Geometry/geometrical aspects

There are several geometrical aspects of interface mean curvature that are particularly important when the interfacial energy is isotropic and the curvature becomes a driving force for mass transport. We present several equivalent cursory statements regarding mean curvature that have rigorous counterparts in differential geometry [6]. 

Coxeter schemes is based on the idea of looking at coset geometries of schemes cf. [43 Theorem E]. However, in this monograph, we shall ignore this geometric aspect completely. 

The characterization of the geometric aspects of a reaction mechanism (definition of the RC, geometry of TS, intermediary complexes, etc.) performed on the basis of in vacuo calculations alone is a risky strategy. It is by far safer to rely on methods giving G (R) directly. This is why we consider it advisable to use accurate continuum methods or combined QM/MM simulations, where the QM part is treated at a sufficient degree of accuracy, as the hybrid QM/MM description of the solute can give. 

A PCET reaction is described by four separate transfer sites derived from a donor and an acceptor for both an electron and a proton [5]. This four state description of PCET gives rise to two important considerations. A geometric aspect to PCET arises when considering the different possible spatial configurations of the four transfer sites. A HAT reaction comprises just one possible arrangement - where the electron and proton transfer sites are coincidental - however this need not be the case for PCET in general. In addition, the two-dimensional reaction space spanned by the four PCET states shown in Fig. 17.1 encompasses infinite mechanistic possibilities (i.e., pathways) for the coordinated transfer of an electron and a proton. These two issues of geometry and mechanism must be taken into account... 

In general, the inertial coordinates q] described in Section 1.2.4 do not fulfill the second condition, and the necessary rotation from these to coordinates that do satisfy condition (2) has been described by Eckart for molecules that rotate and vibrate simultaneously [11]. The procedure has recently been applied to the analysis of static molecular distortions [12]. Once a reference geometry is fixed, the Eckart procedure has the advantage of not being iterative. Whereas for molecules in motion Wj is usually chosen as the alternative choice w,- = I emphasizes the purely geometric aspects of the comparison between structures. 

For ionic systems such as silicates and metal oxides, the interactions between physisorbed molecules and the surface atoms are dominated by electrostatic and repulsive terms. Therefore, empirical potentials are very usefol to gain an overview of possible adsorption geometries. This is particularly appropriate for zeolites, where geometric aspects related to the pore sizes and channel widths provide a first screening. In the case of molecules adsorbed or chemisorbed on metals and for achieving quantitative predictions, one has to resort to quantum mechanical methods. At present, this has been demonstrated only for atoms and rather small molecules such as CO and H2O interacting with surfaces. 

We concentrate here on the structural aspects of helical canal inclusion compounds, primarily because this field of chemical inclusion is still at the relatively juvenile stage of establishing geometry and geometrical Variables. Comments on structure-property relationships for the chemical systems and on structure-function relationships for the biochemical systems are made wherever possible. 

---