Structure factor, X-ray

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

The Bragg peak intensity reduction due to atomic displacements is described by the well-known temperature factors. Assuming that the position can be decomposed into an average position, ,) and an infinitesimal displacement, M = 8R = Ri — (R,) then the X-ray structure factors can be expressed as follows ... 

L. Bosio, S.-H. Chen, and J. Teixeira, Isochoric temperature differential of the X-ray structure factor and structural rearrangements in low-temperature heavy water. Phy. Rev. A 27(3), 1468-1475 (1983). 

The elements of P may now be considered to be experimental parameters obtained simply by an experimental fit to the measured X-ray structure factors (Equation (1)). 

Before we examine in more detail the dynamics of a super-cooled melt of coarse-grained chains and of PB chains, respectively, let us first compare the structure of these two glass-forming systems. Structure is obtained experimentally from either the neutron or the X-ray structure factors. The melt (or liquid) structure factor is given as110... 

FIG. 5.9 Phase angles in an acentric X-N analysis phase angle as calculated with spherical-atom form factors and neutron positional and thermal parameters tpx is the unknown phase of the X-ray structure factors which must be estimated for the calculation of the vector AF. Use of FX — FN introduces a large phase error. Source Coppens (1974). 

The electron diffraction study was complemented by an all-electron theoretical calculation of Lu, Wei, and Zunger (LWZ) (1992), using the local density approximation for the exchange and correlation terms in the Hamiltonian. They find agreement within x0.6% between the calculated and dynamic structure factor values for the lowest three reflections, (100), (110), and (111). But for (200), with sin 0/A = 0.3464 A-1, the discrepancy is as large as 1.7%. The discrepancy is attributed to insufficiently accurate knowledge of the temperature factors in this diatomic crystal, which affect the derivation of the X-ray structure factor from the electron diffraction measurement, as well as the calculation of the dynamic theoretical structure factors needed for the comparison with experiment. For the monoatomic Si crystal for which the B values are well known, the agreement is... 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

The calculation of the X-ray structure factor of distorted polymer structures should be based on a strategy of where to place approximations. As shown by the literature this is a difficult task. 

X-ray structure factors from the theoretical density. Using the theoretical structure factors, one can carry out multipolar refinement in parallel to experimental X-ray data and make in depth comparison of the topological properties from the two sets. 

The same type of calculations have been performed using experimental X-ray structure factors on crystalline phosphoric acid, 7V-acetyl-a,P-dehydrophenyl-alamine methylamide, and N-acetyl-1 -tryptophan methylamide by Souhassou [60] on urea, 9-methyladenosine, and imidazole by Stewart [32] and on 1-alanine [61] and annulene derivatives [62] by Destro and co-workers. The latter authors collected their X-ray data at 16 K [63]. Stewart [32] showed that the positions of the (3, -1) critical points from the promolecule are very close to those of the multipole electron density, but that large differences appear in comparing the density, the Laplacian maps, and the ellipticities at the critical points. Destro et al. [67] showed that the results obtained may be slightly dependent on the refinement model. 

The space group was assigned to P2i, which is monoclinic for Y Cs2. The experimental data were analyzed in an iterative way of a combination of Rietveld analysis (Rietveld, 1969) and the maximum entropy method (MEM) (Bricogne, 1988 Collins, 1982). The MEM can produce an election density distribution map from a set of X-ray structure factors without using any structural model. By the MEM analysis (Kumazawa et al., 1993 Sakata and Sato, 1990), the Rj becomes as low as 1.5% for Y Cs2-... 

WebEmaps (U of Illinois). General TED, Bloch wave simulation, CBED, X-ray structure factors, draw crrystal structures, etc. http //emaps.mrl.uiuc.edu/emaps.asp. 

A Floating Spherical Gaussian Orbital (FSGO) model for polymers calculation of X-ray diffraction structure factors. They have evaluated electron densities and related X-ray structure factors for polyethylene. 

Juretschke, H. J. Invariant-phase information of X-ray structure factors in the two-beam Bragg intensity near a three-beam point. Phys. Rev. Lett. 48, 1487-1489 (1982). 

