Iterative submatrix diagonalisation for large configuration interaction problems

The Davidson method has been highly successful for solving for eigenpairs of the large matrices that are common in quantum chemical simulations. Electronic structure simulations, however, can still easily generate matrices that are too large for current computational resources to handle. Therefore, many strategies have arisen to obtain eigenpairs of sufficient accuracy without considering the full Hamiltonian matrix. This article introduces one such strategy by creating a systematic series of submatrix approximations to the full matrix using natural orbitals. By solving for eigenpairs in this series, the eigenvalue accuracy can be gradually increased until a convergence threshold is reached. Importantly, this allows the series to terminate without ever reaching the full matrix, resulting in lower computational costs and reduced memory demands. Application of the method to the full configuration interaction problem for ground states, excited states and potential energy scans of various systems shows that the iterative submatrix diagonalisation method can systematically control eigenvalue errors and provide substantial cost-savings. This method is therefore expected to be highly useful for large-scale diagonalisation problems in electronic structure theory.