Classical mechanics adiabatic states

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

A detailed description of the nonadiabatic AIMD surface hopping method has been published elsewhere [15, 18, 21, 22] it shall only be summarized briefly here. We have adopted a mixed quantum-classical picture treating the atomic nuclei according to classical mechanics and the electrons quantum-mechanically. In our two-state model, the total electronic wavefunction, l, is represented as a linear combination of the S0 and 5) adiabatic state functions, < 0 and [Pg.267]

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

Another interesting idea to generahze the adiabatic states arises from Eq. (2.9). By replacing the quantmn mechanical momentum operators for nuclei (Pfc) with their classical coimterparts (P/t), one obtains another electronic Hamiltonian... 

Despite of its usefulness, however, incorporating quantum and classical mechanics in a consistent manner is a difficult problem. In the simplest case where nonadiabatic interactions are neghgible, the electronic state evolves adiabatically (i.e. keeps on a single PES) and the nuclear dynamics reduces to a Newtonian dynamics driven by the gradient of PES. This is what is called ab initio molecular dynamics (AIMD) approach (see Refe. [450, 451] for examples of chemical use of AIMD). 

In the current understanding of PCET reactions, both electron and proton are treated quantum-mechanically, and therefore the tunnelling probability must be accounted for both particles. In fact, concerted processes can be described as double tunnelling (proton and electron), with a single transition state. " For a description of the reaction coordinate, four adiabatic states (reactants, products and intermediates) described by paraboloids, are usually considered. The expression for the semi-classical rate constant in this case incorporates elements derived from electron and proton transfer theories... 

Further improvement of TST requires a critical analysis of its fundamental assumptions. The development of this theory was based on the following approximations (i) the distribution of energy in the reactants and transition state follows the Boltzmann distribution law, (ii) every system that crosses the transition state becomes a product and (iii) the movement along the reaction coordinate is adiabatic and can be described by classical mechanics. Next, we analyse each one of these approximations to identify the conditions in which they fail and see how the theory can be improved in terms of both generality and accuracy. 

---