Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D₃ invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.     

Collision-induced rovibrational energy transfer in small polyatomic molecules: the role of intramolecular perturbations

ABSTRACT

Various forms of time-resolved optical double-resonance spectroscopy facilitate rotationally resolved measurements of collision-induced intramolecular vibration-to-vibration (V–V) energy-transfer processes, which take a gas-phase polyatomic molecule from one distinct rovibrational energy level to another. Of longstanding mechanistic interest are questions concerning the extent to which such V–V energy transfer (ET) may be influenced by intramolecular perturbations – notably Fermi resonance (and other anharmonic mixing effects) and Coriolis coupling – within polyatomic molecular rovibrational manifolds of interest. It is evident that quantum-mechanical interference effects can arise, either inhibiting or enhancing the probability of collision-induced ET in perturbed rovibrational manifolds of certain small gas-phase polyatomic molecules, notably CO₂, D₂CO and C₂H₂. This article focuses on a blend of high-resolution rovibrational spectroscopy (characterising initial and final molecular levels and their intramolecular perturbations) and collision dynamics (with colliding molecules defined in terms of isolated-molecule spectroscopic basis states). It aims to offer fresh insights and to consider some apparent mechanistic anomalies (e.g. collision-induced quasi-continuous background effects in the 4ν_CH rovibrational manifold of C₂H₂). Various reported experiments and related theoretical treatments are critically re-examined, in order to pose and address mechanistic questions some of which still challenge detailed understanding.     

The behavior of null geodesics in a class of rotating space-time homogeneous cosmologies

The null geodesics are investigated in a class of open space-time homogeneous cosmological models with rotating sheared matter (Ozsváth classIII). Gödel's model is a special case. A stationary polar coordinate system is employed in which matter rotates rigidly. The geodesic equations are solved. Caustics and global rotation are discussed.     

Equivalent rotations associated with the permutation inversion group revisited: symmetry projection of the rovibrational functions of methane

In this work the analysis of the equivalent rotations from the permutation inversion group formalism is revisited. We emphasize that explicit knowledge of changes in the Euler angles are not required in order to determine the transformation that a given symmetry operation causes to the rotational functions when dealing with the permutation inversion group formalism. Indeed, matrix elements of the equivalent rotations are provided by a single Wigner's D ^(j)(R) function. Taking advantage of this, we propose a symmetry projection approach to build the rovibrational functions of methane. This approach focuses on the relevance of the isomorphism between permutations and equivalent rotations. In our method, symmetry adapted functions are obtained by simultaneous diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner's D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits us to obtain in a systematic fashion an orthonormal set of symmetry-projected functions, with good total angular momentum, and carrying the irreducible representations of the molecular symmetry group.     

Mass and stability of rigidly rotating relativistic polytropes by energy method

The energy of a rigidly rotating star is written in the first and second post-Newtonian approximation. Conditions for equilibrium and stability are derived, as well as evolutionary paths for stars shedding angular momentum or mass. For polytropic index n > 1, these analytic results agree with exact numerical results to within a few percent as long as the general relativity index (P/c²)_c < 0.1. The energy method is applied to low mass white dwarfs, to semirelativistic neutron stars and to supermassive stars.     

Permutation-inversion symmetry of C70 fullerite in the high-temperature phase

The permutation-inversion symmetry group of C₇₀ fullerite in its high-temperature phase is constructed with allowance for the rotation of its constituent molecules, and the local symmetry group of a rotating molecule in the crystal is identified. Irreducible representations of these groups are constructed that are compatible with the principle of wave-function symmetry with respect to permutations of identical nuclei. A group-theoretic classification is made of the quantum states of a rotating molecule and of the crystal in the high-temperature phase of C₇₀ fullerite. Selection rules are derived for electronic, vibrational, and rotational spectra in terms of irreducible representations of the permutation-inversion symmetry group of the crystal. Fiz. Tverd. Tela (St. Petersburg) 39, 1895–1901 (October 1997)     

