Vector-coupling coefficients

Boys, S. F., and Sahni, R. C., Trans. Roy. Soc. [London) A246, 463, (ii) Electronic wave functions. XII. The evaluation of the general vector-coupling coefficients by automatic computation. ... 

Note For the triplet states the functions are given only for the Ms = 1 spin component. For all degenerate levels the a components (as given by the vector coupling coefficients of Table 2) are listed first, and these transform as +1 under oxz and the b components as - 1 under the same operation. 

Let us now consider the overlap between the spherical and the Stark basis. For the latter, the momentum space eigenfimctions, which in configuration space correspond to variable separation in parabolic coordinates, are similarly related to alternative hyperspherical harmonics [2]. The connecting coefficient between spherical and 5 torA basis is formally identical to a usual vector coupling coefficient (from now on n is omitted from the notation) ... 

The overlap between spherical and Zeeman states, was originally derived as a sum of the product of two vector coupling coefficients [3] ... 

Table 59 Overview of the vector coupling coefficients for the R3 group...

Table 60 Vector coupling coefficients within a point group G ...

For algebraic evaluation of the 3j- and 6j-symbols the Racah formulae are useful (Table 58). Various vector coupling coefficients met so far are reviewed in Table 59. In point groups analogous coefficients occur (Table 60). 

If classical coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and vector coupling coefficients are properly set to the Hartree-Fock case, this operator is reduced to Equation (4.160). 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

The one- and two-body density matrices are formed by contraction of the vector coupling coefficients with the left and right eigenvectors of Eq. (10) (assuming CSC = 1)... 

Open-Shell Vector Coupling Coefficients for the in - o> States Represented by a Slater Determinant of Two-Component Spinors... 

Following Sugano, Tanabe and Kamimura [1] the vector coupling coefficients (otherwise known as Clebsch-Gordan coefficients) shown in Table 1 inform us how the decomposition products are constructed from the initial functions. 

An important use of vector coupling coefficients lies in the calculation of matrix elements of the operators in the vibronic Hamiltonian. Knowing the symmetry properties of the basis functions and of the operators, the ratio of the matrix elements can be deduced by inspection of the vector coupling coefficients. Without resorting to complicated formulae, a restricted use of the Wigner Eckart theorem may be illustrated as follows. First let us reduce Table 1 to those columns involving only the decomposition products of E symmetry (Table 2). 

Table 1 E (g) E vector-coupling coefficients with cubic bases. The symmetry labels 6 and s are...

The Hamiltonian in (36) operates on the ground state vibronic wavefunctions and I g which can be expressed as products of electronic functions and an expansion of the two dimensional vibrational states cpi of appropriate symmetry [41], The linear combinations are found using the E g e and H2 g e vector coupling coefficients (the Hi g e coefficients are of course trivial) following the same procedure as used to construct the vibronic Hamiltonian in (15). It may then be readily shown that for strong linear coupling, p 0 and q 1/2 [10,41]. Note however, for second order coupling q can take values less than 1/2 [42]. 

The vector coupling coefficients of Eq. (1) are not symmetric in the -numbers. Therefore, in order to stress the inherent symmetry, the highly symmetric function, V [here denoted V(j)], was defined by Fano and Racah. It is related to the vector coupling coefficient by the expression... 

The irreducible representations may be classified according to whether j is an integer or half of an odd integer. We shall consider here the former, which are the potentially real representations 22 b, p. 287). These representations in contrastandard form 4) can be transformed into real form by a constant unitary matrix, i.e. the same matrix for every element in the group. The elements of the constant matrix will be chosen such that the contrastandard, self-conjugate sets which form the bases for the potentially real representations in complex standard form are connected to the sets of the usual real spherical harmonics which form the bases for the real standard representations. From the constant matrix, the vector coupling coefficients pertaining to the real functions will be deduced. 

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