Adiabatic Electronic Wave Functions Research Articles

Multiple groups in the symmetry classification of adiabatic electronic wavefunctions

It was originally shown by Longuet-Higgins and colleagues that when the electronic Schrödinger equation is solved as a function of the nuclear coordinates the adiabatic electronic wavefunction can undergo a change of sign after completing a closed circuit. This geometric phase occurs for a circuit around a conical intersection, and in particular around a conical intersection corresponding to a linear Jahn—Teller effect. The adiabatic wavefunctions are classified here under a group called the adiabatic multiple group, which is a generalization of the ‘vibronic double group’ of C 3v introduced by Hougen, and is distinct from the familiar electron-spin double group. Although the real electronic wavefunctions can be only double-valued, the groups can have higher multiplicity because of the possibility of different circuits. For a number of symmetric- and spherical-top point groups, the adiabatic multiple group is shown to be the direct product of the point group with a phase group. The adiabatic multiple group can be applied to individual adiabatic orbitals, and so to configurations built from these orbitals. This leads to the rule that the linear Jahn—Teller effect vanishes in the single-configuration approximation for configurations containing non-degenerate electrons plus an even number of e electrons. There does not appear to be any cancellation effect for electron configurations of cubic molecules containing f electrons.

Multiple groups in the symmetry classification of adiabatic electronic wavefunctions

Multiple groups in the symmetry classification of adiabatic electronic wavefunctions

Theory of Rapidly Oscillating Electron Angular Distributions in Slow Ion-Atom Collisions

A general expression for the ionization amplitude in slow ion-atom collisions is derived. The expression is inverted to obtain adiabatic electronic wave functions at complex values of the internuclear distance. It is shown that beating between $\ensuremath{\sigma}$ and $\ensuremath{\pi}$ components of electronic wave functions gives rise to rapid oscillations of electron angular distributions with ion velocity $v$. These rapid oscillations measure the real part of that eigenvalue whose imaginary part gives the well-known Wannier exponent.

Diabolical conical intersections

In the Born-Oppenheimer approximation for molecular dynamics as generalized by Born and Huang, nuclei move on multiple potential-energy surfaces corresponding to different electronic states. These surfaces may intersect at a point in the nuclear coordinates with the topology of a double cone. These conical intersections have important consequences for the dynamics. When an adiabatic electronic wave function is transported around a closed loop in nuclear coordinate space that encloses a conical intersection point, it acquires an additional geometric, or Berry, phase. The Schr\"odinger equation for nuclear motion must be modified accordingly. A conical intersection also permits efficient nonadiabatic transitions between potential-energy surfaces. Most examples of the geometric phase in molecular dynamics have been in situations in which a molecular point-group symmetry required the electronic degeneracy and the consequent conical intersection. Similarly, it has been commonly assumed that the conical intersections facilitating nonadiabatic transitions were largely symmetry driven. However, conical intersections also occur in the absence of any symmetry considerations. This review discusses computational tools for finding and characterizing the conical intersections in such systems. Because these purely accidental intersections are difficult to anticipate, they may occur more frequently than previously thought and in unexpected situations, making the geometric phase effect and the occurrence of efficient nonadiabatic transitions more commonplace phenomena. [S0034-6861(96)00404-7]

Adiabatic and diabatic representations of potential energy curves for the (NaRb) + system

Potential energy curves of (NaRb) + are investigated by an ab initio one-electron approach. The inner electrons are modelled by non-empirical relativistic pseudopotentials. The core polarization potential is included and the deviation of the core-core interaction from 1 R has been corrected. The spectroscopic constants of the ground (1 2Σ +) and lowest excited bound states ( 2 2Σ +, 3 2Σ +, 4 2Σ +, 1 2Π ) are derived. From an analysis of the adiabatic electronic wavefunctions a qualitative diagram of diabatic electronic energies in the range of internuclear distances (4−40 a 0) is made. A diabatic electronic basis for the (NaRb) + molecular system is defined and electrostatic coupling matrix elements have been calculated.

