Curve-crossing Problems Research Articles

Dynamics of Nonadiabatic Chemical Reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.

Rayleigh–Schrödinger perturbation theory for coupled oscillators. III. Curve-crossing problems

The Rayleigh–Schrödinger perturbation theory approach developed previously for evaluating nonadiabatic corrections to the adiabatic energy levels of a system of two coupled oscillators is generalized to the case of the so-called “mixed” representations which arise from the diabatic representation of a given problem by performing a unitary transformation on the diabatic potential energy matrix (the adiabatic representation is obtained as a special case with a purely diagonal potential energy matrix). Different representations provide different coupling conditions and, consequently, different bases for evaluation of the perturbation corrections. This is reflected, quite generally, in the convergence and summability properties of the perturbation series and can thus be used to improve the accuracy and stability of the perturbation calculations. The latter possibility is especially important in the case of closely coinciding levels. Model calculations have revealed that changing representations may allow the determination of the energies of these levels to a high degree of accuracy even in the case of strong perturbation resonances.

A wavepacket — path integral method for curve-crossing problems: application to resonance Raman spectra and photodissociation cross sections

A multidimensional Gaussian wavepacket dynamics — path integral method (GWD-PI) is utilized to study time evolution on two nonadiabatically coupled anharmonic potential energy surfaces. The proposed technique relies on single surface propagations in which the nuclear coordinate wavepacket hops from one potential surface to the other at selected times. Appropriate sequences of hopping times are considered and the output wavepackets associated with each sequence are summed coherently. Good agreement is obtained between GWD-PI and exact quantum dynamics for a two-dimensional model of photodissociation, even in the case of strong and spatially dependent nonadiabatic coupling.

Tunneling dynamics in dissipative curve-crossing problems.

We have studied the real-time dynamics of dissipative tunneling between two coupled diabatic surfaces. The interplay of tunneling, vibrational relaxation, dephasing, and resonances leads to various regimes of population dynamics, from coherent oscillations between the surfaces to exponential decay described by a rate constant. To describe these phenomena, we exploited recent efficient discretized-path-integral techniques; the two-surface system is treated exactly using a numerically constructed propagator, while the effects of the dissipative environment are incorporated through an influence functional. This procedure results in a low-dimensional integral, which is evaluated by quadrature. Use of a discrete-variable representation (DVR) allows us to replace continuous paths by paths that sample a small set of coordinate points; in particular, the conventional two-level-system approximation is shown to correspond to the limit of two DVR points. The specific features pertinent to nonadiabatic tunneling are discussed.

Beyond the time-dependent Hartree grid approximation for curve-crossing problems

A new ‘‘configuration-interaction’’ method is presented which extends the single-surface time-dependent Hartree grid (TDHG) approximation into a formally exact algorithm for obtaining multidimensional quantum wave-packet dynamics on nonradiatively coupled electronic potential surfaces. As a numerical example, photofragmentation cross sections are computed for a two-degree-of-freedom model of direct dissociation. For systems prepared in vibrationally excited states of the ground electronic potential the TDHG approximation fails due to ‘‘direct correlation’’ effects, while our method provides accurate results.

Applications of the generator coordinate approximation to diatomic systems. III. Curve-crossing problems

The generator coordinate approximation, previously applied to vibration–rotation levels near potential-energy minima, is now worked out for curve-crossing situations. We define the weak and strong adiabatic coupling limits. For weak adiabatic coupling both the adiabatic and generator coordinate approximations become exact. In the strong adiabatic coupling limit the adiabatic approximation breaks down, whereas the generator coordinate approximation again reproduces the exact solutions. These theoretical results are confirmed by calculations for a Hamiltonian modeled to the EF,GK 1Σ+g curve crossing in the electronic spectrum of the hydrogen molecule.

Quantum time evolution of vibrational states in curve-crossing problems

The quantum time evolution in a two-state curve-crossing situation can be computed with the split-operator FFT method. We present an improved version of that scheme, based on the analytic exponentiation of two-by-two matrices. We use this approach to investigate the vibrational dynamics of the coupled b′, c′ 1Σ + u states of the N 2 molecule.

