IV. Symmetry and its Implications

Abstract

Significance of symmetry in representing real‐ and reciprocal‐space properties of a crystal and of its constituents is discussed in terms of symmetry eigenfunctions. The formation of the site‐symmetrized multipole expansion for the 32 crystal point symmetries is shown, and the mutual relations between the real‐ and reciprocal‐space expansions are discussed. The origin of the symmetry‐based phase relations, equivalences and extinction rules of structure amplitudes is pointed out. Distinction is made between the accurate and approximate ones arising from the true space‐group symmetry and from the approximate free‐particle symmetry of the constituents, respectively. Finally, the symmetry imposed by reality of charge density and its violation through anomalous dispersion are discussed in terms of symmetric and antisymmetric contributions of atomic charge densities.

X-Ray Diffraction Study of the Electron Density and Anharmonicity in K2PtCl6

“…In addition to the standard procrystal model defined by positional and anisotropic displacement parameters, scale factor and secondary extinction parameters [6], the program provides the Gram-Charlier expansion of anharmonic motion [7] up to 4th order, and a representation of the asphericity of the atoms by means of a sum of multipolar deformation functions Q"i m± = P n , m + r" exp (-a r) C, m ± y t m + [8], which may be chosen to be equivalent to the deformation functions of Hirshfeld [9], The terms y lm± are real spherical harmonic functions [10], the C, m± are their normalization constants, and r is the distance from the atomic center. For Pt and K, cubic spherical harmonics are defined by y K 3 = y32-> yK4 = ^40 + + y^+l^ [10]. The population factors P nlm± and the radial exponents a are adjustable parameters.…”

“…The population factors P nlm± and the radial exponents a are adjustable parameters. The indices of the site-symmetric functions are: -anharmonic parameters 1111, 1122 for Pt; 123, 1111, 1122 for K; 111, 122, 1111, 2222, 1122, 2233 for CI; -spherical harmonics lm± = 00, K4 for Pt; 00, K3, K4 for K; 00, 10, 20, 30, 40, 44+ for CI [10].…”

Three-Dimensional Reciprocal Form Factors and Momentum Densities of Electrons from Compton Experiments I. Symmetry-Adapted Series Expansion of the Electron Momentum Density

“…It was first considered in connection with solutions to the Laplace and the Helmholtz equation with polyhedral boundary conditions (Klein [46], Goursat [47], Pockels [48], Poole [49], Hodgkinson [50], Laporte [51]), in connection with the problem of term splitting when passing from spherical symmetry to another point symmetry (Bethe [52], see also the whole literature on ligand-field Unauthenticated Download Date | 5/11/18 1:52 PM theory), and in connection with the rotational-vibrational spectrum of the methane molecule (Ehlert [53], Jahn [54,55], Hecht [56], Moret-Bailly [57], Fox and Ozier [58]). Systematic group-theoretical studies for a larger number of point groups are due to Laporte [51], Meyer [59], Melvin [60], Altmann [61,62], Döring [63], Bradley and Cracknell [64], and Kurki-Suonio et al [65,66,43]. Owing to the long period over which the literature is spread, the focus of interest and the formalisms are very heterogeneous.…”

“…Because of the additional rotational axes between the pole and the equator on the unit sphere, the point groups T h , O h and Y h require polyhedral (disdodecahedral, hexoctahedral and icosahedral) harmonics X L which are linear combinations of Sf with fixed / and variable m. The whole family of X L for the cubic crystal classes T, T h , O, T d and O h is called "cubic harmonics" by Kurki-Suonio and coworkers [65,66,43], consistent with our nomenclature, while Fox and Krohn [97] restrict it so far to T d (also "tetrahedral harmonics" [58]) and O h , and Mueller and Priestley [98] even to O h alone. Von der Lage and Bethe [70] use the term "kubic harmonics" for the lattice harmonics A kL of the space group 0\ = im 3m for k = 0 and k=(n/a)e z .…”

The structure factor of orientationally disordered crystals: the case of arbitrary space, site and molecular point group

“…SAF's for all crystallographic point groups may be obtained from Bradley & Cracknell (1972). 'Index picking rules' for the selection of linear combinations of Elm for crystallographic point groups have also been given by Kurki-Suonio (1977). One should, however, bear in mind molecules with non-crystallographic point groups, e.g.…”