Tensorial calculations in molecular spectroscopy

1979 ◽  
Vol 57 (11) ◽  
pp. 2030-2044
Author(s):  
Jean-Louis Féménias
Keyword(s):  
Angular Momentum ◽  
Molecular Spectroscopy ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Intensity Factors ◽  
Molecular Physics ◽  
Mechanical Formulation ◽  
The Matrix ◽  
Tensor Operators

A strict quantum mechanical formulation of Van Vleck's concept of Reversed Angular Momentum is given. This generalization leads to a method suitable for the calculation of the matrix elements of all kinds of tensor operators in molecular physics. Applications and examples are described for the diatomic case. Various interactions—in the non-rotating or in the rotating molecule—are studied, and some intensity factors are derived in an original way. In all cases, the simplicity, generality, rapidity, and consistency of the method are tested and, where relevant, compared with other techniques.

SYMMETRY DECOUPLING IN LINEAR ALGEBRAIC VARIATIONAL SCATTERING PROBLEMS

1995 ◽  
Vol 06 (01) ◽  
pp. 105-121
Author(s):  
MEISHAN ZHAO
Keyword(s):  
Angular Momentum ◽  
Variational Principle ◽  
Total Angular Momentum ◽  
Numerical Calculations ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Scattering Problems ◽  
Identical Particles ◽  
Generalized Newton

This paper discusses the symmetry decoupling in quantum mechanical algebraic variational scattering calculations by the generalized Newton variational principle. Symmetry decoupling for collisions involving identical particles is briefly discussed. Detailed discussion is given to decoupling from evaluation of matrix elements with nonzero total angular momentum. Example numerical calculations are presented for BrH2 and DH2 systems to illustrate accuracy and efficiency.

Interpolation of matrices and matrix-valued densities: The unbalanced case

2018 ◽  
Vol 30 (3) ◽  
pp. 458-480 ◽  
Author(s):  
YONGXIN CHEN ◽  
TRYPHON T. GEORGIOU ◽  
ALLEN TANNENBAUM
Keyword(s):  
Open Quantum Systems ◽  
Wasserstein Distance ◽  
Quantum Systems ◽  
Quantum Mechanical ◽  
Frobenius Norm ◽  
Spatial Dependency ◽  
Lindblad Equation ◽  
Optimal Mass Transport ◽  
Mechanical Formulation ◽  
The Matrix

We propose unbalanced versions of the quantum mechanical version of optimal mass transport that is based on the Lindblad equation describing open quantum systems. One of them is a natural interpolation framework between matrices and matrix-valued measures via a quantum mechanical formulation of Fisher-Rao information and the matricial Wasserstein distance, and the second is an interpolation between Wasserstein distance and Frobenius norm. We also give analogous results for the matrix-valued density measures, i.e., we add a spatial dependency on the density matrices. This might extend the applications of the framework to interpolating matrix-valued densities/images with unequal masses.

On a Flexible Double Minimum Potential

1979 ◽  
Vol 34 (9) ◽  
pp. 1106-1112 ◽  
Author(s):  
J. Bohmann ◽  
W. Witschel
Keyword(s):  
Harmonic Oscillator ◽  
Molecular Spectroscopy ◽  
Additional Term ◽  
Numerical Control ◽  
Upper And Lower Bounds ◽  
Parabolic Cylinder ◽  
Matrix Elements ◽  
The Matrix ◽  
Minimum Potential ◽  
Parabolic Cylinder Functions

Abstract The Gauss-perturbed harmonic oscillator, a customary double minimum potential of molecular spectroscopy, is made more flexible by addition of a term α4 · exp(-γ4X4). The matrix elements of the additional term are calculated in the harmonic oscillator basis in terms of parabolic cylinder functions. A sum rule for matrix elements serves as an independent numerical control. The eigenvalues can be given by straightforward diagonalization of the Hamilton-matrix. In addition, upper and lower bounds are given for the partition function.

Méthode du moment angulaire inversé. Partie II : Application aux problèmes hyperfins diatomiques

1975 ◽  
Vol 53 (5) ◽  
pp. 542-552 ◽  
Author(s):  
J-L. Féménias ◽  
C. Athénour
Keyword(s):  
Angular Momentum ◽  
Experimental Verification ◽  
Experimental Results ◽  
Matrix Elements ◽  
Intensity Factors ◽  
Method Of Calculation ◽  
Straightforward Method ◽  
Mi Theory

We present a straightforward method of calculation in rotational molecular problems based on the theory of reversed angular momentum (MI theory) previously discussed. After some general remarks and calculations, we test this method by calculating some well known hyperfine matrix elements in diatomic problems, showing thus its easy handling. The first experimental verification relative to intensity factors in hyperfine diatomic transitions that the method enabled us to calculate, is presented with a brief survey of the recent experimental results obtained from the analysis of the [Formula: see text] system of NbN.

