Computational Chemistry 4.23 - Hartree-Fock-Roothaan Equations

Short lecture on the Hartree-Fock-Roothaan equations for orbitals and energies of molecular systems.

Once we have applied the Born-Oppenheimer approximation (stationary nuclei) to simplify nuclear energy terms and we have applied the Hartree-Fock approximation (mean field) to simplify the electron repulsion many-body problem, there still remains one issue to resolve in order to solve for the Hartree-Fock orbitals and energies of an arbitrary molecule. We introduce a finite atomic orbital basis set to express the Hartree-Fock equations as a set of matrix equations (Hartree-Fock-Roothaan equations). Using these K basis functions gives use 3 K by K matrices: F the Fock matrix, S the overlap matrix, and C the coefficient matrix. These gives us the equation FC = SCE, where E is a matrix which when diagonalized provides the orbital energies for the molecule. The coefficient matrix elements tell us the coefficient for each basis function (row) in each molecular orbital (column). The primary difficulty in solving this equation is that the form of F depends on the form of C, thus the equations are non-linear and must be solved iteratively.

Notes Slide: https://i.imgur.com/WOYfjcL.png

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