Electron density distribution, calculations

Electrophilic Aromatic Substitution. The Tt-excessive character of the pyrrole ring makes the indole ring susceptible to electrophilic attack. The reactivity is greater at the 3-position than at the 2-position. This reactivity pattern is suggested both by electron density distributions calculated by molecular orbital methods and by the relative energies of the intermediates for electrophilic substitution, as represented by the protonated stmctures (7a) and (7b). Stmcture (7b) is more favorable than (7a) because it retains the ben2enoid character of the carbocycHc ring (12). 

Present knowledge of the details of the conformation of proteins is based almost exclusively on results of studies of protein crystals by x-ray diffraction. Protein crystals contain anywhere from 20 to 80% solvent (1 ) (dilute buffer, often containing a high molarity of salt or organic precipitant). While some solvent molecules can be discerned as discrete maxima of the electron density distribution calculated from the x-ray results, the majority of the solvent molecules cannot be located in this manner most of the solvent appears to be very mobile and to have a fluctuating structure perhaps similar to that of liquid water. Many additional distinct locations near which a solvent molecule is present during much of the time have been identified in the course of crystallographic refinement of several small proteins (2,3,4,5, 6), but in all cases the description of solvent structure in the crystal is incomplete probably because only a statistical description is inherently appropriate. 

The structure factors derived from the measurements differed substantially from the theoretical values for the (111), (222), (331), and (422) lines. The electron density distributions calculated from the structure factors by a three-dimensional Fourier synthesis for the ayO, xyV4, and xxz planes are shown in Fig. la-c. 

Figure 2. A summary of the properties of the eleetron density distribution for the skeletal SiOSi dimers in a set of H6Si207 moleeules with geometries fixed at those observed for the Si207 dimers in eoesite. In (a), (b) and (e), respectively, the individual SiO bond lengths, R(SiO), observed for eoesite are plotted against Ai, A2 and A.i, the curvatures of the electron density distribution calculated for the molecules at their saddle points Fc. In (d), R(SiO) is plotted against the magnitude of the electron density, p(rc). In (e) R(SiO) is plotted against G r /p tc) where G(rc)/p(rc) is the kinetic energy density and in (f) R(SiO) is plotted vs. ellipticity, f, of the bonds.

Among the most important requirements in the theory of chemical bonds is the development of a unified method for the description of the chemical interaction between atoms, which would be based on the structure of the atomic electron shells and in which one would utilize the wave functions and the electron density distributions calculated for isolated (free) ions on the basis of the data contained in Mendeleev s periodic table of elements. This unified approach should make it possible to elucidate the interrelationship between the various physical properties and the relationship between the equilibrium and the excited energy states in crystals. In contrast to the study of chemical bonds in a molecule, an analysis of the atomic interaction in crystals must make allowances for the presence of many coordination spheres, the long- and short-range symmetry, the long- and short-range order, and other special features of large crystalline ensembles. As mentioned already, the band theory is intimately related to the chemi-... 

Fig. 3.21 shows the difference between the total electron density in UN MT spheres and the superposition of electron densities of free U and N atoms compared with similar data on HfN, TaN. The calculations show that there is an excessive electron charge ( 0.3e) in the N sphere, whereas the charge in the U MT sphere is nearly equal to the one for a free U atom. This result differs greatly from the electron density distribution calculated by the LMTO method (see below). When the MT model is used in KKR calculations, a rather large part of total electron density is located in the intersphere region of the unit cell (3.1c of UN). 

The most common way of including relativistic effects in a calculation is by using relativisticly parameterized effective core potentials (RECP). These core potentials are included in the calculation as an additional term in the Hamiltonian. Core potentials must be used with the valence basis set that was created for use with that particular core potential. Core potentials are created by htting a potential function to the electron density distribution from an accurate relativistic calculation for the atom. A calculation using core potentials does not have any relativistic terms, but the effect of relativity on the core electrons is included. 

The alkali metals tend to ionize thus, their modeling is dominated by electrostatic interactions. They can be described well by ah initio calculations, provided that diffuse, polarized basis sets are used. This allows the calculation to describe the very polarizable electron density distribution. Core potentials are used for ah initio calculations on the heavier elements. 

In principle, it is possible to calculate the detailed three-dimensional electron density distribution in a unit cell from the three-dimensional x-ray diffraction pattern. 

The Hellmann-Feynman theorem demonstrates the central role of p, the electron density distribution, in understanding forces in molecules and therefore chemical bonding. The main appeal and usefulness of this important theorem is that it shows that the effective force acting on a nucleus in a molecule can be calculated by simple electrostatics once p is known. The theorem can be stated as follows ... 

Figure 6.7 shows the calculated electron density distributions for the H2 and N2 molecules in their equilibrium geometry together with the standard deformation densities. There is clearly a buildup of electron density in the bonding region in both molecules. In the N2 molecule there is also an increase in the electron density in the lone pair region and a de-... 

The 327-670 GHz EPR spectra of canthaxanthin radical cation were resolved into two principal components of the g-tensor (Konovalova et al. 1999). Spectral simulations indicated this to be the result of g-anisotropy where gn=2.0032 and gi=2.0023. This type of g-tensor is consistent with the theory for polyacene rc-radical cations (Stone 1964), which states that the difference gxx gyy decreases with increasing chain length. When gxx-gyy approaches zero, the g-tensor becomes cylindrically symmetrical with gxx=gyy=g and gzz=gn. The cylindrical symmetry for the all-trans carotenoids is not surprising because these molecules are long straight chain polyenes. This also demonstrates that the symmetrical unresolved EPR line at 9 GHz is due to a carotenoid Jt-radical cation with electron density distributed throughout the whole chain of double bonds as predicted by RHF-INDO/SP molecular orbital calculations. The lack of temperature... 

Equations (50) and (51) show that for 0 < 6 < 1 the potential well for the electron near the donor site is more shallow than that in the initial equilibrium configuration. This leads to the fact that the radius of the electron density distribution in the transitional configuration is greater than in the initial equilibrium one (Fig. 3). A similar situation exists for the electron density distribution near the acceptor site. This leads to an increased transmission coefficient as compared to that calculated in the approximation of constant electron density (ACED). 

---