Generalizations of the vector coherent state method

The introduction of a set of intrinsic coordinates to give an explicit construction of the intrinsic states of vector coherent state theory has greatly simplified earlier attempts to generalize this theory to include the construction of vector coherent state realizations of operators other than the group generators. The group U(3)⊇ U(2)×U(1) is used as a prototype. The construction of irreducible tensor operators with specific shift properties is illustrated with a number of examples. These show how the Wigner calculus for a higher symmetry group can be expressed solely in terms of the recoupling coefficients of the core subgroup and the simple K‐matrix elements of vector coherent state theory.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87293/2/27_1.pd

Summation relation for U(N) Racah coefficients

A summation relation is given for U(N) Racah coefficients which has the form of an orthogonality relation, or a composition of recoupling transformations, except that the summation over column indices (for fixed row indices) is over multiplicity labels only. In the recoupling matrix for [f1] × [f2] × [f3] → [f], U(N) irreducible representations [f2] and [f3] are limited to be elementary, [11…10…0]≡[1k], or totally symmetric [k], or of the form [kN−1]. Results are tabulated as functions of the axial distances in [f] for [f2]=[1N−1], [1N−2], or [2N−1]; [f3]=[1], [12], or [2]; all cases which arise in the evaluation of squares of matrix elements of one‐ and two‐body operators averaged over irreducible representations of U(N).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70621/2/JMAPAQ-15-12-2148-1.pd

SU(3) techniques for angular momentum projected matrix elements in multi‐cluster problems

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87456/2/518_1.pd

Branching rules for the subgroups of the unitary group

Expressions are given in terms of simple matrices d for the reduction of the Kronecker (outer) product of two or more irreducible representations which can be characterized by Young patterns. These are then used to obtain practical formulas for branching rules. The needed matrices d can be constructed by a very efficient recursive process.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69691/2/JMAPAQ-19-3-720-1.pd

Lowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphs

Normalized lowering and raising operators are constructed for the orthogonal group in the canonical group chain O(n) ⊃ O(n − 1) ⊃ … ⊃ O(2) with the aid of graphs which simplify their construction. By successive application of such lowering operators for O(n), O(n − 1), … on the highest weight states for each step of the chain, an explicit construction is given for the normalized basis vectors. To illustrate the usefulness of the construction, a derivation is given of the Gel'fand‐Zetlin matrix elements of the infinitesimal generators of O(n).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70553/2/JMAPAQ-8-6-1233-1.pd

Representation of the five‐dimensional harmonic oscillator with scalar‐valued U(5) ⊇ SO(5) ⊇ SO(3)–coupled VCS wave functions

Vector coherent state methods, which reduce the U(5) ⊇ SO(5) ⊇ SO(3) subgroup chain, are used to construct basis states for the five‐dimensional harmonic oscillator. Algorithms are given to calculate matrix elements in this basis. The essential step is the construction of SO(5) ⊇ SO(3) irreps of type [v,0]. The methodology is similar to that used in two recent papers except that one‐dimensional, as opposed to multidimensional, vector‐valued wave functions are used to give conceptually simpler results. Another significant advance is a canonical resolution of the SO(5) ⊇ SO(3) multiplicity problem. © 1995 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70418/2/JMAPAQ-36-9-4711-1.pd

Hindered Rotation in Molecules with Relatively High Potential Barriers

The theory of hindered rotation has been applied to the type of asymmetric molecule in which the hindering barrier is high enough so that the hindered rotation splittings of the energy levels are small compared with the rotational energies but yet large enough to be observable in the microwave spectrum. The specific type of molecule considered consists of a rigid asymmetric component which may undergo a hindered rotation about the symmetry axis of a rigid symmetric component where the symmetric component is in addition assumed to have threefold symmetry and the asymmetric component at least a plane of symmetry containing the symmetry axis of the symmetric component. An example might be the acetaldehyde molecule, CH3CHO.In principle, the theory developed by Burkhard and Dennison can be used directly but in practice the method is difficult to apply to such a molecule since the matrix elements of the Hamiltonian used previously do not degenerate naturally or easily to those for the rigid asymmetric rotator in the infinite barrier limit. In the present treatment a transformation is made on the Hamiltonian whereby this complication is avoided and the resulting calculations are greatly simplified.It is found that the spectrum is essentially that of the rigid rotator with the important exception that all the strong lines are split into two components. For the low J transitions specific formulas have been derived for these splittings which are relatively simple functions of the barrier height, the principal moments of inertia, and two additional parameters involving the molecular dimensions and the masses. The barrier height can thus be deduced from the observed splittings without the use of the somewhat cumbersome machinery needed in the general case.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69966/2/JCPSA6-26-1-31-1.pd

On the Wigner Supermultiplet Scheme

Calculation of Wigner and Racah coefficients for the group SU(4)⊃[SU(2)×SU(2)]SU(4)⊃[SU(2)×SU(2)] make it possible to perform the spin—isospin sums in the cfp (fractional parentage coefficients) expansion of the matrix elements of one‐ and two‐body operators in the Wigner supermultiplet scheme. The SU(4) coefficients needed to evaluate one‐ and two‐particle cfp's, the matrix elements of one‐body operators, and the diagonal matrix elements of two‐body operators are calculated in general algebraic form for many‐particle states characterized by the SU(4) irreducible representations [yy0], [y y − 1 0], [yy1], [y11], [y y − 1 y − 1], [y10], [yy y − 1], [y00], and [yyy], whose states are specified completely by the spin and isospin quantum numbers (y = arbitrary integer). Applications are made to the calculation of the matrix elements of the complete space‐scalar part of the Coulomb interaction and the space‐scalar part of the particle‐hole interaction for nucleons in different major oscillator shells.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70485/2/JMAPAQ-10-9-1571-1.pd

Vibration‐Hindered Rotation Interactions in Methyl Alcohol. The J=0→1 Transition

The hindered rotation fine structure of the J=0→1, K=0→0 transition which has been observed by Venkateswarlu, Edwards, and Gordy in normal methanol as well as in five additional isotopic species can be understood only qualitatively on the basis of earlier investigations of the theory of hindered rotation in methanol. It has been shown that the frequency separations between the various torsional transitions and the splitting of each of these can be explained quantitatively by including in the theory the effects of the vibration‐hindered rotation interactions during the rotation of the whole molecular framework in space. The effects of the asymmetry of the rigid hindered rotator, the Coriolis interactions, and the centrifugal distortion of the molecule are discussed separately. A frequency formula for the transition is derived which contains essentially only four new rotational constants. Three of these depend solely upon the known structure of the molecule and the elastic force constants and can therefore be calculated from a knowledge of the vibrational spectrum. Since this latter has never been analyzed in more than a rough way some small adjustments have been made in the indicated values of the elastic constants which are within the limits of uncertainty. This adjustment is made for the normal molecule after which the three rotational constants are calculated for the remaining isotopic species without further adjustment. The fourth constant in the frequency formula describes the dependence of the barrier height upon the normal coordinates and is the only constant which must be determined empirically for each isotopic species. It has thus been possible to predict the 30 observed separations and splittings with the aid of essentially only six empirical constants. The agreement with experiment is remarkably good with one possible exception where the theory predicts for the fully deuterated methanol a very large splitting of the normal state line whereas the line in question is observed to be single. It is not improbable, however, that the large splitting actually exists and that the second component lay too far away to be recognized.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69972/2/JCPSA6-26-1-48-1.pd