Electron density energy

Reactivity of azides towards acetylenedicarboxylates is very dependent on their electron density (energy HOMO). Thus, strongly electron-deficient 3,5-dicyano-2,4,6-triazidopyridine 1039 reacts slowly with dimethyl acetylenedicarboxylate to give triazole derivative 1038 in 34% yield with most of the starting material recovered unchanged. Under comparable conditions, less electron-deficient 3,5-dichloro-2,4,6-triazidopyridine 1040 reacts with dimethyl acetylenedicarboxylate to provide 2,6-bis(l,2,3-triazol-lyl)pyridine derivative 1041 in 75% yield (Scheme 171) <2001CHE861>. 

I-ig. d. Benz> 1 MO s ami excitation processes for the six and eight electron species. Blackened portions and numbers indicate LCAO MO electron densities. Energies in the center column are in units of the absolute value of fj with, an isolated p-orbitul taken as the energy zero,... 

Parallel-plate reactor Dissociation of CH4 for amorphous carbon Well mixed Mean electron density energy distribution measured with Langmuir probe 82... 

The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

This gives the total energy, which is also the kinetic energy in this case because the potential energy is zero within the box , m tenns of the electron density p x,y,z) = (NIL ). It therefore may be plausible to express kinetic energies in tenns of electron densities p(r), but it is by no means clear how to do so for real atoms and molecules with electron-nuclear and electron-electron interactions operative. 

Wang L-W and Teter M P 1992 Kinetic-energy functional of the electron density Phys. Rev. B 45 13 196-220... 

Weinert M, Wimmer E and Freeman A J 1982 Total-energy all-electron density functional method for bulk solids and surfaces Phys. Rev. B 26 4571-8... 

For each configuration of the nuclei, minimization of tlie total energy with respect to the electron density yields the instantaneous value of a potential energy fiinction V(/ ), and the corresponding forces on the nuclei. In principle,... 

Figure C2.13.1. Electron energies and electron densities for different plasmas.

E is tire density of states between E and E + AE. A simpler way of calculating n is to represent all tire electron states in tire CB by an effective density of states at tire energy E (band edge). The electron density is tlien simply n = NJ (Ef. 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

The electrostatic potential at a point r, 0(r), is defined as the work done to bring unit positive charge from infinity to the point. The electrostatic interaction energy between a point charge q located at r and the molecule equals The electrostatic potential has contributions from both the nuclei and from the electrons, unlike the electron density, which only reflects the electronic distribution. The electrostatic potential due to the M nuclei is ... 

Fig. 3.17 The two possible sets of standing waves at the Brillouin zone boundary. Standing wave A concentrates electron density at the nuclei, whereas wave B concentrates electron density between the nuclei. Wave A thus has a lower energy than wave B.

Lee C, W Yang and R G Parr 1988. Development of the Colle-Salvetti Correlation Energy Formula into a Functional of the Electron Density. Physical Review B37 785-789. 

The simplest molecular orbital method to use, and the one involving the most drastic approximations and assumptions, is the Huckel method. One str ength of the Huckel method is that it provides a semiquantitative theoretical treatment of ground-state energies, bond orders, electron densities, and free valences that appeals to the pictorial sense of molecular structure and reactive affinity that most chemists use in their everyday work. Although one rarely sees Huckel calculations in the resear ch literature anymore, they introduce the reader to many of the concepts and much of the nomenclature used in more rigorous molecular orbital calculations. 

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