Spherical Tensor Formalism Research Articles

Algebraic solutions for [formula omitted] quantum phase transitions in the proton-neutron interacting boson model

A simple systematic procedure to construct the proton-neutron unitary, usdπν(12), orthogonal, osdπν(12), and quasi-spin susdπν(1,1) algebras of the sd bosonic system is presented. New algebraic substructures of these algebras are discussed and the explicit formulae for their generators and Casimir operators are given in the spherical tensor form. The complementarity relationship of the Casimir operators of the susdπν(1,1) and osdπν(12) is derived. The exact algebraic solutions of the quantum phase transition Hamiltonian between the osdπν(12) and usπν(2)⊗udπν(10) limits have been considered, for the first time, in the framework of affine susdπν(1,1) Lie algebra. The low lying energy spectra of the 70Ge, 76−78Se, 96−98Mo, and 100−102Ru isotopes are calculated using the osdπν(12)↔usπν(2)⊗udπν(10) transition Hamiltonian. The good agreement of our computation with empirical result in these isotopes emphasizes the importance of usπν(2)⊗udπν(10) limit. With this addition, symmetry can be extended to many nuclei.

Thomson Scattering above Solar Active Regions and an Ad Hoc Polarization Correction Method for the Emissive Corona

Thomson-scattered photospheric light is the dominant constituent of the lower solar corona’s spectral continuum viewed off-limb at optical wavelengths. Known as the K-corona, it is also linearly polarized. We investigate the possibility of using the a priori polarized characteristics of the K-corona, together with polarized emission lines, to measure and correct instrument-induced polarized crosstalk. First we derive the Stokes parameters of the Thomson scattering of unpolarized light in an irreducible spherical tensor formalism. This allows forward synthesis of the Thomson-scattered signal for the more complex scenario that includes symmetry-breaking features in the incident radiation field, which could limit the accuracy of our proposed technique. For this, we make use of an advanced 3D radiative magnetohydrodynamic coronal model. Together with synthesized polarized signals in the Fe xiii 10746 Å emission line, we find that an ad hoc correction of telescope- and instrument-induced polarization crosstalk is possible under the assumption of a nondepolarizing optical system.

Generalised descriptions on orientational order parameters and mean field theory for uniaxial and biaxial nematic liquid crystals

ABSTRACTWe present a complete set of the orientational and pseudo-vector (or polar) order parameters and a general potential of mean torque responsible for the formation of a lowest symmetry of triclinic biaxial nematic phases. By adopting the irreducible spherical tensor formalism, twenty-five orientational and nine pseudo-vector (or polar) order parameters are explicitly expressed in terms of both the Saupe ordering supermatrix and the Wigner rotation matrices. For various biaxial nematic phases and molecules, the corresponding independent order parameters are uniquely classified, including the orthorhombic monoclinic , and triclinic symmetries as well as the uniaxial nematic phase. These works could also describe the order parameters for the smectic C phases and the polar biaxial phases composed of symmetry.

ChemInform Abstract: Tensors and Rotations in NMR

AbstractReview: 26 refs.

Concept of effective Hamiltonians for transitions in multi-level systems

Employing the concept of effective Hamiltonians, an analytical theory is introduced to describe transitions in a multi-level system in nuclear magnetic resonance (NMR) spectroscopy. Specifically, the discussion is centered towards the treatment of selective and non-selective excitations in static quadrupolar spin (I > 1/2) systems. To this end, effective radiofrequency (RF) Hamiltonians based on the spherical tensor formalism are proposed for describing transitions in both integral (I = 1, 2 and 3) and half-integral (I = 3/2, 5/2 and 7/2) quadrupolar spins. The optimum conditions desired for selective excitation in a multi-level system are derived pedagogically from first principles and presented through analytical expressions. Employing suitable model systems, the derived optimum conditions are substantiated through rigorous numerical simulations based on the spherical tensor formalism. The theory presented provides a framework for describing selective and non-selective RF pulses and could improve our understanding of multiple-pulse experiments involving quadrupolar nuclei.