Vibrational and rotational excitation effects of the N(2D) + D2(X1Σg +) → ND(X3Σ+) + D(2S) reaction

The effects of the rovibrational excitation of reactants in the N(²D) + D₂(X¹Σ_g⁺) → ND(X³Σ⁺) + D(²S) reaction are calculated in a collision energy range from the threshold to 1.0 eV using the time-dependent wave packet approach and a second-order split operator. The reaction probability, integral cross-section, differential cross-section and rate constant of the title reaction are calculated. The integral cross-section and rate constant of the initial states v = 0, j = 0, 1, are in good agreement with experimental data available in the literature. The rotational excitation of the D₂ molecule has little effect on reaction probability, integral cross-section and the rate constant, but it increased the sideways and forward scattering signals. The vibrational excitation of the D₂ molecule reduced the threshold and broke up the forward–backward symmetry of the differential cross-section; it also increased the forward scattering signals. This may be because the vibrational excitation of the D₂ molecule reduced the lifetime of the intermediate complex.     

Trigonally-compressed octahedral geometry of hexaamminecobalt(III) complex cation in aqueous solution investigated from the electronic spectra

Electronic spectra of hexaamminecobalt(III) complex cation in aqueous solution were analyzed to obtain spectral components. Subsequently, based on the spectral components, the coordination geometry around the cobalt(III) ion was investigated, using the reverse angular overlap model method. The result indicates that the geometry is a trigonally compressed octahedron with the polar angle of 57.9?±?1.0° under D_3d symmetry, where the polar angle is the angle between the trigonal axis and the Co–N bond. From this angle, the top and side N–Co–N bond angles are calculated as 94.4° and 85.6°, respectively. The density functional theory computation supported this trigonally compressed structure in aqueous solution.     

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.     

Rotational quantum friction in superfluids: Radiation from object rotating in superfluid vacuum

The friction experienced by a body rotating in a superfluid liquid at T=0 is discussed. The effect is analogous to the amplification of electromagnetic radiation and spontaneous emission by a body or black hole rotating in the quantum vacuum, first discussed by Zel’dovich and Starobinsky. The friction is caused by the interaction of the part of the liquid which is rigidly connected with the rotating body and thus represents a comoving detector, with the “Minkowski” superfluid vacuum outside the body. The emission process is the quantum tunneling of quasiparticles from the detector to the ergoregion, where the energy of quasiparticles is negative in the rotating frame. This quantum rotational friction caused by the emission of quasiparticles is estimated for phonons and rotons in superfluid ⁴He and for Bogoliubov fermions in superfluid ³He. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 4, 257–262 (25 February 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

FTIR spectrum of vinyl fluoride near 3.6 μm: rovibrational analysis of the ν4+ν7 band and modelling Coriolis resonances in a seven-level polyad

ABSTRACT

The Fourier transform infrared (FTIR) spectrum of vinyl fluoride, H₂C=CHF, has been widely investigated in the region of the ν₄+ν₇ combination band around 2800 cm^?1 at a resolution of 0.005 cm^?1. This vibration of A' symmetry gives rise to an a/b-hybrid band with a predominant a-type component. The rovibrational structure is strongly perturbed and the analysis has been rather complicated since this combination band is involved at least in a seven-level interacting polyad, including the ν₈+2ν₁₀, 2ν₈+ν₁₀, 2ν₇+ν₉, ν₇+ν₈+ν₁₂, ν₅+ν₉+ν₁₀ and ν₇+ν₁₀+ν₁₂ vibrational states. The study has been further complicated by the absence of transitions coming from the perturbers that were considered as dark states. The spectral analysis resulted in the identification of 936 transitions with J" ≤ 46 and K_a" ≤ 11, all belonging to the a-type component. Most of the assigned data have been fitted using the Watson's A-reduction Hamiltonian in the I^r representation and proper Coriolis perturbation operators. The model employed includes seven different resonances within a complex polyad resonant system and a set of spectroscopic constants for the ν₄+ν₇ combination band, for the dark states, and Coriolis coupling coefficients have been determined.     

