Approximating the Exchange-Correlation Functional

The development of exchange correlation approximations are shown on Jacobs Ladder (Figure 2.2), where the bottom is HF theory and the top being the exact form of the exchange correlation term (or heaven of chemical accuracy) [1, 2].

Figure 2.2: Jacob’s Ladder of DFT exchange-correlation functionals. Adapted from [1, 2].

One of the first ways that computational chemists attempted to approximate the form of E_xc[ρ]

was through the local density approximation (LDA) (Rung 1 on Jacobs Ladder). The LDA attempts to calculate E_xc[ρ] by assuming that it can be evaluated using only the magnitude of the electron density at specific points in space, as shown in the equation below. LDAs are based upon the idea of a Homogeneous or Uniform Electron Gas (UEG) [22, 3, ?].

E_xc^LDA[ρ] = Z

ρ(r)xc(ρ)dr (2.34)

In the above equation xc(ρ) is the exchange correlation energy per particle of a UEG of density ρ.

The problem with LDA functionals is in the assumption that only the magnitude of the electron density matters which results in an overbinding of systems. This assumption is valid in the case of a UEG, and gives good results, however for many real systems LDA functionals cause errors greater than those obtained when HF theory is used. Nevertheless LDA functionals are still sometimes utilised for describing systems where there is little variation in the electron density such as in bulk metals [22].

In an attempt to remedy the problems of LDA functionals, the next rung of the Jacobs Ladder, Generalised Gradient Approximation (GGA) functionals were developed. These again take the ideas from a UEG but instead of using just the magnitude of the electron density they also use the gradient of the electron density. This is achieved partly through a correction parameter, one of the most successful and popular of these being the Lee-Yang-Parr (LYP) GGA functional which was parameterised based upon the helium atom [35]. This corrects the problem of overbinding that LDAs have, unfortunately these still do not give very accurate results when studying chemical processes and reactions. The problem with GGA functionals is that they overestimate the effect of the Coulomb interaction, effectively resulting in electrons avoiding each other too often [22, 3, 36]. Therefore one of the biggest issues when using GGA functionals to model chemical processes is that barrier heights obtained are generally too high [22, 3]. To further build upon GGA functionals, meta-GGA functionals (Rung 3) were introduced, which attempted to increase the accuracy of calculations by also using the second derivative of the density, an example of a m-GGA being the TPSS xc-functional [37]. However despite the extra correction of the second derivative of the density, m-GGA functionals still produce barrier heights which are too large. The mathematical form of both GGA and m-GGA functionals are shown below in Eqns 2.35 & 2.36.

E_xc^GGA[ρ] =

Hybrid functionals (Rung 4) are perhaps the most commonly used exchange correlation func-tionals, especially the Becke 3-parameter with LYP correlation (B3LYP) functional [38, 35, 39].

Hybrid functionals are constructed using a different approach than that taken in the construc-tion of LDA, GGA and m-GGA funcconstruc-tionals. Hybrid funcconstruc-tionals are constructed by combining

percentages of the lower rung DFT functionals with exact exchange from the HF system.

Using the Hellman-Feynman theorem [40] it can be shown that the exchange correlation term can be calculated as [3]

Exc[ρ] = Z 1

0

hΨ(λ)|V_xc(λ)|Ψ(λ)idλ (2.37)

where λ describes the degree of interelectronic interaction, a value of 0 describing the KS non-interacting system and a value of 1 describing the exact system. Evaluation of Eqn 2.37 is perhaps best shown graphically, as in Figure 2.3.

Figure 2.3: Graphical schematic of computation of Eqn 2.37 adapted from [3]. The area of rectangle A is calculated as the expectation value of the HF exchange operator acting on the non-interacting system, Ψ(0).

The result of integrating Eqn 2.37 is that the area under the curve in Figure 2.3 is obtained.

Whilst little is known about how Ψ and V_xc change with λ, what is known is the value at the

left hand limit. This is the value of exact exchange of the non-interacting system, rectangle A in Figure 2.3 therefore represents exact exchange. The area of rectangle B is hΨ(1)|Vxc(1)|Ψ(1)i − E_x^HF and thus the area under the curve is some fraction z of this area. Although z is not known it can be optimised empirically as in Eqn 2.38.

Exc= E_x^HF + z(E_xc^{DF T} − E_x^HF) (2.38)

which is often displayed using the value a = (1 − z) [3];

E_xc= (1 − a)E_xc^{DF T} + aE_x^HF (2.39)

Eqns 2.38 & 2.39 are the simplified forms for hybrid functionals, the popular B3LYP functional, which contains three parameters (a, b, c), has the form;

E_xc^{B3LY P} = (1 − a)E_x^LDA+ aE^HF_x + b∆E_x^B+ (1 − c)E_c^LDA+ cE_c^{LY P} (2.40)

where a, b & c equal 0.20, 0.72 and 0.81 respectively [38, 39, 35].

Hybrid functionals improve upon GGAs and m-GGAs in the majority of cases as they include some HF exchange, 20% in the case of B3LYP. Including this HF exchange helps reduce the self-interaction error, discussed later in Section 2.4. In the case of modelling reactions, as pure DFT functionals tend to overestimate barrier heights and HF underestimates, hybrid function-als see almost a cancelling of these errors and give generally good agreement with experiment [22, 3].

The development of these different functionals, particularly rungs 3 & 4, has seen DFT become the most popular method amongst chemists and its importance was recognised by the award

of the 1998 Nobel Prize in Chemistry to W. Kohn for ”development of the Density Functional Theory and Computational Methods in Quantum Chemistry” [22, 34, 41].