Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 170

Hartree-Fock equations concern a self-consistent and nonlinear system: the eigenvectors |φn〉 and eigenvalues of the density operator are solutions of an eigenvalue equation (87) which itself depends on . This situation is reminiscent of the one encountered with the zero-temperature Hartree-Fock equations, and, a priori, no exact solutions can be found.

As for the zero-temperature case, we proceed by iteration: starting from a physically reasonable density operator , we use it in (84) to compute a first value of the Hartree-Fock potential operator. We then diagonalize this operator to get its eigenkets and eigenvalues . Next, we build the operator that has the same eigenkets, but whose eigenvalues are the . Inserting this new operator in (84), we get a second iteration of the Hartree-Fock operator. We again diagonalize this operator to compute new eigenvalues and eigenvectors, on which we build the next approximation of , and so on. After a few iterations, we may expect convergence towards a self-consistent solution.