Rotation Matrix Elements Research Articles

Theoretical study of charge exchange dynamics in He+ + NO collisions

We investigate the charge transfer mechanism in the collisions of helium ions on nitric oxide using a molecular description framework with consideration of the orientation of the projectile toward the target. The anisotropy of the collision process has been analysed in detail in connection with the non-adiabatic interactions around avoided crossings. Potential energy curves, radial and rotational coupling matrix elements have been determined by means of ab initio quantum chemical methods. The collision dynamics is performed in the [1.-25.] keV collision energy range using a semiclassical approach, and the total electron transfer cross sections are analysed with regard to available experimental data.

Adjustment of Born-Oppenheimer electronic wave functions to simplify close coupling calculations

Technical problems connected with use of the Born-Oppenheimer clamped-nuclei approximation to generate electronic wave functions, potential energy surfaces (PES), and associated properties are discussed. A computational procedure for adjusting the phases of the wave functions, as well as their order when potential crossings occur, is presented which is based on the calculation of overlaps between sets of molecular orbitals and configuration interaction eigenfunctions obtained at neighboring nuclear conformations. This approach has significant advantages for theoretical treatments describing atomic collisions and photo-dissociation processes by means of ab initio PES, electronic transition moments, and nonadiabatic radial and rotational coupling matrix elements. It ensures that the electronic wave functions are continuous over the entire range of nuclear conformations considered, thereby greatly simplifying the process of obtaining the above quantities from the results of single-point Born-Oppenheimer calculations. The overlap results are also used to define a diabatic transformation of the wave functions obtained for conical intersections that greatly simplifies the computation of off-diagonal matrix elements by eliminating the need for complex phase factors.

Effective potential energy curves of the ground electronic state of CH+

This study presents effective (mass-dependent) potential energy curves for the methylidyne cation, which reproduce highly accurately all the available spectral data and allow for evaluation of reliable ro-vibrational wavefunctions of the probed isotopomers. The ro-vibrational wavefunctions are then used to average ab initio calculated radial functions of the rotational g-factor and spin-rotation constants yielding rotational and vibrational matrix elements of these properties for specific ro-vibrational states or transition moments for all isotopomers. The results can be of use in answering open questions concerning the formation/destruction of CH(+) in the interstellar medium and in the assignment of Zeeman or hyperfine splittings in rotational spectra of CH(+).

Straley Model of Biaxial Nematics Extended

In 1974 Straley [Phys. Rev. A 10, 1881 (1974)] presented the mean-field model of interacting particles similar to hard rectangular blocks. The orientation-dependent potential was built from the four basic functions composed from the standard rotation matrix elements with j = 2. As there is only one basic function composed from the standard rotation matrix elements with j = 3, it is interesting to find out whether it has any influence on the known phase diagrams. We show that the j = 3 term can enlarge the biaxial nematic phase range. The article also discusses an extension of the Romano model mimicking shape amphiphilicity [Phys. Lett. A 333, 110 (2004)]. It is showed that the j = 3 interaction term can create the tetrahedral phase with the second order transition to the isotropic phase. What is more, the first order transition from the biaxial nematic phase to the isotropic phase or to the tetrahedral phase is predicted. Finally, the possible presence of the j = 3 term in other theories is discussed.

New analytical expressions, symmetry relations and numerical solutions for the rotational overlap integrals

AbstractIn this article, extremely simple analytical formulas are obtained for rotational overlap integrals which occur in integrals over two reduced rotation matrix elements. The analytical derivations are based on the properties of the Jacobi polynomials and beta functions. Numerical results and special values for rotational overlap integrals are obtained by using symmetry properties and recurrence relationships for reduced rotation matrix elements. The final results are of surprisingly simple structures and very useful for practical applications. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

Direct fit of experimental ro-vibrational intensities to the dipole moment function: Application to HCl

A dipole moment function (DMF) for hydrogen chloride (HCl) has been obtained using a direct fit approach that fits the best available and appropriately weighted experimental data for individual ro-vibrational transitions. Combining wavefunctions derived from the Rydberg–Klein–Rees (RKR) numerical method and a semi-empirical DMF, line intensities were calculated numerically for bands with Δ v=0, 1, 2, 3, 4, 5, 6, 7 up to v′=7. The results have demonstrated the effectiveness of inclusion of rotational dipole moment matrix elements and appropriate weighting of the experimental data in the DMF fitting. The new method is shown to be superior to the common method of fitting only the rotationless dipole moment elements, while at the same time being simple to implement.

