Radial Coupling Matrix Elements Research Articles

Theoretical investigation of electron capture processes in slow O7+–H collisions

Aims. The total and n-, l-, and S -resolved single-electron-capture cross sections for the collisions of O7+ with atomic hydrogen are studied in the energy region from 10−3eV u−1−5keV u−1. Methods. These state-selective cross sections were calculated by employing the full quantum-mechanical molecular-orbital close-coupling (QMOCC) method. The ab initio multireference single- and double-excitation configuration interaction approach, with optimized atomic basis sets to accurately describe the highly excited states, was used to obtain the adiabatic potentials and the radial and rotational coupling matrix elements that are required in the QMOCC calculation. Results. Our results are compared with other available theoretical and experimental data. The n = 5 manifold is the dominant reaction channel for the chasrge-transfer process for this collision system over the entire energy range, and our results agree better with the experimental data than the other theoretical results in the energy region in which they overlap because we included the necessary highly excited states in the expansion basis set. These charge-exchange cross-section data are useful for understanding and modeling the X-ray emission in astrophysical environments.

A computational study of the non-adiabatic coupling among low-lying doublet states of the CN radical

First principle electronic structure calculations on low-lying doublet states of the CN radical have been performed using the multi-reference configuration interaction method and large all-electron basis sets. Beside the adiabatic interatomic potentials, the non-adiabatic spin-orbit and L-uncoupling electronic matrix elements were evaluated for the mutually perturbed (2)B2Σ+∼(1–4)2Π and (1)A2Π∼(1–3)2Σ+, (1)2Σ−, (1,2)2Δ pairs of states. The radial coupling matrix element was calculated between the X2Σ+ and B2Σ+ states. The reliability of the derived A∼X and A∼B matrix elements was confirmed by comparison with their empirical counterparts. Both local and regular perturbations in the A∼B∼X complex were found to depend mostly on spin-orbit and electronic-rotational interactions between the lowest A2Π, and B2Σ+ states. Excited states, in all their diversity, have little effect on the fine structure of the A∼B complex, except for the qA Λ-doubling function. The ab initio coupling functions can be crucial to providing a global deperturbation treatment for the A, B and X states on a spectroscopic level of accuracy.

Ab initio Calculations of the Lowest $$^{1}\Sigma _{g}^{ + }$$ States of the Na2 Dimer

The ground and 10 lowest excited $$^{1}\Sigma _{g}^{ + }$$ adiabatic electronic states of the Na2 dimer are calculated using the pseudopotential method. The use of the basis [7s6p5d4f] of atomic orbitals makes it possible to extend the range of available internuclear distances up to 1.7–50 A. It is found that the theoretical values of the Te and De constants are in a good agreement with the experimental ones. Herein you will find the sample calculations of the radial non-adiabatic coupling matrix elements enable to transform the basis of the adiabatic states to quasidiabatic one. It is found also that the Le Roy modified radius scales down the left boundary of an asymptotic range for the electronic state with the (3s + 5p) dissociation limit and for the higher states.

Ab Initio Calculation of the Lowest Singlet and Triplet Excited States of the N2 Molecule

Ab initio calculations of the adiabatic potential curves and matrix elements of radial nonadiabatic coupling of the N2 molecule for the states related to dissociation limits I–V were performed. The most important spectral characteristics of the adiabatic states agreed well with the available experimental and theoretical data. The diabatic states were constructed. The diabatic quantum defects and radial matrix elements of the configuration interaction of the dissociative and Rydberg configurations whose states converge to the ground state \(X^{2}\Sigma{_g^+}\) and the first electronically excited state A2Πu of the \(\rm{N_2^+}\) ion were calculated. The possibility of using the results for calculating the cross sections and rate constants of dissociative recombination and associative ionization within the framework of the multichannel quantum defect theory was discussed.

