Quantum reaction dynamics, electronic states equation

The potential energy surface is the central quantity in the discussion and analysis of the dynamics of a reaction. Its determination requires the solution of the many-body electronic Schrodinger equation. While in the early days of theoretical surface science quantum chemical methods had a significant impact, nowadays electronic structure calculations using density functional theory (DFT) [20, 21] are predominantly used. DFT is based on the fact that the exact ground state density and energy can be determined by the minimisation of the energy functional E[n ... 

The electronic states associated to the cyclic and oscillatory reactions, reactions with instabilities, etc. are obeying for the discrete state dynamics to the quantum general evolution equation (QME quantum master equation) (see Gray Scott, 1990 van Kampen, 1987 Gardiner, 1994 Risken, 1984 Haken, 1978, 1987, 1988) ... 

From a theoretical point of view, in the study of atom-atom or atom-molecule collisions one needs to solve the Schrodinger equation, both for nuclear and electronic motions. When the nuclei move at much lower velocities than those of the electrons inside the atoms or molecules, both motions (nuclear and electronic) can be separated via the Born-Oppenheimer approximation. This approach leads to a wave function for each electronic state, which describes the nuclear motion and enables us to calculate the electronic energy as a function of the intemuclear distance, i.e. the potential energy V r). Therefore, V r) can be obtained by solving the electronic Schrodinger equation for each inter-nuclear distance. As a result, the nuclear motion, which we shall see is the way chemical reactions take place, is a dynamical problem that can be solved by using either quantum or classical mechanics. 

In this chapter we present the time-dependent quantum wave packet approaches that can be used to compute rate constants for both nonadiabatic and adiabatic chemical reactions. The emphasis is placed on our recently developed time-dependent quantum wave packet methods for dealing with nonadiabatic processes in tri-atomic and tetra-atomic reaction systems. Quantum wave packet studies and rate constants computations of nonadiabatic reaction processes have been dynamically achieved by implementing nuclear wave packet propagation on multiple electronic states, in combination with the coupled diabatic PESs constructed from ab initio calculations. To this end, newly developed propagators are incorporated into the solution of the time-dependent Schrodinger equation in matrix formulism. Applications of the nonadiabatic time-dependent wave packet approaches and the adiabatic ones to the rate constant computations of the nonadiabatic tri-atomic F (P3/2, P1/2) + D2 (v = 0,... 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

At the most fundamental level one follows the time development of the system in detail. The reactants are started in a specific initial (quantum) state and the equation of motion are propagated to give the final state. The equation of motion of the system is the time dependent Schroinger equation, or, if the atoms involved are heavy enough (not H or Li) Newtons equation. The starting point is the adiabatic potential energy surface on which the process takes place. For some reactions electronic excitations during the reaction are important and must be included in addition to the electronically adiabatic dynamics. 

Theoretical studies of the properties of the individual components of nanocat-alytic systems (including metal nanoclusters, finite or extended supporting substrates, and molecular reactants and products), and of their assemblies (that is, a metal cluster anchored to the surface of a solid support material with molecular reactants adsorbed on either the cluster, the support surface, or both), employ an arsenal of diverse theoretical methodologies and techniques for a recent perspective article about computations in materials science and condensed matter studies [254], These theoretical tools include quantum mechanical electronic structure calculations coupled with structural optimizations (that is, determination of equilibrium, ground state nuclear configurations), searches for reaction pathways and microscopic reaction mechanisms, ab initio investigations of the dynamics of adsorption and reactive processes, statistical mechanical techniques (quantum, semiclassical, and classical) for determination of reaction rates, and evaluation of probabilities for reactive encounters between adsorbed reactants using kinetic equation for multiparticle adsorption, surface diffusion, and collisions between mobile adsorbed species, as well as explorations of spatiotemporal distributions of reactants and products. 

This confusion does not avoid that, for the essential problems of structural chemistry, it remains relatively simple for us to avoid the dynamic problems (the models for all sciences since the eighteenth century) and base the entire structure of chemical systems essentially in the steady state solutions of the Schrodinger equation. This is a good enough approximation since the spacing between electronic levels is sufficiently high for, at the temperatures common at the Earth s surface, we only have the fundamental electronic levels filled (Bent, 1965). In fact, apart from the explanatory discourse between levels 1 and 2 (the structural or quan-tum/electronic level and the reactional or statistical/molecular level), particularly for the description of transition states, photochemical reactions, the ground quantum level will be sufficient for the more intricate thematic descriptions. 

In this chapter, we have described the fundamental parameters that should be obtained when characterising an electronic, singlet or triplet, excited state and how to determine them experimentally including methodologies and required equipment. These characteristics include electronic energy, quantum yields, lifetimes and number and type of species in the excited state. Within this last context, i.e., when excited state reactions give rise to additional species in the excited state we have explored several excited state kinetic schemes, found to be present when excimers, exciplexes are formed and (intra and intermolecular) proton transfer occurs. This includes a complete formalism (with equations) for the steady-state and dynamic approaches for two and three-state systems, from where all the rate constants can be obtained. Additionally, we have explored additional recent developments in photophysics the competition between vibrational relaxation and photochemistry, and the non-discrete analysis (stretched-exponential) of fluorescence decays. 

---