The matrix elements of tensor operators for configurations of three equivalent electrons

1959 ◽  
Vol 250 (1263) ◽  
pp. 562-574 ◽  
Keyword(s):  
Irreducible Representations ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Quantum Numbers ◽  
Fractional Parentage ◽  
The Matrix ◽  
Tensor Operators

The formulae of Redmond are used to construct expressions for the fractional parentage coefficients relating the configurations l 3 and l 2 . The explicit occurrence of godparent states is avoided for the quartet states of f 3 and also for a sequence of doublet states. The latter are defined by the set of quantum numbers f 3 WUSLJJ 2 , where W and U are irreducible representations of the groups R 7 and G 2 . Matrix elements of the type ( f 3 WUSL || U k || f 3 W'U'SL' ), where U k is the sum of the three irreducible tensor operators u k corresponding to the three f electrons, are tabulated for k = 2, 4 and 6 and for all values of W, U, S and L .

Méthode du moment angulaire inversé. Partie I : La théorie MI

1975 ◽  
Vol 53 (5) ◽  
pp. 533-541 ◽  
Author(s):  
J-L. Féménias ◽  
C. Athénour
Keyword(s):  
Angular Momentum ◽  
Wave Functions ◽  
Matrix Elements ◽  
Intensity Factors ◽  
Normal Theory ◽  
Commutation Relations ◽  
Minus Sign ◽  
Mi Theory ◽  
Optimal Set ◽  
General Method

In order to present a general method of calculating wave functions, matrix elements, and intensity factors in molecular rotational problems, we deal here with a theory of reversed angular momentum (MI theory), i.e. momentum whose components obey commutation relations with an anomalous minus sign. This theory is built following the principal corresponding steps of the normal theory and a simple correspondence is established between both of them allowing us to perform calculations in the former and to give the results using the formalism of the latter.In particular we dwell here on the consistency of conventions and notations and on a rigorous presentation of spherical tensor operators in MI theory together with the choice of the optimal set of them leading to simplest calculations.

scholarly journals Über der Kopplung von Schwingungsfreiheitsgraden an die Koordinate intramolekularer Umlagerungen / Coupling of Vibrational Degrees of Freedom lo the Coordinate of Intramolecular Rearrangements

1973 ◽  
Vol 28 (11) ◽  
pp. 1759-1781 ◽  
Author(s):  
J. Brickmann
Keyword(s):  
Degrees Of Freedom ◽  
Energy Surface ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Damping Term ◽  
Intramolecular Rearrangements ◽  
Quantum Numbers ◽  
The Matrix ◽  
Localized Quantum States ◽  
Internal Degrees Of Freedom

Intramolecular rearrangements A ⇌ B are investigated, which can be described in terms of the motion of an effective quantum mechanical ‘‘particle’’ on an energy surface with at least two minima. We regard the energy surface as a function of a limited number of relevant internal degrees of freedom. The rate of isomerization is calculated from the matrix elements of a transition operator W with respect to the localized quantum states of the two isomers, and the coupling to the inter- and intramolecular degrees of freedom, not explizitely considered in the energy surface. It is shown that the matrixelements of W be reduced to integrals over functions of the adiabatic reaction coordinate of the isomerization, and selection rules for the vibrational quantum numbers for the motion perpendicular to this coordinate. The degrees of freedom not relevant for the reaction are summarily taken into account by introducing a heath bath in thermodynamic equilibrium and a simple damping term. Applications are discussed.

Tensor operators under semi-simple groups

1961 ◽  
Vol 57 (3) ◽  
pp. 460-468 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Simple Groups ◽  
Matrix Elements ◽  
Coupling Coefficients ◽  
The Matrix ◽  
Casimir Operators ◽  
Tensor Operators

ABSTRACTTensor operators under any group are defined and the theory is developed for semi-simple continuous groups. Coupled tensor operators are introduced and the matrix elements of tensor operators are expressed in terms of the coupling coefficients. The structure of generalized Casimir operators is investigated.

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