Spherical harmonic orientation functions

Clearly, these have the same shape as the 2pz-orbital. but are oriented along the x-and v-axes, respectively. The threefold degeneracy of the p-orbitals is very clearly shown by the geometric equivalence of the functions 2px,2py and 2pz, which is not obvious from the spherical harmonics. The functions listed in Table 7.1 are, in fact, the real forms for atomic orbitals, which are more useful in chemical applications. All higher p-orbitals have analogous functional forms j /(r), yf r) and zf (r) and are likewise threefold degenerate. 

Here j and k label two of the N scattering particles, a it) is a sum of second-order spherical harmonics (all functions of the orientation of particle k at time t) that determines the scattered depolarized field due to particle k at time t, q is the scattering vector, and ry (0) and rk t) are the positions of particles j and k at times 0 and t, respectively. 

The distribution of orientation of the structural units can be described by a function N(0, solid angle sin 0 d0 dtp d Jt. It is most appropriate to expand this distribution function in a series of generalised spherical harmonic functions. 

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

Expressions (3.42) and (3.43) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same /, in. This is summarized in the statement that the spherical harmonic functions are Fourier-transform invariant. It means, for example, that a dipolar density described by the function dl0, oriented along the c axis of a unit cell, will not contribute to the scattering of the (hkO) reflections, for which H is in the a b plane, which is a nodal plane of the function dU)((l, y). 

The explicit expansion of the spherical harmonic functions describing the orientation of the z/th internuclear vector leads to the following expression,... 

The sudden approximation is easy to implement. One solves the onedimensional Schrodinger equation (3.43) for several fixed orientation angles 7, evaluates the 7-dependent amplitudes (3.47), and determines the partial photodissociation amplitudes (3.46) by integration over 7. Because of the spherical harmonic Yjo(x, 0) on the right-hand side of (3.46), the integrand oscillates rapidly as a function of 7 if the rotational... 

The use of real spherical harmonics is particularly bothersome. It has been demonstrated convincingly that the notion of geometrical sets of oriented real atomic angular momentum wave functions is forbidden by the exclusion principle. The use of such functions to condition atomic densities therefore cannot produce physically meaningful results. The question of increased density between atoms must be considered as undecided, at best. 

Here, the following assumptions are made the radial function Ri r) is the same for all basis functions of the same ul quantum number , and its dependence on a shell quantum number n is of no consequence. The coefficients a/ describe the contribution of s, p, d,. .. character to the hybrid, and the bim govern the shape and orientation of that contribution. Sim are the real surface harmonics, defined in terms of the spherical harmonics (Y m). 

The orientation of bonds is strongly affected by local molecular motions, and orientation CF reflect local dynamics in a very sensitive way. However, the interpretation of multimolecular orientation CF requires the knowledge of dynamic and static correlations between particles. Even in simple liquids this problem is not completely elucidated. In the case of polymers, the situation is even more difficult since particules i and j, which are monomers or parts of monomers may belong to the same chain or to different Chains. Thus, we believe that the molecular interpretation of monomolecular orientation experiments in polymer melts is easier, at least in the present early stage of study. Experimentally, the OACF never appears as the complicated nonseparated function of time and orientation given in expression (3), but only as correlation functions of spherical harmonics... 

Figure 1 Structural (left column) and dynamical (right column) properties of the systems investigated. Upper left centre-of-mass radial pair distribution function gooo( ) lower left spherical harmonic expansion coefficient g2oo(r) upper right angular velocity correlation function lower right orientational correlation function. Dotted lines CO, 80 K, 1 bar thin lines CS2, 293 K, 1 bar thick lines CS2, 293 K, 10 kbar.

The nonlinear Smoluchowski-Vlasov equation is calculated to investigate nonlinear effects on solvation dynamics. While a linear response has been assumed for free energy in equilibrium solvent, the equation includes dynamical nonlinear terms. The solvent density function is expanded in terms of spherical harmonics for orientation of solvent molecules, and then only terms for =0 and 1, and m=0 are taken. The calculated results agree qualitatively with that obtained by many molecular dynamics simulations. In the long-term region, solvent relaxation for a change from a neutral solute to a charged one is slower than that obtained by the linearized equation. Further, in the model, the nonlinear terms lessen effects of acceleration by the translational diffusion on solvent relaxation. 

Figure 3.8 The cr-type group orbitals on the vertices of an O3 structure orbit exhibiting D31J point symmetry displayed on the elliptical projections of Figure 3.7. The circular icons, filled and open circles, identify cr-oriented orbital components at the vertices, sized to reflect the coefficients of the linear combinations, equation 3.20, for the spherical harmonics in Table 3.11. The icons and identify the distinct group orbitals transforming as the irreducible components of the reducible character over the decorated orbit, unnecessary repetitions of these components and central functions for which no group orbital can be constructed owing to the locations of the decorated vertices of the orbit in the Cartesian coordinate system.

Cockroft, J. K., Fitch, A. N., and Simon, A. Powder neutron diffraction studies of orientational order-disorder transitions in molecular and molecular-ionic solids use of symmetry-adapted spherical harmonic functions in the analysis of scattering density distributions arising from orientational disorder. In Collected Papers. Summerschool on Crystallography and its Teaching. Tianjin, China. Sept 15-24, 1988. (Ed., Miao, F.-M.) p. 427. Tianjin Tianjin Normal University (1988). 

Figure 2.50. The illustration of the complex distribution of reciprocal lattice vectors modeled using a spherical harmonic preferred orientation function for the (100) reflection.

The orientations of linear molecules, relative to the global frame, can be specified by two Euler angles (oP = 8P, /> the symmetry-adapted functions G icop) that occur in the intermolecular potential [Eq. (15)] reduce to Racah spherical harmonics C (0/>, />). If the molecules possess a center of inversion such as N2 (when we disregard the occurrence of mixed isotopes l4Nl5N, the natural abundance of l5N being only 0.37%),... 

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