What Is? 

Within the context of Kohn-Sham Density Functional Theory, the exchange-correlation (xc) functional is a term in the electronic potential that captures the portion of the energy which cannot be described by Coulombic effects and classical (Hartree) electron-electron interactions. As such, the xc functional is the portion of the DFT potential that captures all of the complex effects arising from the phenomena of electron exchange and correlation. The exact xc functional cannot be known, and its construction is one of the few fundamental approximations of DFT. As such, the design of approximate xc functionals is of great importance to formulate more accurate and efficient DFT schemes. Early approximations to the xc functional considered only the electron density at the point in space where the functional is evaluated. Later approximations incorporated the gradient of the electron density. Many modern xc functionals are semi-empirical–they are not derived directly from first principles, and incorporate a portion of the Hartree-Fock exchange energy. In many common applications, the most sophisticated modern xc functionals rival CCSD(T) in terms of chemical accuracy, at a lower cost.       

History 

The exchange-correlation functional has its roots in the Kohn-Sham formalism of DFT. In 1965, Kohn and Sham published a landmark paper which proposed that the many-body wavefunction could be rewritten into a sum of one-electron wavefunctions without loss of generality, by including a term in the 1-electron Hamiltonians which captures exchange and correlation effects. In the same paper, they proposed an early approximation for the exchange correlation potential as a function of the local electron density (LDA). While the theoretical achievement by Kohn and Sham was remarkable, DFT with the LDA is generally not accurate enough to be used in quantum chemistry calculations.
As the need for more accurate xc functionals became apparent, there was a divergence between functionals which were constructed using only first principles, versus those that are semi-empirical. The Perdew-Burke-Ernzerhof (PBE) functional, developed in the late 1980’s, demonstrates the appeal of first-principles functionals clearly. Compared to other gradient approximations, it performs reasonably well across all systems, but doesn’t excel in any one case. By contrast, semi-empirical functionals are often more accurate for specific systems at the cost of transferability. A breakthrough in semi-empirical functionals came with the development of the Becke, Lee, Yang, and Parr 3-parameter functional (B3LYP) in the early 1990’s. The functional, in addition to considering effects from the local electron density and its gradient, incorporates a portion of the exact Hartree exchange energy. The addition of a portion of the Hartree exchange helps to mitigate errors caused by electron self-interaction. B3LYP was the first functional considered accurate enough to be used in the broader quantum chemistry field, providing a significant improvement over Hartree-Fock at a much more favorable cost.  

Details 

The exchange-correlation functional may be written as a term in the Sham potential, where the Sham potential V_s[ρ] written as 

where represents the xc functional. The LDA assumes that the xc energy may be written as only a function of the local electron density: 

where reflects the local electron density at each point as if it arises from a uniform, homogeneous electron gas. The exact form of will depend on the specific choice of LDA functional. If greater accuracy is desired, especially when the electron density is varying rapidly, one should consider using a generalized gradient approximation (GGA) functional, where takes on the form 

In this case, the electron density is a function of both the local electron density and its gradient. In the case of PBE, a popular GGA, the functional looks like  

where reflects the uniform electron density as a function of the position from the LDA, and is an 'expansion factor' that incorporates information about the gradient.  

In the case of B3LYP, the xc functional is a hybrid--a weighted average of the exact Hartree exchange energy and a GGA approximation. The functional is a weighted average of the LDA, the Becke 3-parameter exchange functional, the Lee-Yang-Parr correlation functional, and the exact Hartree exchange term: 

where the relative weights on each component were determined to empirically fit small molecule experimental data. This functional provides a meaningful improvement to Hartree-Fock in most cases, but struggles to correctly model long-range dispersion. Corrected functionals, such as B3LYP-D3, have been developed to mitigate this problem. 

When thinking about how different xc functionals relate to each other, it can be helpful to think of Jacob's ladder, a conceptual tool developed by Perdew and Schmidt that references the biblical artifact of the same name. At higher 'rungs' on Jacob's ladder, additional components are considered in the xc functional, and we get closer to the 'heaven' of the exact DFT solution. At lower rungs, we are closer to the 'earth' of the Hartree-Fock approximation. 

Example 

Let's look at the accuracy of a variety of density functionals as implemented in the Quantum Chemistry Toolbox in Maple when compared with Hartree-Fock theory. As an example, we can compute the dipole moment of the hydrogen fluoride molecule. 

>

 

>	mol := MolecularGeometry("hydrogenfluoride");

 

(4.1)

 

whose 3D ball-and-stick structure is  

>	PlotMolecule(mol);

 

To establish a minimum standard of accuracy, we can compute the dipole moment using Hartree-Fock theory:  

>

 

>

 

(4.2)

 

Selected References