Jerome Karle

Structural Chemistry and Quantum Mechanics


Abstract

An area of research will be presented that brings together two different disciplines, structure determination, which comes under the heading of crystallography, and the theoretical calculations of quantum mechanics. The joining of these two disciplines is very natural since X-rays, which are commonly used for crystal structure determination, are sensitive to the distribution of electrons around atoms, and the theoretical description of electron distribution is readily obtained from quantum mechanical theory. In actual practice we did not equate the electron densities from the experiment and theory. Instead, we equated the so-called structure factor magnitudes obtainable independently from experiment and theory. The structure factor magnitudes are proportinal to the square root of the measured X-ray intensities of scattering.

We thus have a set of structure factor magnitudes from experiment and we need to set up a quantum mechanical model that, with some adjustment, can be made to match the experimental data, as well as accuracy permits. The quantum mechanical model involves the use of linear combinations of atomic orbitals which are used to describe an electron density distribution. In order to obtain theoretical values for the structure factor magnitudes, it is necessary to take the Fourier transform of the electron density distribution. With accurate experimental data and a quantum mechanical model that has a sufficiently complex basis set, it should be possible to obtain reasonable agreement between the experimental and theoretical structure factor magnitudes.

It is possible to make adjustments to the quantum mechanical model in order to optimize the agreement between experimental and theoretical values of the structure factor magnitudes. This takes advantage of the structure of the quantum mechanical model that includes a mathematical quantity called a single projector matrix, Gaussian orbitals and atomic coordinates. Adjustments for optimizing the agreement can be made on the elements of the projector matrix, the coordinates and the constants in the exponents of the Gaussian functions that represent the atomic orbitals.

In the course of applying our initial approach to quantum crystallography to our test molecule, maleic anhydride, certain features were included that resulted in increased accuracy and a more efficient procedure. Such features include the use of quantum mechanics to remove systematic errors and eliminate thermal motion effects. Some applications will be discussed.


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