Spherical harmonics generalized

The allowed wavefunctions for the particle on a sphere are called spherical harmonics, generally written... 

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... 

We may readily derive a general expression for the spherical harmonic Yim(9, (p) which results from the repeated application of L+ to Yi-i(9, tp). We begin with equation (5.38a) with m set equal to — /... 

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, 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 vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

In this section, we present an alternative proof of the derivative rule, which provides an expression for the transmission matrix element from an arbitrary tip state expanded in terms of spherical harmonics. In the previous sections, we have expanded the tip wavefunction on the separation surface in terms of spherical harmonics. In general, the expansion is... 

The general expression for the tunneling current can be obtained using the explicit forms of tunneling matrix elements listed in Table 3.2. To put the five d states on an equal footing, normalized spherical harmonics, as listed in Appendix A, are used. The wavefunctions and the tunneling matrix elements are listed in Table 5.1. 

Now we are ready to define the spherical harmonic functions. In Section 1,6 we gave examples for t = 0, 1, 2 here is the general definition. 

In general, the Q m are complex quantities.1- The Y(m are spherical harmonics an asterisk designates the conjugate complex. Equation 2.41... 

It is common to express a general ligand field potential as a superposition of spherical harmonics (34) and the expressions for the octahedral potential,... 

The Onk and Q k are operators which are respectively linear combinations of spherical harmonics and expansions in terms of Cartesian coordinates, 1 = 2 for d-orbitals, 3 for /-orbitals. The parameters B k and A k are, of course, specialized forms of the general form given in equation (2), but including the evaluation of the relevant radial integrals. 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

One vexed question concerning polarization sets is the number of functions in a given shell, as discussed elsewhere. The early quantum chemistry codes employed Cartesian Gaussian functions, so that a d set actually comprises the five spherical harmonic d functions and a 3s function generally termed a contaminant. The reason for this emotive terminology is that with multiple polarization sets the contaminants,... 

The rotational coordinates are Q 2 and Q 5. The rotational motion can be visualized by mapping the trough onto the surface of a 2D sphere the rotation is governed by the usual polar coordinate definitions, 6 and . This is also shown in equation (7) which has the usual form for a rotator with spherical harmonic solutions Ylm. The solutions will be written in the form I i//lo, hn ). For the high spin states case, it was found that l must be odd in order to obey the Pauli s exclusion principle and preserve the antisymmetric nature of the total wavefunctions at any point on the trough under symmetric operations [26]. In the current case, similar arguments show that l must be even. This is because the electronic basis is even under inversion and the whole vibronic wavefunction must also be even under inversion. A general mathematical proof can be found in Ref. [23],... 

The general matrix equation of the problem that determines the amplitudes <2 /" is obtained by substitution of the spherical harmonic expansion (4.319) into the Fokker-Planck equation (4.313). After that the result is multiplied from... 

In Eq. (4.323) notation of the type Y(a,b) means that a spherical harmonic is built on the components of a unit vector a in the coordinate system whose polar axis points along the unit vector b. The functions. + and, S4 in Eq. (4.324) are the equilibrium parameters of the magnetic order of the particle defined, in general, by Eqs. (4.80)-(4.83). 

The symmetry adapted spherical harmonics (also referred to as generalized harmonics), X hf kM), satisfy the symmetries of the molecular point group [51] and are defined as ... 

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