Group Theoretical Treatment Research Articles

Single‐crystal Raman studies on nickel iodate dihydrate, Ni(IO3)2·2H2O

AbstractSingle‐crystal Raman spectra in the range 4000–35 cm−1 and the results of group theoretical treatment including Cartesian symmetry coordinates of nickel iodate dihydrate, Ni(IO3)2 · 2H2O, (space group Pbca, Z = 4) are presented and discussed with respect to the strength of the HO—HOIO3 hydrogen bonds and the extent of intramolecular (intraionic) coupling of the stretching vibrations of both the hydrate H2O molecules and the iodate IO3− ions. The energies and relative intensities of the IO stretching modes in the various species of the unit‐cell group D2h reveal that the iodate ions of the title compound behave as if they possess pseudo C2v site symmetry with a at least partly uncoupled IO stretch due to the significantly shortest (175.5 pm) IO arm and a symmetric (νs) and an antisymmetric stretching vibration (νas) due to the remaining IO2 group (rI–O = 184.2 and 186.0 pm). The respective site‐group modes decrease in the order ν3a ↔ νIO(1) (mean value of the respective unit‐cell group modes: 804 cm−1) > ν1 ↔ νs (772 cm−1) > ν3b ↔ νas (744 cm−1) (ν1 and ν3 as for distorted C3v IO3− ions). For the IO3− bending modes (430–300 cm−1) and the lattice modes below 300 cm−1 the true vibrational modes are not known. The normally used assignment to a symmetric (ν2) and an asymmetric bending vibration (ν4) as for C3v IO3− ions is at least questionable. The hydrogen‐bond distances calculated from the wavenumbers of the OD stretching modes of matrix‐isolated HDO molecules do not fully agree with the data of a neutron diffraction study performed in 1972. Copyright © 2001 John Wiley & Sons, Ltd.

Torsional Splittings in Small-Amplitude Vibrational Fundamental States of Methanol-Type Molecules

Group-theoretical methods are used to show that inverted torsional splittings in fundamental levels of small-amplitude vibrations of methanol-like molecules can be parameterized and understood in terms of the energy level patterns induced when a pair of high-barrier torsionally split components of given vt and tA+tE symmetry species in the molecular symmetry group G6 is allowed to interact with small-amplitude vibrational modes of symmetry vE. Such doubly degenerate vE vibrational modes arise rather naturally in G6 (isomorphic with the point-group C3v) for those methyl-group vibrations in point-group-Cs asymmetric tops such as CH3–CHO that are analogs of the degenerate methyl-group stretch, bend, and rocking vibrations in point-group-C3v symmetric tops such as CH3–C≡C–H. The present group-theoretical treatment is somewhat different from, but (as a comparison of model parameters shows) still fundamentally similar to, the recent local mode explanation of inverted torsional splittings in the C–H stretching fundamental region in methanol.

Raman spectroscopic studies on palladium and platinum hydrido complexes

Raman spectra of platinum metal hydrido complexes, Na2PtH4 (I4/mmm), K2PtH4, K2PtD4, Rb2PtH4, Rb2PtD4, Rb2PdD4 (Fm3m) and K3PtH5 (P4/mbm), are presented and discussed with respect to the influence of the different strengths of Pt—H and Pd—H bonds and of the relative volume of the reduced (per formula unit) unit cells. The ‘antisymmetric’ MD (M = Pt,Pd) stretching mode (species B1g of free, square MX4 units) decreases from 1436 to 1291 cm−1 on going from K2PtD4 to Rb2PdD4. Likewise, the PtH stretch of Na2PtH4 at 2026 cm−1 decreases to 2001 cm−1 for K2PtH4 but increases to 2039 cm−1 for K3PtH5. The totally symmetric MH stretches (A1g), which are of lower intensity, are shifted by about 50 cm−1 to higher wavenumbers compared with the asymmetric stretches. The results of a group theoretical treatment of the respective site group and unit-cell group modes are given. Copyright © 1999 John Wiley & Sons, Ltd.

Group Theory Approach to Band Structure: Scarf and Lamé Hamiltonians

The group theoretical treatment of bound and scattering state problems is extended to include band structure. We show that one can realize Hamiltonians with periodic potentials as dynamical symmetries, where representation theory provides analytic solutions, or which can be treated with more general spectrum generating algebraic methods. We find dynamical symmetries for which we derive the transfer matrices and dispersion relations. Both compact and noncompact groups are found to play a role.

