Coupled-Channels Quantum Theory of Electronic Flux Density in Electronically Adiabatic Processes: Fundamentals

Abstract

The Born-Oppenheimer (BO) description of electronically adiabatic molecular processes predicts a vanishing electronic flux density (j(e)), <j(e, NCM) (x,t) >=1/2∫dR[Δ(b) (x;R) - Δ(a) (x;R)]<j(b,a) (R,t)> even though the electrons certainly move in response to the movement of the nuclei. This article, the first of a pair, proposes a quantum-mechanical "coupled-channels" (CC) theory that allows the approximate extraction of j(e) from the electronically adiabatic BO wave function . The CC theory is detailed for H(2)(+), in which case j(e) can be resolved into components associated with two channels α (=a,b), each of which corresponds to the "collision" of an "internal" atom α (proton a or b plus electron) with the other nucleus β (proton b or a). The dynamical role of the electron, which accommodates itself instantaneously to the motion of the nuclei, is submerged in effective electronic probability (population) densities, Δ(α), associated with each channel (α). The Δ(α) densities are determined by the (time-independent) BO electronic energy eigenfunction, which depends parametrically on the configuration of the nuclei, the motion of which is governed by the usual BO nuclear Schrödinger equation. Intuitively appealing formal expressions for the electronic flux density are derived for H(2)(+).

An open‐source framework for analyzing N‐electron dynamics. I. Multideterminantal wave functions

“…As the electronic states are real‐valued, the diagonal terms , that is, the adiabatic flux density, vanish. The same argument holds for all matrix elements in eq.…”

Electronic and nuclear flux analysis on nonadiabatic electron transfer reaction: A view from single‐configuration adiabatic born–huang representation

“…It is not a difficult task to quantify in terms of Γ( r , Q , t ) how much electrons are migrated in these transitions. However, it is far from a trivial problem to construct electron flux to represent the charge migration, which can happen even in the Born–Oppenheimer dynamics, remaining . As for the “electronic flux” j el , conversely, it always holds j el ( r , Q , t ) = 0 for the single‐configuration Born–Oppenheimer state.…”

“…Þ. [39][40][41][42][43][44] As for the "electronic flux" j el , conversely, it always holds j el (r, Q, t) = 0 for the single-configuration Born-Oppenheimer state. Also, it is noticed by comparison j el (r, Q, t) in Figure 4 and J nu (r, Q, t) in Figure 3b for the full nonadiabatic calculations that the former is far smaller than the latter down to the order of 10 −2 .…”

“…Since this state gives a standing (stationary) wave, information about the time‐direction and thereby direction of electron flow is already canceled away. Diestler and coworkers have developed their coupled‐channel theory to detect the electronic‐state variation synchronously transported by the nuclear motion in general coordinate systems including those of the laboratory‐frame . More recently, Pohl and Tremblay and Engel and coworkers have given extensive discussions from symmetry‐breaking and a flux–flux reflection principle, respectively.…”

“…Diestler and coworkers have developed their coupled-channel theory to detect the electronic-state variation synchronously transported by the nuclear motion in general coordinate systems including those of the laboratory-frame. [39,40] More recently, Pohl and Tremblay [41] and Engel and coworkers [42] have given extensive discussions from symmetry-breaking and a flux-flux reflection principle, respectively. Back in 2009, Okuyama and Takatsuka have proposed time-shift flux, in which time information is externally introduced in terms of molecular deformation.…”