Electron density systems

The condensation of aldehydes or ketones with secondary amines leads to "encunines via N-hemiacetals and immonium hydroxides, when the water is removed. In these conjugated systems electron density and nudeophilicity are largely transferred from the nitrogen to the a-carbon atom, and thus enamines are useful electroneutral d -reagents (G.A. Cook, 1969 S.F. Dyke, 1973). A bulky heterocyclic substituent supports regio- and stereoselective reactions. 

The system electron density p(r) and hence the one-electron probability distribution p(r) = p(r)/N, that is, the density per electron or the shape factor of p, are determined by the first-order density matrix y in the AO representation, also called the charge and bond order (CBO) matrix,... 

Prior to the advent of computer graphics systems, electron density maps were, in actuality, produced as described above. Each section was printed on paper as a field of numbers, each... 

R. McWeeny, in Molecules in Physics, Chemistry, and Biology, J. Maruani, Ed., Reidel, Dordrecht, 1989, p. 1. The Molecule as a Many-Electron System Electron Densities and Molecular Properties. 

A). Hydrogen atoms of the PCP ligand are omitted for clarity, (b) Fragment of molecular graph of the system. Electron density at the (3,-1) bond critical point and bond order (as a delocalization index [21], in bold italic) are reported for W-H H-Ni and C-H H-Ni contacts. Color codes -bond critical points, -ring critical points. 

Now there remains to choose the only form of the electronic density to obtain an analytical formula for electronegativity. Taking into accoimt the quantum wave natirre of valence/peripheial electrons behaving asymptotically on the long-range interaction, in accordance with the basic postulate of quantum mechanics, see the Volume I of the present five-volume book (Putz, 2016a), there is natural to consider the exponential form on the elee-tronic levels for the fermionic system/electronic density in question, that is ... 

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

The so-called Flohenberg-Kolm [ ] theorem states that the ground-state electron density p(r) describing an A-electron system uniquely detemiines tlie potential V(r) in the Flamiltonian... 

In DFT, the electronic density rather than the wavefiinction is tire basic variable. Flohenberg and Kohn showed [24] that all the observable ground-state properties of a system of interacting electrons moving in an external potential are uniquely dependent on the charge density p(r) that minimizes the system s total... 

Total spin den sity reflects th e excess probability of fin din g a versus P electrons in an open-shell system. Tor a system m which the a electron density is equal to the P electron density (for example, a closed-shell system), the spin density is zero. 

The electron density p(r) at a point r can be calculated from the Bom interpretation of the u a Vv tu iiclion as a sum of squares of the spin orbitals at the point r for all occupied molecular orhital-i. For a system of N electrons occupying N/2 real orbitals, we can write ... 

Local spin density functional theory (LSDFT) is an extension of regular DFT in the same way that restricted and unrestricted Hartree-Fock extensions were developed to deal with systems containing unpaired electrons. In this theory both the electron density and the spin density are fundamental quantities with the net spin density being the difference between the density of up-spin and down-spin electrons ... 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

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. 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

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