Addition theorem for spherical harmonics

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... 

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... 

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... 

A binomial Taylor expansion of IR + r2 — and subsequent application of an addition theorem for regular spherical harmonics [82] factorise the electronic (rj, T2) and geometric (R) coordinates as follows ... 

We now turn to the Cl expansion in STOs (7.3.1) for the totally symmetric singlet ground state of the helium atom. In Exercise 7.1, we invoke the addition theorem for the spherical harmonics [6]... 

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. 

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)... 

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. 

The real version of the irreducible tensor method, related to the complex representations as mentioned, is highly useful in the ligand-field theory, as will be shown in the second part of this paper. An additional reason for this is the fact that expansion theorems concerning functions and operators achieve apt forms when the tensor method is applied to real spherical harmonics. 

Other advantages of working in terms of spherical harmonic functions are that for cases with fibre symmetry, the Legendre addition theorem can be used, and affords considerable algebraic simplifications (see for example Ref. 25), and that for lower symmetries, the treatment can readily be generalised. It should be mentioned that the exact definitions of P2 cos9), etc., and p 9), can differ in different treatments due to the adoption of different normalisation procedures (see, for example. Chapter 5, Section 5.2).)... 

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