Set Of Orthogonal Operators Research Articles

Induced operator orthogonality across an atomic shell

The establishing of operator orthogonality for atomic configurations C by means of group theory is extended to operators Hi requiring arbitrary multiplicity labels to distinguish them. By including the zero-electron operator in the set of orthogonal operators, each N-electron operator can be constructed to belong to the irreducible representation (irrep) (1N041+2-2N-1N) of SU(41+2), with the result that: (a) operator orthogonality for C=1n is equivalent to a relation between 3-j symbols for the group SU(41+2), and (b) the effect of electron-hole conjugation is simplified and no quasispin ranks of mixed parity appear. A second kind of orthogonality relation, in which the running index is the operator label i, yields new relations satisfied by the matrix elements of the orthogonal operators. The dependence of operator normalisation on N and n is derived from a group-theoretical formula of Hecht (1975) and confirmed by an approach using fractional parentage. Generalisations to other atomic configurations are briefly considered.

Energies of N equivalent electrons expressed in terms of two-electron energies and independent three-electron parameters: a new complete set of orthogonal operators. III. Ab initio calculations

For pt.I see ibid., vol.17, p.23-75 (1984). As an illustration of the theory developed in the first paper in this series, a set of five orthogonal parameters is used to describe all electrostatic two-particle excitations in the 3dN configurations completely to second order. These parameters represent the term energies of d2 and, consequently, the 3dN configurations are perceived as a collection of 1/2N(N-1)d2 pairs, the total energy of the configuration being the sum of the corresponding pair energies. As regards their effects, the parameters are partitioned into a structural and an additive part; the structural part is found to behave as a linear function of N, whereas the additive part is proportional to the average energy of the configuration. In order to take into account all electrostatic one-particle effects as well, two orthogonalised three-body parameters T1 and T2 are used, the first corresponding to the well known Trees parameter and the second covering the remaining class of one-electron excitations. With a least-squares fitting procedure, both the d2 energies and the three-electron parameters are determined for the dN ground configuration in the III, IV, V and VI spectra of the iron group elements. Surprisingly, the effects of T2 turn out to be important, especially for the lower ionisation stages. Amongst other things, old problems concerning the far-off position with respect to calculations of the 12D term in the 3d6, and parameter irregularities in the 3d7 configurations, seem to be solved with the introduction of T2. As a result, the mean error of the fit in some of these cases is reduced by factors of three to four. Using the above set of parameters, a prediction of the 3d4 ground configuration of Mn IV is made.

Constraints on three-body parameters for the f3configuration

The transformation properties of the complete set of orthogonal operators for the three-electron configuration f3, are used to find constraints on parameters needed to fit the energy levels of that configuration. These constraints are checked against ab initio calculations of those parameters for Np3+ and Pr2+. Excellent agreement is found in both cases.

Energies of N equivalent electrons expressed in terms of two-electron energies and independent three-electron parameters: a new complete set of orthogonal operators. I. Theory

The lN configuration is described as a collection of 1/2N(N-1) electron pairs, with the angular momenta of the electrons of each pair being coupled to the (2l+1) SL terms of l2. In this way, a set of parameters is defined which can be visualised as the energies of these particular SL terms. The parameters represent the effects of two-particle excitations on the lN configuration to second order. Moreover, they turn out, after a minor modification, to form an orthogonal set. The number of pairs multiplied by the average pair energy proves to be equal to the average energy of the full lN configuration. Inner products of general two-particle operators are calculated and conditions for their orthogonality are derived. For the dN configurations, two orthogonal three-body operators are added in order to obtain a complete set of electrostatic parameters. A simple general procedure is sketched that allows for the determination of the contributions of physical excitations to the various orthogonal parameters. This is illustrated by the s to d case. Values of all the matrix elements used are given.

Nuclear spin–lattice relaxation of coupled spin systems in liquids

A new method based upon the Zwanzig–Mori projection operator technique has been introduced to treat the spin–lattice relaxation problem of coupled nuclear spin systems in liquids, with emphasis on high resolution NMR spectroscopy. The important quantities required for the use of the method are a set of orthogonal operators which can readily be constructed and given physical meaning in the eigenstate representation for any particular spin system. Some of the expectation values of these orthogonal operators may be related to the measurable quantity in high resolution NMR experiments. The time rate of change of each observable may then be formulated in terms of the trace of various operator functions, allowing the calculation to be carried out in any representation. The computational advantage of using simple spin-product functions is obvious. The prescription for constructing irreducible orthogonal tensor operators is given, and a variety of such tensor operators are explicitly given for several common spin systems. In the present method each orthogonal operator has a definite symmetry property under spin inversion which can be used to separate the coupled equations into a symmetric and an antisymmetric set. The problem of multiexponential decay in the time evaluation of the longitudinal magnetization, owing to effects of interferences between pairwise dipolar interactions, becomes clear in the present formalism. The present paper is a generalization of earlier work on the relaxation behavior of multiplet spectral structure in high resolution NMR.