Spherical harmonics 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 ... 

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)... 

The derivation [2] starts with an expansion of 1 /r.., just as the Cartesian formulation does, but this time we use the expansion in terms of spherical harmonics (the spherical harmonic addition theorem) [3]. As before, we write R = B — A, and obtain... 

The conversion of the two-electron repulsion potential e /rij to a form involving each electron separately is accomplished by expansion in a series of Legendre polynomials (cos co) where co is the angle subtended by the two electrons. Each P/, can be expressed in turn in a series of products of spherical harmonics by the spherical harmonic addition theorem ... 

Consider the special case in which the reactants form a long-lived collision complex, whose duration is sufficient so that the system has no memory of the initial directions of the vectors. Then the breakup of the complex is independent of its formation so that the dihedral angle between the (v, J) and (V, J) planes will be randomly distributed. Application of the spherical harmonic addition theorem yields... 

Here pa and pb are the charge densities of molecules A and B, respectively. Using the spherical-harmonic addition theorem [94], this energy can be rewritten in terms of the molecular multipole moments defined with respect to the molecule s local frame ... 

Pt(cos (u) can be expanded further by the spherical harmonic addition theorem (Griffith 1961) which expresses the angle to between R and r, in terms of the polar angles (0, (p) and (0 ) characterizes the angular distribution of the disturbing charges and (0/, (Pi) represents the angular position of electron i ... 

Express the partial-wave expansion (7E.3.1) in products of one-electron functions, using the spherical-harmonic addition theorem. 

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... 

All three of ea>, e(2 e(3] can be expressed in terms of vector spherical harmonics. Thus, in addition to the nonlinear B cyclic theorem, the following linear relations occur... 

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)... 

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. 

It is a mathematical property of spherical harmonic functions Yem(6, ) that they obey an addition rule which is known as the biaxial theorem if ri and r2 are two vectors, with directions described by the polar angles (6i, 4> ) and (62,4>2), and if a is the included angle between the two vectors, then... 

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. 

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