Introduction

Most standard textbook approaches to solving systems of linear equations or diagonalizing matrices are described as ``direct'' methods, and they typically require a fixed number of mathematical operations which depends on the dimensions of the problem. These methods generally require access to matrix elements in random order, which poses serious difficulties for the very large matrices typically encountered in computational quantum chemistry: random access of large disk files becomes prohibitively expensive, and often the matrices are too large even to store on disk! In such cases, one may avoid the need for random access to individual matrix elements by turning to iterative techniques, which require only the repeated evaluation of matrix-vector products. Unfortunately, iterative methods are not guaranteed to converge, and they can have difficulties when the matrix is not diagonally dominant or when there are nearly degenerate solutions.

The well-known Davidson method [1] for the iterative solution of the lowest few eigenvalues and eigenvectors of large, symmetric matrices combines some of the features of direct and iterative techniques. Although only matrix-vector operations are required, and there is no need to explicitly store the Hamiltonian matrix, Davidson's method also uses direct methods to diagonalize a small Hamiltonian matrix formed in the subspace of all trial CI vectors that have been considered up to the present iteration. The current estimates of the eigenvalues of the full Hamiltonian matrix are obtained as the eigenvalues the small Hamiltonian matrix, and the current CI vectors are obtained as the linear combinations of the trial vectors whose coefficients are given by the eigenvectors of the small Hamiltonian matrix. Pople and co-workers later used related ideas to iteratively solve the large systems of linear equations occuring in the coupled-perturbed Hartree-Fock method [2].

In 1980, Pulay published a somewhat similar method [3] known as the direct inversion of the iterative subspace (DIIS). Like the Davidson method, DIIS applies direct methods to a small linear algebra problem (now a system of linear equations instead of an eigenvalue problem) in a subspace formed by taking a set of trial vectors from the full-dimensional space. Pulay found that DIIS could be useful for accelerating the convergence of self-consistent-field (SCF) procedures and, to a lesser extent, geometry optimizations.