Quantum-“classical” correspondence in a nonadiabatic transition system

Abstract

A nonadiabatic-transition system which exhibits "quantum chaotic" behavior [Phys. Rev. E 63, 066221 (2001)] is investigated from quasi-classical aspects. Since such a system does not have a naive classical limit, we take the mapping approach by Stock and Thoss [Phys. Rev. Lett. 78, 578 (1997)] to represent the quasi-classical dynamics of the system. We numerically show that there is a sound correspondence between the quantum chaos and classical chaos for the system. Nonadiabatic transition (NT) is a very fundamental concept in physics and chemistry [1,2]. In atomic, molecular, and chemical physics literature, NTs occur as a breakdown of the Born-Oppenheimer (BO) approximation, which is essentially an adiabatic approximation to solve quantum systems with many degrees of freedom. It is still a tough problem to analyze the properties of NTs especially for multidimensional systems. One reason preventing us from deeper understanding of NT is the lack of a naive quantum-classical correspondence for NT like tunneling phenomena [3,4]. One way to get the classical picture for a NT system is to go back to the original system before the BO approximation: Fuchigami and Someda investigated dynamical properties of H + 2 from classical points of view by treating an electron and nuclei as dynamical variables [5]. Though there is a full quantum study for such a small system to compare with [6], this "purist" way cannot be easily applied to much more "complex" systems.The mapping method recently advocated by Stock and Thoss [7] can circumvent this deficiency. (This is reminiscent of the Meyer-Miller method [8].) Their method is as follows: After the BO approximation, the discrete electronic degrees of freedom are mapped onto the Schwinger bosons [9]. Since all the degrees of freedom become just bosons, the total system is rather easily treated semiclassically or quasi-classically. Using this method semiclassically, one can obtain, e.g., absorption spectra even for a pyrazin molecule with 24 degrees of freedom [10]. One can use it quasi-classically by solving the equations of motion derived from a mapping Hamiltonian. This is a very easy way to simulate NT systems because the additional number of degrees of freedom for electronic parts is rather small. Using the periodic orbit theory [11] or the adiabatic switching method [12], one can obtain even quantum eigenenergies and eigenstates, in principle [13].On the other hand, multidimensional NT systems like Jahn-Teller molecules [14] are known to show "quantum chaotic" behavior [11]. Fujisaki and Takatsuka investigated this problem deeply employing the two-mode-twostate (TMTS) system which is considered as a system * Electronic address: fujisaki@bu.edu with two vibrational modes and two electronic states [15]. They calculated the statistical properties of the eigenenergies and eigenfunction for the TMTS system, and found that the system becomes strongly "quantum chaotic" under a certain condition. In addition, they showed that the chaos is not just a reflection of the lower adiabat...

Quantum-classical mechanics as an alternative to quantum mechanics in molecular and chemical physics

“…[98, 99]), there is an idea of the so-called quantum chaos, which has been actively studied, mainly theoretically, for several decades now (see Refs. [103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119]). Dozy chaos differs fundamentally in physical nature from quantum chaos.…”

“…Dozy chaos differs fundamentally in physical nature from quantum chaos. The term “quantum chaos” is generally understood to comprise all problems concerning the quantum mechanical behavior of classically chaotic systems [103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119]. In other words, systems, whose underlying classical dynamics is chaotic due to nonlinear interactions [120, 121, 122, 123], exhibit signatures of the chaos in their quantum mechanics.…”

“…Summarizing, we can say that the emergence of quantum chaos, for example, in elementary chemical processes, is theoretically considered within the framework of quantum mechanics [112, 113, 114, 115, 116, 117, 118, 119], and quantum chaos becomes a classical chaos when the transition from quantum mechanics to classical mechanics occurs. (Note that in this case of elementary chemical processes with quantum chaos, no clear explanations for or comparisons with the experiment were discussed.)…”