Electron density, distribution function

Molecular structure is theoretically intimately related to electron-density distribution functions. In quantum-chemical analysis this density is synthesized as a molecular orbital, by a linear combination of real atomic orbitals, and minimized as a function of total energy. Crystallographically the unit cell density is represented by a Fourier sum over periodic electron wave functions... 

The structure of the unit cell of a crystal may be described in terms of the electron density distribution in the cell. The x,y,z coordinates of the maxima of the electron density function correspond to the positions of the atoms. The electron density distribution function p (x,y,z) may be represented by a three-dimensional Fourier series... 

The theory of AIM allows one to study the concept of chemical bond and the bond strength in terms of electron density distribution function [6, 193]. It exploits the topological features of electron density and thereby a definition of chemical bonding through bond path and bond critical point (BCP). A BCP (it is a point at which gradient vector vanishes, Vp(r) = 0) is found between the... 

Which fundamental properties X could we be interested in Realizing that the electron density distribution function contains all the information about the system in the ground state (Hohenberg and Kohn theorems), its response to several perturbations is certainly of fundamental importance. Other properties also provide valuable information, such as the energy and the electronic chemical potential of the system. We will consider all of these and try to find analytical expressions for their response to, or resistance against, changes in N or v(r). 

The Fukui function is normalized 4=1 [9]). This follows from the normalization condition for the electron density distribution function ... 

The discussion of this technique has been kept short because the diffraction spectra are very similar to Debye - Scherrer patterns the method is becoming very important for molecular structure analysis. An electron beam in a high vacuum (0.1-10 Pa) collides with a molecular beam, and the electrons are diffracted by the molecules. The film reveals washed-out Debye-Scherrer rings, from which a radial electron density distribution function for the molecule can be derived. Together with spectroscopic data, the distribution makes it possible to infer the molecular structure 129]. 

For heteroatomic materials (f f )n the electron density distribution function has been determined by Warren and Gingrich (1934) whereby... 

The Hohenbeig and Kohn theorem (1964 Leach, 1996, pp. 528-533 Levine, 2013) states that all the properties of a molecular system in its ground state can be derived from the electron density distribution function. The total eneigy may be expressed as the sum of kinetic, potential, and exchange/correlation terms as in Equation 5.2, where p is to be understood as a function of the internal coordinates, symbolized by the vector r ... 

Experimentally determined and theoretically calculated electron density distribution functions of the fluoroquasisilatrane 84 have been investigated [214]. 

Silatranes are known since more than 50 years [287, 288] but are stiU fascinating molecules in the focus of an ongoing scientific interest. Fluoro-substituted quasisilatranes have been synthesized [214, 289-291]. Experimental and theoretically calculated electron density distribution functions in the crystal structure of 84 have been investigated [214]. Properties of chemical bonding in silatranes have also been studied in 1-hydrosilatrane [218] and 1-fluorosilatrane [219]. 

A different scheme must be used for determining polarization functions and very diffuse functions (Rydberg functions). It is reasonable to use functions from another basis set for the same element. Another option is to use functions that will depict the electron density distribution at the desired distance from the nucleus as described above. 

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. 

Krijn, M.P.C.M. and Feil, D. (1988), A local density-functional study ofthe electron density distribution in the H20 dimer, J. Chem. Phys., 89, 5787-5793. 

If we multiply the probability density P(x, y, z) by the number of electrons N, then we obtain the electron density distribution or electron distribution, which is denoted by p(x, y, z), which is the probability of finding an electron in an element of volume dr. When integrated over all space, p(x, y, z) gives the total number of electrons in the system, as expected. The real importance of the concept of an electron density is clear when we consider that the wave function tp has no physical meaning and cannot be measured experimentally. This is particularly true for a system with /V electrons. The wave function of such a system is a function of 3N spatial coordinates. In other words, it is a multidimensional function and as such does not exist in real three-dimensional space. On the other hand, the electron density of any atom or molecule is a measurable function that has a clear interpretation and exists in real space. 

In the following chapter we show how the topology of an important function of p, the Laplacian, enables us to obtain additional information from the analysis of the electron density distribution. 

As for the one-dimensional case, the function L makes features emerge from the electron density that p itself does not clearly show. What then does the function L reveal for the spherical electron density of a free atom Because of the spherical symmetry, it suffices to focus on the radial dimension alone. Figure 7.2a shows the relief map of p(r) in a plane through the nucleus of the argon atom. Figure 7.2b shows the relief map of L(r) for the same plane, and Figure 7.2c the corresponding contour map. Since the electron density distribution is... 

The crystallinity of organic pigment powders makes X-ray diffraction analysis the single most important technique to determine crystal modifications. The reflexions that are recorded at various angles from the direction of the incident beam are a function of the unit cell dimensions and are expected to reflect the symmetry and the geometry of the crystal lattice. The intensity of the reflected beam, on the other hand, is largely controlled by the content of the unit cell in other words, since it is indicative of the structural amplitudes and parameters and the electron density distribution, it provides the basis for true structural determination [32],... 

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