Generating function for rotation matrix elements

AbstractThe rotation matrix elements are expressed in terms of the Jacobi, Hypergeometric, and Legendre polynomials in the literature. In this study, the generating function is presented for rotation matrix elements by using properties of Jacobi polynomials. In addition, some special values and Rodrigues' formula of rotation matrix elements are obtained by using the generating function. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

Ab initio calculations of charge transfer in He+ colliding with Ne

The charge-transfer cross sections for He+ colliding with Ne atoms are calculated for the incident energy ranging from 20 eV to 5 keV by a quantum-mechanical molecular-orbital close-coupling method, which utilizes ab initio adiabatic potentials, radial and rotational coupling matrix elements obtained with a multireference configuration interaction method. Total and state-selective cross sections and rate coefficients are presented. It is found that for the collision energy E < 0.8 keV, the main charge-transfer channels are capture to the 1s2s and 1s2p excited states of He, while for E > 0.8 keV, the capture to the 1s2 ground state becomes dominant. It is also found that rotational coupling becomes important for E > 0.6 keV.

Ab initio calculation of the 66 low-lying electronic states of HeH+: adiabatic and diabatic representations

Using the quantum chemistry package MOLPRO and an adapted basis set, we have calculated the adiabatic potential energy curves of the first 20 1Σ+, 19 3Σ+, 12 1Π, 9 3Π, 4 1Δ and 2 3Δ electronic states of the HeH+ molecular ion in CASSCF and CI approaches. The results are compared with previous works. The radial and rotational non-adiabatic coupling matrix elements as well as the dipole moments are also calculated. The asymptotic behaviour of the potential energy curves and of the various couplings between the states is also studied. Using the radial couplings, the diabatic representation is defined and we present an example of our diabatization procedure on the 1Σ+ states.

Evaluation of recurrence relationships and numerical solutions for finite rotation matrix elements by using auxiliary functions

In this article, we derived several new recurrence relations of the rotation matrix elements by using Gauss’ recurrence formulas for hypergeometric functions and auxiliary functions\(A_{m^{\prime}, m}^j (\beta)\) , and \(B_{i, l}^k (\beta)\) . The aim of this contribution is to obtain general algorithm to compute the rotation matrix elements, paying attention to the use recurrence relationships of the auxiliary functions that allow the treatment of the functions with high angular momentum quantum numbers.

Ab initiostudy of charge transfer inB2+low-energy collisions with atomic hydrogen

Charge transfer processes due to collisions of ground state B{sup 2+}(2s {sup 2}S) ions with atomic hydrogen are investigated using the quantum-mechanical molecular-orbital close-coupling (MOCC) method. The MOCC calculations utilize ab initio adiabatic potentials and nonadiabatic radial and rotational coupling matrix elements obtained with the spin-coupled valence-bond approach. Total and state-selective cross sections and rate coefficients are presented. Comparison with the existing experiments shows our results to be in good agreement. When E 400 eV/u, inclusion of rotational coupling increases the total cross section by 50%-80%, improving the agreement between the current calculations and experiments. For state-selective cross sections, rotational coupling induces mixing between different symmetries; however, its effect, especially at low collision energies, is not as important as had been suggested in previous work.

Charge transfer ofO3+ions with atomic hydrogen

Charge transfer processes due to collisions of ground state ${\mathrm{O}}^{3+}{(2s}^{2}2p{}^{2}P)$ ions with atomic hydrogen are investigated using the quantum-mechanical molecular-orbital close-coupling (MOCC) method. The MOCC calculations utilize ab initio adiabatic potentials and nonadiabatic radial and rotational coupling matrix elements obtained with the spin-coupled valence-bond approach. Total and state-selective cross sections and rate coefficients are presented. Comparison with existing experimental and theoretical data shows our results to be in better agreement with the measurements than the previous calculations, although problems with some of the state-selective measurements are noted. Our calculations demonstrate that rotational coupling is not important for the total cross section, but for state-selective cross sections, its relevance increases with energy. For the ratios of triplet to singlet cross sections, significant departures from a statistical value are found, generally in harmony with experiment.

