The absence of any exchange or correlation between electrons in the Hartree method leave this technique too inaccurate for performing modern electronic structure calculations.

Hartree-Fock calculations, which include the exchange interaction between electrons, are most useful for performing calculations on relatively small systems as they are considerably more computationally expensive than Hartree and DFT-LDA calculations, due to the non-local exchange term. Even for atoms, however, Hartree-Fock theory is not ideal. For example, H is predicted to be unstable in contradiction to reality.

Various improvements to Hartree-Fock theory have been attempted. Unrestricted Hartree-Fock theory ignores some of the simplifying restrictions which are normally applied to Hartree-Fock wavefunctions. The exchange interaction is allowed to make the spatial parts of spin up and spin down electron wavefunctions different for the same state. However, although for some systems this results in an improvement [13], especially for open shell systems, it also sometimes produces worse results than conventional Hartree-Fock theory [14]. In general, Hartree-Fock theory is most useful as a tool for providing qualitative answers. It is also used as the starting point for methods, such as some Quantum Monte Carlo calculations (see chapter ).

The success of the local density approximation is currently understood to be due to two points. (i) The sum rule on the exchange-correlation hole is conserved, i.e. within the LDA, the exchange-correlation hole contains exactly an equal and opposite amount of charge to the electron it surrounds. (ii) The exchange-correlation energy only depends on the spherical average of the exchange-correlation hole, i.e.

where is the
spherical average of about
for each . Therefore, the fact that in
the LDA, is constrained to be spherically symmetric about
is not a handicap. However, in strongly correlated
systems, i.e. those containing *d* and *f* orbitals, the correlations
may change the whole nature of the ground state and the Local Density
approximation, derived from homogeneous electron gas results, is not
successful. For example, the high superconductor is an anti-ferromagnetic insulator but the LDA finds it to
be metallic. Also FeO, MnO and NiO all have Mott metal-insulator
transitions but the LDA predicts that they are either semiconductors
or metals. The LDA is only expected to be accurate for systems with
slowly varying electronic charge densities, which is not the case in
most real systems, but despite this it has been surprisingly
successful. Other failings of the LDA are that it tends to overbind
atoms in solids, that it finds stable negative ions to be unstable
and that it predicts iron to be *fcc* paramagnetic, when it is
actually *bcc* ferromagnetic.

The main problem with Hartree, Hartree-Fock and LDA methods is approximations they introduce in the process of reducing the many-body problem to a one-electron problem. Hartree and Hartree-Fock calculations do not, in general, provide satisfactory results and are best used as a qualitative guide to the expected ground state properties. The Configuration Interaction method, while in principle exact, is in practice, only useful for small systems; for condensed matter systems it is not of practical value.

Density Functional theory within the LDA provides the current staple
method of performing electronic structure calculations and for many
purposes gives good results. However, it fails for highly correlated
systems and tends to underestimate band gaps and overestimates
cohesive energies and hence is not ideal. Many-body approaches have
been successful in some calculations, particularly of band gaps, but
they are difficult to implement and it is hard to go beyond the low
order *GW* [15] approximation.

It is therefore clear that there is room for a straightforward, accurate approach to many-body systems: the Quantum Monte Carlo method!

Tue Nov 19 17:11:34 GMT 1996