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PWPB95-D3: 50% HF exchange + 50% Re-Fit PW91 GGA exchange + 73.1% Re-Fit B95 meta-GGA correlation + 26.9% OS MP2 correlation with DFT-D3(0) tail 309 5.3.3 Correlation Functionals References and Further Reading 5.3.5 Specialized Functionals. CHEM6085 Density Functional Theory 4 Local Density Approximation (LDA).Assume that the exchange-correlation energy density at every position in space for the molecule is the same as it would be for the uniform electron gas (UEG) having the same density as found at that position.The expression for the exchange energy is simple.

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The Kohn-Sham equations in 2.30 are thus far exact: no approximations have yet been made; we have simply mapped the fully interacting system onto an auxiliary non-interacting system that yields the same groundstate density.

As mentioned earlier, the Kohn-Sham kinetic energy is not the true kinetic energy; we may use this to define formally the exchange-correlation energy as

where and are the exact kinetic and electron-electron interaction energies respectively. Physically, this term can be interpreted as containing the contributions of detailed correlation and exchange to the system energy. The definition above is such that it ensures that the Kohn-Sham formulation is exact. However, the actual form of is not known; thus we must introduce approximate functionals based upon the electron density to describe this term. There are two common approximations (in various forms) in use: the local density approximation (LDA) [47], and the generalised gradient approximation (GGA) [48]. The simplest approximation is the LDA: this assumes that the exchange-correlation energy at a point is simply equal to the exchange-correlation energy of a uniform electron gas that has the same density at the point . Thus we can write

(2.33)

so that the exchange-correlation potential may be written

What Is Exchange Correlation Functional

with

(2.35)

where in the last equation the assumption is that the exchange-correlation energy is purely local. The most common parametrisation in use for is that of Perdew and Zunger [49], which is based upon the quantum Monte Carlo calculations of Ceperley and Alder [50] on homogeneous electron gases at various densities; the parametrisations provide interpolation formulae linking these results.

The LDA ignores corrections to the exchange-correlation energy due to inhomogeneities in the electron density about . It may seem surprising that this is as successful as it is given the severe nature of the approximation in use; to large extent, it appears [40] that this is due to the fact that the LDA respects the sum rule, that is, that exactly one electron is excluded from the immediate vicinity of a given electron at point . The LDA is known to overbind, particularly in molecules. It is for this reason that in this study we have neglected it in favour of the GGA.

Exchange Correlation Functional In Dft

The GGA attempts to incorporate the effects of inhomogeneities by including the gradient of the electron density; as such it is a semi-local method. The GGA XC functional can be written as

where is known as the enhancement factor. Unlike the LDA, there is no unique form for the GGA, and indeed many possible variations are possible [48,52,53,54], each corresponding to a different enhancement factor. The GGA succeeds in reducing the effects of LDA overbinding [51], and is significantly more successful when applied to molecules. In this work, the PW91 GGA due to Perdew and Wang is used [48].
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