RoothaanHartreeFockmetoder

Roothaan-Hartree-Fock methods are a class of self-consistent field approaches in quantum chemistry that solve for molecular orbitals within the Hartree-Fock approximation using a matrix formulation introduced by Roothaan. They are typically used to describe closed-shell molecules with a finite basis set of atomic orbitals.

In the linear combination of atomic orbitals approach, each molecular orbital is expressed as a linear combination of basis functions. For closed-shell systems, restricted Hartree-Fock (RHF) is common, while unrestricted Hartree-Fock (UHF) handles open-shell cases. The Roothaan equations, written as F C = S C ε, replace the original differential Hartree-Fock equations with a matrix eigenvalue problem. The Fock matrix F contains one-electron terms and two-electron terms that depend on the density matrix, while the overlap matrix S accounts for non-orthogonality of the basis functions. The coefficient matrix C and orbital energies ε are obtained iteratively until self-consistency is reached.

Basis sets, often built from Gaussian-type orbitals, are used to approximate atomic orbitals and to facilitate efficient integral evaluation. The choice of basis set influences accuracy, convergence, and computational cost. The Roothaan-HF method provides a foundational starting point for many post-Hartree-Fock and hybrid methods, with the resulting orbitals and energies used in correlated techniques such as MP2, CCSD, or as references in density functional theory calculations.

Limitations include the neglect of dynamic electron correlation, dependence on the chosen basis set, potential spin contamination in UHF, and convergence challenges for large systems. Historically, the method was introduced by Louis Walter Roothaan in 1951, building on the Hartree-Fock framework to enable practical, matrix-based solutions for molecular systems.