Robert F. Stewart

Bio: Robert F. Stewart is an academic researcher from Carnegie Mellon University. The author has contributed to research in topics: Scattering & Charge density. The author has an hindex of 23, co-authored 39 publications receiving 8397 citations.

A vibrational correction to coherent x‐ray scattering intensities of diatomic molecules

Abstract: Born’s theory for the vibrational averaging of coherent x‐ray scattering has been implemented with the use of generalized x‐ray scattering factors. Estimates at 300 and 2000 °K for H2, N2, and CO are reported, where the generalized x‐ray scattering factors are truncated at the quadrupole level. The vibrational effect gives a scattered intensity which oscillates about the intensity of the nonvibrating molecule with increasing scattering angle. The estimated vibrational correction is no larger than 1% for N2 and CO over the range of the scattering vector (S?15.5 bohr−1 or sinϑ/λ?2.33 A−1) used in this work. For H2, however, the correction is as large as 5%.

Vibrational force constants for diatomic molecules from rigid pseudoatoms

Abstract: The quadratic vibrational force constant for a diatomic molecule can be evaluated from generalized x‐ray scattering factors if the expansion contains at least dipole scattering factors and if the first derivative of these functions with respect to R are known. Rigid pseudoatoms are defined as the finite multipole set, [L‖K], of generalized x‐ray scattering factors determined at Re. For this case, the quadratic force constant estimate depends on the pseudoatom charge density on the neighboring nucleus and on the pseudoatom contribution to the field gradient about the neighboring nucleus as shown by Bader and Bandrauk [J. Chem. Phys. 49, 1666 (1968)]. A previous estimate for H2 is shown to be in error. The estimates of ke for N2, CO, BF, and FH from [2‖2] rigid pseudoatoms are calculated and prove to be too large by more than 50%. The radial relaxation terms reduce these values to the correct ke.

Vibrational Studies of X-Ray Molecular Form Factors and Intensities

Abstract: The vibrational average of \(I_{e1}^{XR}\left( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{S};\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Q}\right)\) is an observable. It is not clear that a corresponding vibrational average of \(F\left( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{S};\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Q}\right)\) is an observable. Vibrationally averaged X-Ray and electron scattering intensities are determined theoretically for the diatomic molecules \({{N}_{2}}\left( {}^{I}\sum _{g}^{+} \right),CO\left( {}^{I}\sum _{g}^{+} \right),BF\left( {}^{I}\sum _{g}^{+} \right)\) and \(FH\left( {}^{I}\sum _{g}^{+} \right)\). These are averages for ground vibrational states. The intensifies are also modeled with generalized X-Ray scattering factors (pseudoatoms). It is found that anharmonic effects are about ten times larger than non-rigid effects. Generalized X-Ray scattering factors are extracted from \( \) and compared to the same functions from \(F\left( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{S};\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}\right)\). If anharmonic terms are included in the pseudoatom model, then deconvolution is found to be excellent. Vibrational force constants for the above diatomics are determined from \(\left[2\left|2 \right. \right]\) rigid pseudoatoms. In all cases k e is too large (by as much as 58%). The \(\left\{{\partial {{f}_{a}}}/{\partial R}\; \right\}\) and \(\left\{ {\partial {{f}_{b}}}/{\partial R}\; \right\}\) nctions necessarily give the correct k e when included in the pseudo atom model.

Multipolar Expansion of One-Electron Densities

Abstract: One and two-center atomic orbital products can be expanded with a finite number of multipoles on the several centers to high accuracy. The problem is formally solved for the charge density of a diatomic molecule by the method of least squares. In this case radial scattering factors for the two centers are directly determined from functional equations. Both the molecular form factor and the elastic x-ray scattering intensity are accurately given for a small set of multipoles on each center. The generalized x-ray scattering factors are not a property unique to the molecule, but a large number of static charge molecular properties are correctly given. Several applications of generalized x-ray scattering factors, with restricted radial functions, to x-ray diffraction data are briefly discussed.

Self‐consistent molecular orbital methods. XX. A basis set for correlated wave functions

Abstract: A contracted Gaussian basis set (6‐311G**) is developed by optimizing exponents and coefficients at the Mo/ller–Plesset (MP) second‐order level for the ground states of first‐row atoms. This has a triple split in the valence s and p shells together with a single set of uncontracted polarization functions on each atom. The basis is tested by computing structures and energies for some simple molecules at various levels of MP theory and comparing with experiment.

Self‐Consistent Molecular‐Orbital Methods. IX. An Extended Gaussian‐Type Basis for Molecular‐Orbital Studies of Organic Molecules

Abstract: An extended basis set of atomic functions expressed as fixed linear combinations of Gaussian functions is presented for hydrogen and the first‐row atoms carbon to fluorine. In this set, described as 4–31 G, each inner shell is represented by a single basis function taken as a sum of four Gaussians and each valence orbital is split into inner and outer parts described by three and one Gaussian function, respectively. The expansion coefficients and Gaussian exponents are determined by minimizing the total calculated energy of the atomic ground state. This basis set is then used in single‐determinant molecular‐orbital studies of a group of small polyatomic molecules. Optimization of valence‐shell scaling factors shows that considerable rescaling of atomic functions occurs in molecules, the largest effects being observed for hydrogen and carbon. However, the range of optimum scale factors for each atom is small enough to allow the selection of a standard molecular set. The use of this standard basis gives theoretical equilibrium geometries in reasonable agreement with experiment.

Natural population analysis

Abstract: A method of ‘‘natural population analysis’’ has been developed to calculate atomic charges and orbital populations of molecular wave functions in general atomic orbital basis sets. The natural analysis is an alternative to conventional Mulliken population analysis, and seems to exhibit improved numerical stability and to better describe the electron distribution in compounds of high ionic character, such as those containing metal atoms. We calculated ab initio SCF‐MO wave functions for compounds of type CH3X and LiX (X=F, OH, NH2, CH3, BH2, BeH, Li, H) in a variety of basis sets to illustrate the generality of the method, and to compare the natural populations with results of Mulliken analysis, density integration, and empirical measures of ionic character. Natural populations are found to give a satisfactory description of these molecules, providing a unified treatment of covalent and extreme ionic limits at modest computational cost.

AB INITIO Molecular Orbital Theory

Abstract: Describes and discusses the use of theoretical models as an alternative to experiment in making accurate predictions of chemical phenomena. Addresses the formulation of theoretical molecular orbital models starting from quantum mechanics, and compares them to experimental results. Draws on a series of models that have already received widespread application and are available for new applications. A new and powerful research tool for the practicing experimental chemist.

Self‐consistent molecular orbital methods 25. Supplementary functions for Gaussian basis sets

Abstract: Standard sets of supplementary diffuse s and p functions, multiple polarization functions (double and triple sets of d functions), and higher angular momentum polarization functions (f functions) are defined for use with the 6‐31G and 6‐311G basis sets. Preliminary applications of the modified basis sets to the calculation of the bond energy and hydrogenation energy of N2 illustrate that these functions can be very important in the accurate computation of reaction energies.