Spherical harmonics, orientation

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 model of non-correlated potential fluctuations is of special interest. First, it can be solved analytically, second, the assumption that subsequent values of orienting field are non-correlated is less constrained from the physical point of view. The theory allows for consideration of a rather general orienting field. When the spherical shape of the cell is distorted and its symmetry becomes axial, the anisotropic potential is characterized by the only given axis e. However, all the spherical harmonics built on this vector contribute to its expansion, not only the term of lowest order... 

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 ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

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 applicability of Eq. (21) rests on the validity of the assumption that the averages over internal and external variables are uncorrelated and thus can be calculated separately. Furthermore, theexpression of Eq. (21) emphasizes the close similarity of the irreducible Cartesian representation to the expression of the problem in terms of polar angles and the normalized 2nd rank spherical harmonics Y (see Eq. (7)). The corresponding polar angles ( (1), (t)) and (C(t), (t)), shown in Fig. 2B, describe the orientation of the internuclear vector and the magnetic field relative to the arbitrary reference frame, respectively. The different representations are related according to the following relationships.37... 

In order for the induced dipole moment, ft, to transform as a vector, the spherical harmonics describing the various orientations have to be coupled in an appropriate way. We write the induced dipole components of a system of two molecules of arbitrary symmetry, according to [141]... 

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... 

As we are dealing with spherical harmonics, and as we are trying to model the aspherical atomic electron density, the orientation of the local atom centered coordinate system is, in principle, arbitrary, appropriate linear combinations always giving the same result. However, in practice it is helpful to choose a local coordinate system such that the multipoles are oriented in rational directions, and thus the most important multipole populations will lie in directions that would be expected to represent chemical bonds or lone pairs [2,20], e.g. for an sp2 hybridized atom, defining one bond as the x direction, the trigonal plane as the xy plane, and z perpendicular defines three lobes of the 33+... 

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 0 and are the spherical polar angles (only two angles are required to define the orientation of the vector r in space). Since these operators are the same as the infinitesimal rotation operators, all the results of the previous sections apply. The eigenfunctions of L2 and Lz are known as the spherical harmonics,... 

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. 

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