Adiabatic Basis Research Articles

A theoretical study of resonance Raman scattering from molecules. II. Temperature effect

The temperature effect on the resonance Raman scattering from molecules is investigated. Considering a total system consisting of a molecular system, a heat bath, and the weak radiation field, a general expression for the relaxation processes is derived utilizing the Liouville operator technique for the time evolution of the density matrix of the total system. By using the perturbation method with respect to the radiation and matter up to second order, rate constants for zero photon (radiationless processes), one-photon and two-photon processes are derived after taking the long time limit in the Markoff approximation. Assuming a weak interaction between the molecular system and heat bath, a temperature-dependent resonance Raman scattering cross section is formulated in the adiabatic basis set for the molecular systems. In the displaced harmonic oscillator model, we obtain an analytical expression for the temperature-dependent Raman scattering cross section for the multimode case, which is suitable for both weak coupling and strong coupling cases. Analytical expressions for the Stokes and anti-Stokes bands of the jth order vibrational transition in the one oscillator model can easily be obtained from the general expression for the resonance Raman scattering cross section derived. An approximate Stokes to anti-Stokes ratio is given up to the second order of the dimensionless displacement Δ. The resonance Raman scattering cross section and the excitation energy profiles are numerically calculated as a function of physical constant such as temperature, molecular vibrational frequency and Δ. It is shown that the excitation profiles of the anti-Stokes band of the jth order vibrational transition have the similar envelope of the corresponding Stokes band shifted by jh/ω. The inverse Raman effect which is closely related to the ordinary Raman effect is also discussed.

How good is the adiabatic basis in heavy ion collisions?

Discusses the optimal choice of the basis for calculations of inner-shell excitations in the collision of heavy ions. Starting from a variational principle which minimises the time-integrated coupling strength, a relation between a model two-centre distance and the actual internuclear separation is derived. For 1s sigma excitations in Pb-Pb collisions it is found that the optimal basis is almost identical to the adiabatic one. For higher quantum states the adiabatic treatment becomes less adequate.

The adiabatic distorted wave infinite order sudden (ADWIOS) approximation for the calculation of inelastic molecular collision cross sections

A new approximate method is presented for the rapid calculation of vibrationally–rotationally inelastic molecular collision cross sections. The method treats the coupling between the rotational states, belonging to the same vibrational level, using the sudden approximation. The vibrational states are treated using an adiabatic basis. With this basis it is found that a first order distorted wave approximation gives good agreement with fully converged close coupling calculations. Some previously developed techniques are extended to permit reliable analytic expressions to be formulated for the angle fixed S matrix elements which appear in the theory. The great computational efficiency of the method is based on these analytic formulas which eliminate the need for the numerical solution of differential equations and for the numerical evaluation of difficult integrals. The angle fixed S matrices which result are then treated within the framework of the recently developed sudden approximation to yield inelastic cross sections. The method is applied to the calculation of vibrationally–rotationally inelastic cross sections for the He+H2 system and comparisons are made with close coupling calculations which are fully converged in both rotational and vibrational basis. Analysis of the results shows that for the He+H2 system the sudden approximation fails significantly within the n=1 vibrational manifold at total energies below 1.1425 eV but the validity of the approximation clearly improves with increasing energy. The He+He2 system was used in the present work as it was the only suitable one for which exact calculations were available in the literature for comparison purposes. For heavier systems the sudden approximation will be valid at much lower energies.

The use of rotationally and orbitally adiabatic basis functions to calculate rotational excitation cross sections for atom-molecule collisions

Four schemes for constructing rotational-orbital basis sets for rotational close coupling calculations for atom-molecule. collisions are considered and are applied to calculate 0 → 1, 0 → 2, and 0 → 3 rotational excitation cross sections for He+ HF at energies 0.05 and 0.017 eV. Adiabatic basis sets are shown to converge faster than conventional basis sets in all cases; in some cases the difference is very dramatic. Further, l-dominant bases are shown to be useful for the 0 → 2 and 0 → 3 cross sections.