用代数方法精确研究HF分子B1Σ的振转能谱

基于代数方法(algebraic method)可以获得双原子分子的包含最高振动能级在内的所有高阶振动能级的精确数值这一事实,又提出了一种新的代数方法(algebraic method 2)来精确研究双原子分子电子态的振转能谱和转动常数. 以HF分子的B¹Σ电子态为例,应用algebraic method 2方法研究了该电子态的振动量子数从ν=0到ν=10的所有振转能谱,获得了与实验值符合得非常好的理论结果. 关键词： 代数方法 双原子分子 振转能级     

A refined potential energy surface and the rovibrational states for the electronic ground state of ozone

A potential energy surface for the electronic ground state of ozone has been optimized by using a variational procedure with the exact vibrational Hamiltonian in bond length-bond angle coordinates. In the optimization, the ab initio force field of Borowski, P., Andersson, K., Malmquist, P.-A., and Roos, B. O., 1992, J. chem. Phys., 97, 5568 is taken as the starting point, and the recent observed vibrational band origins up to 4900 cm^-1 reported by Floud, J.-M., Barbe, A., Camy-Peyret, C., and Plateaux, J. J., 1996, J. molec. Spectrosc., 177, 34 are involved. The root mean square error of this fit for the 39 observed vibrational energy levels is 0.83 cm^-1. In order to test the refined potential, the rovibrational energy levels up to J = 15 are calculated and compared with the observed values.     

Entanglement of a Single Spin-1 Object: An Example of Ubiquitous Entanglement

Using a single spin-1 object as an example, we discuss a recent approach to quantum entanglement. [A.A. Klyachko and A.S. Shumovsky, J. Phys: Conf. Series 36, 87 (2006), E-print quant-ph/0512213]. The key idea of the approach consists in presetting of basic observables in the very definition of quantum system. Specification of basic observables defines the dynamic symmetry of the system. Entangled states of the system are then interpreted as states with maximal amount of uncertainty of all basic observables. The approach gives purely physical picture of entanglement. In particular, it separates principle physical properties of entanglement from inessential. Within the model example under consideration, we show relativity of entanglement with respect to dynamic symmetry and argue existence of single-particle entanglement. A number of physical examples are considered.      

Specific heat of the non-centrosymmetric superconductor Re3W

The alloys of non-centrosymmetric superconductor, Re 3 W, which were reported to have an α-Mn structure [P. Greenfield and P. A. Beck, J. Metals, N. Y. 8, 265 (1959)] with T c = 9 K, are prepared by arc melting. The values of ac susceptibility and the low-temperature specific heat of these alloys are measured. It is found that there are two superconducting phases coexisting in the samples with T c1 ≈ 9 K and T c2 ≈ 7 K, which are both non-centrosymmetric in structure as reported previously. By analysing the specific heat data measured in various magnetic fields down to a temperature of 1.8 K, we find that the absence of the inversion symmetry does not lead to an obvious deviation from an s-wave pairing symmetry in Re 3 W.     

On Relativistic Mass Spectra of a Two-Particle System

Abstract

A relativistic two-particle system with time-asymmetric scalar and vector interactions in the two-dimensional space-time is considered within the frame of the front form of dynamics using the dynamical symmetry approach. The mass-shell equation may be represented in terms of the nonlinear canonical realization of the Lie algebra of the group SO(2, 1). This allows us to quantize the system and to obtain a closed form for the mass spectrum.     

