The Concept of Molecular Geometry

When nuclei in molecules are treated classically, the concept of molecular geometry emerges in a natural way. To be more precise, the equilibrium geometry is defined as the set of internal coordinates r for which the ground-state eigenvalue q(r) of the electronic Hamiltonian attains a local minimum. Different minima in q(r) correspond to equilibrium geometries of isomeric species with identical compositions. Needless to say, this naive picture is inevitably lost in a fully quanturn-mechanical treatment. 

Molecular geometries measured with condensed-phase techniques such as X-ray diffraction or NMR cannot be regarded as inherent properties of isolated species. Similarly, as the determination of molecular geometries from microwave spectra involves collation of data pertaining to many spectroscopic states of species differing in isotopic compositions, such geometries are merely collections of fitting parameters that cannot be viewed as quantum-mechanical observables. 

All the information about positions of nuclei that can be obtained for a given spectroscopic state r s contained in the corresponding nuclear probability density PNJMn(R), 

These complications notwithstanding, molecular geometries can be derived solely from wavefunctions of spectroscopic states (or even nonstationary states with zero linear momentum) under favorable conditions. This can be accomplished in principle [21] by constructing the functions 

Several points concerning the above prescription should be emphasized. First of all, it is an arbitrary construction that is not derivable from the postulates of quantum mechanics. Second, since the presence of a sufficient number of the aforementioned maxima cannot be guaranteed in general, this prescription is by no means universal. Third, since atoms and molecules have infinite extends, similar considerations cannot be employed in a definition of molecular shape. In summary, although isolated molecules possess neither classical structures nor shapes [3], their geometries can be defined under certain conditions. 

---

Actually, the nuclei are not stationary, but execute vibrations of small amplitude about equilibrium positions it is these equilibrium positions that we mean by the fixed nuclear positions. It is only because it is meaningful to speak of (almost) fixed nuclear coordinates that the concepts of molecular geometry or shape and of the PES are valid [12]. The nuclei are much more sluggish than the electrons because they are much more massive (a hydrogen nucleus is about 2,000 more massive than an electron). 

Extension of the concepts of molecular geometry and aggregate structure has led to its use in predicting not only the structure to be expected (the shape to be expected (micelle, vesicle, extended bilayer, etc.), but also the size, size distribution, shape (spherical, ellipsoidal, disk, or rod-shaped), dispersity (or size distribution), critical micelle concentration, average aggregation number, and other such characteristics. The rules of association derived from the geometric analysis of molecular structure are summarized in Figure 15.11. 

More recently, electrostatic theory has been revived due to the concept of molecular electrostatic potentials. The potential of the solute molecule or ion was used successfully to discuss preferred orientations of solvent molecules or solvation sites 50—54). Electrostatic potentials can be calculated without further difficulty provided the nuclear geometry (Rk) and the electron density function q(R) or the molecular wave function W rxc, [Pg.14]

Similarity is one of the fundamental concepts in chemistry as well as in biochemistry. Recently, several monographs dealt with a special kind of similarity, with molecular similarity [1, 2]. The concept of molecular similarity makes it possible to compare and classify the isolated molecules based on their individual properties, such as molecular geometry, dipole moment, charge distribution, etc. From the mathematical point of view, the relation that two molecules are similar to each other by some of their properties is an equivalency relation on the set of isolated molecules. 

This short treatise is intended to provide the reader with a concise summary of the current theoretical status of the molecular structure and geometry concepts. A fully quantum-mechanical treatment of molecules is employed where necessary. The relevance of stationary states of isolated molecules is discussed and the notion of molecular geometry is contrasted with that of molecular structure. 

The molecular basis for the left- and right-handedness of distinct crystals of the same chemical substance and the associated differences in optical rotation was developed from the hypothesis of Paterno (1869) and Kekule that the geometry about a carbon atom bound to four ligands is tetrahedral. Based on the concept of tetrahedral geometry, Van t Hoff and LeBel concluded that when four different groups or atoms are bound to a carbon atom, two distinct tetrahedral molecular forms are possible, and these bear a non-superimposable mirror-image relationship to one another (Fig. 3). This hypothesis provided the link between three-dimensional molecular structure and optical activity, and as such represents the foundation of stereoisomerism and stereochemistry. 

The concept of molecular shape-selective catalysis is based on the action of catalytically active sites internal to the zeolitic framework, to diffusivity resistance either to reactant molecules or to product molecules or to both and to void limitation to reaction intermediates.This implies an intimate interaction between the shape, size and configuration of the molecules and the dimension, geometry and tortuosity of the channels and cages of the zeolite. Several types of effects exist ... 

The Bom-Oppenheimer approximation says that in a molecule the nuclei are essentially stationary compared to the electrons. This is one of the cornerstones of computational chemistry because it makes the concept of molecular shape (geometry) meaningful, makes possible the concept of a PES, and simplifies the application of the Schrodinger equation to molecules by allowing us to focus on the electronic energy and add in the nuclear repulsion energy later. 

Throughout the realm of molecular modeling, the concept of molecular shape arises over and over in one form or another. Just what do scientists mean by a molecule s shape, and how can one use three-dimensional shape in modeling. In Chapter 5, Professor Gustavo A. Arteca examines these issues and delineates the hierarchical levels of molecular shape and shape descriptors. He explains molecular shape in terms of mathematical descriptors of nuclear geometry, connectivity, and molecular surfaces. Of special note are his comments on shape dynamics of flexible molecules and descriptors of relative shape. 

Only recently were we able to show that O-silylhydroxylamines form intramolecular donor acceptor bonds between Si and N centers, separated by one oxygen atom only [1]. This type of secondary bonds between p-block donor and acceptor atoms in P-positions relative to one another had so far been unprecedented or neglected. For the prediction of molecular geometries of compounds having electropositive and electronegative atoms in the p-position to one another, this interaction turns out to be an important factor and should be considered in structure prediction in addition to the general VSEPR concept [2] and Bartell s two bond radii model [3]. 

Contents B. T Sutcliffe The Concept of Molecular Structure. - 0. E. Polansky Topology and Properties of Molecules. -J.RDahl Symmetry in Molecules. -L D. Barron Chirality of Molecular Structures - Basic Principles and their Consequences. - J. E. Boggs Interplay of Experiment and Theory in Determining Molecular Geometries. A. The Experiments. -... 

When constructing more general molecular wave functions there are several concepts that need to be defined. The concept of geometry is inhoduced to mean a (time-dependent) point in the generalized phase space for the total number of centers used to describe the END wave function. The notations R and P are used for the position and conjugate momenta vectors, such that... 

In Chapter 7, we used valence bond theory to explain bonding in molecules. It accounts, at least qualitatively, for the stability of the covalent bond in terms of the overlap of atomic orbitals. By invoking hybridization, valence bond theory can account for the molecular geometries predicted by electron-pair repulsion. Where Lewis structures are inadequate, as in S02, the concept of resonance allows us to explain the observed properties. 

---