Irreducible Tensor Operators Research Articles

Preliminary modeling of CH3D from 4000 to 4550 cm−1

A new study of 12CH3D line positions and intensities was performed for the upper portion of the Enneadecad polyad between 4000 and 4550cm−1. For this, FTIR spectra were recorded with D-enriched methane samples (at 80K with a Bruker 125 IFS at 0.005cm−1 resolution and at 291K with the McMath-Pierce FTS at 0.011cm−1 resolution, respectively). Line positions and intensities were retrieved by least square curve-fitting procedures and analyzed using the effective Hamiltonian and the effective dipole moment expressed in terms of irreducible tensor operators adapted to symmetric top molecules. Initially, only the cold spectrum was used to identify quantum assignments and predict 12CH3D relative intensities in this region. To assign higher quanta up to J equal 14, additional line positions and intensities were obtained from two room temperature spectra. In the final stage, measured intensities from both the cold and the room temperature data were normalized to corresponding values at 296K and then averaged. Combining the two temperature datasets confirmed the assumed quantum assignments and also demonstrated the relative accuracies to be better than ±0.0002cm−1 for line positions and at least ±6% for intensities so that ∼1160 features were selected. Including additional assignments from the room temperature spectra alone permitted 1362 line intensities of 11 bands (involving 23 vibrational symmetry components) to be reproduced with an RMS of 9%. Over 4085 selected positions for 12 bands were modeled to 0.008cm−1. Nevertheless a number of known assignments could not be modeled to within our experimental precisions. More work is needed to obtain a complete characterization of this complex polyad.

Complete nuclear motion Hamiltonian in the irreducible normal mode tensor operator formalism for the methane molecule

A rovibrational model based on the normal-mode complete nuclear Hamiltonian is applied to methane using our recent potential energy surface [A. V. Nikitin, M. Rey, and Vl. G. Tyuterev, Chem. Phys. Lett. 501, 179 (2011)]. The kinetic energy operator and the potential energy function are expanded in power series to which a new truncation-reduction technique is applied. The vibration-rotation Hamiltonian is transformed systematically to a full symmetrized form using irreducible tensor operators. Each term of the Hamiltonian expansion can be thus cast in the tensor form whatever the order of the development. This allows to take full advantage of the symmetry properties for doubly and triply degenerate vibrations and vibration-rotation states. We apply this model to variational computations of energy levels for (12)CH(4), (13)CH(4), and (12)CD(4).

Vector correlations in the F + HD reaction

A theoretical study of the orientation of product rotational angular momenta for two chemical reaction channels: F + HD(ν r = 0, j r = 0) → HF(ν, j) + D and F + HD(ν r = 0, j r = 0) → DF(ν, j) + H at a E coll = 78.54 meV collision energy was performed. Angular momentum orientation was described on the basis of irreducible tensor operators (state multipoles) expressed through anisotropy transfer coefficients, which contained quantum-mechanical scattering T matrices determined on the basis of exact solutions to quantum scattering equations obtained using the hyperquantization algorithm. The possibility of the existence of substantial orientation of the angular momentum of reaction products j in the direction perpendicular to the scattering plane was demonstrated. The dependences of differential reaction cross sections and state multi-poles on the ν and j quantum numbers were calculated and analyzed. A experimental scheme based on the multiphoton ionization method was suggested. The scheme can be used to detect predicted reaction product angular momentum orientation.

A unified calculation of the optical spectral band positions and electron paramagnetic resonance spectral data for Yb 3+ in InP semiconductor

The complete energy matrix for a 4 f 13 ion in cubic crystals and under an external magnetic field is established by means of the irreducible tensor operator and/or equivalent operator methods. By diagonalizing the energy matrix, four optical spectrum band positions and three spin-Hamiltonian parameters [ g factor and hyperfine structure constants A( 171Yb 3+) and A( 173Yb 3+)] for Yb 3+ in InP semiconductor are calculated together. The calculated results are in good agreement with the experimental values. The signs of hyperfine structure constants A( 171Yb 3+) and A( 173Yb 3+) are determined from the calculations. The results are discussed.

One-Dimensional Integrable Spinor BECs Mapped to Matrix Nonlinear Schrödinger Equation and Solution of Bogoliubov Equation in These Systems

In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schr\"odinger equation, and solve the Bogoliubov equation of these systems. A map of spin-$n$ BEC is constructed from the $2^n$-dimensional spinor representation of irreducible tensor operators of $so(2n+1)$. Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.