Vibrational spectroscopy of tolane; Coriolis coupling between Raman-active modes of g symmetry

ABSTRACT

Vibrational spectroscopy of tolane (diphenylacetylene), which has 66 normal modes, has been advanced. Anharmonic wavenumber predictions were made with the quartic potential energy surface obtained with B3LYP/cc-pVTZ model and the second-order perturbation theory (VPT2). Infrared (IR) intensity and Raman activities were computed at the harmonic level. The IR spectrum of the crystal and Raman spectra of the liquid and the crystal tolane were newly recorded. The lingering problem of an excess of polarised Raman bands at wavenumbers appropriate for fundamentals, other than a_g modes, has now been attributed to Coriolis coupling within modes of g symmetry species. Consequently, D_2h point symmetry group has been confirmed for a planar tolane molecule. Assignments for almost all fundamentals of tolane are now secure. The assignment for ν₃₂ remains questionable. Remaining unassigned fundamentals are: ν₃₄ and ν₃₅, which, as a_u symmetry species, are IR- and Raman-inactive transitions, and ν₅₉(b_2u), which is predicted to have a very low wavenumber.     

The symmetry properties and collinearity of the magnetogyric,hyperfine splitting and magnetic susceptibility tensors

The symmetry properties of g and A cannot be elicited using invariance properties, due to their special nature, and are found using the properties of a Kramers doublet. It is shown that the collinearity of g and A depends on the molecule having at least C ₂ with σ _v or C ₂′ symmetry elements. The symmetry properties of x are found from those of g. The conditions for collinearity of the tensors and the consequences of non-collinearity are discussed, with examples.     

Symmetry ascent eigenvectors in finite point groups and an expanded matrix element selection rule

A formalism is proposed in which wave-functions quantized in a finite point group may be unambiguously labelled by their relative behaviour under the mapping of individual components from the generic point group R ₃. Such a mapping is only possible into highly symmetric finite groups including O _h and D _6h . Mapping to lower point groups by conventional symmetry descent creates ambiguities which can be removed by retaining the effect of discriminating virtual operators as parity labels for components. With such labelled wave functions, the formation of unambiguous direct products is possible with the introduction of Symmetry Ascent V Coefficients. By quantizing the wave functions about the desired n-fold axis in complex space, a commutative set of components is obtained. This allows component combination rules similar to those for 3-j symbols to be stated, modified to accommodate the possible mappings in finite groups and to retain the effect of parity. Hamiltonian operators in complex tensor form are treated similarly. The spin-orbit matrix elements for finite groups can thus be written in fully labelled form which when expanded as a scalar product of elements reflects all of the relevant selection rules pertaining to both the representations and components. With the violation of any one such rule, the matrix element vanishes. The electronic symmetry of these systems therefore is higher than that implied by the molecular geometry. The formalism further implies that during descent in symmetry, the number of selection rules for matrix elements can only increase.     

Rovibrational laser jet-cooled spectroscopy of the NH3–Ar complex in the ν2 umbrella region of NH3: comparison between new infrared data and an ab initio calculated spectrum*

ABSTRACT

Five ortho and para bands of the ν₂ umbrella mode of the NH₃–Ar van der Waals complex have been recorded at high resolution using jet-cooled infrared laser spectroscopy. A rovibrational analysis provides accurate band centres and upper state rotational constants for the Π_s(j?=?1,k?=?0)?←?Σ_a(j,k?=?0) and Σ_s(j?=?1,k?=?0)?←?Σ_a(j,k?=?0) ortho bands. The puzzling para bands observed in the region of the lower and upper components of the inversion splitting doublet have been assigned by comparison with rovibrational and tunnelling levels and transitions calculated ab initio. The latter calculations are based on the four-dimensional potential energy surface reported by Loreau et al. [J. Chem. Phys. 141, 224303 (2014)], which takes explicitly into account the umbrella motion of the ammonia molecule. The very good agreement found between Π_s/a,lower(j?=?1,k?=?1)?←?Σ_a(j?=?1,k?=?1) and Π_s/a,upper(j?=?1,k?=?1)?←?Σ_s(j?=?1,k?=?1) experimental and calculated transitions has been exploited to determine precisely two different inversion splittings in the ν₂ state (32.003(1) and 36.008(1)?cm^?1) from extrapolated Q(0) line frequencies and to obtain a qualitative picture of Coriolis couplings present in both the ν₂?=?0 and ν₂?=?1 states.