HANLE EFFECT FOR STELLAR DIPOLES AND QUADRUPOLES

We derive exact expressions for the degree of lineal polarization over a resolved or integrated stellar disc due to resonance scattering and the Hanle effect from a dipolar or quadrupolar distribution of magnetic fields. We apply the theory of scattering polarization within the formalism of the spherical tensors representation for the density matrix and radiation field. The distribution of linear polarization over the stellar disk for different configurations of the magnetic field is studied and its topology discussed. For an unresolved dipole, the resulting polarization can be expressed in terms of just three functions (of the inclination angle and effective dipole strength), that are calculated numerically and their behaviour discussed. Dipolar and (aligned) quadrupoles are considered in some detail, but the techniques here ---in particular, the extensive use of the spherical tensor formalism for polarization---, can easily be applied to more general field configurations.

Tensors and rotations in NMR

AbstractThe transformation of second‐rank Cartesian tensors under rotation plays a fundamental role in the theoretical description of nuclear magnetic resonance experiments, providing the framework for describing anisotropic phenomena such as single crystal rotation patterns, tensor powder patterns, sideband intensities under magic‐angle sample spinning, and as input for relaxation theory. Here, two equivalent procedures for effecting this transformation—direct rotation in Cartesian space and the decomposition of the Cartesian tensor into irreducible spherical tensors that rotate in subgroups of rank 0, 1, and 2—are reviewed. In a departure from the standard formulation, the explicit use of the spherical tensor basis for the decomposition of a spatial Cartesian tensor is introduced, helping to delineate the rotational properties of the basis states from those of the matrix elements. The result is a uniform approach to the rotation of a physical system and the corresponding transformation of the spatial components of the NMR Hamiltonian, expressed as either Cartesian or spherical tensors. This clears up an apparent inconsistency in the NMR literature, where the rotation of a spatial tensor in spherical tensor form has typically been partnered with the inverse rotation in Cartesian form to produce equivalent transformations. © 2011 Wiley Periodicals, Inc. Concepts Magn Reson Part A 38: 221–235, 2011.

Second‐order collision‐induced light scattering: a spherical tensor approach

AbstractCollision‐induced light scattering (CILS) is a rotational–translational Raman scattering occurring in pairs or larger clusters of interacting molecules. A general explicit expression for the dipole correlation function relevant to second‐order CILS is derived in the spherical tensor formalism. A Mathematica program is given to calculate this autocorrelation function and hence the scattered intensity and depolarization ratio. Some examples are given for interacting atoms and tetrahedral molecules. These developments will aid in the interpretation of the weak, high‐frequency wings of induced spectra. Copyright © 2011 John Wiley & Sons, Ltd.

Vector correlations in photodissociation of polarized polyatomic molecules beyond the axial recoil limit

We present the full quantum mechanical theory of the angular momentum distributions of photofragments produced in photolysis of oriented/aligned parent polyatomic molecules beyond the axial recoil limit. This paper generalizes the results of Underwood and Powis(28,29) to the case of non-axial recoil photodissociation of an arbitrary polyatomic molecule. The spherical tensor approach is used throughout this paper. We show that the recoil angular distribution of the angular momentum polarization of each of the photofragments can be presented in a universal spherical tensor form valid for photolysis in diatomic or polyatomic molecules, irrespective of the reaction mechanism. The angular distribution can be written as an expansion over the Wigner D-functions in terms of the set of the anisotropy-transforming coefficients c(K(i)q(i))(K) (k(d), K(0)) which contain all of the information about the photodissociation dynamics and can be either determined from experiment, or computed from quantum mechanical theory. An important new conservation rule is revealed through the analysis, namely that the component q(i) of the initial reagent polarization rank K(i) and the photofragment polarization rank K onto the photofragment recoil direction k is preserved in any photolysis reaction. Both laboratory and body frame expressions for the recoil angle dependence of the photofragment angular momentum polarization are presented. The parent molecule polarization is shown to lead to new terms in the obtained photofragment angular distributions compared with the isotropic case. In particular, the terms with |q(i)| > 2 can appear which are shown to manifest angular momentum helicity non-conservation in the reaction. The expressions for the coefficients c(K(i)q(i))(K) (k(d), K(0)) have been simplified using the quasiclassical approximation in the high-J limit which allows for introducing the dynamical functions and the rotation factors which describe the decreasing of the photofragment angular momentum orientation and alignment due to the rotation of the molecular axis during photodissociation. In this case, the resultant recoil angle dependence is also presented in a form where the anisotropy of the parent molecular ensemble is expressed in terms of the molecular axis distribution, rather than in terms of the molecular density matrix.

