Optimization techniques in energy calculations involving the Hartree-Fock density matrix

Three optimization procedures are examined for their utilization in determining the optimum density matrix for a single determinant wave function. The total energy of a molecular system is written as a function of the density matrix and then optimized subject to the constraints of idempotency and total electron population. This direct calculation of the density matrix (DCDM) method was studied in an attempt to have a formalism which would avoid convergence problems associated with the self-consistent field (SCF) cycle, and which would be applicable to large molecular systems. The optimization procedures studied were the Powell algorithm, Gauss-Jordan reduction, and dynamic programming. Computational factors studied include convergence criteria, stepsize, and weight factors for constraint equation penalty functions. The application considered is for HF. An ab-initio SCF method was used to obtain initial values for the density matrix, and its SCF results were compared to corresponding DCDM predictions. Approximations of the quadratic two-electron energy contributions will be necessary to apply dynamic programming, but this appears to be the method most applicable to large molecular systems if an acceptable approximation can be found. Gauss-Jordan reduction is an applicable technique, but probably not for large molecular systems. BOTM appears to be the method most applicable without introducing approximations, but weight factors for the penalty functions will have to be more efficiently determined as Lagrange multipliers, and this addition would result in a technique probably not applicable to large molecular systems.This paper is based in part on the PhD Thesis of C. A. Waggoner, Department of Chemistry, Mississippi State University. Partial support was furnished by the National Science Foundation, Grant No. Rii-89-02064, the State of Mississippi, and Mississippi State University. Other support was furnished by the Physical and Biological Sciences Institute of MSU and the Office of the Dean, College of Arts and Sciences, MSU. Computer time was furnished by the Thomas E. Tramel Computing Center.