Quantum mode

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore... 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

Ulstrup J. and Jortner J. (1975), The effect of intramolecular quantum modes on free energy relationships for electron transfer reactions , J. Chem. Phys. 63,4358-4368. 

Here, a single effective quantum mode, cO , is assumed to contribute to i,. The Huang-Rhys factor, 5 = Xjfi(a, is a measure of the electron-vibrational coupling interaction. 

Rate formulations that treat the inner-sphere mode(s) quantum mechanically and the outer sphere modes classically are used rather widely. The rate expression for a single harmonic quantum mode is... 

An important achievement of the early theories was the derivation of the exact quantum mechanical expression for the ET rate in the Fermi Golden Rule limit in the linear response regime by Kubo and Toyozawa [4b], Levich and co-workers [20a] and by Ovchinnikov and Ovchinnikova [21], in terms of the dielectric spectral density of the solvent and intramolecular vibrational modes of donor and acceptor complexes. The solvent model was improved to take into account time and space correlation of the polarization fluctuations [20,21]. The importance of high-frequency intramolecular vibrations was fully recognized by Dogonadze and Kuznetsov [22], Efrima and Bixon [23], and by Jortner and co-workers [24,25] and Ulstrup [26]. It was shown that the main role of quantum modes is to effectively reduce the activation energy and thus to increase the reaction rate in the inverted... 

We next consider the case in which the electron-transfer terms are small so that the first-order perturbation theory can be applied and assuming that all the phonon modes are classical - quantum modes will be considered below. In this case, the coordinates of the oscillators can be treated as external parameters. It is then convenient to include the electron-phonon coupling - the last term in Eq. (12) - into the electronic energy of the reactant, and rewrite the first and last term as... 

When quantum modes are reorganized during the electron exchange, they have to be treated separately from the classical modes. We consider explicitly the case in which one such mode is present the generalization to many modes is straightforward, though it results in rather cumbersome formulae. 

The situation is simplest if the frequency ft), of the quantum mode is so high that its quantum of energy is much higher than the thermal energy, ficoi / > kT, so that it is not excited in the initial state. In this case, we only have to consider transitions from the ground state 0 to the nth excited state. Let... 

When the quantum of energy is of the same order of magnitude as the thermal energy, fico kT, we have to consider transitions from excited initial states m to final states n, and perform a thermal average over the initial states of the quantum modes ... 

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