Wavefunctions derived from experiment. I. Motivation and theory

Abstract

An experimental wavefunction is one that has an assumed form and that is also fitted to experimental measurements according to some well defined procedure. In this paper, the concept of extracting wavefunctions from experimental data is critically examined and past efforts are reviewed. In particular, the importance of scattering experiments for wavefunction fitting schemes is highlighted in relation to the more familiar model, the Hamiltonian paradigm. A general and systematically improvable method for fitting a wavefunction to experimental data is proposed. In this method, the parameters in a model wavefunction are determined according to the variational theorem but subject to an imposed constraint that an agreement statistic between the calculated and observed experimental data has a certain acceptable value. Advantages of the method include the fact that any amount of experimental data can be used in the fitting procedure irrespective of the number of parameters in the model wavefunction, the fact that a unique answer is obtained for a given choice of the model wavefunction, and the fact that the method can be used to model different experiments simultaneously. The wavefunction fitting method is illustrated by developing the theory for extracting a single-determinant wavefunction for a fragment of a molecular crystal, using data obtained from elastic X-ray scattering data. Effects due to thermal motion of the nuclei, secondary extinction of the X-ray scattering and different choices for the crystal fragment are treated.

Hirshfeld Atom Refinement

“…This would take into account shortcomings of the theoretical model used to calculate EDs in HAR. In the present form, the term X-ray wavefunction refinement (XWR) refers to the subsequent execution of an iterative HAR, as introduced in this study, and an X-ray constrained wavefunction fitting procedure, introduced earlier by Jayatilaka (1998) and Jayatilaka & Grimwood (2001). The idea of XWR was first introduced by Grabowsky et al (2012), and put into context with other methods by Grabowsky et al (2013) and Schmøkel et al (2013).…”

Modern charge density studies: the entanglement of experiment and theory

“…It is also obvious that equation [18] could be merged with [17] proposed by Jayatilaka and coworkers (53), who connected the Hamiltonian and the diffracted intensities by means of a Lagrangian multiplier. This means imposing a constraint to the wave function: the wave function is calculated in such a way that on the one hand it minimizes the electronic energy (the normal variational principle) and on the other hand it minimizes the difference between the FT of the calculated electron density and the observed structure factors.…”

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.…”

“…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.…”