Resonance Raman scattering from molecules with vibronically coupled excited states

Resonance Raman scattering (RRS) from molecules with an excited state perturbed by a nontotally symmetric vibration is theoretically studied in the adiabatic basis set. Analytical expressions for the RRS cross section and for depolarization ratio are derived in a displaced harmonic potential model.

The EF and GK1Σ g+ states of hydrogen : Calculation of nonadiabatic coupling

The mutual vibronic coupling between the EF and GK states of hydrogen has been computed in an adiabatic basis of electronic-vibrational states. The electronic matrix elements 〈 EF|▿ R| GK〉 and 〈 EF|Δ R| GK〉 are presented over the relevant range 2.0 ≤ R ≤ 4.0 a.u. and the vibronic coupling matrix is displayed for H 2. Nonadiabatic vibroic energies and B values are calculated for H 2, HD, and D 2 and compared with experimental data. The irregularities of the observed vibronic EF and GK progressions are well reproduced by the calculation. The remaining errors of the calculated energies grow systematically as the dissociation limit is approached, indicating the importance of additional nonadiabatic interactions with higher states and possibly of R-dependent errors of the Born-Oppenheimer energies of the EF and GK states at R values far from the potential minima.

A theoretical study of resonance Raman scattering from molecules

An expression for the resonance Raman scattering (RRS) cross section from large molecules is derived by means of a time independent Green function method. This expression is obtained in the adiabatic basis set, and both the radiative and nonradiative damping effects are explicitly considered, assuming the statistical limit in the excited vibronic states. Expanding the nuclear coordinate dependence of the transition moment, an analytical expression for the RRS cross section is derived for a displaced potential energy surface model. The expression obtained is applicable to the resonance Raman scattering not only in the weak coupling but also in the strong coupling case, and consists of the sum of the nth order vibrational transitions. It is shown that in the weak coupling case, the fundamental cross section may be explained as a succession of the absorption to the lowest level in the excited electronic state followed by the emission from it. Multimode effect on the RRS cross section which has not been reported before is studied and the analytical expression for the RRS cross section including this effect is obtained. The dimensionless displacement Δ dependence on the excitation energy profile is calculated. The effect of the nonadiabatic coupling (or the Born–Oppenheimer coupling) on the RRS cross section is also investigated. It is shown that using the adiabatic basis set, the BO coupling appears only in the high order approximations in most cases.

Two-state approximation in the adiabatic and sudden-perturbation limits

The properties of the two-state approximation are considered from the point of view of atomic collision theory in the limit of large and small values of a characteristic collision time $T$. For large $T$ (the adiabatic limit) asymptotically exact expressions are obtained for the elastic-scattering phase shifts and for the nonadiabatic transition probability due to the pseudocrossing of terms. This approximation is carried out under fairly general assumptions about the Hamiltonian, enabling us to consider such processes as transitions between $\ensuremath{\Sigma}\ensuremath{-}\ensuremath{\Pi}$ terms caused by rotation of an internuclear axis. Such general problems of the adiabatic approximation as the applicability of adiabatic perturbation theory, the introduction of a dynamical basis, and the properties of the electronic wave functions in the pseudocrossing region are discussed. For small $T$ (the sudden-peturbation limit) the evolution operator to zeroth and first order in $T$ is calculated. We introduce a general and unambiguous definition of an adiabatic basis as a basis of eigenvectors of the evolution matrix to zeroth order in $T$.

Rotationally and orbitally adiabatic basis sets for electron-molecule scattering

We have applied the close coupling method with conventional, l-dominant, and adiabatic basis functions to electron-molecule elastic scattering and rotational excitation. We show that accurate results may be obtained with small adiabatic basis sets.

Charge transfer in C6+-H(1s) collisions at low velocity

The cross section for charge transfer to fully stripped carbon ions in collision with ground-state hydrogen atoms is calculated in an adiabatic molecular-orbital basis. In contrast to previous work by Salop and Olson (1976), the cross section obtained by retaining only the dominant 4f sigma -5g sigma radial coupling is oscillatory. The inclusion of rotational coupling to other close-lying states damps out these oscillations to provide a smoothly varying cross section whose magnitude is considerably enhanced over that obtained from the two-state calculation.

