Reduction of lower symmetry terms to cos 3τ′ and sin 3τ′ for vibration-internal rotation-overall rotation interactions in −XY3 molecules

In the calculation of the S_t, S_tS_t, dependence of the inertia tensor and the S_t, dependence of the Coriolis coupling for internal rotation in −XY3 molecules, terms of lower symmetry than cos 3τ and sin 3τ appear when curvilinear internal coordinates are used for the internal rotation and other vibrations. This paper shows how these lower symmetry terms are reduced to zero after transformation to symmetry coordinates for the internal rotation and other vibrations.     

The symmetry properties of the normal vibrations of C-15 type superconductor HfV2

The symmetry species of the normal vibrations of HfV₂ are analysed. The symmetry vectors at Г, δ and X, and phonon dispersion ω(δ) are given. The irreducible representation Г'₂ is responsible for the structural transition O_h⁷ → C_2v²⁰.     

Vibrational assignments and force field calculations for trimethylarsenic dichloride,dibromide and their deuterated analogs

Raman and infrared spectra have been recorded for trimethylarsenic dichloride, dibromide and their deuterated analogs and vibrational assignments were made for the skeletal modes of these compounds. The spectra of the dichloride and its deuterated analog were interpreted in terms of a trigonal bipyramidal structure, D_3h symmetry, whereas the spectra of the dibromides were interpreted using the ionic model, [(CH₃)₃AsBr]⁺Br⁻, having C_3v symmetry. Using a generalized valence force potential field, normal coordinate analyses data were obtained for the dichloride and the dibromide. Included in this data are potential energy distributions for a set of symmetry coordinates. The force fields obtained for the dichloride and dibromide were used to calculate frequencies for the corresponding deuterated compounds. The discrepancies between the observed and calculated frequencies for the deuterated compounds are discussed in terms of the assumptions made.     

Modes of vibrations due to (U-center + additive anion impurities) in KC1—II

Defect modes due to U-centers in potassium chloride containing additive anion impurities have been computed by Green's functions techniques. Local vibrations due to U-centers in alkali halides have cubic symmetry, but when one of the twelve nearest neighbour anions is replaced by an additive impurity, the site symmetry of the system reduces from O_h to C_2v and gives rise to new vibrational modes. The phonon Green's functions have been analysed according to the irreducible representation of the point group symmetry, pertaining to the substitutional impurity. We have considered the vibrations of the U-center, additive anion impurity and their nearest neighbours i.e. 36-dimensional defect space. Using group theory, symmetry coordinates were constructed and the 36 x 36 dimensional matrix was block diagonalised into various irreducible representations. The computed local mode frequencies have been conpared with the experimental measurements. Reasonably good agreement is found between them. The computed results for resonant mode frequencies are also displayed.     

Variational Calculation of Energies of Highly Excited Rovibrational States of CO2

Accurate variational calculations of energies of highly excited rovibrational states of ¹²C¹⁶O₂ using a Lanczos recursion are presented. In a first step, we use experimental rovibrational transition frequencies to determine by a least-square fitting procedure a potential energy surface for the CO₂ molecule. This potential energy surface is expressed as a multidimensional power series expansion in the normal coordinates. It is then used to determine all the rovibrational energies for symmetry e levels up to a rotational number J=200, a vibrational energy of 13 000 cm⁻¹, and a vibrational angular momentum l=13.     

Analysis of the rotational spectrum of C3v molecules by using factorization and diagonalization of the energy matrix: Application to CH3C15N

A method is given for the analysis of the rotational spectrum in the ground and excited states of C_3v molecules; it consists in a direct diagonalization of the energy matrix including all elements whose contribution can become significant for the analysis up to the sixth order of approximation.The method of factoring the energy matrix into four submatrices A₁, A₂, E, E, according to the symmetry species of the full point group C_3v, is given. The programm enables the calculation of the rotational frequencies and also carries out by a least-squares method the fitting of the molecular constants for vibrational states v = 0 (ground state) and v_E = 1, 2, 3, and 4, separately or simultaneously over several of these states.The analysis of the rotational spectrum of CH₃C¹⁵N in the v₈ = 0, 1, 2 states is given as an example.     