Excitation cross sections in low-energy hydrogen-helium collisions

Quantum chemical results for the lowest four potentials of the HeH molecule and the corresponding rotational and radial nonadiabatic coupling matrix elements are reported. Close-coupling calculations of the integral cross sections for the excitation processes $\mathrm{H}(1s)+\text{He}\ensuremath{\rightarrow}\phantom{\rule{4pt}{0ex}}\mathrm{H}(2s,2p)+\text{He}$ are performed on this basis. The calculated cross sections are in a reasonable agreement with the experimental data.

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.

Ab initio many-electron study for the low-lying states of the alkali hydride cations in the adiabatic representation

An ab initio multireference single- and double-excitation configuration interaction (CI) study is carried out for the ground and excited electronic states of alkali-hydride cations (LiH(+), NaH(+), KH(+), RbH(+), and CsH(+)). For all alkali-metal atoms, the first inner-shell and valence electrons (nine active electrons, three for Li) are considered explicitly in the ab initio self-consistent-field and CI calculations. The adiabatic potential energy curves, radial and rotational couplings are calculated and presented. Short-range (∼3 a.u.) potential wells produced by the excitation of the inner-shell electrons are found. The depths of the inner potential wells are much greater than those of the outer wells for the CsH(+) system. The computed spectroscopic constants for the long-range potential well of the 2 (2)Σ(+) state are very close to the available theoretical and experimental data. The electronic states of alkali-hydrogen cations are also compared with each other, it is found that the positions of the potential wells shift to larger internuclear distances gradually, and the depths of these potential wells become greater with increasing alkali-metal atomic number. The relationships between structures of the radial coupling matrix elements and the avoiding crossings of the potential curves are analyzed. From NaH(+) to CsH(+), radial coupling matrix elements display more and more complex structures due to the gradual decrease of energy separations for avoided crossings. Finally, the behavior of some rotational couplings is also shown.

Sympathetic cooling of the Ba+ion by collisions with ultracold Rb atoms: Theoretical prospects

State-of-the-art {\em ab initio} techniques have been applied to compute the potential energy curves of the (BaRb)$^+$ molecular ion in the Born-Oppenheimer approximation. The long-range coefficients describing the electrostatic, induction, and dispersion interactions at large interatomic distances are also reported. The electric transition dipole moments governing the X$^1\Sigma \to ^1\Sigma, ^1\Pi$ have been obtained as the first residue of the polarization propagator computed with the linear response coupled-cluster method restricted to single and double excitations. Nonadiabatic radial and angular coupling matrix elements, as well as the spin-orbit coupling matrix elements have been evaluated using the multireference configuration interaction method restricted to single and double excitations with a large active space. With these couplings, the spin-orbit coupled (relativistic) potential energy curves for the 0$^+$ and 1 states relevant for the running experiments have been obtained. Finally, relativistic transition moments and nonadiabatic coupling matrix elements were obtained from the nonrelativistic results and spin-orbit eigenvectors. The electronic structure input has been employed in the single channel scattering calculations of the collisional cross sections between the Ba$^+$ ion and Rb atom. Both nonrelativistic and relativistic potentials were used in these calculations. Our results show that the inelastic cross section corresponding to the charge transfer from the Rb atom to the Ba$^+$ ion is much smaller than the elastic one over a wide range of energies up to 1 mK. This suggests that sympathetic cooling of the Ba$^+$ ion by collisions with ultracold Rb atoms should be possible.

Quantum dynamics of the charge transfer in C+ + S at low collision energies

Following a recent semiclassical investigation by Bacchus-Montabonel and Talbi (2008 Chem. Phys. Lett. 467 28), the C+(2s22p)2P + S(3s23p4)3P charge transfer process involved in the modellization of the interstellar medium chemistry and its reverse reaction are revisited by combining a wave packet approach and semiclassical dynamics in a quasimolecular approach for doublet and quartet states. New radial non-adiabatic coupling matrix elements have been calculated and the mixed treatment gives access to new precise values of the rate coefficients for the direct and reverse charge transfer processes. For this system, quantum and semiclassical results are in good agreement even at low collision kinetic energies. The dominance of the quartet states in the process is confirmed. In the quantum treatment, the collision matrix elements are extracted from wave packets by the flux method with an absorbing potential. The formation of resonances due to a centrifugal barrier is illustrated.

