Introduction

The major disadvantage of DFT is that there is no systematic way to improve a density functional. Thus, quality information on the performance of a density functional is essential in determining how to improve functionals. It is common to compare the exchange-correlation energies from a density functional with those determined by configuration interaction (or the related perturbation or coupled-cluster theories) on a single determinant reference wavefunction. Of greater use in the design of new functionals would be the ability to examine a functional's performance for various subsets of electrons in a molecule. These partitions are common in conventional theory. For example, the correlation energy associated with inner-shell electrons is often separated and ignored (the frozen-core approximation). Also, it is often possible (particularly at the simplest MP2 level) to separate correlation between electrons of parallel ( $\alpha \alpha$ or $\beta \beta$) and antiparallel ( $\alpha \beta$) spin. With MP2 theory the total correlation energy can be expressed as a sum of electron-pair components, making such partitions straightforward. Another useful partition is the splitting of density into core and valence densities, allowing the examination of core-electron/core-electron, core-electron/valence-electron and valence-electron/valence-electron contributions to the correlation energy.

This chapter, following earlier work of Stoll et al. [152,153] and Perdew et al. [154], presents similar partitions of DFT exchange-correlation energies to their conventional counterparts. This is carried out by examining electron correlation relative to the Kohn-Sham single determinant reference wavefunction. Correlation energies are obtained from conventional theory and DFT. Each may be partitioned by dividing the occupied spin orbitals into non-overlapping sets, corresponding to a division of the density into two parts, allowing the resulting energy components to be compared.