Resonances in the cumulative reaction probability for a model electronically nonadiabatic reaction

The cumulative reaction probability, flux–flux correlation function, and rate constant are calculated for a model, two-state, electronically nonadiabatic reaction, given by Shin and Light [S. Shin and J. C. Light, J. Chem. Phys. 101, 2836 (1994)]. We apply straightforward generalizations of the flux matrix/absorbing boundary condition approach of Miller and co-workers to obtain these quantities. The upper adiabatic electronic potential supports bound states, and these manifest themselves as ‘‘recrossing’’ resonances in the cumulative reaction probability, at total energies above the barrier to reaction on the lower adiabatic potential. At energies below the barrier, the cumulative reaction probability for the coupled system is shifted to higher energies relative to the one obtained for the ground state potential. This is due to the effect of an additional effective barrier caused by the nuclear kinetic operator acting on the ground state, adiabatic electronic wave function, as discussed earlier by Shin and Light. Calculations are reported for five sets of electronically nonadiabatic coupling parameters.

Nonadiabatic perturbations and fine structure splittings in the 1,2 3Πg states of B2: An analysis based on adiabatic and rigorous diabatic states

Nonadiabatic effects and fine structure splittings in the 1,2 3Πg states of B2 were studied. Adiabatic electronic wave functions Ψa(J 3Πg) were computed at the multireference configuration interaction level. The interstate derivative couplings 〈Ψa(1 3Πg)‖(d/dR)Ψa(2 3Πg)〉 were computed and used to construct ‘‘rigorous’’ diabatic electronic states. The spin–orbit interactions responsible for the fine structure splitting in the 1,2 3Πg states were determined, as were the 1,2 3Πg∼1 1Πg spin–orbit interactions. These electronic structure data were used to determine the vibrational structure of the 1,2 3Πg states in both coupled adiabatic and coupled diabatic states bases. As a result of an avoided crossing of the 1,2 3Πg states the fine structure splitting constants exhibit a marked geometry dependence. The fine structure splitting constants for the (1 3Πg, v=0–4) levels were found to be −4.2(−4.4), −3.7, −2.8, −1.8, and −0.83 cm−1, respectively. The (1 3Πg, v=0) value is in good agreement with a recent experimental determination given parenthetically. While the diabatic basis provides illuminating qualitative insights into the electronic structure of the states in question the adiabatic basis is preferred computationally.

Semiempirical determination of electronic-vibro-rotational radiative transition probabilities in diatomic molecules I. Theory

The theory of electronic-vibro-rotational radiative transitions is considered within the framework of the perturbation theory. For the case of regular (monotonic) perturbations caused by relatively far lying terms of the molecule the simple analytical expressions for the line strengths of the2s+1A′→2s+1A″ transitions are derived in the second order of the perturbation theory. Most distributed schemes of angular momenta coupling — Hund’s cases “a” and “b” — are considered. It is shown that the line strengths can be expressed in terms of a finite number (usually small) of the parameters (certain combinations of matrix elements describing the effects of perturbations and vibration-rotation interaction) which may be determined either from the experimental data in a semiempirical approach or from numerical calculations of adiabatic electronic and vibrational wave functions.

Differential cross section for molecular ionization by electron impact in the adiabatic approximation.

Molecular ionization by electron impact is examined in the Born-Oppenheimer approximation where the rotational and vibrational motions of the nuclei are considered separable. Expressions for the differential cross section, applicable to an exact treatment of the adiabatic electronic wave functions, are derived in terms of a momentum-transferred summation for three levels of experimental resolution, spatially unpolarized, spatially unpolarized and rotationally unresolved, and spatially unpolarized and rotationally and vibrationally unresolved. Several possible advantages of a momentum-transferred summation relative to a partial-wave summation are discussed. Most importantly, the expected relatively rapid convergence of the momentum-transferred summation effectively reduces the number of transition matrix elements which must be calculated. Additionally, the momentum-transferred summation involves an ''incoherent'' summation which, in general, is more easily computed. Lastly, we present explicit expressions for the transition probabilities in the plane-wave and distorted-wave Born approximations.

Collision induced transitions between 2Π and 2Σ states of diatomic molecules: Quantum theory and collisional propensity rules

We develop the exact quantum description, free of any dynamical approximations, of rotationally inelastic collision induced transitions between 2Π and 2Σ electronic states of a diatomic molecule. An explicit connection is made between the matrix elements of the electrostatic coupling, described in an asymptotically exact diabatic basis, and the results of an ab initio calculation of the appropriate atom–molecule adiabatic electronic wave functions of A′ and A″ symmetry. Analysis of the quantum close-coupled equations demonstrates that the use of Franck–Condon approximations in the description of E → E energy transfer is unjustified and, furthermore, that in collisions involving homonuclear diatomic molecules the s/a permutation-inversion symmetry of the molecular wave functions will be rigorously conserved. The extension of the infinite-order sudden approximation to electronically inelastic 2Π → 2Σ processes allows us to predict two new collisional propensity rules: (a) When Δ J=0 the cross sections will become vanishingly small for transitions which conserve the e/f symmetry index of the molecular wave function. (b) In a high-J Hund’s case (b) limit transitions from either the F1 or F2 2Π-state manifolds will populate only one of the Σ-state spin-doublet levels, consistent with a physical model in which the electronic spin S is a spectator so that the relative orientation of N and S is preserved during the collision.