Ab initiocalculation ofH+He+charge-transfer cross sections for plasma physics

The charge-transfer in low-energy ($0.25$ to $150$ eV/amu) $\mathrm{H}(\mathit{nl})+{\mathrm{He}}^{+}(1s)$ collisions is investigated using a quasimolecular approach for the $n=2,3$ as well as the first two $n=4$ singlet states. The diabatic potential energy curves of the HeH${}^{+}$ molecular ion are obtained from the adiabatic potential energy curves and the nonadiabatic radial coupling matrix elements using a two-by-two diabatization method, and a time-dependent wave-packet approach is used to calculate the state-to-state cross sections. We find a strong dependence of the charge-transfer cross section on the principal and orbital quantum numbers $n$ and $l$ of the initial or final state. We estimate the effect of the nonadiabatic rotational couplings, which is found to be important even at energies below $1$ eV/amu. However, the effect is small on the total cross sections at energies below $10$ eV/amu. We observe that to calculate charge-transfer cross sections in an $n$ manifold, it is only necessary to include states with ${n}^{\ensuremath{'}}\ensuremath{\leqslant}n$, and we discuss the limitations of our approach as the number of states increases.

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.

Quasi-relativistic treatment of the low-lying KCs states

The potential energy curves, permanent and transition dipole moments as well as spin-orbit and angular coupling matrix elements between the KCs electronic states converging to the lowest three dissociation limits were evaluated in the basis of the spin-averaged wavefunctions corresponding to pure Hund’s coupling case ( a). The quasi-relativistic matrix elements have been obtained for a wide range of internuclear distance by using of small (9-electrons) effective core pseudopotentials of both atoms. The core-valence correlation has been accounted for a large scale multi-reference configuration interaction method combined with semi-empirical core polarization potentials. The static dipole polarizabilities of the ground X 1 Σ + and a 3 Σ + states were extracted from the closed-shell coupled-cluster energies by the finite-field method. Among the singlet and triplet Σ + states manifold the pronounced avoided crossing effect between repulsive walls of the (2,3) 3 Σ + states has been discovered and analyzed by finite-difference calculation of radial coupling matrix elements. The resulting transition dipole moments and potentials were used to predict radiative lifetimes and emission branching ratios of excited vibronic states while the calculated angular coupling matrix elements were transformed to Λ -doubling constants of the (1,2) 1 Π states and magnetic g -factor of the ground state. The accuracies of the present results are discussed by comparing with experimental data and preceding calculations.

Calculations of state-selective differential cross sections for charge transfer in collisions between O3+ and H2

The non-dissociative charge-transfer processes in collisions between O3+ and H2 are investigated by using the quantum-mechanical molecular-orbital coupled-channel (QMOCC) method. The adiabatic potentials and radial coupling matrix elements utilized in the QMOCC calculations are obtained with the spin-coupled valence-bond approach. Electronic and vibrational state-selective differential cross sections are presented for projectile energies of 0.1, 1.0 and 10.0 eV/u in the H2 orientation angles of 45° and 89°. The electronic and the vibrational state-selective differential cross sections show similar behaviours: they decrease as the scattering angle increases, and beyond a specific angle the oscillating structures appear. Moreover, it is also found that the vibrational state-selective differential cross sections are strongly orientation-dependent, which provides a possibility to determine the orientations of molecule H2 by identifying the vibrational state-selective differential scattering processes.

Quantum-mechanical calculations of vibrationally resolved cross sections for non-dissociative charge transfer of O3+ with H2

