Irreducible Tensor Formalism Research Articles

Synthesis, characterization, and magnetic properties of new complexes based on self-assembled homotrinuclear units CuII–CuII–CuII

Three new supramolecular entities of CuII have been synthesized and characterized: [Cu2(H2O)2(tmen)2(μ-Cu(H2O)(pba))](PF6)2 (1), [{Cu2(H2O)(tmen)2(μ-Cu(pba))}(μ-SeCN)]n(ClO4)n·2nH2O (2) and [{Cu2(H2O)(tmen)2(μ-Cu(pba))}(μ-SeCN)]n(PF6)n·2nH2O (3), where pba = 1,3-propanediylbis(oxamato) and tmen = N,N,N′N′-tetramethylethylenediamine. The crystal structures of 1 and 2 were solved. Complex 1 is the precursor of 3; complex 2 is derived from the ClO4− analogue of 1. The two complexes have a central core in common: a trinuclear CuII complex with the two terminal CuII ions blocked by N,N,N′N′-tetramethylethylenediamine and with the same environment in the CuII central atom. In the case of complex 1, the structure is a double chain of trinuclear entities linked by hydrogen bonds owing to coordinated water molecules. In the case of complex 2, the selenocyanate ligands produce intermolecular linkages giving a one-dimensional system. The counter anions PF6− and ClO4− do not participate directly in the formation of the supramolecular entities. For complex 3 we propose a one-dimensional structure such as found for complex 2. The magnetic properties of the three complexes were studied by susceptibility measurements vs. temperature; a fit was made by the irreducible tensor formalism (ITO). The values obtained were: J1 = −342.1 cm−1, J2 = 0.61 cm−1 for 1, J1 = −282.2 cm−1, J2 = 2.6 cm−1 for 2 and J1 = −245.4 cm−1, J2 = 1.3 cm−1 for 3.

Noncentral interactions in inelastic scattering of nucleons on nuclei: The case of12C(p,p′)12C*(1+)

An irreducible tensor formalism to analyze inelastic scattering of nucleons on nuclei with arbitrary spins is outlined and the nature of noncentral interactions involved therein is studied. In the particular case of inelastic scattering of nucleons on ${}^{12}\mathrm{C}$ leading to the ${(1}^{+})$ excited state at 15.11 MeV, the relative importance of vector and tensor amplitudes is examined using the DW81 and DREX codes.

Pion production inNNcollisions

The recently proposed irreducible tensor formalism for hadron scattering and reactions is extended to pion production in $\mathrm{NN}$ collisions and a form akin to that of Wolfenstein for elastic $\mathrm{NN}$ scattering is derived together with exact partial wave expansions for the amplitudes. Explicit formulas are derived for spin observables relevant to differential as well as total cross section measurements employing a polarized beam on a polarized target which are currently of experimental interest.

Irreducible tensor phenomenology for pd3He+-

A phenomenology based on the irreducible tensor formalism is developed for the reaction pd3He+-, to study incisively the P-wave dominance noticed in the recent kinematically complete experiment at c.m. excess energy of 70 MeV at the MOMO facility.

Multipolar molecules and multipolar fields: probing and controlling the tensorial nature of nonlinear molecular media

A general class of multipolar molecules is introduced in the context of quadratic nonlinear optics by way of extension of the more specific cases of dipolar and octupolar molecules. An adequate irreducible tensor formalism permits us to define rotationally invariant molecular features that couple to corresponding field tensor components, thereby enabling us to account for a variety of coherent and noncoherent processes such as harmonic light (hyper-Rayleigh) scattering, coherent second-harmonic generation in electrically poled media, and the recently proposed optical poling scheme. Experiments in both harmonic light scattering in solution (for some multipolar molecules) and optical poling (in Disperse Red 1–methyl methacrylate thin films) are analyzed in light of this model. A general tensorial permutation lemma of broad validity allows nonlinear light–matter interactions to be condensed in a statistical medium in compact rotationally invariant expressions: The main tensorial symmetry features for both molecular susceptibility and read–write field polarization tensors that jointly drive these interactions are clearly revealed.

