A theory of molecular structure

By appealing to the theorem of structural stability of Palis and Smale one can show that only two kinds of structural instabilities or catastrophe points can arise and that there are therefore only two basic mechanisms for structural change in a chemical system. 

Palis and Smale s theorem of structural stability when used to describe structural changes in a molecular system predicts a configuration XeR to be structurally stable if p(r, X) has a finite number of critical points such that  

The immediate consequence of the theorem is that a structural instability can be established through only one of two possible mechanisms which correspond to the bifurcation and conflict catastrophes. A change in molecular structure—the making and breaking of chemical bonds—can only be caused by the formation of a degenerate critical point in the electronic charge distribution or by the attainment of an unstable intersection of the submanifolds of bond and ring critical points. 

The full usefulness of the classification using V Pb must await the development of the quantum mechanical aspects of the theory. The Laplacian of the charge density appears in the local expression of the virial theorem and it is shown that its sign determines the relative importance of the local contributions of the potential and kinetic energies to the total energy of the system, A full discussion of this topic is given in Section 7.4. 

These examples demonstrate that molecular structure and its stability are predicted by a theory which uses only the information contained in the quantum mechanical state function and that the static and dynamic properties of a bond can be characterized in terms of the properties of the charge density at the bond critical point. The values of Pb, ab, e, and V Pb enable one to translate the predicted electronic effects of orbital models into observable consequences in the charge distribution. 

---

Bader et al. have developed a theory of molecular structure [8], based on the topological properties of the electron density p(r). In this theory, a molecule may be partitioned into atoms or fragments by using zero-flux surfaces that satisfy the condition... 

To go from experimental observations of solvent effects to an understanding of them requires a conceptual basis that, in one approach, is provided by physical models such as theories of molecular structure or of the liquid state. As a very simple example consider the electrostatic potential energy of a system consisting of two ions of charges Za and Zb in a medium of dielectric constant e. 

After the discovery of quantum mechanics in 1925 it became evident that the quantum mechanical equations constitute a reliable basis for the theory of molecular structure. It also soon became evident that these equations, such as the Schrodinger wave equation, cannot be solved rigorously for any but the simplest molecules. The development of the theory of molecular structure and the nature of the chemical bond during the past twenty-five years has been in considerable part empirical — based upon the facts of chemistry — but with the interpretation of these facts greatly influenced by quantum mechanical principles and concepts. 

So far only a start has been made (mainly by G. E. K. Branch and G. Schwarzenbach) on the problem of correlating the acidity or basicity of a substance with its resonating electronic structure. It should be possible to develop the theory of molecular structure to such an extent as to permit the reliable prediction of the behavior of substances with respect to this property and other physical and chemical properties. 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

In the first place, we have been studying those parts of the theory of the electronic structure of molecules which bear on a theory of valence. That is we are not attempting to present a theory of molecular electronic structure, but an approximate theory of valence. The latter is but a small part of the former. In particular we are (in the main) concerned with localised systems of electrons in their ground states the theory... 

Proc. Roy. Soc. (London) A202, 166 (1950) and Hurley, A. C. On Orbital Theories of Molecular Structure, Thesis, Trinity College, Cambridge, England 1952. 

At this point, we have completed the presentation of the key equations which will be crucial to the development of a predictive theory of molecular structure. These equations will form the basis for determining the relative stability of isomers, the relative stabilization of a cationic, radical or anionic center by substituents, etc. On the other hand, the differential expressions (9) to (12) will form the basis for determining how substitution affects the relative stability of isomers, the relative stabilization of cationic, radical and anionic centers, etc. It is then obvious that a working knowledge of Eqs. (1) to (6) presupposes a great familiarity with the key quantities involved in these equations, namely, orbital energies and interaction matrix elements. 

One further theoretical method that merits consideration at this point is the topological theory of molecular structure exemplified by Bader (1985, 1990). In this method a topological description of the total electron density in the molecule is used. A major advantage of this method is that it allows the total interaction between various centres to be probed. Cremer et al. (1983) used the Bader method to examine the homotropylium cation [12] and concluded that it was indeed homoaromatic. 

The theory of molecular structure based on the topology of molecular charge distribution, developed by Bader and co-workers (83MI2 85ACR9), enables certain features to be revealed that are characteristic of the systems with aromatic cyclic electron delocalization. To describe the structure of a molecule, it is necessary to determine the number and kind of critical points in its electronic charge distribution, i.e., the points where for the gradient vector of the charge density the condition Vp = 0 is fulfilled. 

AS is well known, the quantity known as the overlap integral is of considerable importance in the theory of molecular structure. Although the existing literature1 contains a number of formulas and some numerical values for overlap integrals, it was thought worth while to carry through the much more systematic and comprehensive study whose results are presented below. 

The quantum theory of molecular structure developed here and the standard BO approach rely on the separability between electronic and nuclear configurational degrees of freedom. However, the way this is achieved differs radically between the approaches. In the treatment described here, the nuclei are seen to be trapped by an attractor generated by the stationary electronic wave function (nuclei follow the electronic states ) the electronic wave function does not depend upon the instantaneous positions of the nuclei as early proposed by this author [4] a change of electronic state, characterizing a chemical reaction with reactants and products in their ground electronic states, is described as a Franck-Condon like process. 

The topology of the electron density also leads to the identification of a chemical bond with a line linking neighboring nuclei along which the electron density is a maximum.190 This identification leads to a definition of molecular structure that is remarkable in its ability to recover all chemical structures. The dynamics of the density, as occasioned by nuclear displacements and analyzed by means of the mathematics of qualitative dynamics, leads to a complete theory of structural stability, one that clarifies the meaning of the making and breaking of a chemical bond. 

The molecular concept has become so central in chemistry that understanding of chemical events is commonly assumed to consist of relating experimental observations to micro events at the molecular level, which means changes in molecular structure. In this sense molecular structure is a fundamental theoretical concept in chemistry. As the micro changes are invariably triggered by electron transfer, the correct theory at the molecular level must be quantum mechanics. It is therefore surprising that a quantum theory of molecular structure has never developed. This failure stems from the fact that physics and chemistry operate at different levels and that grafting the models of physics onto chemistry produces an incomplete picture. 

Then in Chapter 3, before we come to the theory of molecular structure, we shall introduce you to the experimental techniques of finding out about molecular structure. This means studying the interactions between molecules and radiation by spectroscopy—using the whole electromagnetic spectrum from X-rays to radio waves. Only then, in Chapter 4, will we go behind the scenes and look at the theories of why atoms combine in the ways they do. Experiment comes before theory. The spectroscopic methods of Chapter 3 will still be telling the truth in a hundred years time, but the theories of Chapter 4 will look quite dated by then. 

Pople JA (1951) Molecular association in liquids II. A theory of the structure of water. Proc R Soc 205A 163-178... 

---