Electronuclear basis for three-dimensional electronic nonadiabatic chemical reactions

The quantum mechnical theory of electronic nonadiabatic atom–diatomic molecule reactive collisions in three dimensions is discussed with emphasis on the reaction F(2P3/2,2P1/2)+H2(1Σ+g,v,j). It is known that the first two excited electronic adiabatic surfaces of the FH2 system are very close to the ground state surface at large F–H2 distances. Only the ground state surface, however, can lead to reaction at subdissociative energies. The importance of considering the exchange of flux between these surfaces is discussed. An electrorotational basis set {ΩJMα} is formulated for use in a quantum mechanical, reactive scattering calculation for systems of this type. These basis functions are formed by starting with good total angular momentum (nuclear plus electronic) kets in the space-fixed frame. Then, these functions are written in terms of the body-fixed natural collision (NCC) coordinates, and the Euler angles that orient the body frame with respect to the space frame. One feature of the {ΩJMα} setting them apart from other basis sets typically used in scattering calculations is that they cannot be factored into products of nuclear and electronic parts. It is then shown how the ΩJMα may be written in terms of adiabatic electronic basis functions used in the ’’diatomics-in-molecules’’ (DIM) treatment of atom–diatom systems. By writing the {ΩJMga} in terms of DIM basis functions, matrix elements of the type 〈ΩJMα′‖Ĥel‖ΩJMα〉 are evaluated as linear combinations of pure electronic matrix elements of Ĥel in the DIM basis. Four nonmixing parity types are found for the ΩJMα. Plots of selected matrix elements 〈ΩJMα′‖Ĥel‖ΩJMα〉 as functions of the reaction coordinate s and vibrational coordinate ρ are presented for F+H2. Matrix elements of T̂rot (the nuclear rotational kinetic energy operator expressed in NCC) are also evaluated in the electrotational basis. The method of conversion from the electrotational basis to an adiabatic electronic basis is formulated. This transformation should be useful after the system passes the zone where electronic transitions are significant.

Nonadiabatic interactions in unimolecular decay. II. Simplified formalism

Simplified semiclassical formulas are derived for the study of the nonadiabatic transition probability between two electronic states. They are expressed in terms of the coupling function g (R) equal to the matrix element of the d/dR operator between adiabatic states. It is shown that if g (R) keeps the same sign for all values of R, the integration of the classical-trajectory equations in the adiabatic basis depends on a particular function T (S) only, which is the well-known Massey parameter. This function plays a role similar to the t (s) function introduced by Delos and Thorson in their ’’close coupling model.’’ The present method is an extension of the treatment suggested by Rosen and Zener and assumes that ΔE, the energy difference between adiabatic energies, can be considered as constant. The significance of this is investigated. In simple cases, the transition probability can be expressed in terms of an adiabatic parameter χ only, defined by Eq. (32). Formulas are given for the isotopic effect, and for the influence of the excitation energy on the transition probability. In a polyatomic molecule, different kinds of couplings must be distinguished, depending on the nature of the internal coordinates which vary along the nuclear trajectory.

Asymptotic form of effective potentials of the Coulomb three-body problem in the adiabatic representation

For the three-body problem with a Coulomb interaction (two positively charged particles and one negatively charged particle) in the adiabatic basis mod i>, the first three coefficients in the asymptotic expansion of the effective potentials Uij(R) in powers of R-1 at R>>1 and leading terms in the expansion of Uij(R) in powers of R at R<<1 are obtained, where R is the distance between the positively charged particles.

On the non-adiabatic excitation of localized tunneling centers in solids

Tunneling centers almost rigidly move with their surrounding, whence a moving basis is adequate. An appropriately modified adiabatic basis is established, which is related to a well defined moving center. The non-adiabatic terms in the Hamiltonian are given. They cannot be neglected off-hand and contain momentum coupling terms to the lattice.

