Symmetry-adapted bases over Liouville space

Abstract

The irreducible representation under for spin clusters of identical I_i ≥ 1 spin systems are explicitly derived over |IMα> Hilbert space for the range of F _z subspaces by using the combinatorial aspects of each ‘weight’ set (M ₁ – M ₄). This defines the invariance properties for each specific F_z , and consequently for the complete system over ∑F_z . Our investigations show that there are limiting , i.e. independent of I_i , for F_z > I_zi for integer spins, or F_z > 2I_zi , for half integer spins.

Similar considerations apply to the irreducible representations that characterize like k_i - | kqv≫ Liouville space for q > (k ^max – k_i ), where k and k_i refer to the tensor ranks of the total system and the I_i spin, respectively. The invariance properties of clusters under for I = 1, 3/2 and 5/2 are derived for both | IMα> and | kqv≫ spaces. A general {[Xtilde] _i } form over the total Liouville space is given for arbitrary I_i .

In the context of the multiquantum N.M.R. of spin clusters, these properties allow one to construct a density operator strictly under the dual groups; such tensors provide a valid -partitioned quantum-Liouville description, of both evolution and relaxation processes, which is applicable to non-magnetically equivalent spin systems, such as [AX]₄ under . For bicluster systems specific aspects of recoupling theory, implicit in the evaluation of reduced matrix elements (RME), are examined; explicit basis and RME forms under are considered.

Combinatorics are shown to play a large role in characterizing the various spaces, |IMα: S_n >, | kqv: S_n ≫ and , by allowing direct evaluations of the pertinent invariances and components under .