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.

Wavefunctions derived from experiment. I. Motivation and theory.

There are two schools of thought in chemistry: one derived from the valence bond and molecular orbital models of bonding, the other appealing directly to the measurable electron density and the quantum mechanical theorems that determine its behavior, an approach embodied in the quantum theory of atoms in molecules, QTAIM. No one questions the validity of the former approach, and indeed molecular orbital models and QTAIM play complementary roles, the models finding expression in the principles of physics. However, some orbital proponents step beyond the models to impose their personal stamp on their use in interpretive chemistry, by denying the possible existence of a physical basis for the concepts of chemistry. This places them at odds with QTAIM, whose very existence stems from the discovery in the observable topology of the electron density, the definitions of atoms, of the bonding between atoms and hence of molecular structure. Relating these concepts to the electron density provides the necessary link for their ultimate quantum definition. This paper explores in depth the possible causes of the difficulties some have in accepting the quantum basis of structure beginning with the arguments associated with the acceptance of a "bond path" as a criterion for bonding. This identification is based on the finding that all classical structures may be mapped onto molecular graphs consisting of bond paths linking neighboring atoms, a mapping that has no known exceptions and one that is further bolstered by the finding that there are no examples of "missing bond paths". Difficulties arise when the quantum concept is applied to systems that are not amenable to the classical models of bonding. Thus one is faced with the recurring dilemma of science, of having to escape the constraints of a model that requires a change in the existing paradigm, a process that has been in operation since the discovery of new and novel structures necessitated the extension of the Lewis model and the octet rule. The paper reviews all facets of bonding beginning with the work of Pauling and Slater in their accounting for crystal structures, taking note of Pauling's advocating possible bonding between large anions. Many examples of nonbonded or van der Waals interactions are considered from both points of view. The final section deals with the consequences of the realization that bonded quantum atoms that share an interatomic surface do not "overlap". The time has come for entering students of chemistry to be taught that the electron density can be seen, touched, and measured and that the chemical structures they learn are in fact the tracings of "bonds" onto lines of maximum density that link bonded nuclei. Matter, as we perceive it, is bound by the electrostatic force of attraction between the nuclei and the electron density.

Definition of molecular structure: by choice or by appeal to observation?

Determination of electronic wave functions from electronic densities is discussed. A systematic new method based on the constrained‐search formulation of density functional theory is developed, and the method is tested for Be, Ne, and Ar. The results show that the Kohn–Sham kinetic energy and the Kohn–Sham orbitals can be obtained readily and accurately by this method. Since Kohn–Sham orbitals are very close to Hartree–Fock orbitals, the problem of finding a physical meaningful wave function from an electron density is thereby solved.

Constrained‐search method to determine electronic wave functions from electronic densities

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, eg, 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

Quantum crystallography and the use of kernel projector matrices

It is demonstrated that the approximate kinetic energy density calculated using the second-order gradient expansion with parameters of the multipole model fitted to experimental structure factors reproduces the main features of this quantity in a molecular or crystal position space. The use of the local virial theorem provides an appropriate derivation of approximate potential energy density and electronic energy density from the experimental (model) electron density and its derivatives. Consideration of these functions is not restricted by the critical points in the electron density and provides a comprehensive characterization of bonding in molecules and crystals.

The mapping of electronic energy distributions using experimental electron density.

The method of the one-electron density matrix determination from X-ray diffraction data is developed. It takes into account the translational symmetry and is applicable to covalent crystals. The crystal energies and directed Compton profiles are calculated in impulse approximation for silicon and diamond. The agreement with the experiments is rather good.



Die Methode der Ein-Elektronen-Dichtematrixbestimmung aus Rontgenbeugungsdaten wird entwickelt. Sie berucksichtigt die Translationssymmetrie und ist auf kovalente Kristalle anwendbar. Die Kristallenergien und gerichteten Compton-Profile werden in Impulsnaherung fur Silizium und Diamant berechnet. Die Ubereinstimmung mit den Experimenten ist recht gut.

Yu. V. Aleksandrov

Papers

The Crystal Electron Energy and Compton Profile Calculations from X-Ray Diffraction Data