Potential energy surface basic concepts

Before getting into a deeper analysis of the concept of resonance, we must define precisely what we understand by chemical structure . One of the most basic concepts in molecular quantum mechanics is the one of potential energy surface (PES). It allows us to define a molecular structure as an arrangement of nuclear positions in space. The definition of molecular structure depends on the validity of the Bom-Oppenheimer approximation for a given state. Actually, its validity is limited to selected portions of the entire Bom-Oppenheimer PES. When a state is described by one PES, we call it an adiabatic state. It is clear that the concept of chemical structure , depends on the existence of a previously defined molecular structure . Only adiabatic states have a molecular structure . From now on, we will always be dealing with adiabatic states. 

Force Fields. The basic assumption underlying molecular mechanics is that classical physical concepts can be used to represent the forces between atoms. In other words, one can approximate the potential energy surface by the summation of a set of equations representing pairwise and multibody interactions. These equations represent forces between atoms related to bonded and nonbonded interactions. Pairwise interactions are often represented by a harmonic potential - 6q) ]... 

In addition we will consider the possibility to obtain reliable theoretical information on the preferred attach sites for proton and metal cations and on the potential energy surfaces (PES) that cannot be determined experimentally even with the most modern and sophisticated mass-spectrometric instruments [22,23j. Furthermore, we will propose the way to rationalize some of chemical properties by using the concepts of hardness, softness and other reactivity indices (Fukui functions) for which an exact definition exists only in the framework of DFT 1111. These last fascinating tools can contribute to increase funhemiore the DFT use going in the "core" of molecules to predict and explain basic chemical concepts. 

A POTENTIAL ENERGY SURFACE is an effective potential function for molecular vibrational motion or atomic and molecular collisions as a function of intemuclear coordinates. The concept of a potential energy surface is basic to the quantum mechanical and semiclassical description of molecular energy states and dynamical processes. It arises from the great mass disparity between nuclei and electrons (a factor of 1838 or more) and may be understood by considering electronic motions to be much faster than nuclear motions. (When we say nuclear motions and nuclear degrees of freedom in this article, we refer to motions of the nuclei considered as wholes, i.e., to atomic motions.) This difference in timescales leads to the so-called electronic adiabatic approximation and to... 

In Chap. 2, the basic concepts relevant for the description of photochemical processes are presented. The molecular Schrddinger equation and the Bom-Oppenheimer approximation are first introduced. Then, the notions of vibronic coupling and conical intersection are discussed and the diabatic representation for the electronic states is introduced. Finally, a review of the methodology used in this thesis for molecular electronic structure calculations, and their use in the exploration of potential energy surfaces, is presented. 

In this chapter, the main concepts relevant for the theoretical study of elementary photochemical processes are briefly reviewed. The notions of vibronic coupling and conical intersection are first introduced. The main basic tools from the molecular electronic structure theory and their use for the exploration of potential energy surfaces are then presented. 

Divergent couplings ai e a nuisance for the computational treatment of the nuclear dynamics. In cases of exact or near degeneracy of electronic potential-energy surfaces it is therefore preferable to introduce an alternative electronic representation, the so-called diabatic (or quasi-diabatic) representation, which avoids singular coupling elements. The basic concept of diabatic states has been introduced in early descriptions of atomic collision processes and vibronic-coupling phenomena in molecular spectroscopy. ... 

Given that quantum chemistry calculations directly provide electronic energies, which formally correspond to zero temperature and pressure, ways for connecting to finite, realistic temperature and pressure are needed. One method is first-principles thermodynamics (FPT), the basic concept of which is that the thermodynamically prevailing state of a surface is the one that minimizes the surface free energy, y, subject to external conditions such as temperature and the chemical potentials of the various components of the system ... 

If the density of holes Ps at the surface - or equivalently the quasi-Fermi level Ep p — are equal at the surface of an n- and p-semiconductor electrode, then the same reaction with identical rates, i.e. equal currents, takes place at both types of electrodes (Fig. 15). Since holes are majority carriers in a p-type semiconductor, the position of the quasi-Fermi level Ep,p is identical to the electrode potential (see right side of Fig. 15), and therefore-with respect to the reference electrode - directly measurable. The density of p can easily be calculated, provided that the positions of the energy bands at the surface are known. The measurements of a current-potential curve also yields automatically the relationship between current and quasi-Fermi level of holes. The basic concept implies that the position of the quasi-Fermi level Ep,p at the surface of an n-type semiconductor and the corresponding hole density Ps can be derived for a given photocurrent, since the same relationship between current and the quasi-Fermi level of holes holds. 

---