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# Quantum Monte Carlo Calculations

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!

Next: Layout of Thesis Up: Introduction Previous: Kohn-Sham Equations

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996