Tensors double

In general, the double tensor W[see (14.30)] is related to the triple tensor... 

Operator 1 ), by (15.56), is proportional to the quasispin operator, and operator W<0kK by (15.59), to double tensor with an odd sum of ranks k + k. These tensors, as shown in Chapter 14, are generators of the Sp4i+2 group, and so the above selection of generator subsets corresponds to the reduction of the Rsi+4 group on the direct product of two subgroups ... 

The irreducible tensorial products of the double tensors can be found... 

Considering the tensorial properties of the electron creation and annihilation operators in quasispin space, we shall introduce the double tensor... 

Thus, the double tensor, defined by (23.52), is a convenient standard quantity for studies of mixed configurations. Let us turn now to the properties of the tensorial products of two-shell operators (23.52). Proceeding in the same way as in derivation of (23.11), we arrive at... 

Using the properties of the tensors T(ja)(7>) and of the 3n/-coefficients (see Chapter 6), we find the additional relationships for the sums of double tensors... 

Equations (23.56)-(23.58) hold for any 71 and j2, but for some specific values of 71 and j2 these relationships are simplified, thereby enabling us to obtain analytical expressions for individual tensorial products of double tensors. It is worth noting that such expressions can only be derived at 71 = 1/2 (j2 is arbitrary). 

The reduced matrix elements of a double-tensor operator (formed as a scalar product of tensor operators of rank k and 1) become... 

The full set of af operators forms a double tensor of rank s = 1/2 in spin space and rank / = 1 in orbital space. The al,smi. operator directly coincides with the msmv component of this af tensor. The corresponding annihilation operators also form a double tensor of ranks s = 1/2 and V = 1. In this case the msmv tensor component, commonly denoted as is related to the... 

C. Hattig and B. A. Hess, Mol. Phys., 81, 813 (1994). Calculation of Orientation-Dependent Double-Tensor Moments for Coulomb-Type Intermolecular Interactions. 

Here the reduced matrix element of the double tensor operator vSL V( 11 > v S L ) occurs it becomes expressed as follows (k = 1)... 

The basic expressions for the matrix elements of the spin-orbit coupling operator have been derived by Griffith [48]. A double tensor operator is Xf[y where, for fixed y and varying M, X fy is an irreducible tensor operator with respect to spin variables. Similarly, for fixed S and varying y, Xf[y is an irreducible tensor operator with respect to space variables. A matrix element of a double tensor operator is now reduced as follows... 

Defining the unit double tensor (Racah, 1942b)... 

A tensor operator is thus created from U and V, and the analogy to the coupling of angular momentum eigenfunctions is underlined by the presence of the Clebsch-Gordan coefficient in (43) or, equivalently, the 3/ symbol in (45). If, on the other hand, we are given a double tensor, this may be represented as a sum of tensor products by... 

The special significance of this equation derives from the fact that some of the interaction operators considered here contain double tensors, while the general formulae used in the following refer to tensor products of the form X. ... 

We emphasize that we will here deal, in general, with double tensor fields, which are defined with respect to 3 and surface S(t) embedded in 3. Thus, we will deal with the quantities which obey the transformation law for a tensor under the following groups of transformations ... 

Clearly, the definition of the order of a singular surface may be expressed alternatively in terms of the covariant derivatives with respect to material variables X, that is,

substitute double tensors of the type in... 

Substitution (101) appears to be the simplest way in which spin-dependent phenomena can be incorporated into the theory. The operator (100) commutes with S and is thus consistent with effects produced by such spin-independent interactions as the Coulomb and crystal-field terms in the Hamiltonian. Moreover, the effect of (101) turns out to be equivalent to adding to each reduced matrix element of F " a part (proportional to c,) that involves the reduced matrix element of the double tensor where = s (Judd 1977b). Such double tensors are straightforward to evaluate furthermore the proportionality = /4 holds for all terms of... 

For the ss interaction, the reduced matrix elements between the core states comprise an odd symmetrical double tensor (u + k odd) and are therefore the same for (nl) n l and (n/) n /. The soo interaction is composed of two parts The first, comprising the factors k and s behaves under conjugation like the spin-orbit interaction. It is represented by an even symmetrical double tensor (u + k even) and therefore changes sign the second part, comprising ( and s is represented by an odd symmetrical operator, and therefore remains invariant under conjugation. 

From a physical point of view this relativistic model is also based on the perturbation approach, and at the second order, similarly as in the case of the standard J-O Theory, the crystal field potential plays the role of a mechanism that forces the electric dipole/ t—>f transitions. The only difference is that now the transition amplitude is in effectively relativistic form, as determined by the double tensor operator, but still of one particle nature. Furthermore, the same partitioning of space as in non-relativistic approach is valid here. The same requirements about the parity of the excited configurations are expected to be satisfied. As a final step of derivation of the effective operators, the coupling of double inter-shell tensor operators has to be performed. This procedure is based on the same rules of Racah algebra as presented in the case of the standard J-O theory. However, the coupling of the inter-shell double tensor operators consists of two steps, for spin and orbital parts separately. Thus, the rules presented in equations (10.15) and (10.16) have to be applied twice for orbital and spin momenta couplings, resulting in two 3j— and two 6j— coefficients. 

---