Potential energy surface hypersurfaces

The fitting parameters in the transfomi method are properties related to the two potential energy surfaces that define die electronic resonance. These curves are obtained when the two hypersurfaces are cut along theyth nomial mode coordinate. In order of increasing theoretical sophistication these properties are (i) the relative position of their minima (often called the displacement parameters), (ii) the force constant of the vibration (its frequency), (iii) nuclear coordinate dependence of the electronic transition moment and (iv) the issue of mode mixing upon excitation—known as the Duschinsky effect—requiring a multidimensional approach. 

Variational transition state theory (VTST) is formulated around a variational theorem, which allows the optimization of a hypersurface (points on the potential energy surface) that is the elfective point of no return for reactions. This hypersurface is not necessarily through the saddle point. Assuming that molecules react without a reverse reaction once they have passed this surface... 

Potential energy hypersurfaces form the basis for the complete description of a reacting chemical system, if they are throughly researched (see also part 2.2). Due to the fact that when the potential energy surface is known and therefore the geometrical and electronical structure of the educts, activated complexes, reactive intermediates, if available, as well as the products, are also known, the characterizations described in parts 3.1 and 3.2 can be carried out in theory. 

Polyurethane foam 8, 27, 46, 72 Potential energy hypersurfaces (see Potential energy surfaces)... 

As briefly stated in the introduction, we may consider one-dimensional cross sections through the zero-order potential energy surfaces for the two spin states, cf. Fig. 9, in order to illustrate the spin interconversion process and the accompanying modification of molecular structure. The potential energy of the complex in the particular spin state is thus plotted as a function of the vibrational coordinate that is most active in the process, i.e., the metal-ligand bond distance, R. These potential curves may be taken to represent a suitable cross section of the metal 3N-6 dimensional potential energy hypersurface of the molecule. Each potential curve has a minimum corresponding to the stable... 

This expression constitutes the basis of current interpretations of electron transfer processes in biological systems. From Eq. (9), the functions Hg, (Q) and Hbb (Q) represent potential energy surfaces for the nuclear motion described by Xav and Xbw respectively, if the weak diagonal corrections Taa and T b are neglected. Then, the region Q Q where Xav and Xbw overlap significantly corresponds to the minimum of the intersection hypersurface between Hga (Q) and Hbb (Q)- Referring to definition (5), this implies ... 

The first step to making the theory more closely mimic the experiment is to consider not just one structure for a given chemical formula, but all possible structures. That is, we fully characterize the potential energy surface (PES) for a given chemical formula (this requires invocation of the Born-Oppenheimer approximation, as discussed in more detail in Chapters 4 and 15). The PES is a hypersurface defined by the potential energy of a collection of atoms over all possible atomic arrangements the PES has 3N — 6 coordinate dimensions, where N is the number of atoms >3. This dimensionality derives from the three-dimensional nature of Cartesian space. Thus each structure, which is a point on the PES, can be defined by a vector X where... 

Aside from simple processes such as cis-trans isomerization, the first example of a potential energy surface derived for a photochemical rearrangement or reaction was in 1967. This was work by the Zimmerman group 44-47) in which the hypersurface for the Di-u-Methane rearrangement of barrelene to semibullvalene was obtained note Fig. 13. 

Figure 7-1. Three-dimensional potential energy surfaces (a) Energy hypersurface for FSSF SSF2 isomerization (detail). Reproduced with permission [19] copyright (1977) American Chemical Society (b) Potential energy surface of the molecular rearrangement of Agl3, with the corresponding structures indicated on the sides [20], Copyright (2005) American Chemical Society.

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