Clebsch-Gordan expansion

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

The expansion coefficients on the right hand side of Eq. (1.25) are the Clebsch-Gordan coefficients.2 The eigenfunctions of the angular momentum, which can be written, abstractly, using Dirac notation 11, m >, satisfy the equations (h=1)... 

Here, the subscript (c) is short for the set of expansion parameters (c) = (2i, 22, A, L, oi, u2) r, is the vibrational coordinate of the molecule i R is the separation between the centers of mass of the molecules the Q, are the orientations (Euler angles a, jS y,) of molecule i Q specifies the direction of the separation / the C(2i22A M[M2Ma), etc., are Clebsch-Gordan coefficients the DxMt) are Wigner rotation matrices. The expansion coefficients A(C) = A2i22Al u1u2(ri,r2, R) are independent of the coordinate system these will be referred to as multipole-induced or overlap-induced dipole components - whichever the case may be. 

The expansion parameters that appear in Eqs. 4.7 and 4.8 as arguments of the Clebsch-Gordan coefficients satisfy the triangular relationship,... 

It will represent a two-shell multi-configuration wave function in which the coefficients of expansion in configurations are the conventional Clebsch-Gordan coefficients. 

For the coupling of two angular momenta a and b to the resulting value c with the corresponding magnetic quantum numbers a, P and y, the Clebsch-Gordan coefficients (atxbp cy) are defined as expansion coefficients in the relation... 

The expansion coefficients are the Clebsch—Gordan coefficients. We often abbreviate the meaning of (3.85) by saying that it expresses the coupling of the angular momenta 71 and 72 to 7. 

The reduced Lippmann—Schwinger equations are obtained by expanding all the amplitudes of (6.73) according to (7.36) and again using the orthonormality relations (3.71,3.89) to eliminate the integral over k and the sum over Clebsch—Gordan coefficients in the expansion of the projection operator... 

Determinantal representation of CSFs. Expansion of CSFs in the unitary group approach in terms of spin-orbital determinants. The coefficients are determined as products of factors,/k, determined from the Clebsch-Gordan coefficients and phase factors of Table II. The coefficient sparseness of the determinants is predictable and corresponds to the allowed area principle. 

When we substitute this expansion into the RS second-order expression, we get a numerator that contains two Clebsch-Gordan coupled products of transition matrix elements on A and B. They are of the type... 

The second transformation matrix U(123) is an expansion of the matrix of the previous generation U(l2) and Clebsch-Gordan coefficients for a new set of... 

Recall section 4.4.3 and Eq. (4.143) for the definition of the Clebsch-Gordan coefficients (vector coupling coefficients) and Wigner-3)-symbols. Of course, the Clebsch-Gordan coefficients finally define the expansion coefficients of the pre-contracted CSF in Eq. (8.99). 

Harmonics of given total (ordinary) angular momentum / = / + / > (not to be confused with the grand orbital momentum L) are constructed by Clebsch-Gordan coupling of the HH (5.10). One should further restrict the expansion (5.6) to these HH having the desired permutation properties. 

The 85th Edition includes updates and expansions of several tables, such as Aqueous Solubility of Organic Compounds, Thermal Conductivity of Liquids, and Table of the Isotopes. A new table on Azeotropic Data for Binary Mixtures has been added, as well as tables on Index of Refraction of Inorganic Crystals and Critical Solution Temperatures of Polymer Solutions. In response to user requests, several topics such as Coefficient of Friction and Miscibility of Organic Solvents have been restored to the Handbook. The latest recommended values of the Fundamental Physical Constants, released in December 2003, are included in this edition. Finally, the Appendix on Mathematical Tables has been revised by Dr. Daniel Zwillinger, editor of the CRC Standard Mathematical Tables and Formulae it includes new information on factorials, Clebsch-Gordan coefficients, orthogonal polynomials, statistical formulas, and other topics. 

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