Convergence study of a Schrödinger-equation algorithm and structure-factor determination from the wavefunction

Abstract

The algorithm [Bethanis, Tzamalis, Hountas & Tsoucaris (2002). Acta Cryst. A58, 265-269] which reformulates the quantum-mechanical problem of solving a Schrödinger (S) equation in a crystallographic context has been upgraded and tested for many aspects of convergence. The upgraded algorithm in reciprocal space aims at determining a wavefunction Phi(H) such that (a) Phi(H) fulfils the S equation within certain precision and (b) Phi(H) minimizes by least squares the differences between the calculated structure factors from the wavefunction and the observed ones. Calculations have been made with three molecules (11, 41 and 110 non-H atoms in the asymmetric unit) for different numbers of initially given phases. Three main questions have been addressed: (I) Does the iterative calculation of the wavefunction converge? (II) Do the calculated wavefunctions converge to a unique set of Phi(H) values independent of the initial random set of Phi(H)? (III) Is the calculated Phi(H) set a good approximation of a wavefunction able to produce within certain errors the correct values of the phases of the structure factors? Concerning questions (I) and (II), our results give a strong hint about fast convergence to a unique wavefunction independent of the arbitrary starting wavefunction. This is an essential prerequisite for practical applications. For question (III) in the case closer to the ab initio situation, the final mean phase error, respectively, for the three structures is 3, 26 and 28 degrees. The combination of (a) and (b) in the upgraded algorithm has been proved crucial especially for the results concerning the larger structures.