Electron Density in a Molecule

Knowledge of the wave function permits calculation of the various quantities which characterize the molecule. In particular the total electron density appears in the form  

---

The preceding analysis of the topology of the electron density in a molecule enables us to to define both the atoms and the bonds in a molecule. 

As well as these permanent dipole moments, random motion of electron density in a molecule leads to a tiny, instantaneous dipole, which can also induce an opposing dipole in neighbouring molecules. This leads to weak intermolecu-lar attractions which are known as dispersive forces or London forces, and are present in all molecules, ions and atoms - even those with no permanent dipole moment. Dispersive forces decrease rapidly with distance, and the attractions are in proportion to 1/r6, where r is the distance between attracting species. 

A promising simplification has been proposed by Bader (1990) who has shown that the electron density in a molecule can be uniquely partitioned into atomic fragments that behave as open quantum systems. Using a topological analysis of the electron density, he has been able to trace the paths of chemical bonds. This approach has recently been applied to the electron density in inorganic crystals by Pendas et al. (1997, 1998) and Luana et al. (1997). While this analysis holds great promise, the bond paths of the electron density in inorganic solids are not the same as the more traditional chemical bonds and, for reasons discussed in Section 14.8, the electron density model is difficult to compare with the traditional chemical bond models. 

Here, we seek to obtain wave functions - molecular orbitals - in a manner analogous to atomic orbital (AO) theory. We harbour no preconceptions about the chemical bond except that, as in VB theory, the atomic orbitals of the constituent atoms are used as a basis. A naive, zeroth-order approximation might be to regard each AO as an MO, so that the distribution of electron density in a molecule is simply obtained by superimposing the constituent atoms whose AOs remain essentially unaltered. But since there is inevitably an appreciable amount of orbital overlap between atoms in any stable molecule - without it there would be no bonding - we must find a set of orthogonal linear combinations of the constituent atomic orbitals. These are the MOs, and their number must be equal to the number of AOs being combined. 

DFT has led to a substantial simplification of quantum-chemical computations. Like the Hellmann-Feynman theorem it expresses the reasonable assumption of a reciprocal relationship between potential energy and electron density in a molecule. In principle this relationship means that all ground-state molecular properties may be calculated from the ground-state electron density p(x, y, z), which is a function of only three coordinates, instead of a many-parameter molecular wave function in configuration space. The formal theorem behind DFT which defines the electronic energy as a functional of the density function provides no guidance on how to establish the density function p r) without resort to wave mechanics. 

These computer-generated electrostatic potential maps use color to show the calculated electron density in a molecule. Electron-rich regions are red and electron-poor regions are blue. 

It is clear that the entire electronic density in a molecule has the role of determining the nuclear distribution hence bonding, consequently, chemical bonding cannot be confined to lines in space. It is well understood that bond diagrams represent only an oversimplified, "short-hand" notation for the actual molecular structure, nevertheless, as most successful notations do, chemical bonds as formal lines have acquired an almost unquestioned reputation of their own as if they were truly responsible for holding molecules together. 

We have given this primitive modelling of the density to stress the importance of finding localized descriptions of the electron density in a molecule (or solid, especially amorphous materials, e.g. Si, where the periodicity that is so helpful in a crystalline solid no longer is present). 

A dipole moment ii associatod With a permanent uneven sharing of electron density in a molecule. It is a vector quantity that depends on the size of the partial charges in the molecule and the distance between them, and is measured in units of Debye ([if For example, chlorobenzene has a dipole moment of 1 75 D oriented towards the chlorine, in 1,4-dichlorobenzene, the individual moments directed towards the dilorine atoms are equal and opposed to each other, Consequently, the dip ok momenl of this molecuk is zurc. 

The distribution of electron density in a molecule can be shown using an electrostatic potential map. These maps are color coded to illustrate areas of high and low electron density. Electron-rich regions are indicated in red, and electron-deficient sites are indicated in blue. Regions of intermediate electron density are shown in orange, yellow, and green. 

When comparing two maps, the comparison is useful only if they are plotted using the same scale of color gradation. For this reason, whenever we compare two plots in this text, they will be drawn side by side using the same scale. It will be difficult to compare two plots in different parts of the book, because the scale may be different. Despite this limitation, an electrostatic potential plot is a useful tool for visually evaluating the distribution of electron density in a molecule, and with care, comparing the electron density in two different molecules. 