Three-dimensional natural coordinate asymmetric top theory of reactions: Application to H + H2

A particular partitioning of the Hamiltonian in natural collision coordinates is shown to lead to the use of hindered asymmetric top basis functions to represent all rotational motion (triangle tumbling plus internal bending) during reaction. These functions (along with perturbed Morse oscillator functions) are used as an adiabatic basis for expansion of the scattering wavefunction. The theory is discussed for both one and two reaction path potentials. The close coupled equations for the translational wavefunctions are then solved for the H + H2 reaction at total angular momentum J = 0. Wavefunction bifurcation and matching at the reactant–product boundary surface is considered in detail. Finally, numerical results (reaction probabilities, probability conservation, detailed balance, energy dependence of reactive S-matrix elements, probability density, wavefunction real part, and flux) are presented and comparisons are made with other quantum mechanical, semiclassical, and statistical reaction studies.

Nonadiabatic effects on vibronic intensities of larger polyatomic molecules in the preresonance regime

The approximations used to simplify the perturbation based theory of nonadiabatic interactions and their effects on vibronic intensities in the preresonance regime are examined for the larger polyatomic molecules. They include employment of a crude adiabatic basis set and certain approximate sum rules.

The exponential approximation for collinear reactive scattering

A representation of the S-matrix for reactive scattering is derived on the basis of the exponential approximation. The theory differs from previous distorted wave studies of reactive scattering by using an adiabatic basis set, defined in a reaction path coordinate system. Trial calculations are reported on H + H2 and the results compared with converged close coupling calculations. The comparison shows that the range of applicability of this approximation extends to higher energies when this type of basis is used.

Line shape of a molecular resonance

In this paper we consider some implications of intramolecular electronic relaxation on the optical line shape of large molecules in the statistical limit. General expressions for the line shape were derived utilizing the Green's function formalism, which account both for interference with background absorption and for interference between resonances. The energy dependence of the line-width functions was elucidated. We have demonstrated that the line shape for an ‘isolated’ resonance is a fanian; however, as the line profile index is determined by the ratio of the level spacing to the non-radiative width, the corrections to the lorentzian line shape are small in this case. We have established the equivalence of the physical description utilizing a three-level crude adiabatic basis and a two-level adiabatic basis. In the case of overlapping resonances the line shape may be recast in terms of a modified Fano-type formula where the line profile index is energy dependent. The nature of interference effects in ...

Intramolecular Nonradiative Transitions in the ``Non-Condon'' Scheme

In this paper we consider the nonradiative intramolecular decay of a large molecule utilizing Born-Oppenheimer wavefunctions as a zero order basis, and bypassing the conventional Condon approximation for the calculation of the electronic coupling matrix matrix elements. The electronic adiabatic wavefunctions are expanded in terms of the Wigner-Brillouin perturbation series in the weak electronic-vibrational coupling limit. We have applied a generalized version of Feynman's operator calculus to derive general expressions for the nonradiative decay probability of a statistical harmonic molecule characterized by displaced potential surfaces. Numerical calculations were performed for the decay of the vibrationless excited electronic state in the ``non-Condon'' scheme. The numerical data for the decay rate in a two electronic level system in the weak electronic vibrational coupling limit exceed the results obtained invoking the Condon approximation by 2–3 orders of magnitude; the exact correction factor depends on the molecular parameters, and, in particular, is roughly proportional to the square of the (frequency normalized) electronic energy gap. Finally, the relevant off resonance coupling terms in the adiabatic representation are shown to be appreciably smaller than the near resonance coupling terms, demonstrating the superiority of the adiabatic basis over the crude adiabatic basis in describing electronic relaxation processes.

Transition Probabilities in Molecular Collisions: Computational Studies of Rotational Excitation

Exact numerical solutions are presented for the coupled-state formulation of a rigid-rotor–structureless-atom collision, and are compared with qualitative ideas and semiquantitative decoupling approximations for the system. The advantages of using an adiabatic basis for the internal states of the system are demonstrated, in distorted-wave-type computations and in accounting for the direct part and the resonance energies in slow inelastic collisions, in the absence of curve crossing. The diabatic regime is also considered, and a diabatic basis is used for a curve-crossing problem. For these slow, head-on collisions, the Franck–Condon factors are not exponentially small, and significant (&amp;gt;10−1) transition probabilities are possible.