Moment spherical harmonic

On the other hand, the use of even higher-order DO approximations is not warranted given the current level of the accuracy and availability of radiative properties. In multidimensional systems, moment, spherical harmonics, and hybrid multiflux approximations usually do not yield results with the accuracy or efficiency of the DO approximations. 

Then the moment induced by the electric vector of the incident light is parallel to that vector resulting in complete polarization of the scattered radiation. The A lg i>(CO) mode of the hexacarbonyls provides a pertinent example08. Suppose we have a set of coupled vibrators, equidistant from some origin. Then it must be possible to express the basis functions for the vibrations in terms of spherical harmonics, for the former are orthogonal and the latter comprise a complete set. The polarization of a totally symmetric vibration will be determined by its overlap with the spherically symmetrical term which may be taken as r2 = x2 + y1 + z2. Because of the orthogo-... 

The spherical harmonic density functions are referred to as multipoles, since the functions with 1 = 0, 1, 2, 3, 4, etc., correspond to components of the charge distribution p r) which give nonzero contributions to the monopole (/ = 0), dipole (/ = 1), quadrupole (/ = 2), octupole (/ = 3), hexadecapole (/ = 4), etc., moments of the atomic charge distribution. 

The Hirshfeld functions give an excellent fit to the density, as illustrated for tetrafluoroterephthalonitrile in chapter 5 (see Fig. 5.12). But, because they are less localized than the spherical harmonic functions, net atomic charges are less well defined. A comparison of the two formalisms has been made in the refinement of pyridinium dicyanomethylide (Baert et al. 1982). While both models fit the data equally well, the Hirshfeld model leads to a much larger value of the molecular dipole moment obtained by summation over the atomic functions using the equations described in chapter 7. The multipole results appear in better agreement with other experimental and theoretical values, which suggests that the latter are preferable when electrostatic properties are to be evaluated directly from the least-squares results. When the evaluation is based on the density predicted by the model, both formalisms should perform well. 

In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics c,mp multiplied by r, which defines the spherical harmonic electrostatic moments ... 

The linear relationships between the traceless moments 0 and the spherical harmonic moments lmp are obtained by use of the definitions of the functions clmp. For example, for the quadrupolar moment element xx, we obtain the equality (3x2 — l)/2 = a (x2 — y2)/2 + b(3z2 — 1). Solution for a and b for this and corresponding equations for the other moments leads to... 

An element of an electrostatic moment tensor can only be nonzero if the distribution has a component of the same symmetry as the corresponding operator. In other words, the integrand in Eq. (7.1) must have a component that is invariant under the symmetry operations of the distribution, namely, it is totally symmetric with respect to the operations of the point group of the distribution. As an example, for the x component of the dipole moment to be nonzero, p(r)x must have a totally symmetric component, which will be the case if p(r) has a component with the symmetry of x. The symmetry restrictions of the spherical electrostatic moments are those of the spherical harmonics given in appendix section D.4. Restrictions for the other definitions follow directly from those listed in this appendix. 

As the traceless quadrupole moments are linear combinations of the spherical harmonic quadrupole moments, the corresponding expressions follow directly... 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

Equations (4.30) and (4.31) have been developed and dehned within a time-dependent framework. These equations are identical to Eqs. (35) and (32), respectively, of Ref. 80. They differ only in that a different, more appropriate, normalization has been used here for the continuum wavefunction and that the transition dipole moment function has not been expanded in terms of a spherical harmonic basis of angular functions. All the analysis given in Ref. 80 continues to be valid. In particular, the details of the angular distributions of the various differential cross sections and the relationships between the various possible integral and differential cross sections have been described in that paper. 

The simplest possible case is a non-homonuclear diatomic (non-homonuclear because a dipole moment is required for a rotational spectrum to be observed). In that case, solution of Eq. (9.38) is entirely analogous to solution of the corresponding hydrogen atom problem, and indicates the eigenfunctions S to be the usual spherical harmonics (j)), with eigenvalues given by... 

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 dynamic coupling mechanism predicts that hypersensitivity should be observed when the point group of the lanthanide complex contains Y3m spherical harmonics in the expansion of the point potential. The good agreement between the calculated and observed values for Tj or Q.2 parameters shows that the dynamic coupling mechanism makes a significant contribution to the intensities of the quadrupole allowed f-electron transitions in lanthanide complexes. Qualitatively, the mechanism is allowed for all lanthanide group symmetries in which the electric quadrupole component 6,a,fi and the electric dipole moment p, a transform under a common representation. 

Therefore, the hybrid density moment is simply a properly weighted average of each individual orbital moment. The orthogonality of the spherical harmonics has insured that different radial functions do not mix together. 

All the correlation functions above are normalized, therefore equations (4 and 5) are identical to correlation functions over linear momentum p = mv and angular momentum J — lu, respectively. Note that, in this context I is the moment of inertia tensor The correlation function in equation (6) is calculated over the spherical harmonics. If m = 0, this reduces to time correlation function over Legendre polynomials ... 

The constant A = 3.96 cm-1 has been obtained from the free-molecule zero-field splitting (Mizushima, 1975) and C is a Racah spherical harmonic with 1 = 2. The tensor that describes the interaction between the magnetic dipole moments ge Sp, where ge equals 2.0023 and fxB is the Bohr magneton, can be written immediately as... 

The trigonal planar complexes of the MX3 type, whether of the lanthanide or of the transition metal series, have a particular significance. In an amendment to the crystal field intensity theory, designed to accommodate the hypersensitive f—f transitions, the addition of the first-rank spherical harmonics, Yim to the ligand field was proposed in order to form, with the dipole operator, an effective quadrupole operator within the 4f shell i). Crystal field potentials of the Yim form are restricted, however, to metal complexes with a permanent dipole moment, that is, to the site symmetries Cpv and their subgroups ), and the proposal does not account for the relatively large oscillator strengths of the quadrupole-allowed d—d or f—f transitions of... 

Since the transformation properties of spherical harmonics are well known, the spherical-tensor notation has some advantages, particularly in the derivation of general theorems however, the reality of cartesian tensors also has its attractions, especially for small values of /. Normally, the moments of a particular three-dimensional molecule are most conveniently given in an x,y,z frame. 

For a molecule with atoms of unit mass on a unit sphere the moments of inertia may be expressed in terms of the following spherical harmonic expansions ... 

The success of the moments of inertia as a useful structural parameter is at first surprising, since the mathematical expressions for the moments of inertia of idealised clusters are complex, do not relate to any recognisable feature, and vary from structure to structure. In this analysis, the moments of inertia are used as a parameter to describe only the gross deviation from sphericality. As was illustrated in Sect. 2.2, the shape of the ellipsoid enveloping the atoms is not sufficient to describe clusters as oblate or prolate, information about atomic positions is also required. The splitting of the orbitals, which is defined by the spherical harmonic perturbation, is dependent on the same factors which determine the moments of inertia, the cluster shape and distribution of atoms about the surface. The moments of inertia therefore provide a generalised shape parameter, without referring to specific features of the individual structures. 

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