A contracted Schr{\"o}dinger 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 2-particle reduced density matrix
(2-RDM) from the 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
Schr{\"o}dinger equation. The 1,2-CSE is solved through the
development and implementation of a reduced, linear-scaling
analogue 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
et al. [Phys. Rev. B {\bf 47}, 10891 (1993)] and Daw [Phys. Rev.
B {\bf 47}, 10895 (1993)]. Convergence of the contracted power
method with purification is illustrated with several molecules.
While providing a new tool for semi-empirical, Hartree-Fock, and
density functional calculations, the 1,2-CSE also represents an
initial step towards a linear-scaling algorithm for solving
higher CSEs which explicitly treat electron correlation.
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