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, the atomic distances within a formamide molecule and the intermolecular N···O distance between the hydrogen-bonded molecules are determined by the X-ray diffraction experiment.
Abstract: The molecular and liquid structures of formamide have been studied by means of X-ray diffraction. Relative stabilities of aggregates of formamide molecules have been calculated by using the ab initio LCGO-MO-SCF procedure and minimal basis sets. The atomic distances within a formamide molecule and the intermolecular N···O distance between the hydrogen–bonded molecules are determined by the X-ray diffraction experiment as follows: C=O: 1.24(1) A, C–N: 1.33(1) A, N···O (intramolecular): 2.25(2) A, and N···O (intermolecular): 3.05(5) A. The intermolecular (Remark: Graphics omitted.) bond angle (Θ) is about 120°. It is concluded from the X-ray diffraction experiment that liquid formamide mainly consists of the chain-like hydrogen–bonded structure of formamide molecules by combining through –NH2···O=CH– interactions. The conclusion is supported by the ab initio calculations. Formation of ring-dimers in the liquid formamide has not been confirmed by the X-ray diffraction study, although a possibility of the for...
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TL;DR: In this article, Epstein and Bernal showed that the Ballhausen-Dahl model is not valid for M(h5−C5H5)2L2 systems.
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TL;DR: The results from the topological analysis of these systems confirm the success of the 'best anisotropic' model in providing parameters for the H atoms that give charge densities in agreement with the reference models based on H-atom parameters derived from neutron diffraction.
Abstract: Extensive and precise X-ray diffraction data for xylitol have been used to test different approaches to estimate nuclear parameters for H atoms in charge-density studies. The parameters from a neutron diffraction study of the same compound were taken as a reference. The resulting static charge densities obtained for the different approaches based on a multipole model were subjected to a topological analysis. The comparative analysis led to the following results. The procedure of extending the X-H bond to match bond lengths from neutron diffraction studies provides the best agreement with the neutron positional parameters. An isotropic model for the atomic displacements of H atoms is highly unsatisfactory and leads to significant deviations for the properties of the bond critical points including those that only involve non-H atoms. Anisotropic displacement parameters for H atoms can be derived from the X-ray data that are in agreement with the values from the neutron study, and the resulting charge-density models are in good agreement with the reference model. The anisotropic displacement parameters for H atoms are derived from the X-ray data as a sum of the external (rigid-body) and internal vibrations. The external vibrations are obtained from a TLS analysis of the ADPs of the non-H atoms and the internal vibrations from analysis of neutron diffraction studies of related compounds. The results from the analysis of positional and thermal parameters were combined to devise a 'best anisotropic' model, which was employed for three other systems where X-ray and neutron data were available. The results from the topological analysis of these systems confirm the success of the 'best anisotropic' model in providing parameters for the H atoms that give charge densities in agreement with the reference models based on H-atom parameters derived from neutron diffraction.
90 citations
Cites methods from "Coherent X‐Ray Scattering for the H..."
...In the VALRAY program system (Stewart et al., 1998), the SDS form factor is represented by the Fourier±Bessel transform of a single exponential type function exp ÿ r with exponent of 2.32 bohrÿ1....
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...…of H atoms Reference model R R R N N R H-atom positions and ADPs (adjusted) from neutron diffraction study IAM R R R R SDS H-atom scattering factor (Stewart et al., 1965) HIGH R R R R Data: sin = 0:77 AÊ ÿ1 POL R R R R Polarized scattering factor for H atoms IDEAL IAM IAM E IAM XÐH bond lengths…...
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...Most re®nement programs like SHELXL (Sheldrick, 1997) have the SDS (Stewart et al., 1965) form factor as the default option....
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...The two models give R values similar to the one obtained from the IAM (SHELXL) re®nement that employs the SDS form factor (Stewart et al., 1965)....
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...C and O atoms H atoms Model xyz ADPs Multipoles xyz ADPs Uiso Multipoles Treatment of H atoms Reference model R R R N N R H-atom positions and ADPs (adjusted) from neutron diffraction study IAM R R R R SDS H-atom scattering factor (Stewart et al., 1965) HIGH R R R R Data: sin = 0:77 AÊ ÿ1 POL R R R R Polarized scattering factor for H atoms IDEAL IAM IAM E IAM XÐH bond lengths extended to idealized values ISO:I R R R R H-atom scattering factor: single STF, exponent 2.32 bohrÿ1....
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TL;DR: In this article, the crystal structure of (η3-cyclooctatrienyl)2Ni has been determined by X-ray methods; the two nickel-bonded η3allyl groups are mutually trans.
88 citations
TL;DR: In this article, X-ray crystal structures for dimeric Calcium-bis[bis(trimethylsilyl)amide and DME 2 have been reported, and the calcium atom has the coordination number four with CaNt-bonds of 227 pm and CaO-distances of 240 pm.
Abstract: Die Molekul- und Kristallstrukturen des dimeren Calcium-bis[bis(trimethylsilyl)amids] 1 und des Calcium-bis[bis(trimethylsilyl)amids] · DME 2 werden beschrieben. {Ca[N(SiMe3)2]2}21 kristallisiert in der triklinen Raumgruppe P1 mit {a = 890,9(2); b = 1259,3(4); c = 2158,8(7) pm; α = 88,13(2); β = 88,49(2); γ = 72,25(2)°; Z = 2}. Die Calciumatome sind dreifach koordiniert mit CaNt-Bindungslangen von 227 pm zu den terminalen Liganden und CaNb-Bindungen von 247 pm zu den verbruckenden Amidogruppen. Ca[N(SiMe3)2]2 · DME 2 kristallisiert aus n-Hexan in der monoklinen Raumgruppe C2/c mit {a = 854,9(3); b = 1738,2(6); c = 2042,1(6) pm; β = 101,27(2)°; Z = 4}. Das Calciumatom hat die Koordinationszahl 4 mit CaN-Bindungslangen von 227 pm und CaO-Abstanden von 240 pm; das Molekul von Verbindung 2 weist C2-Symmetrie auf.
Molecular and Crystal Structures of dimeric Calcium-bis[bis(trimethylsilyl)amide] and Calcium-bis[bis(trimethylsilyl)amide] · DME
X-ray crystal structures are reported for dimeric Calcium-bis[bis(trimethylsilyl)amide] 1 and Calcium-bis[bis(trimethylsilyl)amide] · DME 2. {Ca[N(SiMe3)2]2}21 crystallizes in the triclinic space group P1 with {a = 890.9(2); b = 1259.3(4); c = 2158.8(7) pm; α = 88.13(2); β = 88.49(2); γ = 72.25(2)°; Z = 2}. The calcium atoms are three-coordinate with CaNt-bonds of 227 pm to the terminal ligand and CaNb-bonds of 247 pm to the bridging amido groups. Ca[N(SiMe3)2]2 · DME 2 crystallizes from n-hexane in the monoclinic space group C2/c with {a = 854.9(3); b = 1738.2(6); c = 2042.1(6) pm; β = 101.27(2)°; Z = 4}. The calcium atom has the coordination number four with CaN-bonds of 227 pm and CaO-distances of 240 pm. The molecule of compound 2 possesses C2-symmetry.
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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