Multidimensional theory, harmonic

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

Fortunately, it is relatively simple to estimate from harmonic transition-state theory whether quantum tunneling is important or not. Applying multidimensional transition-state theory, Eq. (6.15), requires finding the vibrational frequencies of the system of interest at energy minimum A (v, V2,. . . , vN) and transition state (vj,. v, , ). Using these frequencies, we can define the zero-point energy corrected activation energy ... 

The separability of the Hamiltonian in the normal mode form implies that the dynamics is in some sense trivial. One must only consider the continuum limit of a collection of independent harmonic oscillators and a single parabolic barrier. As described in Sec. III.D, this simple dynamics leads to some important relations between the Hamiltonian approach and the more standard stochastic theories. Multidimensional generalization of the parabolic barrier case will be discussed briefly in Sec. VIII. 

Obviously, the energy Vj, which is the potential energy surface for the nuclear motion, has a minimum value Vej at some equilibrium configuration Rea, where the set a comprises the rigid coordinates. In the vicinity of the minimum, the potential Vj is essentially quadratic, giving rise to a multidimensional harmonic oscillator. According to the standard theory, the frequencies of the harmonic oscillator scale as and the root-mean-square displacements from the Rea scale as k. Born and Oppenheimer have exploited the latter property by introducing new coordinates Qa for the displacement from equilibrium... 

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