Group theoretical perturbative treatment of nonlinear Hamiltonians on the dual of a Lie algebra

Lie algebraic and group theoretical approaches for a perturbative treatment of a Hamiltonian system on the dual of a Lie algebra are discussed. The known case of Weyl group modelling geometrical optics is reviewed and the SU(2) group is also studied.

Should the Casten triangle be a pentagon?

We discuss an often overlooked property of the interacting boson model in its simplest version. From a pure group theoretical treatment the IBM-1 has five instead of three dynamical symmetries. The origin and consequences of the two additional symmetries are presented.

Energy level statistics of the two-dimensional Hubbard model at low filling

The energy level statistics of the Hubbard model for $L \times L$ square lattices (L=3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of the coupling strength. All known symmetries of the model (space, spin and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L=4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L=4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance $\Sigma^2$, and the spectral rigidity $\Delta_3$, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to the Gaussian ensemble statistics stressing the nonperturbative nature of the Hubbard model.

Photonic Bands Using Vector Spherical Waves. III. Group-Theoretical Treatment

The group-theoretical treatment for photonic bands is given. The symmetry-adapted plane-wave bases are given for the eigenvectors of the electric fields and the transformation properties of the vector spherical-waves are clarified. For a fcc array of dielectric spheres, the group-theoretical band assignment is presented and the result is used to clarify the origins of the difference from band to band, concerning the mixing between transverse and longitudinal components, the activity to an incident light in the reflectivity, the mutual orthogonality of the eigenvectors and so on. Promising application of photonic crystal as a useful polarizer of light is proposed.

Microscopic Calculations forO(6) Nuclei by the Interacting Boson Model

We present microscopic calculations of the Interacting Boson Model (IBM) based on the shell model interaction by using the OAI mapping approach. We determine the col­ lective pairs, which correspond to the IBM bosons, by the number conserved Hartree-Fock­ Bogoluibov (HFB) and the proton-neutron Tamm-Dancoff methods, and we take into consid­ eration couplings to the non-collective degrees of freedom. The present realistic calculations are carried out for the Te, Xe and Ba isotopes. Among them, the Xe isotopes are known by numerous phenomenological works to show the 0(6) symmetry. We present the clear 0(6) symmetry in the spectra and the wave function which are microscopically obtained. Low-lying spectra of medium and medium-heavy nuclei show simple and regular structures, although these nuclei consist of many interacting protons and neutrons and their dynamics is intrinsically very complicate. This is known to be the collec­ tive motion. Especially, in the case of even-even nuclei, there are quite simpler and common features in energy levels and electro-magnetic transitions. The understand­ ing of them has been one of the main problems in nuclear structure. Many theories have been advocated, developed and extended. Among them, the Interacting Boson Model (IBM), l)- 6 ) which was first introduced by Arima and Iachello in the 1970's, has shown to be rather successful. In the IBM, nucleon collective pairs are approximated in terms of bosons. This notion can simplify the treatment of the nucleon many-body system. Furthermore this ansatz facilitates the group theoretical treatment. Originally s and d bosons, which are counterparts of S(J = 0) and D(J = 2) nucleon pairs, are introduced as the building blocks of the IBM. Because these bosons do not distinguish the proton and neutron degrees of freedom, the nucleon pair counterparts of these bosons are ambiguous. But, the s and d bosons span a U(6) space and its group chains containing the 0(3) subgroup correspond physically to vibrational, rotational and ''( unstable nuclei as limiting cases. These group chains are U(5), SU(3) and 0(6) limits. Unlike to other collective models, the IBM can give us a clear description of the 0(6) nuclei besides the U(5) and SU(3) nuclei. There are many nuclei, the spectra of which show the pattern of these group theoretical limits. Furthermore the intermediate situations between three limits are easily tractable by diagonalization of the Hamiltonian because the dimension of the original IBM space is at most about one hundred. At this stage, many phenomenological works were carried out, which showed that the low-lying states of many even-even nuclei can be explained by using six parameters of the IBM in a unified way. These parameters are considered to

Effective Boson-Fermion Dynamics for Subfermion Models

Abstract Effective composite particle dynamics can be derived by weak mapping of quantum fields. This method was already applied to derive effective boson or boson-fermion coupling theories from a nonlinear subfermion field. In this paper we present an extension of those calculations to the general group theoretical treatm ent of two-fermion bound states and their coupling to (elementary) fermions within an arbitrary nonlinear spinor-isospinor field model. The resulting effective field equations are com pared with the corresponding phenomenological expressions which for example underly the standard electroweak theory. PACS 11 .10 - Field theory. PACS 12.10 - Unified field theories and models. PACS 12.35 - Composite models of particles.