Another way to define ionic charges consists in partitioning space into elementary volumes associated to each atom. One method has been proposed by Bader [240,241]. Bader noted that, although the concept of atoms seems to lose significance when one considers the total electron density in a molecule or in a condensed phase, chemical intuition still relies on the notion that a molecule or a solid is a collection of atoms linked by a network of bonds. Consequently, Bader proposes to define an atom in molecule as a closed system, which can be described by a Schrodinger equation, and whose volume is defined in such a way that no electron flux passes through its surface. The mathematical condition which defines the partitioning of space into atomic bassins is thus ... 

R.G. Parr and A. Berk, "The Bare-Nuclear Potential as Harbinger for the Electron Density in a Molecule", in Chemical Applications of Atomic and Molecular Electrostatic Potentials, P. Politzer and D.G.Truhlar (Eds,), Plenum, New York, 1981, pp. 51-62. 

As the resolution of the Bragg reflection data is improved, it becomes possible to obtain information on the more minute details of electron density in a molecule. At high enough resolution information can be obtained on the redistribution of electron density (deformation density) around atoms when they combine to form a molecule. Electrons in molecules ma -form bonds or exist as lone pairs, thereby distorting the electron density around each atom and requiring a more complicated function to describe this overall electron density than normally used, in which it is treated as if it were spherically symmetrical (deformed to an ellipsoid in order to account for anisotropic displacements). This assumption is inherent in the use of spherically-symmetrical scattering factors although the elec-... 

Deformation density The difference between the electron density in a molecule, with all its distortions as a result of bonding, and the promolecule density, obtained by forming a molecule with spherical electron density around each atom (free atoms). This map contains effects caused both by the errors in the relative phases of Bragg reflections, experimental errors in the data, and inadequacies in the representations of the scattering factors of free atoms. 

In theoretical terms, the total electron density in a molecule is easily expressed in terms of the occupied molecular orbitals. Additional information is gained from the m.o. approach especially regarding the electronic energy for ground and excited states and the detailed features (e.g. phase) of individual m.o.s. Molecular orbitals are mathematical functions that can be constructed as linear combinations of orbitals of the contributing atoms, in a process where the atoms lose their individuality, except for the respective nuclei and, perhaps, the core electrons. The valence electrons are described by functions which, in general, extend to several atoms or even to the whole molecule. 

Added in Proofs, Combined X-ray and electron diffraction data can be used to analyze the electron density in a molecule. Identification of the square of the wave function with the electron density then gives the coefficients of the basis atomic orbitals in the various molecular orbitals, and therefore permits the evaluation of the electron populations. This method was used for instance with 1,3,5-trimethyl-benzene, and good agreement with theoretical data was obtained [B. H. O Connor,... 

A great deal of evidence has accumulated to show that a metal ion in a chelate ring can alter the electron density in a molecule. One method of obtaining quantitative data on such changes in electron density is to compare the rates of electrophilic attack at a point in the unchelated and chelated molecules as, for example, in the iodination of 8-hydroxyquinoline chelates (Section II,A). Another such kinetic study has been carried out on the diazo coupling of 8-hydroxyquinoline-5-sulfonic acid and its zinc(II) chelate with diazotized sulfanilic acid (141)-... 

R. G. Parr and A. Berk, The bare-nuclear potential as harbinger for the electron density in a molecule,... 

Polarity is related to the symmetry of the arrangement of electron density in a molecule. Polar molecules are those which possess a net dipole moment, which means that the electron density is not symmetrically distributed in all directions. Nonpolar molecules have the electron density distributed in such a way that there is no net dipole moment. Typically this doesn t mean that nonpolar molecules have their electron density distributed evenly over every part of the molecule, but rather that the dipole moments created by an unequal sharing of electrons in each individual bond cancel each other out, so that there is no net direction in which an asymmetry of electron density exists. 

When electrons are shared between two atoms, their distribution is different from that in the individual atoms. In Figure 7.5, we see how electron density in a molecule differs from the spherically symmetrical distribution seen in isolated atoms. In a covalently bonded molecule, there is a buildup of electron density, or a concentration of negative charge, in the region between the two bonded atoms. The increase in electron density need not be particularly large even a slight... 

The electronic density in a molecule is close to a superposition of the atomic electron densities. The bonding electron orbitals are superpositions of atomic valence orbitals. 

The circulation of electron density in a molecule in an applied magnetic field. 

Electrostatic potential (elpot) maps provide a way to visualize the distribution of electron density in a molecule. Electrostatic potential is defined as the potential energy that a positively charged particle would experience in a molecule s presence. The electrostatic potential is made up of two parts. 

Diamagnetic current in NMR (Sections 13.3 and 13.7Q The drculation of electron density in a molecule in an applied magnetic field. 

---