Infrared spectra of solid germane

An IR spectroscopic study has shown that GeH₄ molecules are situated on one set of sites of C, symmetry in phase IV. The III–IV transition at 63 K is apparently a second-order phase transition. In phases II and III the molecules are on sites of C_3v. or C₃ symmetry. The II–III phase transition was observed at 67 K. In solid GeD₄, phase transitions were observed at 68.5 and 77 K. In phases II and III the site symmetry is C_I. The II–III phase transition in GeD₄ is apparently second order. There is evidence that the ν₁, vibration of GeD₄ is IR active in the solid state.     

The contribution to the specific heat from the spin-phonon interaction for (3d)n ions

The contribution, ΔC, to the specific heat arising from the spin-phonon interaction is calculated for general spin values. For a given set of spin-phonon coupling constants and cubic symmetry the ratio ΔC_max/ C(Schottky)_max for ions having S = 2 or $\text{5}\text{2}$ is one or two orders of magnitude greater than that for ions having S = 1 or $\text{3}\text{2}$. The relative low values of ΔC in the latter cases is traced to two distinct reasons, and the differences that the inclusion of a lower symmetry crystals field makes to the ratio is discussed.     

Derivation of the nonrigid rotation-large-amplitude internal motion Hamiltonian of the general molecule

The nonrigid (effective) rotation-large-amplitude internal motion Hamiltonian (NRLH) of the general molecule with one or more large-amplitude vibrations has been derived to the order of magnitude κ²T_VIB. The derivation takes advantage of the idea of a nonrigid reference configuration and uses the contact transformation method as a mathematical tool. The NRLH has a form fairly similar to that of the effective rotation Hamiltonian of semirigid (i.e., normal) molecules. From a careful examination of the Eckart-Sayvetz conditions and of the Taylor expansions of the potential energy surface in terms of curvilinear displacement coordinates, three types of large-amplitude internal coordinates of different physical meaning (effective large-amplitude internal coordinates, real large-amplitude internal coordinates, and reaction path coordinates) are described. To test the ideas and the formulas the effective bending potential function of the C₃ molecule in its ground electronic and ground stretching vibrational state is calculated from the ab initio potential energy surface given by W. P. Kraemer, P. R. Bunker, and M. Yoshimine (J. Mol. Spectrosc. 107, 191–207 (1984)). The calculations were carried out by using either the effective or the real large-amplitude bending coordinate of C₃. The NRLH theory is compared to the nonrigid bender theory at a theoretical level as well as through the results of the test calculations.     

Lattice dynamics of II–VI,III–V compounds

The lattice vibrations of II–VI compound ZnSe and III–V compound InSb have been calculated in the frame work of Banerjee-Varshni's second neighbour ionic [SNI] model utilizing critical-point phonons as an input to determine the required seven parameters and the experimentally determined three elastic constants C₁₁, C₁₂, C₄₄ as restraints on the values of the parameters. A reinvestigation of the experimental elastic constants particularly C₁₁ and C₄₄ for ZnSe has been suggested. Results are presented for the dispersion curves along high symmetry directions. A reasonable agreement with the recently measured inelastic nutron scattering data is observed.     

Withdrawal of Electrodynamical Forbiddance and Peculiarities of the SERS Spectra in Fullerene C<Subscript>70</Subscript>

It is demonstrated that in fullerene C₇₀, which can be considered as a deformed fullerene C₆₀ in some mean sense there is a withdrawal of an Electrodynamical forbiddance of a strong quadrupole light-molecule interaction, which is realized in the fullerene C₆₀. This situation occurs because of the reduction of symmetry of C₆₀ from the icosahedral symmetry group Y_h to the group D_5h. The withdrawal results in appearance of the lines in the SERS spectra of C₆₀, which are forbidden in usual Raman scattering and are active in the infrared absorption spectra. The experimentally measured SERS spectra of C₇₀ demonstrates existence of such lines that strongly confirms our ideas about the dipole-quadrupole SERS mechanism.     