Ab initio CI calculations of the potential curves and nonadiabatic coupling matrix elements for collisions of protons with the ethylene molecule

AbstractThe interaction of ethylene with a single proton has been studied theoretically by means of multireference configuration interaction calculations. Potential energy surfaces have been obtained as well as both radial and rotational nonadiabatic coupling matrix elements. Three different approaches of the proton toward the midpoint of the target molecule have been considered for eight collision channels. Along the π direction a number of deep potential minima have been found and the ground‐state proton affinity is calculated to be 7.25 eV on this basis. There is a strongly avoided crossing in this case between the higher‐energy C2H4/H+ incoming channel and the ground C2H/H singlet state, and the corresponding radial coupling matrix element has a broad shape as a result. For the approaches along the in‐plane perpendicular axes of the molecule there is a potential crossing for these two states due to their different symmetries in these orientations, from which it is clear that the incoming channel is strongly bound whereas the charge‐transfer state is repulsive. Comparison with available experimental results indicates that the computed energy differences between the electronic states are accurate to within 0.1–0.2 eV. For the approach along the CC bond there is a weakly avoided crossing between the lowest triplet state of C2H4 and the C2H state corresponding to ionization from the σ molecular orbital (MO). One of the potential minima occurring for the π approach is seen to result from a doubly excited state in which the two π electrons are transferred to a π* MO and the incoming proton, respectively, so that a relatively stable three‐center bond is formed. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Nonlinear harmonic components of the electric polarization of symmetric-top molecules

The nonlinear dielectric relaxation of an assembly of polar molecules submitted to the action of a dc field superimposed on an ac electric field is studied theoretically in the framework of the noninertial rotational diffusion model. It is also assumed that the permanent dipole moment vector makes an arbitrary angle with respect to the long molecular axis of each rodlike molecule. By calculating the ensemble averages of two Wigner rotation matrix elements 〈D0,01(Ω)〈(t) and 〈D0,11(Ω)〉(t), it is possible to express the first three harmonic components of the electric polarization varying in ω, 2ω and 3ω in analytic form, together with the steady-state frequency-dependent component. Three-dimensional dispersion and absorption plots are presented in order to show the influence of the diffusion anisotropy parameter Δ. Cole–Cole diagrams of the second and third harmonic components are also presented allowing one to distinguish two relaxation processes along the short and the long molecular axes, respectively.

Statistical Theory of Elastic Constants of Cholesteric Liquid Crystals

A statistical theory of cholesteric liquid crystals composed of short rigid biaxial molecules is presented. It is derived in the thermodynamic limit at a small density and a small twist. The uniaxial (biaxial) cholesteric phase is regarded as a distorted form of the uniaxial (biaxial) nematic phase. The chirality of the interactions and the implementation of the inversion to the rotation matrix elements are discussed in detail. General microscopic expressions for the elastic constants are derived. The expressions involve the one-particle distribution function and the potential energy of two-body short-range interactions. It is shown that the elastic constants determine the twist of the phase. The stability condition for the cholesteric and nematic phases is presented. The theory is used to study unary and binary systems. The temperature and concentration dependence of the order parameters, the elastic constants and the twist of the phase are obtained. The possibility of phase separation is not investigated.

3 nj-symbols and harmonic superposition coefficients: an icosahedral abacus

Angular momentum recoupling coefficients of angular momentum theory and matrix elements for basis set transformation of hyperspherical harmonics enjoy properties and sum rules crucial for applications but complicated without the guidance of graphical techniques. These coefficients being related to Racah's polynomials, the graphs also apply to polynomials of the hypergeometric family, their q-analogues and their `elliptic' extensions. A useful `abacus' exploiting the connections with presentations of icosahedral and related symmetries is introduced. Particular and limiting cases, such as those of the semiclassical type, allow a unified view of properties of angular and hyperangular momentum algebra, including relationships among vector coupling coefficients and rotation matrix elements.

Ab initio calculation of predissociation linewidths in the Schumann–Runge bands of the oxygen molecule

Ab initio multireference single- and double-excitation configuration interaction calculations have been carried out for potential curves of the B 3Σu−, Πu1, Πu3, Πu5, and Σu+3 electronic states of the O2 molecule. Spin-orbit interaction matrix elements among the above states and rotational interaction matrix elements between the B 3Σu− and the Πu3 states have been evaluated in order to investigate the observed line broadening in the Schumann–Runge bands. Predissociation linewidths of rovibrational levels of the B 3Σu− state are determined by using the complex rotation method in conjunction with the Gauss–Hermite quadrature procedure. It has been found that the Πu5 and Πu3 states play dominant roles in the predissociation of the B 3Σu− state. The calculated linewidths for rovibrational levels of v=0–13, N=0–36 of the F2 fine-structure components of the B 3Σu− state are in very good agreement with experimental results. Isotopic effects in the predissociation of the B 3Σu− state are studied by means of calculations for O218 and O16O18 and the theoretical results are found to be in good agreement with the observed average predissociation linewidths in each case.

