Solution of Schrödinger’s equation for large systems

Abstract

Iterative diagonalization of the Hamiltonian matrix is required to solve very large electronicstructure problems. Present algorithms are limited in their convergence rates at low wave numbers by stability problems associated with large changes in the Hartree potential, and at high wave numbers with large changes in the kinetic energy. A new method is described which includes the e6'ect of density changes on the potentials and properly scales the changes in kinetic energy. The use of this method has increased the rate of convergence by over an order of magnitude for large problems. I. INTRQDUCTIQNThe motivation for this work has its foundations in attempts to model periodic ce11s containing many atoms while using a plane-wave basis. The number of plane waves required is moderate when modeling a material such as silicon, but is much larger when some of the atoms are first-row or transition-metal atoms. The nonlocal pseudopotentials associated with these atoms are quite deep and require high-wave-number plane waves to describe the resulting wave functions. The dual requirements of simultaneously describing a large cell volume and a high kinetic energy lead to an explosive growth in the number of basis functions. The need for a new algorithm became apparent when simulations had to be abandoned due to computational difhculties related to the 1arge number of plane waves, as well as the charge instabilities associated with large systems.We outline an adaptation to systems which have orthonormality constraints of an iterative conjugate-gradient method known to be eScient in very large minimization problems. The practical use of this method has been limited by the difhculty of determining the exact linear combinations of vectors which minimize the total energy. We describe a simple new technique for minimizing the total energy in a self-consistent manner. II. BACKGROUNDThe method of self-consistent fields commonly used in quantum-mechanical simulation is gracefully nonlinear.In the usual formulation, the wave function of the system to be studied is expressed as a product of single-particle eigenstates. The energy of the system is a function of a set of coeKcients of basis functions. While there exist many stationary points in the energy, for unpolarized electrons there are no false minima. As long as the ground state is not orthogonal to the starting point, the exact ground state is reachable from any starting set of coefticients by following a path of decreasing energy. Since the energy is at least a quartic function of the coe%cients, no direct method of solution exists and approximate solutions must be iterated until they no longer change. To begin the standard procedure, an initial electron-electron potential is determined by some sort of guess and a Hamiltonian matrix is generated. The Harniltonian matrix is diagonalized and the lowest eigenvectors are occupied. An electron density and corresponding electron-electron potential are generated from the solutions, and the process begins again. If one simply uses the outp...

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