Collision-induced absorption byCO2in the far infrared: Analysis of leading-order moments and interpretation of the experiment

The diagrammatic theory, developed recently by the authors [Phys. Rev. A 74, 012732 (2006)], is applied to binary collision-induced properties, with emphasis on induced dipole moments. Assuming rototranslational dynamics to be classical and using irreducible spherical tensor formalism, exact analytical formulas are worked out for the two leading order spectral moments of a collision-induced band by two interacting linear molecules. The formulas are applied to the far infrared absorption by ${\text{CO}}_{2}{\text{-CO}}_{2}$, and permit interpretation of the experiment. This study provides evidence of the adequacy of the electrostatic induction mechanism, provided that hitherto missing vibrational terms of static polarizability are considered.

Intermolecular Raman spectroscopy of long-range interactions: TheCO2−Arcollision-inducedν3CO2band

We report an exhaustive joint theoretical/experimental collision-induced Raman scattering (CIRS) study of the Raman-forbidden ${\ensuremath{\nu}}_{3}$ band of $\mathrm{C}{\mathrm{O}}_{2}$. Original zeroth and second anisotropic spectral moment formulas for $\mathrm{C}{\mathrm{O}}_{2}\text{\ensuremath{-}}\mathrm{Ar}$ are derived, in which complete expressions of Raman amplitudes (derived by the authors with a recently reported universal method [Phys. Rev. A 74, 012723 (2006)]) are input. The method, applicable to any kind of spectroscopy and whatever the number of photons, molecules, or interaction involved, uses the irreducible spherical tensor formalism in conjunction with a Feynman-like diagrammatic technique to describe any long-range induced property mechanism. Experimentally, spectral moments are deduced from careful, absolute-scale, frequency-resolved $\mathrm{C}{\mathrm{O}}_{2}\text{\ensuremath{-}}\mathrm{Ar}$ depolarized CIRS measurements of unprecedented accuracy. From comparison between theory and experiment, we provide quantitative evidence of a substantial contribution of a nonlinear dipole polarization mechanism, predicted theoretically in the preceding paper. In this mechanism, both photons are shown to interact with Ar (which then couples to $\mathrm{C}{\mathrm{O}}_{2}$ via intermolecular interactions), rather than with both colliders that is the case in the standard dipole-induced quadrupole (DIQ) interaction. The effect had thus far escaped notice possibly because of the reduced accuracy of the earlier self-consistent field dipole-quadrupole polarizablity computations along with a lack of CIRS measurements. In light of recent extensive computations by Haskopoulos and Maroulis [Chem. Phys. Lett. 417, 235 (2006)], the improved ab initio data of these properties are found to corroborate our predictions, and confirm that the Raman amplitude owing to the dipole-dipole-quadrupole hyperpolarizability of the atomic perturber strongly and destructively interferes with the (otherwise dominant) DIQ Raman amplitude.

The spherical tensor formalism applied to relaxation in magnetic resonance

Spherical tensor operators are used to simplify the relaxation terms of the Redfield equation. A method for treating relaxation is developed using trace-normalized spherical tensor operators. The Redfield equation is recast as a set of coupled linear differential equations for the expectation values of a complete set of spherical tensor operators. A single spin-1/2 particle is treated, and the correspondence of the equation of motion with the Bloch equation is demonstrated. The advantage of the spherical tensor approach is that both the coherent and relaxation terms of the equation of motion are written as matrices that operate on a common vector consisting of the expectation values of spin variables. The process used to construct the equation of motion helps to organize the various contributions to spin relaxation and can be applied uniformly regardless of spin system size. As an example, the relaxation of a two-spin system by the chemical shift anisotropy and dipolar Hamiltonians is developed using the spherical tensor formalism. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 270–290, 2006

