Slater determinant, Dⁿ, corresponds to the total energy: $E[n]=⟨Ψn|H|Ψn⟩/⟨Ψn|Ψn⟩=⟨Dn|HTC|Dn⟩$ 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.