Matrix form of the Hartree-Fock equations

Same as in previous chapters, we are going to expand the wavefunctions as linear combinations of a finite number of basis states:

$$\phi_k({\bf r}) = \sum_p C_{pk} \chi_p({\bf r}).$$ (109)

For a given basis, we obtain the following matrix equation, which is known as Roothaan equation:

$$FC_k = \epsilon SC_k,$$ (110)

where S is the overlap matrix for the orbital basis, and the matrix $F$ is given by:

$$F_{pq} = h_{pq} + \sum_k \sum_{pq} C^*_{rk}C_{sk} (2 \langle ps\vert g\vert qs\rangle - \langle pr \vert g \vert sq \rangle)$$ (111)

where

$$h_{pq} = \langle p\vert h\vert q\rangle = \int d^3r \chi^*_p... ...rac{Z_n}{\vert{\bf r} - {\bf R}_n\vert}\right]\chi_q({\bf r}),$$ (112)

and

$$\langle pr \vert g\vert qs \rangle = \int d^3r_1 d^3r_2 \chi... ...bf r}_2 - {\bf r}_2\vert} \chi_q({\bf r}_1) \chi_s({\bf r}_2).$$ (113)

As we have seen before, these equations should be solved by a self-consistent iterative procedure.

It is convenient to introduce the density matrix, defined as

$$\rho_{pq} = 2 \sum_k C_{pk}C^*_{qk}$$ (114)

which is the matrix representation for the operator

$$\rho = 2 \sum_k \vert\phi_k\rangle\langle \phi_k\vert.$$ (115)

Using this expression we can rewrite the Fock matrix as:

$$F_{pq} = h_{pq} + \frac{1}{2} \sum_{rs} \rho_{rs}(2 \langle ps\vert g\vert qs\rangle - \langle pr \vert g \vert sq \rangle),$$ (116)

and the energy is given by:

$$E = \sum_{pq} \rho_{pq} h_{pq} + \frac{1}{2} \sum_{pqrs} \rh... ...\rangle - \frac{1}{2}\langle pr\vert g\vert sq\rangle \right].$$ (117)