An alternative view of the coherent x-ray scattering factor with application to the hydrogen atom Am.The x-ray form factors for a bonded hydrogen in the hydrogen molecule have been calculated for a spherical approximation to the bonded atom. These factors may be better suited for the least-squares refinement of x-ray diffraction data from organic molecular crystals than those for the isolated hydrogen atom. It has been shown that within the spherical approximation for the bonded hydrogens in H 2 , a least-squares refinement of the atomic positions will result in a bond length (R. value) short of neutron diffraction or spectroscopic values. The spherical atoms are optimally positioned 0.07 A off each proton into the bond. A nonspherical density for the bonded hydrogen atom in the hydrogen molecule has also been defined and the corresponding complex scattering factors have been calculated. The electronic density for the hydrogen molecule in these calculations was based on a modified form of the Kolos-Roothaan wavefunction for H2• Scattering calculations were made tractable by expansion of a plane wave in spheroidal wavefunctions. 5 Compare K. Ruedenberg, Rev. Mod.

Least-squares representations of Slater-type atomic orbitals as a sum of Gaussian-type orbitals are presented. These have the special feature that common Gaussian exponents are shared between Slater-type 2s and 2p functions. Use of these atomic orbitals in self-consistent molecular-orbital calculations is shown to lead to values of atomization energies, atomic populations, and electric dipole moments which converge rapidly (with increasing size of Gaussian expansion) to the values appropriate for pure Slater-type orbitals. The ζ exponents (or scale factors) for the atomic orbitals which are optimized for a number of molecules are also shown to be nearly independent of the number of Gaussian functions. A standard set of ζ values for use in molecular calculations is suggested on the basis of this study and is shown to be adequate for the calculation of total and atomization energies, but less appropriate for studies of charge distribution.

The one-electron density function for a group of atoms within the asymmetric region of a unit cell is represented by a finite multipole expansion of the charge density about each atomic center. Each atomic expansion is called a pseudoatom. If the pseudoatom charge density is effectively rigid with nuclear motion, then the model may be used for a static charge density analysis of X-ray diffraction data. A valence density multipole model for pseudoatoms is restricted to single exponential radial functions. The representation is rotationally invariant. The model may be used for determination of static charge physical properties as well as aspects of chemical bonding. These results can be a critical test of the X-ray diffraction experiment for the determination of electron density distributions. The pseudoatoms discussed are primarily intended for crystals comprised of first and second-row atoms. The valence scattering model demands extensive data sets (probably at low temperatures) or an independent determination of atomic positions and mean square amplitudes of vibration.

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Least-squares representations of the 3s and 3p Slater-type atomic orbitals by a small number of Gaussian functions are presented. The use of these Gaussian representations in self-consistent molecular orbital calculations extends our previous study to molecules containing second row elements. Calculated atomization energies, electric dipole moments, and atomic charges are shown to rapidly converge (with increasing number of Gaussians) to their Slater limits. Results of valence shell optimization studies on a series of second-row compounds are nearly independent of the level of the Gaussian approximation, and they allow a set of standard molecular ξ exponents to be proposed.

Generalized x-ray scattering factors have been determined from the evaluation of Fourier transforms of atomic-orbital products. Self-consistent-field atomic orbitals for first-row atoms have been studied. Analytical scattering expressions for rapid evaluation on digital computers have been developed. Scattering factors from small GTO expansions have been compared with both Clementi and Huzinaga SCF atomic orbitals. For one-center orbital products, the three-set GTO atomic orbitals agree with full SCF scattering factors to within five parts per hundred or better; the four-set GTO expansions have relative differences of 1.5% or less. For the two-center scattering cases, three or more Gaussians per atomic orbital yield relative errors less than 1%. The generalized x-ray scattering factors can serve as basis functions in the analysis of charge densities from x-ray diffraction data.

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