1. The Hartree-Fock-Roothaan Method

2. Variational Principle

3. Energy of a Slater Determinant Where: And the Hamiltonian operator is given by:

4. First of all, it is useful to note that the Hamiltonian is a sum of three terms: not depend on any electronic coordinates sum of one-electron operators h, so called because each depends only on the coordinates of one electron sum of n(n-1)/2 two-electron contributions, which each depend on the coordinates of two electrons

5. Hamiltonian can thus be written more concisely as a sum of zero-, one- and two-electron terms :: first term in this sum is an integral over a constant, since the nuclear-nuclear repulsion energy does not depend on the electronic coordinates

6. V NN is the potential energy due to nuclear-nuclear Coulombic repulsion. first term in this sum is an integral over a constant, since the nuclear-nuclear repulsion energy does not depend on the electronic coordinates which can be re-written as sums of integrals, leading to:

7. Each one-electron operator h only acts on a very small part of the wavefunction Note that the one-electron operator h includes a part corresponding to the electron's kinetic energyand another to the potential energy created by the attractive Coulombic interaction with the nuclei

8. Te is the electronic kinetic energy, and VNe is the potential energy due to nuclear-electronic Coulombic attraction

9. The second set of integrals, the two-electron terms The first and fourth, and second and third

10. 2 (r) is the probability of finding an electron at a given point in space this integral is often called the Coulomb Integral, written in shorthand as J ij. Because 1/r is always positive, this term contributes a positive energy Exchange Integral, and written as K ij, Exchange Integral comes from the fact that the two electrons exchange their positions from the left to the right of the integral. This suggests, correctly, that it has something to do with the Pauli principle.

11. potential energy due to electronic-electronic Coulombic repulsion for a closed-shell system (= a spin singlet where all the occupied orbitals have two electrons in them), Or:

12. The Hartree-Fock-Roothaan Method