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!