Efficient update of determinants for many-electron wave function overlaps

Abstract The calculation of overlaps between many-electron wave functions at different nuclear geometries during nonadiabatic dynamics simulations requires the evaluation of a large number of determinants of matrices that differ only in a few rows/columns. While this calculation is fast for small systems, its cost grows faster than the alternative electronic structure calculation used to obtain the wave functions. For wave functions that can be written as a CIS expansion, all determinants can be computed using the set of level-2 minors of the reference matrix. However, this is still a costly computation for large systems. In this paper, we provide an algorithm for efficiently calculating all level-2 minors of a matrix by re-utilizing and updating the LU factorization for the determinants of the minors. This approach results in a parallel version of the algorithm that is up to an order of magnitude faster then the current best parallel implementation. The algorithm thus allows the computation of exact wave function overlaps for relatively large systems, with a high density of states, at virtually no cost compared with the electronic structure calculations. Furthermore, the new algorithm opens the path to further investigations in efficient computing of the exact wave function overlaps for complex wave functions such as MR-CIS and MR-CISD. Program summary Program Title: CIS Overlap Licensing provisions: MIT Programming language: FORTRAN 2008, C Nature of problem: Calculation of overlaps between CIS type wave functions at different nuclear geometries during nonadiabatic dynamics simulation requires calculating a large number of connected determinants and scales with the seventh power of the size of the system being studied. Without additional approximations, for large systems this computation becomes more costly than the electronic structure calculation used to obtain the wave functions. Solution method: All of the determinants required for a CIS wave function overlap calculation can be derived from the set of all level-2 minors of the matrix of overlaps between the reference Slater determinants. We developed an algorithm for efficiently computing all level-2 minors of a matrix. This part of the computational process was the bottleneck in previous solutions to the problem and is now an order of magnitude faster resulting in significantly faster overall calculation of the wave function overlaps.

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