**Quantum-classical mechanics: Statistical approach to molecular collisions**

**Published:**28 April 2023 by

**MDPI**in

**The 1st International Online Conference on Mathematics and Applications**session

**Mathematical Physics**

**Abstract:**

As is known, the theory of quantum transitions in quantum mechanics is based on the convergence of a series of time-dependent perturbation theory. This series converges in atomic and nuclear physics. In molecular physics, the series of time-dependent perturbation theory converges only if the Born-Oppenheimer adiabatic approximation and the Franck-Condon principle are strictly observed. Obviously, in real molecular systems there are always at least small deviations from the adiabatic approximation. Within the framework of quantum mechanics, these deviations lead to singular dynamics of molecular quantum transitions. The only way to eliminate this singularity is to introduce chaos into the electron-nuclear dynamics of the transient state. As a result of the introduction of chaos, we no longer have quantum mechanics, but quantum-classical mechanics, in which the initial and final states are quantum in the adiabatic approximation, and the transient chaotic electron-nuclear(-vibrational) state is classical due to chaos, and the transitions themselves are no longer quantum, but quantum-classical [1,2]. This procedure for introducing chaos into the transient state was done in the simplest case of quantum-classical mechanics, namely, in the case of quantum-classical mechanics of elementary electron transfers in condensed media. Chaos is introduced by replacing the infinitely small imaginary additive in the energy denominator of the total Green's function of the "electron + nuclear environment" system with a finite value [1,2]. This chaos is called dozy chaos, and quantum-classical mechanics is also called dozy-chaos mechanics. The analytical results obtained in this new fundamental physical theory make it possible to explain a large number of experimental data, for example, on the shape of the optical bands of polymethine dyes and their aggregates [2]. The simplicity of the case of quantum-classical mechanics of elementary electron transfers in condensed media and the possibility of obtaining the corresponding analytical result are connected, in particular, with the possibility of neglecting local oscillations of nuclei and taking into account only non-local oscillations in the theory. There is another "simple" problem in the quantum-classical mechanics of complex physical systems, where a similar success in the application of analytical methods can be achieved. This problem is the problem of molecular collisions in gases, which has applications to monomolecular reactions at low pressures. If in the problem of elementary electron transfers in condensed media the electronic state changes significantly, then during such molecular collisions in gases, the electronic states of the molecules do not change, and it is only necessary to take into account the redistribution of vibrational energy between local vibrations in colliding polyatomic molecules. In this case, the transient chaotic state of the motion of nuclei that occurs during molecular collisions can be described by statistical methods based on the use of the microcanonical distribution for molecular collisions [3]. Whereas in the problem of elementary electron transfer in condensed media the singular dynamics of the transient state is damped by dozy chaos, in this statistical approach to molecular collisions in gases, the dynamics of energy redistribution between local vibrations in colliding polyatomic molecules is taken into account by separating all modes into active and passive modes. Active modes include low-frequency vibrational modes and rotational modes that rapidly exchange energy at the moment of collision. Passive modes include high-frequency vibrational modes, which are effectively included in the process of energy redistribution after the elementary act of molecular collision has already been completed. Analytical results are obtained for the distribution function of the probability of energy transfer in collisions of molecules (canonical distribution for molecular collisions), as well as for all moments of the N-th order of the distribution function, which have the form of certain polynomials of the N-th order.

[1] Egorov, V.V. Quantum–classical mechanics as an alternative to quantum mechanics in molecular and chemical physics. Heliyon Phys. **2019**, 5, e02579-1–e02579-27.

[2] Egorov, V.V. Quantum–classical mechanics: Nano-resonance in polymethine dyes. Mathematics **2022**, 10(9), 1443-1–1443-25.

[3] Lifshitz, E.M.; Pitaevskii, L.P. *Physical Kinetics (Course of Theoretical Physics, Volume 10)*; Butterworth-Heinemann: Oxford, 2012.

**Keywords:**dozy-chaos mechanics; molecular collisions; monomolecular reactions; gases at low pressures; microcanonical distribution for molecular collisions; canonical distribution for molecular collisions