The Hartree-Fock method

In the previous section we have seen how to apply the variational method to a simple simgle-particle problem. As we treat more complicated problems, such as heavier atoms, molecules, and ultimately, solids, the complexitiy increases as the number of particles, and degrees of freedom increases. In these so-called, many-body problems, we have to consider the motion of the nuclei, the interaction between the protons and electrons, and between electrons themselves. We will consider a general system of $N$ nuclei described by coordinates, ${\bf R}_1,...,{\bf R}_N \equiv {\bf R}$, momenta, ${\bf P}_1,...,{\bf P}_N \equiv {\bf P}$, and masses $M_1,...,M_N$, and $N_e$ electrons described by coordinates, ${\bf r}_1,...,{\bf r}_{N_e} \equiv {\bf r}$, momenta, ${\bf p}_1,...,{\bf p}_{N_e}\equiv {\bf p}$, and spin variables, $s_1,...,s_{N_e}\equiv s$. The Hamiltonian of the system is given by

$${H} = {T}_N({\bf R}) + {T}_e({\bf r}) + V_{NN}({\bf R}) + {V}_{eN}({\bf r}, {\bf R}) + {V}_{ee}({\bf r}),$$

where $T_N$ is the kinetic energy of the nuclei, $T_e$ is the kinetic energy of the electrons, and $V_{NN}$, $V_{eN}$ and $V_{ee}$ contain the nucleus-nucleus, nuclei-electron, and electron-electron interactions, respectively:

$$T_N = \sum_{I=1}^N \frac{{\bf P}_I^2}{2M_I},$$ (49)

$$T_e = \sum_{i=1}^{N_e} \frac{{\bf p}_i^2}{2m_I},$$ (50)

$$V_{NN} = \sum_{I>J}\frac{Z_I Z_J e^2}{\vert{\bf R}_I-{\bf R}_J\vert} = \sum_{I>J} \frac{Z_I Z_J e^2}{r_{IJ}},$$ (51)

$$V_{eN} = -\sum_{i,I}\frac{Z_Ie^2}{\vert{\bf R}_I-{\bf r}_i\vert} = -\sum_{I>J} \frac{Z_I e^2}{r_{Ii}},$$ (52)

$$V_{ee} = \sum_{i,j}\frac{e^2}{\vert{\bf r}_i-{\bf r}_i\vert} = -\sum_{i>j} \frac{e^2}{r_{ij}}.$$ (53)

This is sometimes jokingly referred to as the "equation of everything". Clearly,if we could solve this problem, condensed matter would be a dead field. "Luckily" for us, this equation is extremely complicated, and basically intractable. Therefore, we are forced to make several approximations. The first one is to assume that the nuclei are static. This is justified when when realizes that the mass of the protons and neutrons is much larger that the mass of the electrons by three orders of magnitude. Therefore the time scale for motion of the nuclei is much large than the one for the electrons, that move at much faster speeds. This approach is the so-called Born-Oppenheimer approximation: by taking the position of the nuclei fixed, the main remaining problem is the electronic part.

A second approximation is to assume that the wave-function of the many-electron system takes the form of an antisymmetized product of one-electron wave-functions (remember that electrons are fermions). This simplification transforms the complicated many-body problem into the problem of a single-particle in an effective "mean-field" potential determined by the positions of the other electrons. This is the basic idea behind the Hartree-Fock method.

We can immediatley make two obervations: The first is that we are assuming that the physics can be described by single-particle wave-functions, and therefore, thsi corresponds to approximating the actual ground state by a variational ansatz. As a consequence, all the concepts learned in the previous section will apply here as well. A second observation is that the effective potential feld by the electrons will have to be calculated self-consistently: every time we update or modify the single particle wave-function, the potential will have to be updated as well.

We shall see that in this variational approach, correlations between electrons are neglected to some extent. In particular, Coulomb repulsion between electrons is taken into account in an averaged way.