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On the exchange and correlation partitioning of energies

The terms ``exchange'' and ``correlation'' are like the terms ``nonlinear,'' ``enantioselective,'' and ``structured'' in the terms nonlinear dynamics, enantioselective catalysis, and structured programming respectively -- they represent quantities made notable by their absence in historically prominent methods.

In ab initio methods, exchange refers to the difference in energy between a Slater determinant and a Hartree product wavefunction, while correlation refers to the difference in energy between the exact energy and the Slater determinant energy (Table 6.1). Physically, exchange can be viewed as the effect of adding Pauli repulsion to an independent-electron mean-field model, while correlation can be viewed as the effect of adding instantaneous Coulomb repulsion (as opposed to the Coulomb repulsion that determines the shape of the orbitals) to a mean-field model.

Since exchange and correlation are defined with respect to the levels of approximation within a specific method, we must take care in attempting to compare these quantities across different methods.

Table 6.1: Comparison of terms and physical effects included in ab initio versus density functional versus electron force field methods.
Terms Effects described Method name
Ab initio (wavefunction based)
Hartree product ke + electrostatics + self  
Slater determinant + exchange - self Hartree-Fock
Few Slaters + static correlation Gen. valence bond
Many Slaters + dynamic correlation Config. interaction
Density functional theory (Kohn-Sham)
Hartree product ke + electrostatics + self  
f Slater determinant + f exact exchange - f self  
(1 - f) exch functional + (1 - f) local exchange  
corr functional + dynamic correlation LDA, GGA ( $ \mathrm{f=0}$);
    hybrid ( $ \mathrm{f \neq 0}$)
Electron force field
Hartree product ke + electrostatics  
pairwise exchange + exchange eFF (exch only)
pairwise correlation + dynamic correlation eFF (exch + corr)

In density functional theory, we estimate correlation by integrating over a function of the electron density. Because of the way DFT is formulated, it emphasizes corrections to an independent particle model made when electrons are close to each other, so-called local or dynamic correlations. In contrast, configuration interaction methods most easily correct for longer range static correlations made when electrons delocalize over a longer distance, as in resonance stabilization or bond breaking -- density functional theory neglects these effects. Although for many chemical problems it is acceptable to neglect long-range electron correlation, it is not so acceptable to neglect long-range electron exchange. Modern hybrid density functionals combine a local exchange which is compatible with local correlation functionals with some fraction of longer-range nonlocal Hartree-Fock exchange (exact exchange).

Most work to improve ab initio methods has focused on finding more efficient ways to add dynamic correlation in a consistent way, whether through perturbation theory, configuration interaction, coupled cluster methods, or other schemes. Most work to improve density functional theory has focused on developing ``non-local'' correlation functionals, perhaps based on orbitals [11]; and on correcting the so-called ``self-interaction error,'' caused because in orbital schemes where less than the full quantity of exact exchange is used, some residual self-repulsion of individual electrons against themselves remains.

In the electron force field, we approximate exchange energy as a pairwise sum over same-spin electrons, and correlation energy as a pairwise sum over opposite-spin electrons. Since exchange arises as a consequence of Pauli repulsion, it is straightforward to see why it would be computed as an interaction between same-spin electrons. However, why should we restrict correlation to be an interaction between opposite-spin electrons? We reason that same-spin electrons are already segregated from each other because of the Pauli principle, so that the effects of adding electron correlation to pairs of same-spin electrons should be small compared to the electron correlation that acts between opposite-spin pairs.

With this scheme, both exchange and correlation are treated at the same level, in a consistent way. It should be possible to account for both long- and short-range exchange and correlation, since we are summing over electrons that may be far away from each other. Also, because we compute electrostatic interactions only between different electrons, there is no self-interaction error. Our method of estimating electron correlation as a pairwise sum over orbitals is reminiscent of the independent electron pair approximation (IEPA) methods developed back in the 1960s [12,13]. However, since those methods were developed in a configuration interaction framework, there were some issues with size consistency, which occurred because virtual orbitals could mix and lower the correlation energy even when molecules were infinitely separated [14]. Our method should have no such difficulties.

We consider the sum of kinetic energy, electrostatic potential energy, and pairwise exchange to be the equivalent of a Hartree-Fock calculation; and the further addition of pairwise correlation to be the equivalent of an ``exact calculation,'' comparable to a configuration interaction or B3LYP calculation. There are some difficulties with direct comparisons: Hartree-Fock can be computed in the exact basis limit, while we are limited to a subminimal basis of spherical Gaussians; also there are differences in the way static versus dynamic correlations are handled in CI versus DFT methods. With these caveats, we proceed with our comparisons, and find, remarkably, that the agreement is often quite reasonable.

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Julius 2008-04-29