Parallel implementation of the MAGPACK package for the analysis of high-nuclearity spin clusters

Molecular clusters are formed by a finite number of exchange-coupled paramagnetic centers and they are model systems between molecules and extended solids. In order to simulate their properties and extrapolate to solids, the size of the systems to be treated should be as large as possible. In this context, the use of efficient parallel codes is essential. We present the parallel programs ParAni and ParIso, for anisotropic and isotropic models, that enable the calculation of large energy matrices in parallel and the subsequent computation of the relevant spectral information. The evaluation of the matrix elements is based on the serial package Magpack that uses the irreducible tensor operators technique and takes into account all kinds of anisotropic and isotropic magnetic interactions. To obtain the eigenvalues, the energy matrix is partially diagonalized by means of the SLEPc library. The calculation of eigenvalues and eigenvectors of these spin clusters enables us to evaluate the bulk magnetic properties (magnetic susceptibility and magnetization) as well as the spectroscopic properties (inelastic neutron scattering spectra). The results are encouraging in terms of parallel efficiency and open the way to address very challenging problems.

Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules

A theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators is presented for the first time. Each term of the Hamiltonian expansion can thus be cast in the tensor form in a systematic way using the formalism of ladder operators. Pyramidal XY3 molecules appear to be good candidates to validate this approach which allows taking advantage of the symmetry properties when doubly degenerate vibrational modes are considered. Examples of applications will be given for PH3 where variational calculations have been carried out from our recent potential energy surface [Nikitin et al., J. Chem. Phys. 130, 244312 (2009)].

The Transformation of Irreducible Tensor Operators Under Spherical Functions

The irreducible tensor operators and their tensor products employing Racah algebra are studied. Transformation procedure of the coordinate system operators act on are introduced. The rotation matrices and their parametrization by the spherical coordinates of vector in the fixed and rotated coordinate systems are determined. A new way of calculation of the irreducible coupled tensor product matrix elements is suggested. As an example, the proposed technique is applied for the matrix element construction for two electrons in a field of a fixed nucleus.

DIRAC: A new version of computer algebra tools for studying the properties and behavior of hydrogen-like ions

During recent years, the Dirac package has proved to be an efficient tool for studying the structural properties and dynamic behavior of hydrogen-like ions. Originally designed as a set of Maple procedures, this package provides interactive access to the wave and Green's functions in the non-relativistic and relativistic frameworks and supports analytical evaluation of a large number of radial integrals that are required for the construction of transition amplitudes and interaction cross sections. We provide here a new version of the Dirac program which is developed within the framework of Mathematica (version 6.0). This new version aims to cater to a wider community of researchers that use the Mathematica platform and to take advantage of the generally faster processing times therein. Moreover, the addition of new procedures, a more convenient and detailed help system, as well as source code revisions to overcome identified shortcomings should ensure expanded use of the new Dirac program over its predecessor.

Orbital‐dependent magnetic properties of molecular cluster containing high‐spin Co(II) ions

AbstractTo shed light on magnetic properties of the magnetic exchange molecular cluster containing the constituent metal ion with unquenched orbital angular momentum, as representative examples two octahedrally coordinated high‐spin Co(II)‐cluster complexes were synthesized and characterized: complex (1) [Co3(Hbzp)6][ClO4]2· 2CH3OH·H2O (bzp = 2‐benzoylpyridine) with a linear core Co(III)‐Co(II)‐Co(III) and complex (2) [Co4L(N3)4(CH3CN)4](ClO4)4·2H2O (L = Macrocyclic schiff base of tetra‐2,6‐pyridinedicarboxaldehyde‐tetra‐ethane‐1,2‐diamine) comprised by two nearly independent Co(II)‐dimers. Following Lines' isotropic exchange approximation the magnetic interaction effective Hamiltonian was introduced, which includes the exchange interactions, the spin‐orbit coupling, the low‐symmetry crystal field, and the intercluster exchange interaction. On the basis of the irreducible tensor operator approach to solve Hamiltonian we have coded the software, BJMAG package, and performed the best‐fit procedure for experimentally temperature variable magnetic susceptibilities for the complexes (1) and (2). The best‐fit parameters were obtained as follows: λ = −126 cm−1, Δ = 432 cm−1, κ = 0.96, mJ′ = 2.7 cm−1 for the complex (1) and λ = −165 cm−1, Δ = 960 cm−1, κ = 0.95, J′ = 9 cm−1, mJ′ = −0.1 cm−1 for the complex (2). To further elucidate the individual role of each interaction in the employed model, the effects of the key factors governing the magnetic properties on temperature variable magnetic susceptibilities were examined in detail. The significance of the intercluster exchange interactions at low temperatures was briefly discussed and a self‐consistent molecular field to calculate the thermally average spin of the nearest neighbor in the intercluster interaction was proposed. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Nonlinear nonstationary theory of light propagation through a resonance medium taking into account optical pumping of atoms

