Basic Roothaan-Hall Equations

Secular Equation

c_i|F-ε_iS|c_i = 0

c_i: eigenvector; F: Fock matrix; ε>_i: eigenvalue; S: overlap matrix. The total electronic energy of the system is given by:

E = 1/2P(H+F),

in which P: density matrix; H: one-electron matrix. The general Fock Matrix element is

F_μν^α = H_μν +Σ_λΣ_σ[P_μν^α+β <μν|λσ> - P_μν^α <μλ|νσ>].

Or, spin-free

F_μν = H_μν +Σ_λΣ_σ[P_μν <μν|λσ> - ½P_μν <μλ|νσ>].

The methods all use a minimum basis set consisting of a maximum of one atomic orbital for each angular quantum number. The normal basis set for any atom consists of one s and three p orbitals (p_x, p_y and p_z).