Linear scaling and the 1,2-contracted Schrodinger equation

A contracted Schrodinger equation (1,2-CSE) is derived for the class of Hamiltonians without explicit interactions including those from Hartree-Fock and density functional theories. With cumulant reconstruction of the two-particle reduced density matrix (2-RDM) from the one-particle-RDM (1-RDM), the 1,2-CSE may be expressed solely in terms of the 1-RDM. We prove that a 1-RDM satisfies the 1,2-CSE if and only if it is an eigenstate of the N-particle Schrodinger equation. The 1,2-CSE is solved through the development and implementation of a reduced, linear-scaling analog of the ordinary power method for finding matrix eigenvalues. The power formula for updating the 1-RDM requires fewer matrix operations than the gradient procedure derived by Li [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)]. Convergence of the contracted power method with purification is illustrated with several molecules. While providing a new tool for semiempirical, Hartree-Fock, and density functional calculations, the 1,2-CSE also represents an initial step toward a linear-scaling algorithm for solving higher CSEs which explicitly treat electron correlation. (C) 2001 American Institute of Physics.