The general nonstationary theory of propagation of an elliptically polarized pulse resonantly interacting with the 1/2–1/2 transition is presented. The theory that takes into account relaxation processes is developed in terms of irreducible tensor operator representation allowing diagonalization of the relaxation matrix and introduction of effective decay times governing relaxation of both population and coherence between magnetic sublevels of the resonant system. It is shown that interference of magnetic sublevels in the field of polarized radiation that optically pumps the atoms can result in inversionless amplification of radiation.

Numerically exact and approximate determination of energy eigenvalues for antiferromagnetic molecules using irreducible tensor operators and general point-group symmetries

For small-enough quantum systems numerical exact and complete diagonalization of the Hamiltonian enables one to evaluate and discuss all static, dynamic, and thermodynamic properties. In this article we consider Heisenberg spin systems and extend the range of applicability of the exact diagonalization method by showing how the irreducible tensor operator technique can be combined with an unrestricted use of general point-group symmetries. We also present ideas on how to use spin-rotational and point-group symmetries in order to obtain approximate spectra.

Spin tomography and star-product kernel for qubits and qutrits

Using the irreducible tensor-operator technique, we establish the relation between different forms of spin tomograms. Quantizer and dequantizer operators are presented in simple explicit forms and are specified for the low-spin states. The kernel of the star-product is evaluated for qubits and qutrits and its connection with a generic formula is found.

Angular momentum coupling in NMR: a new view of rotation matrix products via Racah algebra

Choosing the rotation angle-rotation axis parameters instead of the conventional Euler angle parameters {α, β, γ} to parametrise rotations, and a spherical tensor operator basis, Happer was the first to propose a novel multipole operator expansion of the rotation operator [W. Happer, Ann. Phys. (N.Y.) 48, 579 (1968)]. As Happer pointed out, such an expansion in irreducible spherical tensor operators readily lends itself to the methods of Racah algebra, a feature we exploit to simplify the calculus of rotation matrix products. Working with a Clebsch–Gordan coefficient expansion [W. Happer, ibid. cit.; M.S. Marinov, Yad. Fiz. 5, 943 (1967)] of the corresponding rotation matrices , and closure relations for these (2J + 1)-dimensional irreducible representations of the rotation group, we state, and prove for the first time, simple recoupling coefficient expansions for rotation matrix products. With an eye towards applications in NMR, where multiple rotations in spin or coordinate space often require the efficient evaluation of such rotation matrix products, we focus on expansions defining the matrices representing the result of two or three consecutive rotations. We show that the coefficients of these expansions are defined by 6 − j symbols and bipolar spherical harmonics (two rotations), or by 9 − j symbols and tripolar spherical harmonics (three rotations). We show that these recoupling coefficient expansions, in conjunction with Racah algebra, are the cornerstone of an unusually simple and direct method of (1) proving the Euler–Rodrigues (quaternion) parameter composition rules for two and three consecutive rotations and (2) constructing rotational invariants expressed in Weyl-invariant form. These methods are distinguished by their independence on the usual algebraic approaches, either those of operator algebra, or ordinary algebra.

Vector and tensor polarizations of light beams

Generalized Stokes vectors consisting of nine real parameters in terms of irreducible spherical and Cartesian tensor operators are considered to describe the 3D electromagnetic field. Vector and tensor polarization degrees are introduced to characterize the polarized beam. It is shown that the degree of a pure vector polarized light beam is less than 1, i.e., vector polarized light beam is only partially polarized. The geometrical interpretation of vector and tensor polarizations is presented.

Vector and tensor polarizations of light beams

Generalized Stokes vectors consisting of nine real parameters in terms of irreducible spherical and Cartesian tensor operators are considered to describe the 3D electromagnetic field. Vector and tensor polarization degrees are introduced to characterize the polarized beam. It is shown that the degree of a pure vector polarized light beam is less than 1, i.e., vector polarized light beam is only partially polarized. The geometrical interpretation of vector and tensor polarizations is presented.

Matrix product representations for all valence bond states

We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence bond states proposed in the Affleck--Kennedy--Lieb--Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest-neighbor interactions, and the partially dimerized or AKLT/valence bond solid states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions.

