Electron density allocation

As with methods for allocating electron density to atoms, the Mayer method is not necessarily correct, though it appears to be a useful measure of the bond order that conforms to accepted pictures of bonding in molecules. 

An example of quantum mechanical schemes is the oldest and most widely used Mulliken population analysis [1], which simply divides the part of the electron density localized between two atoms, the overlap population that identifies a bond, equally between the two atoms of a bond. Alternatively, empirical methods to allocate atomic charges to directly bonded atoms in a reasonable way use appropriate rules which combine the atomic electronegativities with experimental structural information on the bonds linking the atoms of interest. A widely used approach included in many programs is the Gasteiger-Hiickel scheme [1]. 

It is clear that the Mulliken operator works in the Hilbert space of the basis functions, which has repercussions on the way the electron density is assigned to the nuclei. The basis functions are allocated to the atomic nuclei they are centered on the decision as to which portion of the electron density belongs to which nucleus rests on... 

D20.4 The phase problem arises with the analysis of data in X-ray diffraction when seeking to perform a Fourier synthesis of the electron density. In order to carry out the sum it is necessary to know the signs of the structure factors however, because diffraction intensities are proportional to the square of the structure factors, the intensities do not provide information on the sign. For non-centrosymmetric crystals, the structure factors may be complex, and the phase a in the expression F/m = F w e is indeterminate. The phase problem may be evaded by the use of a Patterson synthesis or tackled directly by using the so-called direct methods of phase allocation. 

Accurate experimental electronic properties can now be obtained in only one day with synchrotron radiation and a charge-coupled device area detection technique. Recently spectacular electron densities were acquired on DL-proline monohydrate at 100 K.130 The accuracy of the data is comparable or even superior to the accuracy obtained from a 6-week experiment on DL-aspartic acid with conventional X-ray diffraction methods. A data acquisition time of one day is comparable to the time needed for an ab initio calculation on isolated molecules. This technique renders larger molecular systems of biological importance accessible to electron density experiments. The impact of the rapid collection of accurate data should not be underestimated. Indeed, if properly allocated a dedicated synchrotron source could now routinely produce accurate experimental densities at a dramatically increased rate. 

RS Mulliken suggested a widely used method for performing population analysis [Mulliken 1955], The starting point is Equation (2.215), which relates the total number of electrons to the density matrix and to the overlap integrals. In the Mulliken method, all of the electron density (P ) in an orbital is allocated to the atom on which is located. The remaining... 

On the other hand, transfer of electron density from the substituent to the antibonding it orbital of the carbonyl renders the C=0 bond weaker and leads to the red-shift. Such shifts are obvious in aryl and vinyl ketones in comparison to their dialkyl counterparts (Figure 12.6). Of course, one has to be cognizant of other electronic effects that can influence the carbonyl group as well. For example, hybridization effects can mask stereoelectronic trends. In particular, strained cycles made of the p-rich banana bonds have to allocate more s-characto in the exocyclrc bonds. Increased s-character makes bonds stronger and shorter, which is reflected in the blue-shifted carbouyl stretching frequencies for the strained cycloalkanones. 

If a heavy atom is included in the crystal, the heavy-atom method can be used to allocate an initial set of phases to the structure factors. Otherwise, direct methods are usually used, which depend on mathematical relationships and probabilities. After the phases of all the structure factors are known, we calculate a 3D electron density map and find the positions of all the atoms in the unit cell. The next step is to refine all the atomic parameters (type of atom, multiplicities, atom positions, and temperature factors) and the scale factors to obtain the best fit of calculated and observed structure factors. Once all the atomic parameters are satisfactorily refined, the analysis goes... 

Table 8 contains chemical analyses representative of each of the nine species given by Foster [1962], plus analyses for kammererite, kotschubeite, the Mn-chlorite pennantite, the Li-Al chlorite cookeite, and a dioctahedral chlorite. The analyses have been allocated to structural formulas, based on 18 (O + OH) atoms, in Table 9. Small amounts of P2O5 and of the large cations K, Na, Ca, or Ba have been excluded from the allocation, where present, on the assumption that they represent impurities. CaO is found by analysis in small amounts in many chlorites, and Belov [1950] has suggested that the Ca might be incorporated in octahedral coordination between the silicate layer and the interlayer. Brown and Bailey [1963] found no evidence for this on electron-density maps of a particular chlorite they investigated.

The large isotropic component is due to the unpaired electron spin density in the carbon 2s orbital, and this value (544MHz) can be used to derive an estimate of the carbon 2 s orbital contribution to the molecular orbital. Since the theoretical isotropic coupling constant for is 3777MHz, then C2s = 544/3777 = 0.144. The anisotropic dipolar part of the hyperfine arises from unpaired spin density in the 2p orbital. However because the dipolar contribution in Equation 1.52 cannot be reduced to zero, this implies that a fraction of the spin density is allocated to the 2p orbital perpendicular to the molecular plane. Therefore, the dipolar component of Equation 1.52 must be further decomposed into two symmetrical tensors oriented along the z and x axes ... 

The threshold excitation phenomenon implies the presence of vacant electronic states at the sample surface that are available for the allocation of both the primary and the excited electrons and directly relates to the local density of the states. The most prominent features of these local states can arise from the methodology used in their determination such as disappearance potential... 

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