Electron density gradients

Figure 9.4 Electron density gradient paths in a plane containing the atoms of the HCN molecule. The solid lines are the intersections of the zero-flux surfaces with the plane. The large black dots are the bond critical points...

One of the most deeply held notions of chemistry is the concept of chemical bond. Such bonds, almost always depicted as formal lines between atomic symbols, have their counterpart within the quantum chemical description of electron densities often [45-51], but not always [54-60], a correspondence can be made between conventional chemical bonds and specific lines defined by the electron density gradient maps. These lines usually interconnect nuclear positions. However, as has been pointed out by Cioslowski [54-60], there are cases where this correspondence fails. 

Figure 24. Electron density gradient paths in the ureido plane in Pnma glycouryl (a) and Cmcm glycouryl (b).

The ab initio approach includes all a- and 71-electrons and seeks to use either analytical or numerical solutions to the integrals that occur in the quantum mechanical problem. This procedure was initially carried out within the framework of the one electron HF-SCF method using the basis sets described above. Subsequently it has been implemented using density functional theory (DFT). If the electron density in the ground state of the system is known, then in principle this knowledge can be used to determine the physical properties of the system. For instance, the locations of the nuclei are revealed by discontinuities in the electron density gradient, while the integral of the density is directly related to the number of electrons present. Ab initio methods are obviously computationally intensive. 

The dendrimer shown in Fig. 3.11 has imide groups that can trap tin ions site-specifically. Interestingly, the tin ions bind to this dendrimer in a stepwise fashion according to the electron density gradient. 

We have used transferable atom equivalent (TAE) descriptors [116,117] that encode the distributions of electron density based molecular properties, such as kinetic energy densities, local average ionization potentials, Fukui functions, electron density gradients, and second derivatives as well as the density itself. In addition autocorrelation descriptors (RAD) were used and represent the molecular geometry characteristics of the molecules, while they are also canonical and independent of 3D coordinates. The 2D descriptors alone or in combination with the latter 3D descriptors were calculated for 26 data sets collated by us from numerous publications. These data sets encompass various ADME/TOX-related enzymes, transporters, and ion channels as... 

Keywords Electron density gradient trajectory bond critical point interaction line atomic... 

EP DRN Electrostatic potential Electron density gradient normal to 0.002 e/au ... 

To correct for the nonuniformity of the electron density, gradients of the density are introduced into the exchange and correlation functionals, creating in this manner nonlocal or semilocal functionals. The first gradient approximation was not successful since it did not fulfill many of the requirements of the exchange-correlation functional it was even worse than the local approximation. 

Measurement of product velocity distributions was not feasible in the D + H2 study however, even in experiments where the recoil velocity. distribution is accurately measured, it is diflBcult to extract a c.m. cross section due to the diflBculty of estimating a resolution that depends on beam shapes, detector geometry, and electron density gradients in the ionizer. 

Another model for the electron density gradient was proposed by Blundell and analyzed by Vonk it consists of a linear density change in the interface [21, 28]. In this model, called the geometric linear model, the smoothing function is of rectangular type (Fig. 19.10) and its Fourier transform is given by... 

Non-Local Density Approximation A density functional theory (DFT) method with the exchange-correlation energy correction containing the electron density gradient. 

In the case of a smooth sharp surface one finds (7 = 0 otherwise this parameter is related to the extent of the electron density gradient around the scattering particle. When the surface shows roughness, the exponent in the denominator can be affected and become smaller than 4. The importance of the Porod approximation is that it contains information on the interface of the particle with the matrix. 

Bader s atoms-in-molecules theory [69, 70] is based on the topology of the electron density p(r) and enables to partition the three-dimensional real space into non-overlapping domains called basins . To achieve such separation, one looks at the so-called gradient paths (GPs), which are the equivalent of field lines in classical electromagnetism. A GP is defined as a curve C such that the electron density gradient pif) (which can stem from theoretical calculations as well as from X-ray diffraction experiments) is tangent to C at every C point. In general (non-nuclear... 

Ji is the electron flux vector, C is the electrochemical potential acting on the electron flux, and Js represents the total entropy density flux vector. We choose this expression, rather than a version based on Eq. (6.1.29), because we wish to treat separately the effects of temperature and of electrochemical potential. The latter involves aU the contributions associated with temperature gradients, electron density gradients, and the externally imposed electrostatic field. It is expedient to introduce a current density vector as = e Ti- Then, along one dimension, we adopt i) = T 3s T + -V( /e) as our dissipation function. For this unidirectional flow pattern, the... 

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