Nonrelativistic quantum theory of the contact inelastic scattering of an x-ray photon by an atom

The nonrelativistic analytical structure of the doubly differential cross section of the contact inelastic scattering of an x-ray photon by a free atom is determined by means of the irreducible tensor operator theory outside the frame of the impulse approximation. For the neon atom in the vicinity of the 1$s$ shell ionization threshold our theory predicts the existence of the distinct fine structure of the cross section caused by transitions of the atomic core electrons into the excited discrete spectrum states. The results of our calculations with inclusion of the effects of radial relaxation, inelastic scattering through the intermediate states, and elastic Rayleigh scattering, are predictions, while at the $22\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$ incident photons they compare well with the synchrotron experiment by Jung et al. [Phys. Rev. Lett. 81, 1596 (1998)].

Irreducible tensor operator method study on molecular magnetism: A practical approach and its use for high-nuclearity magnetic cluster

A new practical approach to molecular magnetism was suggested. First, for the complicated magnetic exchange system with unquenched orbital angular momentum a general form of the effective Hamiltonian including an intercluster exchange interaction of the self-consistent field was introduced. On the basis of the irreducible tensor operator method the practical approach was established, and accomplished by the BJMAG package, where a new strategy to automatically search more good quantum numbers was adopted to solve further block diagonalization in the eigenvalue problem. Finally, potentiality of the approach to isotropic Heisenberg spin clusters was tested through two high-nuclearity clusters, the nonamer cluster and the Mn 7-cluster compound as representative examples.

Intermolecular Interactions: Physical Picture, Computational Methods and Model Potentials

Preface. 1 Background Knowledge. 1.1 The Subject and its Specificity. 1.2 A Brief Historical Survey. 1.3 The Concept of Interatomic Potential and Adiabatic Approximation. 1.4 General Classification of Intermolecular Interactions. References. 2 Types of Intermolecular Interactions: Qualitative Picture. 2.1 Direct Electrostatic Interactions. 2.2 Resonance Interaction. 2.3 Polarization Interactions. 2.4 Exchange Interaction. 2.5 Retardation Effects in Long-Range Interactions and the Influence of Temperature. 2.6 Relativistic (Magnetic) Interactions. 2.7 Interaction Between Macroscopic Bodies. References. 3 Calculation of Intermolecular Interactions. 3.1 Large Distances. 3.2 Intermediate and Short Distances. References. 4 Nonadditivity of Intermolecular Interactions. 4.1 Physical Nature of Nonadditivity and the Definition of Many-Body Forces. 4.2 Manifestations of Nonadditive Effects. 4.3 Perturbation Theory and Many-Body Decomposition. 4.4 Many-Body Effects in Atomic Clusters. 4.5 Atom-Atom Potential Scheme and Nonadditivity. References. 5 Model Potentials. 5.1 Semiempirical Model Potentials. 5.2 Determination of Parameters in Model Potentials. 5.3 Reconstructing Potentials on the Basis of Experimental Data. 5.4 Global Optimization Methods. References. Appendix 1: Fundamental Physical Constants and Conversion Table of Physical Units. Appendix 2: Some Necessary Mathematical Apparatus. A2.1 Vector and Tensor Calculus. A2.1.1 Definition of vector the addition law. A2.1.2 Scalar and vector products triple scalar product. A2.1.3 Determinants. A2.1.4 Vector analysis gradient, divergence and curl. A2.1.5 Vector spaces and matrices. A2.1.6 Tensors. A2.2 Group Theory. A2.2.1 Properties of group operations. A2.2.2 Representations of groups. A2.2.3 The permutation group. A2.2.4 The linear groups. The three-dimensional rotation group. A2.2.5 Point groups. A2.2.6 Irreducible tensor operators. Spherical tensors. References. Appendix 3: Methods of Quantum-Mechanical Calculations of Many-Electron Systems. A3.1 Adiabatic Approximation. A3.2 Variational Methods. A3.2.1 Self-consistent field method. A3.2.2 Methods taking into account the electron correlation. A3.2.2.1 r12-dependent wave functions. A3.2.2.2 Configuration interaction. A3.2.2.3 Coupled cluster method. A3.2.2.4 Density functional theory approach. A3.3 Perturbation Theory. A3.3.1 Rayleigh-Schr odinger perturbation theory. A3.3.2 Moller-Plesset perturbation theory. A3.3.3 Operator formalism and the Brillouin-Wigner perturbation theory. A3.3.4 Variational perturbation theory. A3.3.5 Asymptotic expansions Pade approximants. References. Index.