Quantum and Classical Degrees of Freedom Proton Transfer

QUANTUM AND CLASSICAL DEGREES OF FREEDOM. PROTON TRANSFER 

On the other hand, if AE kT, the system, when going over to excited particles, loses much more in the excitation probability exp(-E/kT) than it gains in the tunneling probability. For such systems, the most advantageous path is the tunneling from the ground state without a preliminary thermal excitation. Such systems are called quantum-mechanical. 

A classical transition is associated with surmounting the barrier. The probability of this transition is determined by the probability of a fluctuation with the required energy. This explains an exponential, activation nature of the temperature dependence of the process rate. Quantum-mechanical transition occurs from the zero level and is independent of temperature. However, a quantum transition requires a preliminary equalization of the levels of the initial and the final states. This equalization requires the surmounting of the barrier of the classical subsystem, which results in 

For all these particles, the transition probability P(E) = 1, while the Boltzmann factor is practically the same. For higher values of E, the Boltzmann factor becomes smaller. 

It should be emphasized that classical or quantum-mechanical nature of a system is not an inherent property of the system under any conditions. There is always a competition between two ways in which the process can occur, and the same system may behave as a purely classical system at sufficiently high temperatures and as a purely quantum system at sufficiently low temperatures. Dogonadze and Kuznetsov[217] explained just in this way the transition from the Arrhenius dependence of the reaction rate to its temperature independence at low temperatures observed for some reactions, in particular, for redox transformations of cytochrome C[244]. For a sufficiently low value of kT, the characteristic energy for all vibrational modes essential for the reaction turns out to be greater than kT, and all transformations occur quantum-mechanically and not classically. The temperature independence of the reaction rate is not the only possible peculiarity of the behavior of low-temperature systems. Other unusual temperature dependences are also possible[217]. Of course, there are also intermediate cases (AE kT) when tunneling dominates though not from the ground level but from excited levels. 

---