Classical bath

We consider a classical equilibrium system of independent harmonic oscillators whose positions and velocities are denoted XjVy = iy, respectively. In fact, dealing with normal modes implies that we have gone through the linearization and diagonalization procedure described in Section 4.2.1. In this procedure it is convenient to work in mass-normalized coordinates, in particular when the problem involves different particle masses. This would lead to mass weighted position and 

5 The normal modes are derived from the mass weighted coordinates, therefore u has the dimen-sionality [Z][m] /2 [jj system of identical atoms it is sometimes convenient to derive the normal modes from the original coordinates so as to keep the conventional dimensionahty of m and u. In this case the Hamiltonian is H = (/w/2) (m + Eqs (6.78) take the forms 

The coefficient in front of the exponent is a nonnalization factor. This leads to 

This bath is characterized by the density of modes function, g(u)), defined such that g(ai)dcei is the number of modes whose frequency lies in the interval Cf).a + day. Let 

Such correlation functions are often encountered in treatments of systems coupled to their thennal environment, where the mode 1 for the system-bath interaction is taken as a product of A or B with a system variable. In such treatments the coefficients Cj reflect the distribution of the system-bath coupling among the different modes. In classical mechanics these functions can be easily evaluated explicitly from the definition (6.6) by using the general solution of the harmonic oscillator equations of motion 

In the second equality we have used the unitarity of T. Using also Jlj (T )jkTig = 1 we get from (6.85) 

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This model permits one to immediately relate the bath frequency spectrum to the rate-constant temperature dependence. For the classical bath (PhoOc < 1) the Franck-Condon factor is proportional to exp( —with the reorganization energy equal to... 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

It is easiest to formulate this problem in the case of a single high-frequency vibrational mode, or chromophore, so let us consider this situation first. For the absorption line shape, which involves only the ground and excited state of the chromophore, a cmcial element is the 0 —> 1 transition frequency and its dependence on the classical bath coordinates. Second, one needs (in the case of IR spectroscopy) the projection of the transition dipole in the direction p of the electric field axis. This projection can depend on bath coordinates in two ways. 

The instantaneous OH frequency was calculated at each time step in an adiabatic approximation (fast quantal vibration in a slow classical bath ). We applied second-order perturbation theory, in which the instantaneous solvent-induced frequency shift from the gas-phase value is obtained from the solute-solvent forces and their derivatives acting on a rigid OH bond. This method is both numerically advantageous and allows exploration of sources of various solvent contributions to the frequency shift. 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which (if we recall that in conventional units it is... 

The energy conservation and time-reversibility of the obtained propagation scheme is tested by perfoming a simulation of a quantum oscillator coupled to a classical bath consisting of 79 argon atoms. The RPS guarantees an improved... 

As an application of this formalism, we consider a two-level quantum system coupled to a classical bath as a simple model for a transfer reaction in a condensed phase environment. The Hamiltonian operator of this system, expressed in the diabatic basis L), P), has the matrix form [43]... 

In order to extend the linearization scheme to non-adiabatic dynamics it is convenient to represent the role of the discrete electronic states in terms of operators that simplify the evolution of the quantum subsystem with out changing its effect on the classical bath. A way to do this was first suggested by Miller, McCurdy and Meyer [28,29[ and has more recently been revisited by Thoss and Stock [30, 31[. Their method, known as the mapping formalism, represents the electronic degrees of freedom and the transitions between different states in terms of positions and momenta of a set of fictitious harmonic oscillators. Formally the approach is exact, but approximations (e.g. semi-classical, linearized SC-IVR, etc.) must be made for its numerical implementation. 

R J. Rossky (1998) Nonadiabatic quantum dynamics simulation using classical baths. In G. Ciccotti B. Berne, and D. Coker, editors, Classical and quantum dynamics in condensed phase simulations. World Scientific, Dordrecht, p. 515... 

A more interesting point was made by Bader and Berne, who noted that the vibrational relaxation rate of a classical oscillator in a classical harmonic bath is identical to that of a quantum oscillator in a quantum harmonic bath [71]. On the other hand, when the relaxation of the quantum system is calculated using the corrected correlation function of the classical bath [Eq. (31)], the predicted rate is slower by a factor of j/3h(i) coth(/3h(o/2), which can be quite substantial for high-frequency solutes. The conclusions of a number of recent studies were shown to be strongly affected by this inconsistency [42,43,72]. Quantizing the solvent by mapping the classical correlation functions onto a quantum harmonic bath corrects the discrepancy. 

Polypropylene (PP) fibers having a nonpolar paraffinic character are generally undyeable by the classical bath-dyeing method and therefore the substantial part of the PP fiber production is colored with pigments (mass dyed fibers). Only a small part of PP fibers is dyed after prelinu-nary modification. 

The dyeing of unmodified PP fibers is very rarely applied in industry. The classical bath-dyeing procedure utilizes some structures of disperse dyes with a long linear hydrocarbon chain, e.g. anthraquinone dye structure with alkyl chain beyond C q-C j units. 

Proteins, however, must function at biological temperatures, and to be useful, the Davydov soliton must survive at these temperatures. The first difficulty faced by the Davydov/Scott model was the question of the thermal stability of the Davydov soliton. The Davydov/Scott Hamiltonian includes two systems one, the amide I vibration, is treated as a quantum mechanical entity and the second, the vibrations of the peptide groups as a whole (or the changes in the hydrogen bond lengths) are very often treated classically, an approximation that shall be designated here as the mixed quantum-classical approximation. The first simulations of the Davydov/Scott model at finite temperature were performed within the mixed quantum/classical model and coupled the classical part of the system to a classical bath. The result was that the localized excitation dispersed in a few picoseconds at biological temperatures. However, this result clashed with another obtained in Monte... 

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