Uniform semiclassical treatment of the radial coupling term in the adiabatic basis: Application to the excitation transfer He(1S)+Ne2+(1D)-->He(1S)+Ne2+(1S).

We present a uniform semiclassical analysis of the radial coupling A(r)(d/dr) in the adiabatic basis. Various approximations are tested for solving the perturbative form of the probability integrals. Our model is applied to the excitation transfer He(/sup 1/S)+Ne/sup 2/+(/sup 1/D)..-->..He(/sup 1/S)+Ne/sup 2 +/(/sup 1/S). The results we obtain are in good agreement among themselves and provide an alternative model in the adiabatic frame to complete the current Landau-Zener-Stueckelberg--like calculations. The dominating role of the turning points in such curve-crossing problems is also demonstrated.

Non-adiabatic transitions between two groups of interesting energy levels

The problem of non-adiabatic transitions between two groups of pseudocrossing potential energy curves is formulated in terms of two interacting quasistationary states. The method developed is applied to the multiple curve-crossing problems appearing in the A-+BC+( nu ) to A+BC( nu ') and A*(n)+A to A*(n')+A reactions. Closed-form expressions for the transition probabilities between two given states are derived, with account being taken of the effect of other states under realistic assumptions for the curve-crossing parameters.

An algorithm for the evaluation of the complex Airy functions

The evaluation of complex Airy functions is required in the approximation of certain second-order linear differential equations arising in the treatment of multiple turning-point and energy curve-crossing problems in quantum mechanics. Pairs of numerically linearly independent solutions throughout the z-plane can be constructed from the fundamental solutions to the complex Airy equation, Ai(z), Bi(z), and Ai( z e ±2 πi/3 ). Integral representations for these complex functions and their derivatives are given, and being of the Stieltjes type, the integrals are evaluated using the generalized Gaussian quadrature method of Shohat and Tamarkin as implemented by Gordon. These integral representations, employed together with the Taylor series for small z and the appropriate connection formulas, allow the creation of an accurate and efficient algorithm to evaluate the complex functions over the entire z-plane. The algorithm is presented in detail at the end of this article.

Semiclassical theory of tunneling and curve-crossing problems: a diagrammatic approach

It is shown how established local solutions of the barrier penetration and curve-crossing problems may be linked together to give simple uniform analytical descriptions of a wide variety of spectroscopic phenomena. Applications to inversion doubling, restricted rotation, predissociation by rotation and by curve-crossing, continuum interactions and molecular perturbations are described. In each case the necessary computations may be reduced to one-dimensional quadratures.

Strong-Coupling Semiclassical Methods: Phase-Corrected Average Approximation for Atom-Atom Collisions

A method is presented for improving the phase of the semiclassical radial functions obtained from the average approximation''. It yields improved phases for problems possessing localized regions of strong coupling, for example in curve-crossing problems. For problems without such regions, the method yields essentially the same results as the average approximation. Results are presented for atomic-scattering situations in both close-coupling and distorted-wave theory. (auth)

Non-adiabatic transitions

Known limitations of the Stuckelberg-Landau-Zener formula are described in terms of an analogy between the curve-crossing and barrier penetration problems. Available perturbation and other correction methods are reviewed. Applications of the theory to the interpretation of molecular predissociation, collision cross-section and kinetic rate constant measurements are outlined.

Role of Potential Curve Crossing in Subexcitation Molecular Collisions: Exact (Two-State) Computations vs Decoupling Approximations for Resonance Positions

Exact numerical solutions for two-state curve-crossing problems in the subexcitation region are presented and discussed in terms of various decoupling schemes. These provide an approximate, single-channel description of the problem. The energies of the bound states in the (decoupled) closed channel are used as approximations for the resonance positions. The transition from a weak to a strong coupling regime is discussed and a treatment of the resonances in the intermediate coupling region using a diabatic basis is provided.