Robert F. Stewart

Self‐Consistent Molecular‐Orbital Methods. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals

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.

Self‐Consistent Molecular Orbital Methods. IV. Use of Gaussian Expansions of Slater‐Type Orbitals. Extension to Second‐Row Molecules

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.

Small Gaussian Expansions of Slater‐Type Orbitals

Small Gaussian expansions of Slater‐type orbitals by the method of least squares are presented. Expansion lengths are from one to six. Slater‐type orbitals 1s through 5g are included. The least‐squares equations were solved by a full‐matrix method. Expectation values, r−2, r−1, r, and r2 for the several expansions are tabulated. Orbital products of the small Gaussian expansions have been Fourier analyzed and are satisfactory for x‐ray scattering analysis of x‐ray diffraction data or of molecular wavefunctions comprised of extended STO basis sets.

Generalized X‐Ray Scattering Factors

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.

Small Gaussian Expansions of Atomic Orbitals

Expansions of Clementi STO SCF AOs for some first‐row atoms with GTOs have been obtained by the method of least squares. Expansion lengths vary from two to five Gaussians for each AO. In addition, Gaussian sets with constraints that the exponential parameters be shared for 1s and 2s and 2s and 2p orbital expansions are reported. For all cases a full matrix least‐squares procedure was employed whereby the expansion coefficients and exponential parameters were simultaneously varied. All orbital expansions were constrained to a normalization condition. These expansions are useful for computing x‐ray scattering factors and can be used in quantum‐chemical studies.

Generalized x‐ray scattering factors in diatomic molecules

Generalized x‐ray scattering factors for atoms (pseudoatoms) in diatomic molecules are determined from a finite multipole expansion of the charge density about each nucleus. The Fourier–Bessel coefficients of the pseudoatom radial density functions are determined by a least squares fit to the molecular form factor. All molecular one‐center averages of the form 〈g (ra) Pj(cosϑa) 〉 are correctly given by the pseudoatom superposition whenever j?J, where J is the highest order multipole included in the a pseudoatom, regardless of the highest multipole order K for the b pseudoatom. To illustrate this property, a number of relationships between moments of the molecular charge distribution and of the pseudoatom are given. In addition, a sum rule relating the molecular form factor to the expectation values 〈ra−(j+1)Pj(cosϑa) 〉 and 〈rb−(k+1)Pk(cosϑb) 〉 is derived. For H2 the theoretical, coherent x‐ray scattering intensity is reproduced to about 1% for J=K=1 and to about 0.1% for J=K=2.

Valence Structure from X‐Ray Diffraction Data: An L‐Shell Projection Method

Generalized x‐ray scattering factors, in a simplified form, have been applied to several organic molecular crystals. Atomic charge parameters for L‐shell scattering factors have been determined from x‐ray structure factors by the method of least squares. L‐shell scattering factors have been computed for products of SCF AOs and of standard molecular STOs (Slater‐type orbitals). Standard STOs are more satisfactory for point‐charge analysis than are SCF AOs. Bias from several sets of thermal parameters is rather small and does not alter general conclusions on valence structure. The experimental x‐ray charges are compared with Mulliken gross atomic populations from INDO and STO‐3G calculations. Agreement with theory is good for the molecular crystals of s‐triazine, cyanuric acid, and uracil. Within the point‐charge approximation, the experimental x‐ray dipole moment for uracil is 4.0 ± 1.3 D. The direction is at an angle 71° ± 12° from the N(1)–C(4) interatomic vector towards atom N(3).

Two-center calculations for x-ray scattering.

Abstract An analytical procedure for the evaluation of two-center integrals in x-ray scattering calculations has been developed. The scattering potential is represented in prolate spheroidal coordinates. The plane wave operator is expanded in spheroidal wave-functions. All integrals are evaluated analytically with the aid of stable recursion relations. It is found that the several infinite sums involved are rapidly convergent, so that a practical computational method is at hand. The method is illustrated with a calculation of the molecular scattering factor for C-H from a near Hartree-Fock wavefunction comprised of an extended basis set of Slater-type orbitals. The coherent x-ray scattering intensity of HZ has been computed from a natural spin orbital expansion of the exact electronic wavefunction for the 1Σg+ state of H2.

Simultaneous variation of multipole parameters and Gram-Charlier coefficients in a charge-density study of tetrafluoroterephthalonitrile based on X-ray and neutron data.

Difficulties encountered in modelling the scattering of fluorine in organic compounds have been investigated through refinements of accurate X-ray and neutron diffraction data measured on tetrafluoroterephthalonitrile, TFT, at 122.4 K. Multipole refinements led to a highly contracted octopole on fluorine. The subsequent analysis revealed that fluorine does not possess a valence octopole but exhibits anharmonic thermal motion that can be modelled by the octopole multipole parameters. The scattering contribution from the octopole shows the same cubic dependence in the scattering vector as the Gram-Charlier expansion of the nuclear displacements to third order. The analysis also showed that refinement of third-order Gram-Charlier coefficients on fluorine requires data to at least 0.93 A(-1) resolution in sinthetas/lambda. The X-ray data extending to 1.27 A(-1) were of sufficient resolution to include third-order Gram-Charlier coefficients for N, F and the cyano C atoms in the refinement, whereas the neutron data only enabled refinement of the third-order Gram-Charlier coefficients for nitrogen. The refinements of the neutron and X-ray diffraction data yielded identical atomic displacement parameters for all the atoms. Though inclusion of anharmonic motion for N and F atoms provides the best model, it does not affect the crystal electron density, and all intramolecular bond critical points have identical features. Application of the anharmonic model, however, leads to small differences in the intermolecular interactions, which is illustrated by the electrostatic potential adjacent to the N atom. The characteristics of the C-F bond were elucidated by the topological analysis of the crystal electron density, which also supported the proposed quinonoid structure of the benzene ring.

Diatomic generalized x‐ray scattering factors: Results from Hartree–Fock electron density functions

Generalized x‐ray scattering factors based on multipole representation of the atoms have been calculated from Hartree–Fock Roothaan one‐electron molecular density functions of first row diatomic hydrides (BH to FH) and the 14 electron series BF, N2, (1Σ)CO, and (3Π)CO. The effect of different truncation of the mutipole expansions is discussed for NH. Results for the other molecules from expansions up to quadrupolar terms on each center are discussed. The monopole scattering factors are virtually the same as Hartree–Fock atomic form factors except at low sinϑ/λ values. The dipole and quadrupole scattering factors are rather specific to the particular molecule for the valence region, but core polarization features are similar. Static charge properties of the molecule are accurately given by the generalized scattering factors. However, charge properties associated with the individual pseudoatoms are strongly dependent on truncation of the multipole expansion. The coherent x‐ray scattering intensities of the ...