Local Electron Correlation

Traditional configuration-based theories of dynamic electron correlation, such as variational configuration interaction(CI), Møller-Plesset perturbation theory(MPn), and coupled-cluster (CC) theory, exhibit steep scaling with both the number of basis functions and the number of electrons. Their overall formal scaling at constant basis set quality varies between O(N 5) and O(N 7) or higher, where N is a measure of the molecular size. Worse, this formal scaling is not reduced automatically for large systems, in the way that the Fock matrix construction in SCF and DFT theory approaches O(N 2) scaling for large systems. The reason for this is the use of canonical molecular orbitals, which are generally delocalized over the whole molecule. Methods with more benign (ultimately linear) scaling must be formulated in terms of more localized quantities: atomic orbitals (AOs or basis functions), or localized molecular orbitals. This is particularly challenging for Møller-Plesset perturbation theory which, in its usual formulation, is defined in terms of canonical orbitals. This talk reviews the development of local electron correlation theories, with emphasis on our own work, and work derived from it. A very important step toward expressing, at least partly, correlation theory in terms of atomic orbitals was Meyer's self-consistent electron pair theory [1]. Although this theory was not aimed at large molecules and low scaling, it became the basis of much further development. Its generator state form [2] provides an unsurpassed concise matrix formulation of CI, MPn and CC. The matrix form not only facilitates numerical computation and implementation but also leads to improved operation count, by up to a factor of 2. Local correlation theories were first seriously proposed in the early and mid 1980s. However, localization becomes effective only for larger molecules (10-15 heavy atoms or more), and the state of computing at that time made it prohibitive to carry out the underlying large SCF calculations, except for a few fortunate researchers with good access to supercomputers. The situation has changed radically in the last decade, and SCF calculations with thousands of basis functions can now be easily performed on inexpensive clusters of personal computers. This has led to a resurgence of interest in local correlation theory. The Pulay-Saebo local correlation theory [3] is based on two simple facts. First, the pair correlation energy between localized orbitals diminishes rapidly with increasing separation between the orbital centroids, asymptotically like R-6. Thus the number of (occupied) orbital pairs with non-negligible contributions grows …