Addition theorems

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

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

The spectroscopic measurements determine values for , Legendre addition theorem which for this situation reduces to... 

For this situation of a transversely isotropic aggregate of transversely isotropic units, the Legendre addition theorem gives... 

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. 

Ap and As are the absorbances measured with p- and s-polarization, respectively, and A0 — (Ap + 2As)/3 is the structural absorbance spectrum that would be measured for an isotropic sample. The order parameter of the main chain can be determined using the Legendre addition theorem (Equation (24)). 

In fluorescence spectroscopy, the orientation distribution of the guest probe is not necessarily identical to the actual orientation of the polymer chains, even if it is added at very small concentrations (i.e., a probe with high fluorescence efficiency). As a matter of fact, it is generally assumed that long linear probes are parallel to the polymer main chain, but this is not necessarily the case. Nevertheless, if the relation between the distribution of the probe axes and those of the polymer axes is known, the ODF of the structural units can be calculated from that of the probe thanks to the Legendre s addition theorem. Finally, the added probe seems to be mainly located in the amorphous domains of the polymer [69] so that fluorescence spectroscopy provides information relative to the noncrystalline regions of the polymeric samples. 

In the case of uniaxial cylindrical symmetry around the MD, the (Pt)m values calculated from different crystal planes can be related to the (JV) coefficients of any of the three crystallographic angles using the Legendre addition theorem (see Section 4). For example, the (Pt)c coefficients corresponding to the crystalline c-axis is given by [81]... 

The two expansions discussed so far appear to be quite different. In the multistate Gaussian model, different functions are centered at different values of AU. In the Gram-Charlier expansion, all terms are centered at (AU)0. The difference, however, is smaller that it appears. In fact, one can express a combination of Gaussian functions in the form of (2.56) taking advantage of the addition theorem for Hermite polynomials [44], Similarly, another, previously proposed representation of Pq(AU) as a r function [45] can also be transformed into the more general form of (2.56). 

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... 

In comparison, if one uses the transformation group of the Schrodinger equation as well as the addition theorem (9.31) for the eigenfunctions, the summation is easy to carry out the whole summation (9.33) is easier to calculate than one single term. 

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)... 

As an example, we calculate the coefficient D2. As from the addition theorem... 

The functions /0 and /j are originally defined in terms of the polar angle that is 0 = arccos(e h). Thus, before performing integration, one needs to transform both integrands to the same coordinate set. Doing this with the aid of the addition theorem for Legendre polynomials, one finds... 

Westerhoff, H. V. Chen, Y. D. How do enzyme activities control metabolite concentrations An additional theorem in the theory of metabolic control. Eur J Biochem 1984,142 425-430. 

Considering a solute molecule of the type i = CH3(CH2) X, where X is a functional group, and applying Martin s additivity theorem we can write... 

The third and final approach to the electron correlation problem included briefly here is density functional theory (DFT), a review of which has been given by Kohn in his Nobel lecture [38], The Hohcnberg Kolin theorem [39] states that there is a one-to-one mapping between the potential V(r) in which the electrons in a molecule move, the associated electron density p(r), and the ground state wave function lP0. A consequence of this is that given the density p(r), the potential and wave function lf 0 are functionals of that density. An additional theorem provided by Kohn and Sham [40] states that it is possible to construct an auxiliary reference system of non-interacting... 

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. 

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