Doping and temperature dependence of inversion symmetry breaking in La2−xSrxCuO4

The doping dependence of the Raman spectra of high quality La_2−xSr_xCu^16,18O₄ polycrystalline compounds has been investigated at low temperatures. It is shown that symmetry forbidden bands peaked at ∼150 cm⁻¹, ∼280 cm⁻¹, and ∼370 cm⁻¹ are activated in the (xx/yy) polarization Raman spectra due to the local breaking of the inversion symmetry mainly at low temperatures and for doping concentrations for which the compound is superconducting. The apparent A₁-character of the activated modes in the symmetry reduced phase indicates a reduction from the D_2h to C_2v or D₂ crystal symmetries, which associates the observed modes to specific IR-active phonons with eigenvectors mainly along the c-axis. The temperature and doping dependence of this inversion symmetry breaking and the superconducting transition temperature are very similar, though the symmetry reduction occurs at significantly higher temperatures.     

Structure,electronic and magnetic properties of hexagonal boron nitride sheets doped by 5d transition metal atoms: First-principles calculations and molecular orbital analysis

A first-principles calculation based on density functional theory is carried out to reveal the geometry, electronic structures and magnetic properties of hexagonal boron nitride sheets (h-BNSs) doped by 5d transitional mental atoms (Lu, Hf, Ta, W, Re, Os, Ir, Pt, Au and Hg) at boron-site (B_5d) and nitrogen-site (N_5d). Results of pure h-BNS, h-BNS with B vacancy (V_B) and N vacancy (V_N) are also given for comparison. It is shown that all the h-BNSs doped with 5d atoms possess a C_3v local symmetry except for N_Lu and N_Hg which have a clear deviation. For the same 5d dopant, the binding energy of B_5d is larger than that of N_5d, which indicates the substitution of a 5d atom for B is preferred. The total densities of states are presented, where impurity energy levels exist. Besides, the total magnetic moments (TMMs) change regularly with the increment of the 5d atomic number. Theoretical analyses by molecular orbital under C_3v symmetry explain the impurity energy levels and TMMs.     

Spectroscopic line parameters of NH3 and PH3 in the far infrared

NH₃ and PH₃ rotation and rotation-inversion line parameters in the far to medium i.r. are calculated for remote sounding purposes of planetary atmospheres; 1607 lines of ¹⁴NH₃, 362 lines of ¹⁵NH₃ and 325 lines of PH₃ are compiled. The absolute intensity formulation has been reviewed in the case of rotation adn rotation-inversion lines of molecules with C_3v symmetry. The justification of the general agreement between the authors, and comparisons with other published expressions are given.     

Static and vibrational properties of equiatomic Na-based binary alloys

The computations of the static and vibrational properties of four equiatomic Na-based binary alloys viz. Na_0.5Li_0.5, Na_0.5K_0.5, Na_0.5Rb_0.5 and Na_0.5Cs_0.5, to second order in local model potential is discussed in terms of real-space sum of Born von Karman central force constants. The local field correlation functions due to Hartree (H), Ichimaru-Utsumi (IU) and Sarkar et al. (S) are used to investigate the influence of the screening effects on the aforesaid properties. Results for the lattice constants C₁₁, C₁₂, C₄₄, C₁₂-C₄₄, C₁₂/C₄₄ and bulk modulus B obtained using the H-local field correction function have higher values in comparison with the results obtained for the same properties using IU- and S-local field correction functions. The results for the Shear modulus (C′), deviation from Cauchy's relation, Poisson's ratio σ, Young modulus Y, propagation velocity of elastic waves, phonon dispersion curves and degree of anisotropy A are highly appreciable for the four equiatomic Na-based binary alloys.     

Correlation between energy levels of linear and bent X2Y2 molecules

The permutation-inversion group developed by Longuet-Higgins is extended to a classification of the vibronic, torsional, and rotational wavefunctions of a nonrigid X₂Y₂ molecule by introducing a symmetry operation $\text{T}\text{?}$, which rotates the top half of the molecule by 2π and, accordingly, the molecule-fixed x axis by π. Since the energy levels of linear (D_∞h) and bent (C_2h, C_2h, and C₂) forms of X₂Y₂ are classified according to a set of common symmetry operations of this extended permutation-inversion group, their energy levels can be correlated, including those of nonrigid forms such as a quasilinear system or a free internal rotor. Nuclear spin weights and selection rules are derived.     