Nearside–farside analysis of differential cross sections using Jacobi functions of the first and second kinds: Application to Ar+N2 rotationally inelastic scattering

We report the first nearside–farside (NF) analysis of angular scattering for an inelastic molecular collision in which the partial wave series for the scattering amplitude is expanded in a basis set of reduced rotation matrix elements dmf,miJ(θ), where θ is the scattering angle, J is the total angular momentum quantum number, and mi,mf are initial and final helicity quantum numbers, respectively. The practical implementation of the NF theory is described in detail; it exploits in an essential way the properties of a function that we denote emf,miJ(θ) and call a reduced rotation matrix element of the second kind. The caustic structure of dmf,miJ(θ) and emf,miJ(θ) is taken into account via a restricted nearside–farside (resNF) decomposition of the scattering amplitude. The NFres theory is used to analyze polarization and degeneracy averaged differential cross sections for the Ar+N2(ji=2,mi=0,±1,±2)→Ar+N2(jf=2,mf=0,±1,±2) collision system, treated as an atom+rigid-rotor. The NFres analysis always provides a clear physical interpretation of the scattering (except sometimes for θ≈0°,180°) for phenomena such as diffraction oscillations and potential rainbows, as well as for more complicated (unnamed) interference effects. We also report results for some approximations to the NFres theory. Mathematical properties of the emf,miJ(θ) required for the NFres analysis are derived.

Rotation matrix elements and further decompositionfunctions of two-vector tesseral spherical tensor operators; theiruses in electron paramagnetic resonance spectroscopy

Matrix elements, Ag,h(k) (k = 1-6), which describe ageneral Euler angle transformation of coordinates to which tesseralspherical tensor operators, ℑk,q, are referred have beencalculated and extended to include matrix elements for odd k. Thematrix elements have been incorporated into a general axis-transformation computer program which relates to parameter sets inany one of the more commonly used tesseral forms, namely,conventional Stevens, normalized Stevens (Racah normalization) andnormalized spherical tensor (Koster and Statz normalization)operators. Tables of decomposition functions of tesseral sphericaltensor operators, ℑk,q(B,J) (J = S,I), are extended to detail decompositions for terms ofdimension BJ7 and, implicitly, for decomposition of any twovector operators ℑk,q(V,W) toexperimentally usable single vector forms where VkVWkW(one of kV, kW unity) is the dimension of a general term in thedecomposition. Tables detailing the symmetry-allowed terms underthe 11 Laue (site) crystal classes are also extended to includetesseral tensorial sets up to rank 8, thus including the new terms.The use of these functions to describe electron paramagneticresonance studies of high-spin nuclear Zeeman interactions is discussed.

General Explicit Pure Rotational Effect for Infrared, Raman and Higher (ΔJ) Spectra for a Diatomic Molecule

The problem of the pure rotational transitions vJ ←→ vJ ' for any spectra |J - J '| ≥ 1 for a diatomic molecule is considered. It is proved that, the wave functions ΨvJ and ΨvJ ' are expanded in terms of the running number m = [J '(J ' + 1) - J(J + 1)]/2 as ΨvJ = Σn = 0 πnmn and ΨvJ ' = Σn = 0 πn(-m)n where πn are expressed in terms of the pure vibrational wave function ϕ0 and its rotational corrections ϕn (defined in the conventional perturbation theory). By using this m-representation of the wave functions the pure rotational matrix elements of the considered transitions are given by MvJvJ ' =⟨ ΨvJ | γ | ΨvJ ' ⟩ = Σn = 0 µ2nm2n where µ2n are simple combinations of simple integrals of the form ⟨ ϕi | γ | ϕn ⟩. This formulation is valid for any potential (either numerical or analytical), any vibrational level v and any operator γ. The numerical application to the Dunham potential of the molecule H2 in the Raman transitions and to the Huffaker potential of the molecule CO in the infrared transitions shows the validity and the high accuracy of the present formulation.