MAPLE symbolic computation of the long‐range many‐body intermolecular potentials: Three‐body induction forces between two atoms and a linear molecule

AbstractGeneral theory of the long‐range intermolecular potentials, resulting from the use of the Rayleigh–Schrödinger perturbation theory at the multipole approximation level and the irreducible tensor formalism, is applied to describe the three‐body induction effects in a system composed of two atoms and a linear molecule. Explicit expressions for the leading van der Waals coefficients are derived by employing the symbolic computation language MAPLE. Both the general geometry and the special case of the isosceles triangular configuration are considered. Final equations are presented using the interface between MAPLE and the typesetting language LATEX. Some technical aspects associated with the use of MAPLE in theory of long‐range intermolecular potentials are briefly discussed. Explicit expressions obtained in this article should be useful in determination of the nonadditive induction effects in van der Waals clusters composed of two atoms and a diatomic molecule. In particular, they may be helpful in explaining the discrepancies between experiment and theory that are observed in high‐resolution microwave studies of the complex Ar ⃛Ar ⃛HCl. © 1993 John Wiley & Sons, Inc.

Contact transformations in tensor formalism. Effective Hamiltonian and dipole moment for the v2, v4 dyad of tetrahedral molecules

Simultaneous unitary transformations of the effective Hamiltonian and the effective dipole moment are studied via an irreducible tensor formalism for the v 2, v 4 dyad of tetrahedral molecules. Large unitary transformations relating resonance and isolated band models are treated by taking into account the non-leading contributions and multiple commutators. The unitary equivalence of different parameter sets of the effective dipole moment is explicitly shown for the v 2, v 4 bands of 12CH 4.

Computer programs for the Boltzmann collision matrix elements

When the distribution function in the kinetic theory of gases is expanded in a basis of orthogonal functions, the Boltzmann collision operators can be evaluated in terms of appropriate matrix elements. These matrix elements are usually given in terms of highly complex algebraic expressions. When Burnett functions, which consist of Sonine polynomials and spherical harmonics, are used as the basis, the irreducible tensor formalism provides expressions for the matrix elements that are algebraically simple, possess high symmetry, and are computationally more economical than in any other basis. The package reported here consists of routines to compute such matrix elements in a Burnett function basis for a mixture of hard sphere gases, as also the loss integral of a Burnett mode and the functions themselves. The matrix elements involve the Clebsch-Gordan and Brody-Moshinsky coefficients, both of which are used here for unusually high values of their arguments. For the purpose of validation both coefficients are computed using two different methods. Though written for hard sphere molecules the package can, with only slight modification, be adapted to more general molecular models as well.

Symmetry selection rules in rotationally inelastic collisions of open-shell diatomic molecules and polyatomic tops: Implications of an irreducible tensor expansion

An irreducible tensor formalism due to Grawert is applied to collisions of open-shell diatomics and polyatomic tops. Explicit examination of the reduced T-matrix elements within the infinite-order sudden and first-order Born limits allows the derivation of several general symmetry selection rules for a rotationally inelastic process. In particular, both differential and integral cross sections will vanish unless the symmetry indices of the initial and final states satisfy certain conditions, for the following types of transitions: J=0→J′M′; J,M=0→J′,M′=0; and JM→J,−M. The M=0→M′=0 selection rule represents the extension, to an arbitrary orientation of the quantization axis, of a selection rule we have derived earlier. A comparison with results of fully quantum studies of collisions of NH3 and H2CO with He confirms the predictive accuracy of these rules. We discuss how both the M=0→M′=0 and JM→J,−M selection rules could be directly verified, in experiments involving laser excitation of a particular symmetry level followed by resolution of the fluorescence polarization.