The prediction of energies and geometries of hydrogen bonded DNA base-pairs via a topological electrostatic potentialElectronic supplementary information (ESI) available: Comparison of the BSSE corrected interaction energies, geometrical parameters and total interaction energies of the 27 base-pair series. See http://www.rsc.org/suppdata/cp/b2/b204485d/

We introduce an anisotropic model of electrostatic interaction based on topological atoms obtained from the gradient vector field partitioning of the electron density. High-order electrostatic moments are computed within the compact spherical tensor formalism from ab initio wave functions of rigid geometry-optimized monomers. When combined with a simple hard-sphere or Lennard-Jones potential this topological electrostatic interaction potential correctly predicts the geometries of 27 DNA base-pairs. With the Lennard-Jones(LJ) potential internuclear distances of frontier atoms between base-pairs differ only by 0.08 Å compared to supermolecular calculations at B3LYP/6-31G(d,p) level. The discrepancy for angles involved in hydrogen bond contacts amounts to only 3.5°. The topological model globally reproduces the correct ranking in intermolecular interaction energy (6.5 ± 6.5 kJ mol−1). Subsequently we compare the interaction energy profiles of the topological model with distributed multipole analysis, Merz–Kollman charges and the natural population analysis at the B3LYP/6-311+G(2d,p) level. The convergence of the topological multipole expansion is somewhat more favorable than that based on DMA but the two models have similar basis set dependence. This work encourages the development of a topological intermolecular potential, which already predicts a reliable geometry of a DNA tetrad in its present form.

Polarization effects in quantum coherences probed by two-color, resonant four-wave mixing in the time domain

We present a combined theoretical and experimental study of the effects of laser polarization on optical coherences produced in two-color, resonant four-wave mixing (TC-RFWM). A time-dependent model incorporating diagrammatic perturbation theory and spherical tensor formalism is used to interpret observations of quantum beats due to molecular hyperfine structure in time-resolved TC-RFWM in nitric oxide. Good agreement is found between the model and the observed time-resolved signals for two distinct excitation schemes and a variety of polarization configurations including both polarization and population gratings. Measured hyperfine energy intervals are reported for the $X{}^{2}{\ensuremath{\Pi}}_{1/2},$ $v=0$ ground state and the $A{}^{2}{\ensuremath{\Sigma}}^{+},$ $v=0$ excited state of NO. The experimental results demonstrate that TC-RFWM can be used to perform state-selective, quantum beat spectroscopy in three-level systems by suitably designing three experimental features: the excitation scheme for the matter-field interaction, the time ordering of the laser pulses, and the polarization of the incident laser beams.

The Lineshape of Saturation-Broadened Transitions in the Gas Phase

The equations of motion for the density matrix are manipulated in spherical tensor form to derive formulas for the lineshapes of saturation-broadened spectroscopic transitions for gas-phase molecules. The formulas, which include consideration of the spatial degeneracy, are given for both plane-polarized and circularly polarized radiation and are in the form of weighted sums of the reciprocals of the eigenvalues of a matrix. In most practical cases, the eigenvalues will be in the form of Lorentz denominators, which expresses the lineshape as a sum of saturation-broadened Lorentz lineshapes. The approximations required to make the lineshapes equal to a sum of Karplus–Schwinger saturation-broadened lineshapes are described. The lineshape formula for plane-polarized radiation is equivalent to the continued-fraction form previously reported (R. H. Schwendeman, J. Chem. Phys. 73, 4838–4847 (1980)).

State-selective quantum beat spectroscopy via coherent control of Liouville-pathway interference in two-colour resonant four-wave mixing

Diagrammatic perturbation theory and the spherical tensor formalism are used to model the signal observed in time-resolved, two-colour resonant four-wave mixing (TC-RFWM). An analysis using the Liouville formalism illustrates how TC-RFWM can be used to perform state-selective quantum beat spectroscopy in three-level systems by suitably designing three experimental features: the excitation scheme for the matter-field interaction, the time ordering of the laser pulses and the polarization of the incident laser beams. In contrast to traditional fluorescence quantum beat spectroscopy, time-resolved TC-RFWM results in a background-free, coherent signal that allows for high-resolution detection of the spectrally unresolved energy structure within a selected material state of the three-level system. The beats are shown to result from the quantum interference between different Liouville-space pathways in the laser-induced evolution of the material density operator and appear as a temporal modulation in the amplitude of the light beam emerging as the output of the mixing process. This type of control of a coherent optical field, based on the selective preparation of superposition states, is reminiscent of other recently developed optical techniques such as coherent population trapping and lasing without inversion.

