Complex States of Simple Molecular Systems

Abstract

A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.When the stationary Born-Oppenheimer description for nearly degenerate state is regarded as "embedded" in a broader dynamic formulation, namely through solution of the time dependent Schrödinger equation, the wave function becomes necessarily complex. The analytic behavior of this function in the complex time-plane can be exploited to gain information about the connection between phases and moduli of the componentamplitudes. One form of this connection are a pair of reciprocal (Kramers-Kronig type) relations in the complex time-domain. These show that certain phase changes in a component amplitude (such as, e.g., lead to a non-zero Berry-phase) require changes in the amplitude-moduli, too, and imply degeneracies of the electronic state.The subject of conical degeneracy, or intersection, of adjacent potential surface is extended to nonlinear nuclear-electronic coupling, which can then result in multiple conical degeneracies on a nuclear coordinate plane. We treat cases of double and fourfold intersections, the latter under trigonal symmetry, and employ an analytic-graphical phase tracing method to obtain the resulting Berry phases as the system circles around some or all of the degeneracy points. These phases can take up values that are all integral multiples of π, with integers that vary with the physical situation (or with the model postulated for it).A further type of invariance, with respect to gauge transformation, is also tied to the complex form of the wave function. For a many-component state this leads to a consideration of tensorial (Yang-Mills type) fields F for molecular systems. It is shown that it is the truncation of the Born-Oppenheimer electron-nuclear superposition that generates the molecular Yang-Mills fields.The equations of motion for the nuclear degrees of freedom are derived from a Lagr...

Variational procedure for time-dependent processes

“…We shall now apply the proposed variational procedure to yield, in one case exactly and in another case approximately, the solution for a (hermitian) Hamiltonian that has a periodic variation. The cases chosen are such that analytical solutions are known exactly ([37]- [39]), so that we can compare to them the variational solutions to be obtained here. Specifically, we consider the time development of a doublet subject to a Schrödinger equation whose Hamiltonian in a doublet representation is…”

The adiabatic‐to‐diabatic transformation angle and the berry phase for coupled jahn–teller/renner–teller systems: The F + H2 as a case study

“…In contrast to the just mentioned ADT treatment by Werner et al for the F + H 2 system, most of the treatments of other molecular systems are done by applying the original BO‐NACTs 1–14, 18–23, 29–47. This study for the F + H 2 system is the first that uses BO‐NACTs for the diabatization of this system.…”

Time‐dependent molecular fields created by the interaction of an external electromagnetic field with a molecular system

“…We complete this section with two comments: The derivations in this section apply as long as the electronic eigen‐functions are analytic functions at every point in the region of interest. In case, certain eigen‐functions are not analytic at some points in this region, for example, at conical intersections (ci), the respective τ ‐matrix elements are singular h, and, therefore, their derivatives are not defined (such points are designated as pathological points). Abelian situations are encountered only at the vicinity of conical intersections, the reason being that close enough to these points the Hilbert spaces are formed by two states only, namely, the two states that form the singular NACT. …”