Representation of Electron Density Distribution

Coppens, X-ray Charge Densities and Chemical Bonding, Oxford University Press, Oxford, 1997. 

Shibata and F. Hirota, in Stereochemical Applications of Gas-Phase Electron Diffraction, I. Hargittai and M. Hargittai, eds., VCH Publishers, New York, 1988, Chap. 4. 

These experimental electron density distributions are in accord with the VB, MO, and DFT descriptions of chemical bonding, but are not easily applied to the determination of the relatively small differences caused by substituent effects that are of primary interest in interpreting reactivity. As a result, most efforts to describe electron density distribution rely on theoretical computations. The various computational approaches to molecular structure should all arrive at the same correct total electron distribution, although it might be partitioned among orbitals differently. The issue we discuss in this section is how to interpret information about electron density in a way that is chemically informative, which includes efforts to partition the total electron density among atoms. These efforts require a definition (model) of the atoms, since there is no inherent property of molecules that partitions the total electron density among individual atoms. 

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In representations of electron densities, the presence or lack of boundaries plays a crucial role. A quantum mechanically valid electron density distribution of a molecule cannot have boundaries, nevertheless, artificial electron density representations with actual boundaries provide useful tools of analysis. For these reasons, among the manifold representations of molecular electron densities, manifolds with boundaries play a special role. 

Because all of these electronic aspects of aromaticity are ultimately derived from the electron distribution, we might ask whether representations of electron density reveal any special features in aromatic compounds. The electron density of the IT electrons can be mapped through the MESP (molecular electrostatic potential, see Section 1.4.5). The MESP perpendicular to the ring is completely symmetrical for benzene, as would be expected for a delocalized structure and is maximal at about... 

In 1993 Glaser and Choy published a full paper on the electron density distributions (at the RHF/6-31G level and including electron correlation at the MP2(full)/ 6-3IG level) in the heterosubstituted systems XN (X = F, HO, H2N) and compared these systems with methanediazonium ion. The results show that the positive charge on the N()ff)-atom is always larger than that on N(a). The N(a)-atom may even carry a negative charge Unconnected structures of electrophile and dinitrogen must be considered for adequate representation of the density distribution in XNi. ... 

Fig. 13.12. A scheme of the SE and TE exchange non-additivides. Panels (a), (b), and (cj shew the SE mechamsnL (a) Hiree non-interacting molecules (schematic representation of electron densities), (b) Pauli deformation of molecules A andB. (c) Electrostatic interaction of the Pauli deformation resulting from single electron exchanges between A and B with the dipole mcxneiit of C. (d) The TE mechanism molecules A and B exchange an electron with the mediation of molecule C. All molecular electrcm density distributions undCTgo PauU defmnation.

The topological analysis of the ELF provides a picture in which the electron density is distributed and localized in different volumes called basins, thus enabling one to discuss the reliability of simplified representations of electron densities in terms of superposition of promolecular densities or resonant Lewis structures. 

Let s take a moment to add just a little more detail to this argument. Figure 14.4 shows a schematic representation of electron density (or electron probability ) versus atomic radius for the Af,Sd, and 6s orbitals. (Electron density here is defined as the function ATTr tf/, sometimes known as the radial distribution function. It gives the probability of finding the electron on the surface of a series of concentric spheres of radius r. You may have studied this in previous chemistry courses but you need not know the mathematical details to understand the following argument.) Note that most of the time the 6s electron, as expected, is found farther away from the nucleus (located at r = 0) than the 4f and 5d electrons. We say the 6s electron is shielded from the nucleus by the intervening 4f and 5d electrons. 

We need ways to visualize electrons as particle-waves delocalized in three-dimensional space. Orbital pictures provide maps of how an electron wave Is distributed In space. There are several ways to represent these three-dimensional maps. Each one shows some important orbital features, but none shows all of them. We use three different representations plots of electron density, pictures of electron density, and pictures of electron contour surfaces. 

A complete and detailed description of molecular structure includes statements concerning the metric coordinates ) of atomic nuclei supplemented by electron density distribution data. Although a large amount of data is involved, such representation is not particularly suitable for the chemically relevant structural features of molecules. 

Most commonly used is certainly the molecular electrostatic potential. It can be derived from any kind of charge distribution. Usually, the MEP is first calculated on a grid and subsequently transformed to the sphere or Gaussian representation. Quite important is the electron density distribution, which closely models the steric occupancy by a molecule. Other approaches utilize artificial fields for physicochemical properties commonly associated with binding, like a field for the hydrophobicity [193] or H-bonding potential [133,194]. 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

Molecular graphics (Henkel and Clarke, 1985) refers to a technique for the visualization and manipulation of molecules on a graphical display device. The technique provides an exciting opportunity to augment the traditional description of chemical structures by allowing the manipulation and observation in real time and in three dimensions, of both molecular structures and many of their calculated properties. Recent advances in this area allow visualization of even intimate mechanisms of chemical reactions by graphical representation of the distribution and redistribution of electron density in atoms and molecules along the reaction pathway. 

In contrast, 2- and 4-pyrones are considered to have relatively little aromatic character. Whereas in an analogous nitrogen series 4-pyridone 5.23 has significant aromatic character (mesomeric representation 5.23a making a considerable contribution to the overall electronic distribution), aromatic mesomeric representation 9.3a makes less of a contribution to the overall electronic structure of 4-pyrone. As with furan, the higher electronegativity of oxygen leads to heterocycles of little aromaticity in cases where delocalisation of electron density from the heteroatom is a prerequisite for that aromaticity. 

A molecule contains a nuclear distribution and an electronic distribution there is nothing else in a molecule. The nuclear arrangement is fully reflected in the electronic density distribution, consequently, the electronic density and its changes are sufficient to derive all information on all molecular properties. Molecular bodies are the fuzzy bodies of electronic charge density distributions consequently, the shape and shape changes of these fuzzy bodies potentially describe all molecular properties. Modern computational methods of quantum chemistry provide practical means to describe molecular electron distributions, and sufficiently accurate quantum chemical representations of the fuzzy molecular bodies are of importance for many reasons. A detailed analysis and understanding of "static" molecular properties such as "equilibrium" structure, and the more important dynamic properties such as vibrations, conformational changes and chemical reactions are hardly possible without a description of the molecule itself that implies a description of molecular bodies. 

The TAE/RECON method, developed by Breneman and co-workers based on Bader s quantum theory of Atoms In Molecules (AIM). The TAB method of molecular electron density reconstruction utilizes a library of integrated atomic basins , as defined by the AIM theory, to rapidly reconstruct representations of molecular electron density distributions and van der Waals electronic surface properties. RECON is capable of rapidly generating 6-31-I-G level electron densities and electronic properties of large molecules, proteins or molecular databases, using TAB reconstruction. A library of atomic charge density fragments has been assembled in a form that allows for the rapid retrieval of the fragments, followed by rapid molecular assembly. Additional details of the method are described elsewhere. ... 

There are a number of ways of monitoring the distribution of electron density in any molecular entity. The total density can be computed at a number of points in space and presented as a contour map or some three-dimensional representation. Shifts are easily examined by density difference maps which plot the difference in density between two different configurations. For example, the density shifts caused by H-bond formation can be taken as the difference between the complex on one hand, and the sum of the densities of the two non interacting subunits on the other, with the two species placed in identical positions in either case. Comparisons with x-ray diffraction data have verified the validity of this ap-proach. Also, the total density of the complex itself can be examined for the presence of critical points that indicate H-bonding interactions . 

Figure 6.1 A schematic representation of the electron density distribution in d orbitals...

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