Charge transfer due to collisions of ground state O3+(2s22p 2P) ions with molecular hydrogen is investigated using the quantum-mechanical molecular-orbital close-coupling (MOCC) method，and electronic and vibrational state-selective cross sections along with the corresponding differential cross sections are calculated for projectile energies of 100，500，1000 and 5000eV/u at the orientation angles of 25°，45° and 89°. The adiabatic potentials and radial coupling matrix elements utilized in the QMOCC calculations were obtained with the spin-coupled valence-bond approach. The infinite order sudden approximation (IOSA) and the vibrational sudden approximation (VSA) are utilized to deal with the rotation of H2 and the coupling between the electron and the vibration of H2. It is found that the distribution of vibrationally resolved cross sections with the vibrational quantum number v′ of H+2(v′) varies with the increment of the projectile energy; and the electronic and vibrational state-selective differential cross sections show similar behaviors: there is a highest platform within a very small scattering angle，beyond which the differential cross sections decrease as the scattering angle increases and lots of oscillating structures appear，where the scattering angle of the first structure decreases as E-1/2p with the increment of the projectile energy Ep; and the structure and amplitude of the differential cross sections are sensitive to the orientation of molecule H2，which provides a possibility to identify the orientations of molecule H2 by the vibrational state-selective differential scattering processes.

POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field

A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multi-channel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented.

Charge transfer and ionization in collisions ofSi3+with H from low to high energy

Charge transfer processes due to collisions of ground state ${\mathrm{Si}}^{3+}(3s\phantom{\rule{0.3em}{0ex}}^{1}S)$ ions with atomic hydrogen are investigated using the quantum-mechanical molecular-orbital close-coupling (MOCC) and classical-trajectory Monte Carlo (CTMC) methods. The MOCC calculations utilize ab initio adiabatic potentials and nonadiabatic radial coupling matrix elements obtained from Herrero et al. [J. Phys. B 29, 5583 (1996)] which were calculated with a full configuration-interaction method. Total and state-selective single-electron capture cross sections are obtained for collision energies from $0.01\phantom{\rule{0.3em}{0ex}}\mathrm{eV}∕\mathrm{u}$ to $1\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}∕\mathrm{u}$. Total and state-selective rate coefficients are also presented for temperatures from $2\ifmmode\times\else\texttimes\fi{}{10}^{3}\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ to ${10}^{7}\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. Comparison with existing data reveals that the total CTMC cross sections are in good agreement with the experimental measurements at the higher considered energies and that previous Landau-Zener calculations underestimate the total rate coefficients by a factor of up to two. The CTMC calculations of target ionization are presented for high energies.

An ab initio study of the excited states of the vinoxy radical

Ab initio potential energy surfaces and radial coupling matrix elements of the vinoxy radical have been calculated in a subspace of two active angles involved in the photodissociation process. A quasidiabatic scheme has been developed in the vicinity of the conical intersection and some implications on the possible photodissociation mechanism of vinoxy from different formation path have been discussed.

Vibrationally resolved charge transfer ofO3+with molecular hydrogen

Charge transfer due to collisions of ground state ${\mathrm{O}}^{3+}(2{s}^{2}2p{\phantom{\rule{0.2em}{0ex}}}^{2}{P}^{o})$ ions with molecular hydrogen are investigated using the quantum-mechanical molecular-orbital (QMO) coupled-channel method. The QMO calculations utilize $ab$ initio adiabatic potentials and nonadiabatic radial coupling matrix elements obtained with the spin-coupled valence-bond approach for a representative range of orientation angles and diatom internuclear separations. Vibrationally resolved cross sections for nondissociative single electron capture are obtained for energies between 0.1 eV∕u and 10 keV∕u for ${\mathrm{H}}_{2}$ in its ground vibrational level using the infinite order sudden approximation (IOSA). Two further approximations are considered in which the electronic radial couplings are assumed to be independent of the diatom stretching. In the first case, vibrational motion is taken into account by multiplying the electronic radial couplings by Franck-Condon (FC) ionization factors while in the second, vibrational motion is completely neglected. We refer to these two approaches as the vibrational sudden approximation (VSA) and the electronic approximation (EA), respectively. In the latter, the resulting cross sections for electronic transitions are multiplied by FC factors to obtain relative vibrationally resolved cross sections which are independent of the collision energy (the centroid approximation). Comparison with existing experimental data for total and electronic state-selective cross sections shows best agreement with IOSA and VSA, but discrepancies for EA. The triplet-singlet electronic cross section ratio reveals a departure at low collision energies from the statistical value.