Atom–molecule reactive scattering and symmetrized cross section for the system containing identical nuclei

A formulation of atom-diatomic molecule (or molecular ion) collision on a single potential surface of ground electronic state of triatomic (or ionic) system containing identical nuclei is presented based on the formal multiarrangement channel collision theory. A properly symmetrized scattering matrix which is unitary over all arrangement channels is introduced. The matrix makes the formulation quite systematic. Furthermore, it enables us to extend the formulation efficiently to more general collision systems containing several species of the identical particles by partitioning the arrangement channels into collections of the indistinguishable ones. This is illustrated in the simple three particle atom–diatom system of the present work. Through the matrix, formulas of the optical theorem satisfied by the symmetrized cross sections are also obtained. The adiabatic electronic wave functions are found to be either symmetric or antisymmetric under interchange of nuclear coordinates. It is determined by the electronic states but is independent of the fermion or boson characteristic of the nuclear spin. Formulas of the properly symmetrized cross sections that arise from the nuclear spin statistics of the indistinguishable nuclei are obtained for the symmetric as well as the antisymmetric electronic wave function of the triatomic system. They are obtained in terms of either resultant nuclear spins of the molecules or individual nuclear spin quantum numbers. These cross sections are the only observable ones in the experiment. Depending on the interchange symmetries of the electronic wave functions with respect to nuclear coordinates, the observable cross sections of the reaction for indistinguishable nuclei result in quite different characteristics as manifested, for example, in the principle of detailed balance. When the scattering takes place through direct and exchange collisions coherently, either resultant nuclear spin conversion or rotational parity conversion between initial and final molecules is an indication that the reaction has taken place via exchange scattering for certain type of collisions. However, for other type of collisions, only the resultant spin conversion is possible in the reactive (exchange) scattering. When a collision proceeds through exchange scattering only, an unusual spin statistical factor appears in the symmetrized cross section. The factor is not only a function of the spin quantum number of the identical nuclei and rotational quantum number of homonuclear molecule, but is also dependent on the symmetry property of the electronic wave function mentioned above. Our discussions are mainly confined to the cases where either the electronic wave function is nondegenerate on the potential surface or no conical intersection is present when it is degenerate so that the reaction takes place through continuous single valued electronic wave function with respect to the nuclear coordinates. The cases where the conical intersection is present are only briefly discussed by applying the recent result to the present formulas.

Perturbative determination of nonadiabatic coupling matrix elements

Equations for the first- and second-derivative coupling matrix elements between the adiabatic electronic wave functions of a canonical Van Vleck approach are derived for the case of a single nuclear coordinate. Expressions for their order-by-order evaluation, analogous to the multidimensional perturbative expansion of the effective Hamiltonian (given elsewhere) are presented. Diagrammatic notations useful for enumerating certain components of the derivatives are introduced, and the diagrams required through first order are shown.

Theory of nonradiative decays in the non-Condon scheme

Without invoking the so-called Condon approximation, a theory is developed for the nonradiative decay rate constant in molecules. The electronic coupling matrix elements obtained by using adiabatic electronic wavefunctions are expressed in terms of commutators [p2, η], where p is the nuclear momentum operator and η is a function of nuclear coordinates. Evaluating commutator [p, η] at a fixed nuclear configuration, we obtain the rate constant within a Condon-like approximation. Assuming harmonic and displaced potential energy surfaces, the rate constant of the nonradiative decays where the promoting mode has a displacement is derived in the low temperature limit. The dependences of the non-Condon correction factors on the vibronic coupling constant γ, the displacement Δ, and the energy gap E are examined by performing model calculations. The correction factors decrease with increasing γ owing to the resonance effect near the potential surface crossing. The calculations also show that the rate constant in the non-Condon scheme exceeds that obtained invoking the Condon approximation by about two orders of magnitude. Finally, a simple upper bound to the decay rate constant is derived by applying a Condon-like approximation to the electronic coupling matrix elements. It is shown that in the weak coupling situation the evaluation of the electronic coupling matrix elements by the use of a Q-centroid method gives an upper bound to the rate constant.