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Next: Bloch's Theorem and Plane Up: First Principles Molecular Dynamics Previous: Density Functional Theory

Exchange, Correlation and the Local Density Approximation

 

In seeking the solutions to the system of equations (3.17), it is found that all quantities are represented as functionals of the electronic charge density. The important point that makes this system easier to solve (or more precisely, require less computation) than, for example the Hartree-Fock equations, is that the effective potential is local (although note Section 3.7 discussing the use of nonlocal pseudopotentials). Therefore there is no more complexity added in solving (3.17) than there is in the Hartree approximation. Of course, this is only true if the exchange-correlation energy can be described as a function of the local charge density. A method of doing so is known as the local density approximation[29] (LDA). In LDA, the exchange-correlation energy of an electronic system is constructed by assuming that the exchange-correlation energy per electron at a point tex2html_wrap_inline5508 in the electron gas, tex2html_wrap_inline5510 is equal to the exchange-correlation energy per electron in a homogeneous electron gas that has the same electron density at the point tex2html_wrap_inline5508 . It follows that

  eqnarray378

so that

  eqnarray387

with

  eqnarray396

where Equation (3.21) is the assumption that the exchange-correlation energy is purely local. Several parameterisations for tex2html_wrap_inline5514 exist, but the most commonly used is that of Perdew and Zunger[37]. This parameterisation is based on the quantum Monte Carlo calculations of Ceperley and Alder[38] on homogeneous electron gases at various densities. The parameterisation uses interpolation formulas to link these exact results for the exchange and correlation energy at many different densities.

In LDA, corrections to the exchange-correlation energy due to the inhomogeneities in the electronic charge density about tex2html_wrap_inline5508 are ignored. Considering this inexact nature of the approximation, it may at first seem somewhat surprising that such calculations are so successful. This can be partially attributed to the fact that LDA gives the correct sum rule to the exchange-correlation hole. That is, there is a total electronic charge of one electron excluded from the neighbourhood of the electron at tex2html_wrap_inline5508 . Attempts to improve on LDA, such as gradient expansions to correct for inhomogeneities do not seem to show any improvement in results obtained by the simple LDA. One of the reasons for this failure is that the sum rule is not obeyed by the exchange-correlation holegif.

A summary of the contributions of electron-electron interactions in N-electron systems is shown in Figure 3.1. It illustrates the conditional electron probability distributions tex2html_wrap_inline5484 of N-1 electrons around an electron with given spin situated at tex2html_wrap_inline5524 . In the Hartree approximation, Figure (a), all electrons are treated as independent, therefore tex2html_wrap_inline5484 is structureless. Figure (b) represents the Hartree-Fock approximation where the N-electron wavefunction reflects the Pauli exclusion principle. Around the electron at tex2html_wrap_inline5524 the exchange hole can be seen where the the density of spins equal to that of the central electron is reduced. Electrons with opposite spins are unaffected. In the LDA, where spin states are degenerate, each type of electron sees the same exchange-correlation hole (the sum rule being illustrated where the size of the hole is one electron). Figure (d) shows electron-electron interaction for non-degenerate spin systems (the local spin density approximation (LSD). It can be seen that the spin degenerate LDA is simply the average of the LSD).

  figure419
Figure 3.1: Summary of the electron-electron interactions (excluding coulomb effects) in (a) the Hartree approximation, (b) the Hartree-Fock approximation, (c) the local density approximation and (d) the local spin density approximation which allows for different interactions for like-unlike spins. 


next up previous
Next: Bloch's Theorem and Plane Up: First Principles Molecular Dynamics Previous: Density Functional Theory

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996