Polyatomic systems subsystems

We consider a microscopic polyatomic system consisting of N nuclei and n electrons (1-4). Let the positions of the nuclei be described by the radius vectors Rx (a = 1,..., N). If the polyatomic system is free of external force, the total linear momentum is conserved and thus its center of mass moves with a constant velocity vector (5). Consequently, a new coordinate system with its origin fixed at the center of mass can be introduced (the center-of-mass coordinate system), where the description of the polyatomic system can be simplified. Since the position of the center of mass of the entire polyatomic system practically coincides with the position of the center of mass of the nuclear subsystem, the number of the degrees of freedom, F, of the nuclei in the center-of-mass system can be reduced by 3 due to the translation of the center of mass, and by 3 connected with the overall rotation about the center of mass (in case of a linear polyatomic system, the reduction due to the overall rotation is only by 2) (5) i.e., the number of independent nuclear coordinates is F = 3N — 6 (3N — 5). The radius vectors Rx can be then expressed in terms of F generalized coordinates Q (5) ... 

Equation (7) represents a crucial result for theoretical chemistry if the electronic wave function q) is stationary while the time evolution of the polyatomic system is in progress, the nuclei move in the field of force, the potential of which is equal to the energy of one of the eigenstates of the electronic subsystem. In this connection, the potential function Wm = Wm(Q) is referred to as the potential energy surface (PES) corresponding to the mth electronic state of the polyatomic system (10-12). 

In this section we will examine conditions implying stationary states of the electronic subsystem during the time the evolution of the entire polyatomic system is in progress. The nuclear motion will be treated classically (20,21). 

If simultaneously W k = 0 and <

= 0 in a region of M, then the nuclei move in the field of force the potential of which is given by the energy Wm of a single state of the electronic subsystem and there is no difference between the description of the electronic subsystem in the adiabatic or diabatic basis sets, i.e., Wam = W, . The corresponding behavior of the polyatomic system is referred to as electronically adiabatic ... 

However, if the region of nonadiabatic behavior is well localized in M, the classical treatment of the nuclear subsystem can be preserved and the evolution of the polyatomic system in the nonadiabatic region (where no potential energy surface is defined ) is to be described by special means. 

The effective-mode construction can thus be employed both for a discrete set of vibrational modes (e.g., in a polyatomic molecule) and for typical system-bath type situations where the spectrum of bath modes is dense. In the latter case, the environment and its coupling to the electronic subsystem are entirely characterized by a spectral density. Approximate spectral densities can be constructed from few effective modes, representing a simplified realization of the true environmental spectral density that is designed to give a faithful representation of the dynamics on short time scales [32,33]. Thus, even a highly structured, multi-peaked spectral density can be reduced to an effective, simplified spectral density on ultrafast time scales. Importantly, the procedure converges, as has recently been shown in Ref. [34]. 

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