Ab initiodetermination of a crystal structure by means of the Schrödinger equation

Abstract

It is shown that the use of the Schro È dinger equation may lead to the ab initio determination of the positions of the nuclei in a crystal, given a limited number of diffracted-beam intensities. In particular, it is shown that an extremely simpli®ed Schro È dinger equation in physical momentum space provides a suf®ciently sound theoretical basis to develop an algorithm using diffraction data alone. This algorithm has been tested with a known 41 atom crystalline structure (not including hydrogen atoms) in space group P1. The extracted information is suf®cient to determine the positions of all atoms. In addition, theoretical developments relevant to the connection between momentum space in quantum mechanics and diffraction theory have been formulated.

CrystTwiV: a webserver for automated phase extension and refinement in X-ray crystallography

“…(2) Preparation of data including parameter estimation and calculation of statistics. (3) A detailed monitoring of the criteria for the correctness of the extended set of phases at each phase-extension step and phase-refinement cycle.…”

“…This FT has a precise physical meaning as the momentum space wave function: its square modulus represents the probability distribution over the momentum in the same way that ρ( r ) represents the probability distribution over the position of a quantum-mechanical particle (2). In the present context, this physical quantum mechanical meaning is not directly involved and the set of Ψ H plays the role of an auxiliary set of variables that determines E H via the left part of Equation (1).…”

High-resolution crystallography and drug design

“…QM/X-ray hybrid Hamiltonians were used by Raul Cachau (National Cancer Institute, SAIC-Frederick, USA) using the program DYNGA (Parker et al, 2003) to refine the structure of some of the phenyl groups in the structure of aldose reductase. The use of QM methods in Xray crystallography opens new possibilities in structure analysis (Bethanis et al, 2002;Jayatilaka and Grimwood, 2001;Schiffer and Hermans, 2003). Detailed inspection of the improved molecular geometries obtained at ultra-high resolution and refined using the more accurate QM/X-ray Hamiltonians may reveal chemical properties of the molecules.…”

“…We can further the analysis of the molecular properties by exploring the actual reactivity parameters of a given atom. In order to do so we must first fit a QM wavefunction to the experimental density (Bethanis et al, 2002). We can best explain the advantages of these procedures by first outlining the way a wavefunction is usually written during the computation of large chemical compounds [equations (1)- (3)]:…”

“…It should in theory be possible to infer the occupation of molecular orbitals, and therefore to determine the reactivity of a given atom, directly from the diffraction data. Such a goal is not yet attained on a regular basis for macromolecules of biological interest, but a number of efforts are being made in this direction (Bethanis et al, 2002;Jayatilaka and Grimwood, 2001). This level of detail would allow the chemical characterization of the interaction between a potential drug and a pharmaceutical target (well beyond the purely geometrical characterization of the interaction), and the identification of the sources of potency and selectivity of lead compounds (e.g.…”