Degenerate four-wave mixing spectroscopy as a probe of orientation and alignment in molecular systems

Degenerate four-wave mixing (DFWM) spectroscopy is shown to provide a facile means for probing angular momentum (or rotational) anisotropy in nonequilibrated ensembles of gaseous molecules, with judicious selection of experimental conditions permitting quantitative determination of population distributions and Zeeman coherences for magnetic sublevels of the target species. A theoretical description of the nonlinear response induced under such circumstances is obtained by incorporating a state multipole expansion of the zero-order density operator into a perturbative (weak-field) treatment for the DFWM interaction. Aside from allowing the effects of incident field polarizations and phase-matching geometries to be considered in detail, this compact spherical tensor formalism provides guidelines for the extraction of spatial information from rovibronically resolved spectral data. Furthermore, these analyses have identified unusual polarization schemes that lead to signal generation only in the presence of rotational anisotropy, thereby suggesting a new class of four-wave mixing measurements that permit the selective detection of molecular orientation and alignment.

Time-resolved two-photon induced anisotropy decay: The rotational diffusion regime

Two-photon excitation (TPE) of randomly oriented chromophores in solution generates an anisotropic distribution. In a previous paper [Chem. Phys. 179, 513 (1994)], the polarization dependence of the TPE signal probed by a secondary spectroscopic transition (fluorescence or transient absorption) was determined. In this paper, the time dependence of anisotropic two-photon induced fluorescence or transient absorption signals due to rotational diffusion is treated in spherical tensor formalism. The two-photon signal in general contains isotropic (orientation independent) and anisotropic (orientation dependent) contributions. The latter decay with up to five exponential components. Four time-dependent anisotropy parameters can be defined and measured, allowing additional information, not available in conventional one-photon fluorescence depolarization measurements, to be determined. The special case of one-color TPE is discussed in particular. It is shown that by measurement of the linear and circular anisotropies r1(t) and r2(t), more than one rotational correlation time can be determined in many cases, providing information on rotational diffusion parameters not readily determined by analogous one-photon methods and leading in some cases to resolution of rotational motion about the principal diffusion axes of the molecule.

Overtone and hot-band assignments in the 2{ν3, ν6} levels of CHD3

Assignments have been carried out for the 2ν3/(ν3+ν6)/2ν6 (triad)←ground state overtone bands and the triad←ν3/ν6 hot-band system of CHD3. A vibrational extrapolation method, using correspondences between a spherical-tensor formalism and the more conventional description of symmetric-rotor spectra, was used to facilitate the assignment and analysis of these bands. The resulting assignments have been confirmed by time-resolved double-resonance measurements.

Time-resolved anisotropic two-photon spectroscopy

Two-photon excitation (TPE) in a randomly oriented liquid sample generates an anisotropic distribution of excitations that can be probed by a secondary spectroscopic transition, e.g. fluorescence or transient absorption. The orientation dependence of the secondary transition dipole (and therefore of the fluorescence or transient-absorption signal) can be exploited to generate information on molecular symmetry and orientation that is not available in two-photon absorption alone. A theoretical formalism is developed here for the orientationally averaged two-photon absorptivity taking into account the orientation of the secondary transition dipole. Spherical tensor formalism is employed to distinguish isotropic and anisotropic components of the two-photon absorptivity, and several two-photon anisotropy parameters are defined. The theory describes time-resolved detection of TPE by transient-absorption or fluorescence decay measurements. Time-resolved measurements with time-resolution that is fast relative to the rotational correlation times are shown to provide new information, including the two-photon anisotropies. Applications to TPE fluorescence anisotropy measurements and TPE induced anisotropic transient absorption are described, and illustrated by experimental measurements on bacteriorhodopsin.