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Quantum crystallography and the use of kernel projector matrices
Author(s) -
Massa L.,
Huang L.,
Karle J.
Publication year - 1995
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560560841
Subject(s) - density matrix , electron diffraction , diffraction , electron density , quantum , projector , physics , matrix (chemical analysis) , structure factor , trace (psycholinguistics) , quantum mechanics , electron , chemistry , condensed matter physics , optics , linguistics , philosophy , chromatography
Quantum crystallography is a developing technique for extracting quantum mechanically valid properties from X‐ray diffraction experiments. Quantum mechanics and crystallography are joined through the fact that the electron distributions around atoms are the source of X‐ray diffraction and electron density distributions are observables that lend themselves readily to quantum mechanical description. Direct contact with the X‐ray diffraction data is made by equating the structure factor magnitudes, which are readily obtained from the measured X‐ray diffraction intensities, with the magnitudes of certain Fourier transforms of the quantum mechanical description of the electron distribution. The quantum mechanical description of the electron density involves molecular orbitals and an associated matrix. By requiring that the associated matrix be a projector with a normalized trace, while optimizing the fit to the experimental structure factor magnitudes, strong constraints are imposed on the relationship between the X‐ray data and the quantum mechanical description of the electron density distribution. The final result should be a wave function that is in good agreement with the X‐ray diffraction information and from which a variety of properties, e.g., electron densities and electrostatic potentials, could be extracted. The method for making the fit to the X‐ray data involves the use of least‐squares calculations in which the defining equations are the structure factor equations and equations that arise from the conditions defining a projector matrix with a normalized trace. The variables are the elements of the projector. It is also possible to refine such parameters as atomic coordinates. The calculations are facilitated by use of good initial projector matrices. For this purpose, a method has been developed for generating projector matrices for large molecules from the sum of kernel matrices. Kernel matrices are obtained from the use of fragments of the known atomic coordinates of the substance of interest and the application of molecular orbital methods in quantum mechanics. The use of fragments is justified by the fact that overlap integrals rapidly approach zero as the distances between atoms increase. © 1995 John Wiley & Sons, Inc.