Group-Theoretical Treatment of Tunneling Splittings in the Methanol Dimer

Tunneling splitting patterns, selection rules, and effective B values for K = 0 rotational levels of the hydrogen-bonded methanol dimer are predicted from group-theoretical considerations, a modified internal axis method, approximate descriptions of possible tunneling paths, and the assumption that K = 0 levels can be treated by themselves. It is shown that 25 different tunneling motions are possible when the two methanol monomers are identical, giving rise to 16 tunneling-rotational levels for K = 0: four nondegenerate A levels, eight doubly degenerate E levels, and four fourfold degenerate G levels. The gross features of a recently observed methanol dimer K = 0 a-type microwave spectrum can be explained using only two of these tunneling motions, namely the internal rotation of each of the two inequivalent methyl groups in the dimer. Finer details of the observed splittings presumably arise from some of the remaining 23 tunneling motions. Tunneling splitting patterns are also predicted for cases in which the two methanols in the dimer are isotopically different, but the C 3 v symmetry of both methyl tops is preserved.

Structure of Berry's phase: Some group-theoretical examples.

The group-theoretical treatment of Berry's aholonomy developed in a paper of Giller, Kosi\ifmmode \acute{n}\else \'{n}\fi{}ski, and Szijmanowski [Int. J. Mod. Phys. A 4, 1453 (1989)] is applied to some specific examples. The general structure of the corresponding Berry matrices is determined. All irreducible representations of SO(n) are found for which the Berry matrices parametrized by the points of ${\mathit{S}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$\ensuremath{\simeq}SO(n)/SO(n-1) are determined by the Riemannian connection on ${\mathit{S}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$.

Vibrational spectra of the cis and trans isomers of the Mo(CO)4(PPh3)2 complex

AbstractRaman and infrared spectra of polycrystalline cis and trans isomers of the Mo(CO)4(PPh3)2 complex were recorded at room temperature. The molecular vibrations around the central Mo atoms and the carbonyl ones were assigned and correlated with the group theoretical treatment. Some changes in the spectra of the triphenylphosphine ligand have also been discussed.

A group theoretic treatment of the geometric phase

We define a new unitary operator in the Hilbert space of a quantum system which parallel transports the state of the system along an arbitrary curve in the projective Hilbert space. This operator is geometrical even for an open curve in the sense that it depends uniquely only on the curve and is independent of the Hamiltonian. Using this, when the curve is closed, the geometric phases discovered by Pancharatnam, Berry and Aharanov-Anandan are obtained.

Ring mediated coupling of the internal motions of the amine and methyl groups in p-methylaniline

The jet-cooled fluorescence excitation spectra of p-toluidine (p-methylaniline), aniline, aniline (ND2), p-toluidine (ND2), and p-fluoroaniline are presented. The spectra strongly suggest that the internal motions of the methyl and amine groups are coupled, the spectral evidence being a splitting of the amine inversion peak. A group theoretical treatment of the problem is presented using molecular symmetry group methods, however, this treatment appears to fail to properly predict selection rules for the methyl torsion. Possible reasons for this failure are discussed. The symmetry of the potential surface along the nonrigid coordinates is also discussed and a model which explains some of the unusual features of the spectra is presented.

Raman spectra and group theoretical treatment of polycrystalline (dppm)Mo(CO) 4 and (dppe)Mo(CO) 4

Laser-excited Raman spectra of polycrystalline (dppm)Mo(CO) 4 and (dppe)Mo(CO) 4 were recorded at room and low temperature. A factor group analysis was carried out for the dppm compound. The predicted correlation splitting due to weak intermolecular forces within the crystals is observed in only a few instances. The molecular mode data can therefore be correlated with solution data and assigned. The lattice mode region could be obtained as close as 5 cm −1 to the exciting line for both Stokes and anti-Stokes scattering. From the Raman results it is concluded that the two samples have different crystal structures.