Crystal field and microscopic spin Hamiltonians approach including spin-spin and spin-other-orbit interactions for d and d ions at low symmetry C3 symmetry sites: V in Al2O3

A new module has been developed within the CFA/MSH computer package, which is applicable for d² and d⁸ ions at sites of trigonal symmetry type I (C_3v,D₃,D_3d) and type II (C₃,C_3i), including the ‘imaginary’ CF term. For the first time the spin-spin (SS) and spin-other-orbit (SOO) interactions have also been included in the Hamiltonian. This module enables to study the contributions to the energy levels and the spin Hamiltonian parameters, i.e. zero-field splitting D and g-factors: g_∥ and g_⊥. The contributions arising from the spin-orbit (SO), SS, and SOO interaction as well as those due to the low symmetry CF effects induced by the distortion angle ?, which describes the difference between C₃ and C_3v symmetry, can be studied. As an application of the new module, calculations have been carried out for V³⁺(3d²) ions in α-Al₂O₃ crystal, taking into account for the first time the SS and SOO interactions, and the low symmetry CF effects. The results show that (i) the contributions from the SS and SOO interactions to the energy levels are larger for free V³⁺ ions than those for V³⁺ ions in α-Al₂O₃ crystal, (ii) both the contributions to the SH parameters and the energy levels arising from the SOO interaction are larger than those arising from the SS interaction, (iii) the contributions due to the low symmetry CF effects induced by the distortion angle ? are in general significant, (iv) D and g_⊥ are sensitive to the distortion angle ?, whereas g_∥ is insensitive to ?, and (v) the influence of the lattice distortions on the spectroscopic properties of V³⁺ ion in α-Al₂O₃ is pronounced. It appears important for similar ion-crystal cases to consider the lattice distortions in detailed calculations, which take into account the relevant contributions from the SO, SS and SOO interactions. A good agreement between the theoretical and experimental results has been obtained.     

The rovibrational levels of ammonia

In this paper we report variational rovibrational studies on ammonia and its isotopomers. We use six internal coordinates (one of which describes the umbrella motion). The expansion functions are products of one-dimensional functions of these internal coordinates, multiplied by appropriate functions of the Euler angles to describe the rotational motion. We use a previously published high accuracy six-dimensional potential energy surface [LEONARD, C., HANDY, N. C., CARTER, S., and BOWMAN, J. M., 2002, Spectrachim. Acta, 58, 825]. We derive the full kinetic energy operator for the rovibrational motion in these coordinates. This operator has been completely checked to give a hermitian secular matrix. All matrix elements are evaluated numerically by quadrature. The symmetry of the expansion functions is fully described in D_3h, C_2v and C_s. It is not possible to perform the calculations in D_3h, but complete degeneracy in the appropriate levels is obtained with the C_2v program. The algebraic complexity of this program has been far greater than for any other variational study we have undertaken for a tetra-atomic molecule.

We present J = 0, 1, 2 energy levels for the experimentally observed band origins of NH₃, and J = 0, 1 energy levels for the ground state and fundamentals of NH₂D, ND₂H and ND₃. For the asymmetric isotopomers, identical results are obtained for both C_2v and C_s, thus confirming the validity of the method. The levels we obtain are completely converged. Agreement with observation is of the order of 0.5% (of course being dependent upon the accuracy of the potential energy surface); therefore the ordering of the rotational levels and their splitting is completely predicted and understood.     

Spin-spin and spin-other-orbit effects of optical spectra and the spin-Hamiltonian parameters for Cr ions in MgO crystals

An extended complete diagonalization method/microscopic spin-Hamiltonian (CDM/MSH) program has been developed, which is applicable for d³ ions at sites of tetragonal symmetry type I (C_4v, D_2d, D₄, D_4h) and trigonal symmetry type I (C_3v, D₃, D_3d). The Hamiltonian includes the spin-spin (SS) and spin-other-orbit (SOO) magnetic interactions besides the spin-orbit (SO) magnetic interaction usually taken into account. Utilizing the extended CDM/MSH program, the optical spectra, the spin-Hamiltonian (SH) parameters of the ground state ⁴B₁, and the splitting δ(²E) of the first excited ²E state for Cr³⁺ (3d³) ions at C_4v symmetry sites in MgO crystals have been successfully investigated. It is found that although the SO magnetic interaction is the most important one, the contributions to the SH parameters and the optical spectra from the SS and SOO magnetic interactions for Cr³⁺:MgO crystals are appreciable and should not be omitted, especially reaching 27.8% for the zero field splitting parameter D.