Exchange kernel of density functional response theory from the common energy denominator approximation (CEDA) for the Kohn–Sham Green's function

Abstract

A complete and explicit expression for the exchange kernel f _xσ of density functional response theory (DFRT) is derived in terms of the occupied Kohn-Sham (KS) orbitals ψ _iσ. It is based on the common energy denominator approximation (CEDA) for the KS Green's function (O. V. Gritsenko and E. J. Baerends, Phys. Rev. A 64, 042506 (2001)). The kernel f _xσ ^CEDA is naturally subdivided into the Slater f _Sσ ^CEDA and the 'response' f _respσ ^CEDA parts, which are the derivatives of the Slater ν _Sσ and response ν _respσ potentials, respectively. While f _Sσ ^CEDA is obtained with a straightforward differentiation of ν _Sσ , some terms of f _respσ ^CEDA are obtained from the solution of linear equations for the corresponding derivatives. All components of f _xσ ^CEDA are explicitly expressed in terms of the products ψ _iσ ^* ψ _jσ of the occupied KS orbitals taken at the positions r ₁ and r ₂, as well as the potentials of these products at r ₃. The coefficients in these expressions are obtained by inversion of the matrix, associated with the overlap matrix of the products ψ _iσ ^* ψ _jσ and ψ _kσ ^* ψ _lσ . Terms are indicated, which generate in an external electric field an ultra-nonlocal potential δν _xσ , counteracting an external field, and possible approximations to f _xσ ^CEDA are considered.