The Electron Density from Experiment

Clearly the form of a deformation density depends crucially on the definition of the reference state used in its calculation. A deformation density is therefore meaningful only in terms of its reference state, which must be taken into account in its interpretation. As we will see shortly, the theory of AIM provides information on bonding directly from the total molecular electron density, thereby avoiding a reference density and its associated problems. But first we discuss experimentally obtained electron densities. 

Unlike the wave function, the electron density can be experimentally determined via X-ray diffraction because X-rays are scattered by electrons. A diffraction experiment yields an angular pattern of scattered X-ray beam intensities from which structure factors can be obtained after careful data processing. The structure factors F(H), where H are indices denoting a particular scattering direction, are the Fourier transform of the unit cell electron density. Therefore we can obtain p(r) experimentally via  

If we are interested only in the determination of a molecular structure, as most chemists have been, it suffices to approximate the true molecular electron density by the sum of the spherically averaged densities of the atoms, as discussed in Section 6.4. A least-squares procedure fits the model reference density preKr)t0 the observed density pobs(r) by minimizing the residual density Ap(r), defined as follows  

The model reference density pref is a good approximation to the dominant part of p appearing very close to the nuclei, and so Ap(r) will be very small everywhere and is assumed to be experimental noise. If the peaks in p are located, then the nuclear positions are known and the structure is resolved. Because they have no core, hydrogen atoms produce only very small maxima, and thus their positions are difficult to locate with any accuracy. If it is important to locate their positions accurately, this can be done by neutron diffraction. Neutrons are scattered by nuclei rather than electrons, and so the positions of the nuclei are obtained directly. Neutron diffraction is particularly important for the accurate determination of the positions of hydrogen atoms. 

In the next section we will see how the theory of AIM enables us to obtain chemical insight directly from the experimental density as determined by experiment or by calculation, thereby avoiding the need for deformation densities. 

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In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

Far-infrared absorption measurements gave an independent determination of the electron density from the position of the Frohlich mode near 9 meV ( 140 jum) a density of 2.3 X 1017 cm-3 was inferred. Other experiments on Sb-doped Ge and pure germanium irradiated at different powers showed appreciable changes in absorption band positions and shapes. Rose et al. 

Along with the study of the single-electron states structure, some papers deal with the electron density distribution in refractory carbides. Thus, the calculations of Neckel et al (1975) show, as experiments do, the definite transfer of the electron density from the metal atoms to the metalloid atoms. The electron density distribution was studied in more detail in the subsequent papers of Neckel et al (1976), Blaha and Schwarz (1983), Mokhracheva, Geld and Tskai (1983), Blaha et al (1985a, 1985b, 1987), Schwarz and Blaha (1984) and Kim and Williams (1988). These papers contain comparisons of calculated results for carbides and nitrides and are therefore discussed in Chapter 3. 

However, the full Sn2 participation by the C2, C3 a-bond can only occur if the electron density from that bond can buildup on the side of the cyclopropane ring remote from the leaving group. Of the two possible disrotatory modes, therefore, the one shown in Fig. 3.9 is likely to be strongly preferred experiment indicates that this is so. 

The electron densities for a spin electrons and for spin electrons are always equal in a singlet spin state, but in non-singlet spin states the densities may be different, giving a resultant spin density. If we evaluate the spin density function at the position of certain nuclei, it gives a value proportional to the isotropic hyperfine coupling constant that can be measured from electron spin resonance experiments. 

Secondly, the influence of the cations on their surrounding space is not negligible. The cations produce strong non-uniform electrostatic fields and the electron density in such a model is far from constant, or even slowly varying. The free electrons experience a very strong attraction when they approach the cations. 

We have performed full-potential calculations on TisSia in its proposed stable crystal structure. The enthalpy of formation obtained from these calculations agrees well with the value deduced from experiment. Due to the low crystal symmetry, the possibility of a more complex bonding character arises. The charge density in this phase differs considerably from that in the hypothetical unstable structure, so two-electron bonds can be excluded in this phase. We have also showed that the opening of a quasigap in the Si DOS has its origin in the Ti-Si interaction. 

How large are orbitals Experiments that measure atomic radii provide information about the size of an orbital. In addition, theoretical models of the atom predict how the electron density of a particular orbital changes with distance from the nucleus, r. When these sources of information are combined, they reveal several regular features about orbital size. 

The shortest cation-anion distance in an ionic compound corresponds to the sum of the ionic radii. This distance can be determined experimentally. However, there is no straightforward way to obtain values for the radii themselves. Data taken from carefully performed X-ray diffraction experiments allow the calculation of the electron density in the crystal the point having the minimum electron density along the connection line between a cation and an adjacent anion can be taken as the contact point of the ions. As shown in the example of sodium fluoride in Fig. 6.1, the ions in the crystal show certain deviations from spherical shape, i.e. the electron shell is polarized. This indicates the presence of some degree of covalent bonding, which can be interpreted as a partial backflow of electron density from the anion to the cation. The electron density minimum therefore does not necessarily represent the ideal place for the limit between cation and anion. 

ELF can be visualized with different kinds of images. Colored sections through a molecule are popular, using white for high values of ELF, followed by yellow-red-violet-blue-dark blue for decreasing values simultaneously, the electron density can be depicted by the density of colored points. Contour lines can be used instead of the colors for black and white printing. Another possibility is to draw perspective images with iso surfaces, i.e. surfaces with a constant value of ELF. Fig. 10.2 shows iso surfaces with ELF = 0.8 for some molecules from experience a value of ELF = 0.8 is well suited to reveal the distribution of electron pairs in space. 

Since the phase angles cannot be measured in X-ray experiments, structure solution usually involves an iterative process, in which starting from a rough estimate of the phases, the structure suggested by the electron density map obtained from Eq. (13-3) and the phase computed by Eq. (13-1) are gradually refined, until the computed structure factor amplitudes from Eq. (13-1) converge to the ones observed experimentally. 

Figure 7. Schematic diagram of a flowing-afterglow electron-ion experiment. The diameter of flow tubes is typically 5 to 10 cm and the length is 1 to 2 meters. The carrier gas (helium) enters through the discharge and flows with a velocity of 50 to 100 m/s towards the downstream end of the tube where it exits into a fast pump. Recombination occurs mainly in the region 10 to 20 cm downstream from the movable reagent inlet, at which the ions under study are produced by ion-molecule reactions. The Langmuir probe measures the variation of the electron density in that region. A differentially pumped mass spectrometer is used to determine which ion species are present in the plasma.

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