Abstract
A uniform derivation of the self-consistent field equations in a finite basis set is presented. Both restricted and unrestricted Hartree–Fock (HF) theory as well as various density functional approximations are considered. The unitary invariance of the HF and density functional models is discussed, paving the way for the use of localized molecular orbitals. The self-consistent field equations are derived in a non-orthogonal basis set, and their solution is discussed also in the presence of linear dependencies in the basis. It is argued why iterative diagonalization of the Kohn–Sham–Fock matrix leads to the minimization of the total energy. Alternative methods for the solution of the self-consistent field equations via direct minimization as well as stability analysis are briefly discussed. Explicit expressions are given for the contributions to the Kohn–Sham–Fock matrix up to meta-GGA functionals. Range-separated hybrids and non-local correlation functionals are summarily reviewed.
Highlights
Electronic structure calculations have become a cornerstone of modern-day research in chemistry and materials physics, allowing in silico modeling of chemical reactions and the first principles design of novel catalysts [1]
We have presented an overview of self-consistent field calculations within a variational basis set formalism, and discussed the solution of the self-consistent field equations arising from Hartree–Fock as well as various levels of density functional approximations using either the traditional fixed-point equations or direct minimization, as well as various conceptual and numerical issues arising in their implementation
No assumptions have been made on the underlying basis set in the present work: the self-consistent field formalism is the same regardless of the form of the basis functions, which can be chosen to be, e.g., Gaussian-type orbitals (GTOs), Slater-type orbitals (STOs), numerical atomic orbitals (NAOs), or finite element shape functions
Summary
Electronic structure calculations have become a cornerstone of modern-day research in chemistry and materials physics, allowing in silico modeling of chemical reactions and the first principles design of novel catalysts [1]. DFT turns out to yield self-consistent field (SCF) equations that assume the same form as in HF but with a different expression for the Fock matrix F. Pople and coworkers reported the equations necessary for solving SCF for DFT in the LCAO context up to generalized gradient approximation (GGA) functionals in [16]; an analogous derivation was presented by Kobayashi et al in [19]. This paper, presents such a derivation, yielding expressions of the DFT contributions to the Kohn–Sham–Fock matrix up to the level of meta-GGA functionals in a consistent way, facilitating the implementation of DFT in new programs.
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