Spherical Addition Theorems

Friedman, B., and Russek, J., 1954, Addition theorems for spherical waves. Quart. Appl. Math. 12 13. 

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. 

We proceed to estahUsh the addition theorem for four-dimensional spherical harmonics. Equation (9.19) is an identity with respect to r. Expanding the integrand in powers of r... 

Making use of the usual addition theorem for three-dimensional spherical harmonics and using the expression (9.21) for one can rewrite (9,31) as... 

To calculate angular distributions for an actual collision, F -fR) must be written in terms of angles with respect to the external frame, that is, with respect to the relative heavy-particle velocity direction. This is done by applying the addition theorem of spherical harmonics... 

Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

To evaluate the averages like those in Eq. (4.78), it is very convenient to pass from cosines ((en)k) to the set of corresponding Legendre polynomials for which a spherical harmonics expansion (addition theorem)... 

Since we shall be interested in the electric field gradient with respect to a molecule-fixed coordinate system, we need to transform (8.492) from space-fixed to molecule-fixed axes the relationships between the two are illustrated in figure 8.53. Denoting molecule-fixed axes with primes, and space-fixed axes without primes, the spherical harmonic addition theorem gives the result ... 

The addition theorem relates spherical harmonics with different arguments. 

Applying the addition theorem, the Legendre polynomials P/(cos0p) can be expressed in terms of products of the spherical harmonics as below... 

When the perturbation from more ligands is to be taken into account, the addition theorem for spherical harmonics is usually used to develop the expression of the electrostatic model when more ligands are involved. This can be done very simply by means of the concepts of the angular-overlap model. 

Eq. (39 a) is in effect the addition theorem for spherical harmonics. This can be seen by rewriting it as... 

The expansion of Eq. (62) can be inverted to express the spherical harmonics in terms of spheroconal harmonics, whose completeness in turn leads to the addition theorem in the form... 

In going from the = 2 to the = 3 eigenfunctions, the raising actions of the p operators is implemented. The translations into spherical or sphero-conal harmonics follow by using coordinate transformation equations and the addition theorem [5, 6]. 

Section 2.6 recognizes that for the hydrogen atom, its Hamiltonian also commutes with and H correspondingly, it also admits solutions with Lame spheroconal harmonics polynomial eigenfunctions. It also shares the same radial eigenfunction with the familiar solution with spherical harmonics, and additionally both can be obtained from a common generating function and both satisfy the addition theorem. 

Important examples of these expressions are the calculations of matrix elements of operators which are expanded by means of the spherical harmonic addition theorem (6, p. 141)... 

We note that the augmented spherical Neumann Ntjl(r/St) and Bessel J (r) functions are in this case defined from a tail of radial dependence (r/S ) rather than (r/S) as in the energy-dependent orbital (8.1). The addition theorem (6.13,8.7) which represents the expansion in the sphere at q of the tail of the muffin-tin orbital centred at q must therefore be changed to include the correct radial dependences, i.e. (r/St). From (6.13,8.7) we find... 

In this addition theorem the spherical harmonics iV are included in N. ... 

The reflection matrix for a closely packed medium composed of wavelengthsized scatterers can be represented as a sum of matrices (10) with the coefficients of the addition theorem (7). These coefficients describe all the details of the field in the vicinity of any scatterer, including the near-field effects [26]. We consider the manifestations of these effects quahtatively using the field configuration near a spherical scatterer as the simplest example. 

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