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The Journal of Chemical Physics

Multipole operators for an arbitrary number of spins

B. C. Sanctuary

Published Jun 1, 1976• 115 citations
Publisher

Abstract

Basis states and operators composed of n, in general, different nuclear spins of integer or half-integer values are explicitly constructed. Various coupling schemes are discussed, and transformations between them derived. Following this, a complete set of irreducible tensor operators T(k){V} is constructed which can be used as a basis for expanding operators which depend on n nuclear spins. It is further shown that decomposition of the tensor’s components T(k)q{V} into a sum of products of two irreducible tensor components involves transformation matrices between different coupling schemes. Various properties and commutation relations of the T(k)q{V}’s are given along with a discussion of their reduced matrix elements.

DOI
10.1063/1.432104

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Basis states and operators composed of n, in general, different nuclear spins of integer or half-integer values are explicitly constructed. Various coupling schemes are discussed, and transformations between them derived. Following this, a complete set of irreducible tensor operators T(k){V} is constructed which can be used as a basis for expanding operators which depend on n nuclear spins. It is further shown that decomposition of the tensor’s components T(k)q{V} into a sum of products of two irreducible tensor components involves transformation matrices between different coupling schemes. Various properties and commutation relations of the T(k)q{V}’s are given along with a discussion of their reduced matrix elements.
A geometric theory of the irreducible tensor operators of quantum spin systems. It is based upon the Maxwell-Sylvester geometric representation of the multipolar electrostatic potential. In the latter, an order-ℓ\ellℓ multipolar potential is represented by a collection of ℓ\ellℓ equal length vectors, i.e. by ℓ\ellℓ points on a sphere, instead of by its components on some fixed (but arbitrary) basis. The geometric representation offers a much more appropriate tool for getting physical insight on specific characteristics of a multipole, such as its symmetries, or its departure from ideal symmetry. We derive explicit expressions enabling to perform any calculations we may need to perform on multipoles. All relevant quantities are eventually expressed in terms of scalar products of pairs of vectors (i.e., in terms of geometric quantities such as lengths and angles). The whole formalism is entirely independent of any particular choice of coordinate, and needs no use of the somehow abstract formalism traditionally used when dealing with angular momenta. The formalism is then applied to treat the problem of the irreducible tensor operators of quantum spin systems. It enables to completely dispense with the calculation and use of the Stevens operators, which can be quite complicated even for moderate values of ℓ\ellℓ. Explicit expressions for the calculation of expectations values of physical observables are derived. They essentially consist in combinations of scalar products of vector pairs. Together with the coherent state representation of the quantum states of spin systems, this provides a complete geometric, coordinate-free, description of the states, dynamics and physical properties of these systems.
The space part of a general spin-spin interaction is expanded in a complete orthonormal set of irreducible tensors constructed from spherical harmonics of the interacting particles. With this expansions, the mmm-scheme matrix element of the spin-spin interaction is represented as a sum of weighted radial integrals along with its graphical representation. The graphical form of the mmm-scheme matrix element enables one to obtain graphically matrix elements between many-particle configurations in any coupling scheme.
Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.
Our recent results [N. L. Manakov et al., Phys. Rev. A 57, 3233 (1998); 61, 022103 (2000)] on the invariant representations of finite rotation matrices (FRM's) of integer rank j (in terms of tensor products of vectors connected with a space-fixed reference frame) are generalized here for the general case of arbitrary (i.e., integer or half-integer) rank j. This extension is carried out by using new spinor representations of FRM's in terms of specially introduced spinor-annihilation operators. We demonstrate that all widely used, standard representations of FRM's follow as special cases of our invariant representation for particular parametrizations of the rotation parameters. As the simplest application of invariant spinor representations of FRM's, the factorized form of Wigner ${\mathbf{d}}^{j}(\ensuremath{\beta})$ matrices with an arbitrary rank j is obtained as a product of two triangular matrices composed of various powers of $\mathrm{cos}(\ensuremath{\beta}/2).$
In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.