The polymorphism of alkali metal formates. Part 4. A Raman study of the phase transition in KHCOO

KHCOO II is orthorhombic at ambient conditions and it is shown that traces of moisture affect the polymorphism of these very hygroscopic crystals. Dry KHCOO II transforms into phase I at 417 K (144 °C), and this phase can be supercooled to room temperature, remaining metastable for several days before transforming back to the orthorhombic phase II. The Raman spectra of phases I and II, as well as of supercooled phase I, are reported in the present study. The absence of some prominent translational modes in the Raman spectra of KHCOO II, compared to NaHCOO II, can be explained on the basis of a group-theoretical treatment. From the temperature dependence of the linewidths of various Raman-active librational and internal modes, activation energies are obtained for intramolecular motions of the formate ions. Fermi resonance occurs between the overtone of the bending mode 2ν5 and the C—H stretching mode ν1 in KHCOO II and the coupling constant W increases with temperature. The Raman and X-ray data show that KHCOO I is structurally different from NaCHOO I, but it is not possible to assign a definite structure to this phase on the basis of a Raman spectrum alone. Key words: Potassium formate, phase transition, Raman spectra.

MF 2 vibrations and librations of water molecules in the series MF 2·4H 2O (M = Fe, Co, Ni or Zn)

The infrared absorption spectra of the series MF 2·4H 2O (M = Fe, Co, Ni or Zn) and of the respective deuterates were recorded at 296 and ∼ 100 K in the 1200-400 cm −1 wavenumber region. Using the known ZnF 2·4H 2O structure as a model, the number of i.r. active librations and MF 2 vibrations was predicted with the aid of a group theoretical treatment. The librations were distinguished from the MF 2 and MO vibrations and assigned, using isotopic ratios and correlations between the unit cell volumes, the uncoupled OH and OD stretch vibrations of HDO and the twisting libration. The six librations are assigned to two types of water molecules with low symmetry and with different hydrogen bond strengths, and are compatible with an orthorhombic Pca2 1 structure. The v 1 and v 3 intramolecular MF 2 vibrations are assigned, using ZnF 2·4H 2O crystal data and matrix-isolated v 1 and v 3 ( v 1.3) values as guide. Shifts of v 1.3 and v 1/ v 3 relative to the matrix-isolated values suggest smaller MF bond lengths. The shifts in the values of v 1.3 and v 1/ v 3 upon deuteration and lowering of the temperature indicate smaller MF bond lengths and FMF angles.

Impurity Distribution on the Sites of β-Mn Structure for Mn(Co), Mn(Fe) and Mn(Ni) Alloys

The occupation probability of impurity atoms, Co, Fe or Ni, on each site of β-Mn structure has been determined by means of neutron diffraction with the single crystal specimens. Almost 90% of the impurity atoms occupy the site I. In the β-Mn(Co) alloy, an atomic long range order of Co impurities on the site I has been discovered below 500 K. The ordered structure is analyzed by introducing the scalar basis functions based on the group theoretical treatment and is discussed on the relation to the icosahedral packing sequence.

Group theoretical treatment of classical n-time correlation functions

The methods of group theoretical statistical mechanics are used to investigate the set of n-time correlation functions in isotropic molecular liquids and in molecular liquids subjected to external electric and magnetic fields. It is found that the group theory corroborates the results of computer simulation where these are available, both for auto- and cross-correlation functions, in the field-off and field-on equilibrium conditions. Many more elements of the time correlation functions are allowed by symmetry in the presence of fields. For the first time, the symmetry and time dependence of higher-order n-time correlation functions are investigated by group theory. This results in a great simplification of the exploratory work of the computer simulator, because the group theory clearly distinguishes between those elements that vanish for all t and those that may exist in the laboratory frame ( X, Y, Z) and the molecule-fixed frame ( x, y, z). Group theoretical statistical mechanics reveal clearly that the statistics of a molecular liquid cannot be Gaussian in general, because three-time and higher-order correlation functions exist by symmetry. These should be observable in future computer simulation. Thus we arrive at the important conclusion that group theory applied to the dynamics of molecules in the liquid state invalidates a large number of literature theories of diffusion, including all those based on the Debye concept of “rotational” diffusion. These theories should be modified accordingly.