Intensities of rotation and vibration‐rotation Raman transitions in asymmetric top molecules

AbstractThe derivation of the intensities of rotation and rotation‐vibration Raman transitions for asymmetric top molecules is given, based on irreducible spherical tensor formalism. A computer program package written to calculate such intensities is described. Its use is demonstrated by reproducing the observed contours of the pure rotational band and the asymmetric stretching vibrational band in the Raman spectrum of the SO2 molecule. The polarizability tensor component ratio required to simulate the pure rotational contour is combined with literature values for mean polarizability and depolarization ratio to obtain the principal values of the SO2 polarizability tensor. A correlation of the polarizability tensor component ratio with band types is discussed for totally symmetric bands which have been reproduced using our program and those of others.

Origin of the second-order optical susceptibilities of crystalline substituted benzene

An analysis of the second-order optical nonlinearities of crystalline substituted benzene is presented. A new irreducible tensor formalism for the decomposition of a crystal tensor as the sum of the contributions of microscopic units is developed and applied to the study of the second-order susceptibility in four benzene derivatives with related structures and substituent radicals ($m$-dinitrobenzene, $m$-nitroaniline, $m$-aminophenol, and resorcinol). It is shown that an additivity approach can precisely account for the crystal nonlinearities and enables one to calculate the contributions of each radical to the nonlinearity of the molecule. these contributions are found to be strongly related to the donor or acceptor character of the radical. A very simple model is proposed to explain this relation.

Magnetic parameters of p2,4, dt22,4, and d2 configurations in axial symmetric crystal fields

The energies and wavefunctions of a p2,4 configuration under the influence of a crystal field possessing axial symmetry are calculated. If the crystal field parameter Δ is smaller than, or of the order of, the Slater-Condon parameter F2, the 3P wavefunctions have the lowest energies. In that case one finds, depending on the sign of Δ, that either an orbital doublet or an orbital singlet has the lowest energy. For the latter, the fine structure parameters, the g values, and the hyperfine parameters are calculated. Using an irreducible tensor formalism, it is shown that the results for p2,4 can be used to find the interactions within dt22,4. Tetragonal as well as trigonal symmetry are treated and it is found that the hyperfine interactions differ substantially for both cases. Finally, the interactions within 3F(d2) under the influence of a crystal field possessing tetragonal symmetry are treated.

Symmetries in Nuclei

A review is given of symmetries in the vector spaces of the nuclear shell model, the goodness of nuclear symmetries, and the irreducible tensor formalism as a calculational tool.

Die irreduzibel-tensor-formulierung der bewegungsgleichungen des statistischen operators eines atomaren systems unter den Bedingungen des optischen pumpens

The dynamical equations for optical pumping are derived by the “regressive” method in the irreducible tensor formalism.

Reduced matrix element of an irreducible spherical tensor operator—New derivation and applications

A new derivation is given of the result that the reduced matrix element of an irreducible spherical tensor operator T( kq) is (gamma;J‖ T (k) ‖γJ) = [(2J + k + 1)!k!/(2J − k)!(2k − 1)!!] 1 2 [f(J)] k , where f(J) = [(2J)(2J + 1)(2J + 2)] − 1 2 (γJ‖ T (1) ‖γJ) ; the reduced matrix is assumed diagonal in γ and J. Applications of this result to problems of molecular structure and spectroscopy and solid state physics are discussed. These are cases in which high order polynomials in angular momentum operators make the use of the irreducible spherical tensor formalism advantageous.

Theory of Long-Range Interatomic Forces. I. Dispersion Energies between Unexcited Atoms

A general theory of second-order dispersion forces between atoms in nondegenerate ground states is first developed by using an irreducible tensor formalism and the theory of angular momentum. This forms the basis for calculations of forces between excited systems, which are the subject of later publications. Attention is given to the interaction of two noble gas atoms where it is assumed that each electron oscillates with simple harmonic motion, and the interaction between two alkali atoms is calculated by considering the electrons to be moving in a Coulomb field. The dominant terms of the dispersion energy between a number of atoms and molecules are tabulated. The results indicate that the hitherto neglected dipole-octupole contributions are in many cases larger than the quadrupole-quadrupole terms.