Inclusion of the geometric phase in quantum reactive scattering calculations: A variational formulation

Abstract

We present a method for including the geometric phase in quantum reactive scattering computations based on the log derivative version of the Kohn variational principle. A new variational functional is developed which includes the influence of the geometric phase through modifications in the momentum operators. The system investigated is a two-dimensional reactive scattering model which includes the vector potential induced by the magnetic field of an infinitely long solenoid. The coordinates used in this model are analogous to Jacobi coordinates used in atom–diatom systems. Some interesting features of this study include the gauge invariance of the scattering probabilities, symmetry adaptation of the wave function, and the behavior of the probability density in the presence of the geometric phase.

Geometric phase effects on transition-state resonances and bound vibrational states of H3 via a time-dependent wavepacket method

“…First, by introducing a vector potential into the nuclear Schro dinger equation such that the electronic wavefunction becomes complex single valued.28h30,37 Such an approach can be extended to the more general conical intersection problem. Secondly, by using a real double-valued basis with the appropriate half-odd integer angular momenta as done by Kuppermann and co-workers14h18, 32,33,36 In this approach the e †ects are incorporated directly into the basis, and the resulting nuclear Schro dinger equation does not contain a vector potential. For the present work, we have adopted the BillingÈMarkovic Ïs special FFT technique, which has been utilized in quantum scattering calculations19 and uses hyperspherical coordinates.…”

The Electronic Adiabatic-Diabatic Transformation Matrix: A Theoretical and Numerical Study of a Three-State System

“…1,2 Recently, however, it has become evident that molecular processes taking place on a given electronic state may be significantly affected by states that are far above that state. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] In particular, two recent studies in which results of single-surface and two-surface scattering processes were compared showed undoubtedly significant discrepancies. 17,18 These studies also showed that singlestate results can be improved by employing an extended version of the ordinary Born-Oppenheimer (BO) single-state equation, 17,19,20 which contains the nonadiabatic coupling terms that are responsible for the effects due to higher states.…”

Regularized Born-Oppenheimer molecular dynamics

“…Importantly, since both the vector potential ∇ζ and the electronic potential are now singular, the nuclear problem is essentially equivalent to the Aharonov-Bohm problem, thereby leading to a particularly relevant Berry phase effect, known as the molecular Aharonov-Bohm effect. In this case, ∇ × ∇ζ is usually represented as a delta function [26,29] and thus the standard Hamilton-Jacobi analogy leads to intractable classical trajectory equations which prevent a molecular dynamics approach and lead to the necessity of solving the Schrödinger equation 4for the entire nuclear wavefunction [26,57]. In this context, a standard approach is to approximate and truncate an expansion of Ω over the basis set provided by Gaussian coherent states [19,33,42].…”