Electron density, determination

According to the Hohenberg-Kohn theorem of the density functional theory, the ground-state electron density determines all molecular properties. E. Bright Wilson [46] noticed that Kato s theorem [47,48] provides an explicit procedure for constructing the Hamiltonian of a Coulomb system from the electron density ... 

Interaction of incident electrons with the electrostatic potential (ESP) gives the possibility to reconstmct the potential from transmission electron diffraction (ED) experiments. ESP and the electron density determine all physical properties of crystals (e.g. energy of electrostatic interaction, characteristics of the electrostatic field in crystals, dipole, quadmple and other momentum of nuclear, diamagnetic susceptibility. 

Abstract This chapter comments on the motivations and the methods of crystallographic studies at low temperature. Cry-crystallography is a brunch of Crystallography, a science that is too often confused with a technique. On the other hand, the scientific background to study crystal phases at low temperature is here provided, together with a survey of many possible techniques that provide complementary or supplementary information. Several apphcations are discussed, in particular in relation with highly accurate studies like electron density determination or phase transition mechanisms. 

Gimarc (83JA1979,86JA4303) proposed a rule of topological charge stabilization which states that heteroatoms prefer to be located at sites that conform to the pattern of relative electron densities determined by connectivity or topology in an isoelectronic, isostructural, homoatomic system that is called the uniform reference frame <83JA1979>. For the series of thienothiophene positional isomers, the pentalene dianion (5) serves as the uniform reference frame ((55) represents pentalene). 

There are instances where theoretical and experimental charge distributions are found to correspond well with protonation behavior, as demonstrated by electron spectroscopy for chemical analysis (ESCA) spectral and complete neglect of differential overlap (CNDO/2) theoretical results. Respective excess electron densities determined by these two methods are uracil N-3 (—0.24, —0.23), uracil N-l ( — 0.21, —0.18), 5-fluorouracil N-3 (—0.19, —0.24), 5-fluorouracil N-l (-0.18, -0.18) (78CJC1555). 

Figure 4. Structure model (left) and isosurfaces of electron densities determined for the sodium cobalt oxide hydrate by MEM analysis (right). Equidensity level 1.0 A 3.

Whereas density functional theory guaranties that for the ground electronic state of molecules the electron density determines the energy, the actual construction of such energy functions from first principles is a problem of considerable complexity. The electron densities computed by the MEDLA method suggest various approximations to the molecular energy of large systems. 

FIGURE 5. Summary of electronic distribution in aniline, (a) Bond distances (A), NBO charges [bracket, in au] and Wiberg indices (parentheses, in au). (b) Topology of the electron density determined from atom-in-molecule calculations p(r) = electron density, L = Laplacian of the density defined as L(r) = —V2p(r) and e = ellipticity of the bond critical point, (c) Laplacian map of the density, (d) Iso-surfaces of the electron localization function, ELF = 0.87 the values are the populations of the valence basins... 

However, products from external quenching are obtained in protonolyses of cyclopropanes that are activated by electron-withdrawing substituents, since in these cases weaker acids (better nucleophiles) can be used to induce the cleavage. This is astonishing and seems to indicate that the polarization of the cyclopropane bonds or the neighborhood of a protonated hetero functionality and not the electron density determines the ease of the proton attack. Examples are given by the formation of 45, 47, 50, 52, " and... 

The sweeping theorem of Hohenberg and Kohn is that, like the wave function, the ground state s electron density determines all the properties of an electronic system [1]. The result is proved in three steps. First, one recalls that the number of electrons is determined from the electron density using Eq. (14). Next, one demonstrates that the external potential can be determined from the ground-state electron density. From N and v(r), we may determine the electronic Hamiltonian and solve Schro-dinger s equation for the wave function, subsequently determining all observable properties of the system. 

A key to this development is the assertion that the ground-state electron density determines the electronic external potential. Stated mathematically, we must demonstrate that the external potential is a functional of the electron density. Just as/is a function of x if and only if no argument of/ x0, corresponds to more than one value of / (which is to say that f(x) is single-valued for all x in the domain of/), the external potential, v(r) is a functional of the ground-state electron density, p(r), if and only if no two external potentials correspond to the same ground-state electron density. 

The general result that the ground-state electron density determines the molecular properties is not really surprising if one takes an information-theoretical approach to the Hohenberg-Kohn density functional model. 

There is a part of the truth in this idealized picture. For most cases, it is sufficient to limit oneself (by concentration on optical excitations and photo-electron ionization, obeying the principle of Franck and Condon or by resignation) to scrutinize the electronic density determined by fixed nuclear positions. The really great success of quantum mechanics was the application to monatomic entities (but some restraining comments are made in section 4.8) where the nucleus is firmly put at the origin (unless below 10 amu, in which case perfectionists start worrying about the center of gravity shifted a trifle by the electrons). 

The first-order change in the electron density determines the change in the dipole moment under the influence of an external perturbation. In the most common case of an electric perturbation, the corresponding property is the frequency-dependent polarizability, which is directly related to (vertical) excitation energies oscillator strengths fi, and transition dipole moments... 

Fig. 13. Electron densities determined applying optical emission spectroscopy along the axis of tube for methane and acetylene plasma system.

There exists a variational principle in terms of the electron density which determines the ground state energy and electron density. Further, the ground state electron density determines the external potential, within an additive constant. 

Dijfraction by single crystals electron density determination... 

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