Coherent X‐Ray Scattering for the Hydrogen Atom in the Hydrogen Molecule
01 May 1965-Journal of Chemical Physics (American Institute of PhysicsAIP)-Vol. 42, Iss: 9, pp 3175-3187
TL;DR: In this paper, the x-ray form factors for a bonded hydrogen in the hydrogen molecule have been calculated for a spherical approximation to the bonded atom, and the corresponding complex scattering factors have also been calculated.
Abstract: 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 H2, a least‐squares refinement of the atomic positions will result in a bond length (Re 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.
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TL;DR: In this paper, it was shown that the dative Si-O bond from the THF molecule, with a length of 1.878(4) A, is significantly longer than ordinary SiO bonds involving two- or three-coordinated oxygen.
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TL;DR: In this paper, the Fourier-Bessel coefficients of the pseudoatom radial density functions are determined by a least squares fit to the molecular form factor, where the moments of the molecular charge distribution and of a pseudoatom are given.
Abstract: 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.
84 citations
TL;DR: In this article, the electron density distribution in the simplest amino acid, α-glycine, was studied using a Gaussian basis set of double-ζ type and a MO-LCAO SCF approach.
Abstract: Accurate x‐ray and neutron diffraction data have been combined to study the electron density distribution in the simplest amino acid, α‐glycine. Ab initio electron density calculations have also been made using an MO‐LCAO SCF approach employing a Gaussian basis set of ``double‐ζ'' type. Highly satisfactory qualitative agreement is found.
83 citations
TL;DR: In this paper, the incrystal molecular dipole moment of the nonlinear optical material 2−methyl−4−nitroaniline has been determined from a charge density analysis of x-ray diffraction data.
Abstract: The in‐crystal molecular dipole moment of the nonlinear optical material 2‐methyl‐4‐nitroaniline has been determined from a charge density analysis of x‐ray diffraction data. The results indicate a considerable enhancement of the free molecule dipole moment, due to the crystal field. The analysis suggests that aspherical pseudoatoms are essential for modeling the charge distribution in a noncentrosymmetric crystal. Careful consideration must also be given to the treatment of hydrogen atoms, in the absence of complementary neutron diffraction data. An analysis of the deformation density and Laplacian of the charge density proves useful for revealing weak hydrogen bonding effects. Ab initio calculations at the Hartree–Fock double‐ζ level are reported for the molecule 2‐methyl‐4‐nitro‐aniline, with and without an applied electric field. In the former case, the magnitude and direction of the applied field were determined by a dipole lattice sum, to assess the magnitude of crystal field effects. The effect was...
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TL;DR: In this article, the reaction between InCl and (PhCH 2 ) 5 C 5 In 1 (1) is solved to final values of R = 0.025 for 5545 observed reflections with F 0 ≥ 4σ (F 0 ).
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TL;DR: In this article, the quantum mechanical wave functions of molecules are discussed and an attempt is made to effect a simultaneous regional and physical partitioning of the molecular density, the molecular pair density, and the molecular energy, in such a way that meaningful concepts can be associated with the density and energy fragments thus formed.
Abstract: The quantum mechanical wave functions of molecules are discussed. An attempt is made to effect a simultaneous regional and physical partitioning of the molecular density, the molecular pair density, and the molecular energy, in such a way that meaningful concepts can be associated with the density and energy fragments thus formed. The origin of chemical binding is interpreted in terms of the concepts formulated in the partitioning process. (T.F.H.)
768 citations
576 citations
TL;DR: The solution of Schroedinger's equation for the normal hydrogen molecule is approximated by the function $C[{e}^{\ensuremath{-}\frac{z({r}_{1}+{p}_{2})}{a}}+{e^{\ensem{-]-{m{e})+{m}−m{n}−n}]$ where m is the distance of one of the electrons to the two nuclei, and r is the distances of one electron to the other electron.
Abstract: The solution of Schroedinger's equation for the normal hydrogen molecule is approximated by the function $C[{e}^{\ensuremath{-}\frac{z({r}_{1}+{p}_{2})}{a}}+{e}^{\ensuremath{-}\frac{z({r}_{2}+{p}_{1})}{a}}]$ where $a=\frac{{h}^{2}}{4{\ensuremath{\pi}}^{2}m{e}^{2}}$, ${r}_{1}$ and ${p}_{1}$ are the distances of one of the electrons to the two nuclei, and ${r}_{2}$ and ${p}_{2}$ those for the other electron. The value of $Z$ is so determined as to give a minimum value to the variational integral which generates Schroedinger's wave equation. This minimum value of the integral gives the approximate energy $E$. For every nuclear separation $D$, there is a $Z$ which gives the best approximation and a corresponding $E$. We thus obtain an approximate energy curve as a function of the separation. The minimum of this curve gives the following data for the configuration corresponding to the normal hydrogen molecule: the heat of dissociation = 3.76 volts, the moment of inertia ${J}_{0}=4.59\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}41}$ gr. ${\mathrm{cm}}^{2}$, the nuclear vibrational frequency ${\ensuremath{
u}}_{0}=4900$ ${\mathrm{cm}}^{\ensuremath{-}1}$.
292 citations
TL;DR: In this paper, a simple wave function for the normal state of the hydrogen molecule, in which both the atomic and ionic configurations are taken into account, was set up and treated by a variational method.
Abstract: A simple wave function for the normal state of the hydrogen molecule, in which both the atomic and ionic configurations are taken into account, was set up and treated by a variational method. The dissociation energy was found to be 4.00 v.e. as compared to the experimental value of 4.68 v.e. and Rosen's value of 4.02 v.e. obtained by use of a function involving complicated integrals. It was found that the atomic function occurs with a coefficient 3.9 times that of the ionic function. A similar function with different screening constants for the atomic and ionic parts was also tried. It was found that the best results are obtained when these screening constants are equal. The addition of Rosen's term to the atomic‐ionic function resulted in a value of 4.10 v.e. for the dissociation energy.
253 citations
133 citations