We now discuss the analysis of the x-ray intensities. The atoms of the C6o molecule are placed at the vertices of a truncated icosahedron. - The x-ray structure factor is given by the Fourier transform of the electronic charge density this can be factored into an atomic carbon form factor times the Fourier transform of a thin shell of radius R modulated by the angular distribution of the atoms. For a molecule with icosahedral symmetry, the leading terms in a spherical-harmonic expansion of the charge density are Koo(fl) (the spherically symmetric contribution) and KfimCn), where ft denotes polar and azimuthal coordinates. The corresponding terms in the molecular form factor are proportional to SS ° (q)ac jo(qR)ss n(qR)/qR and... 

This chapter is structured as follows In Sect. 6.2, a basic introduction to molecular refinement is presented, stressing particularly relevant aspects. Section 6.3 reviews the recent work by Falklof et al., describing how the 2 x 2 x 2 supercell for the lysozyme structure was obtained. Section 6.4 reviews some modern advances in DFT, focusing on dispersion-corrected DFT, while Sect. 6.5 describes the effects of DFT optimization of atomic coordinates on the agreement between observed and calculated X-ray structure factors. The aim is to determine an optimal electronic-structure computational procedure for quantum protein refinement, and we consider only the effects of minor local perturbations to the existing protein model rather than those that would be produced by allowing full protein refinement. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

The first summation constrains the conformational torsion angles (and sometimes the covalent bond angles) to lie near to the expected values derived from single crystal and polymer studies the second summation minimizes the differences between the observed and calculated X-ray structure factor amplitudes the third summation relaxes unfavourable non-bonded interactions and the fourth summation contains constraints Hn whose values are zero when residue connectivity and furanose ring closure have been achieved. 

The liquid structure factor of CCI4 and its derivatives with respect to temperature at fixed pressure or fixed volume, needed by eq. (2), were evaluated by Molecular Dynamics (MD) simulations. We have used the OPLS model for tetrachloromethane [9] In this model, the CCI4 molecules are described as rigid tetrahedra (dc-ci = 1 -769 A) and the intermolecular potentials are atom centered 6-12 Lennard-Jones potentials plus the coulombic interaction with partial charges on C and Cl. We performed NVT simulations with 512 molecules for about 1 ns each. The different x-ray structure factors were obtained from the accumulated partial radial distribution functions [10], using the atomic form-factors from the DABAX database [11]. In order to estimate the partial derivatives of the structure factor, we have used finite differences we considered two different temperatures, Ti = 300 K and T2 = 328 K, and two molar volumes, Vi = 97.3 cm mol and V2 = 100.65 cm mol which are the molar volumes along the liquid-vapor coexistence line for the two temperatures Tj and Tz respectively [12]. Three simulations were then run for the temperature and molar volume conditions (TiiVi), T2,V )... 

FIG. 3 X-ray structure factor of C44H90 compared between a united atom and an explicit atom simulation [8]. 

Recently, MD simulation has been applied as a tool in the refinement of three-dimensional biomolecular structures from X-ray diffraction and two-dimensional NMR data. In the refinement process, MD trajectories are run at elevated temperatures (perhaps several thousand degrees Kelvin) to enhance conformational sampling. The molecular systems are cooled periodically to permit the trajectories to settle into local minimum energy conformations. Constraint terms based on X-ray structure factors or NMR NOE distances are added to the standard potential energy functions, so that the MD trajectories relax to conformations that satisfy the experimental data as the systems are cooled. Thus, the complete potential function in a refinement simulation has the form... 

Notable novelties concerning the structure of ionic liquids came from the contribution of Atkin and co-workers, who showed the existence of pre-peaks in the X-ray structure factors of protic ionic liquids (alkylammonium nitrates) with small side chains [73]. This intermediate-range ordering is likely to be due to different kinds of structural features than those observed in ionic liquids with longer side chains, but demonstrates once more how ionic liquids are full of surprises. Strong pre-peaks in other protic ionic liquids were also described in a number of contributions by Umebayashi and co-workers [60]. 

In principle, it should now be possible to construct a map of electron density within the cell from a set of X-ray structure factors - but there is a major stumbling block. The structure factors are not just numbers but are complex quantities corresponding to sums of wave motions, and therefore have both amplitudes and phases (Eqs 10.8-10.10). All detectors measure intensities integrated over a period of time, so aU we can obtain are the moduli of the structure factors, Ffj i. The so-called crystallographic phase problem is therefore to deduce the phases of the structure factors, as well as the amplitudes. If we manage to do this, we have solved the structure. Note that the term structure solution is used specifically to describe the initial approximate identification of the atom positions within the unit cell, and is distinct from the subsequent refinement of those positions. 

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