On the Physical Origin of the 1/4 Exact Exchange in Density Functional Theory

This talk discusses the physical origin of the widely used 1/4 fraction of exact exchange in density functional theory (DFT)’s hybrid functionals. We show that this 1/4 fraction of exact exchange can be attributed to spin-flip exchange interactions between pair states with antiparallel spin, which arise from the coupling between doubly-excited configurations contributing to the ground state energy. To obtain this result, we compare two models of the exchange energy of an electron pair in an unpolarized electron gas. One model treats the spin states of the electron pair as a statistical ensemble, while the other includes their quantum superposition. The quantum mechanical treatment is shown to possess an exchange energy equal to that of the statistical ensemble, plus an additional 1/4 exact exchange energy due to a spin-flip exchange interaction between electron pair states with antiparallel spin. By showing that the statistical ensemble contains 3/4 of the density-based exchange energy, we can attribute, for each pair of occupied orbitals, the 1/4 exact exchange used in hybrid DFT to the spin-flip exchange processes mentioned above. Implications including the tendency of the LDA to overbind and the performance of hybrids for few-electron systems will be discussed.