Half-electron Model

In the approach of Dewar and co-workers (34), termed the half-electron method , a physical model is considered in which an unpaired electron is replaced by two hypothetical half-electrons of opposite spin. For radicals containing one unpaired electron, the eigenvalue problem of this method is, in our opinion, identical with the method of Longuet-Higgins and Pople (27) ... 

In the free electron model, the electrons are presumed to be loosely bound to the atoms, making them free to move throughout the metal. The development of this model requires the use of quantum statistics that apply to particles (such as electrons) that have half integral spin. These particles, known as fermions, obey the Pauli exclusion principle. In a metal, the electrons are treated as if they were particles in a three-dimensional box represented by the surfaces of the metal. For such a system when considering a cubic box, the energy of a particle is given by... 

It is known that for a set of ID and 2D quantum spin models the exact ground state of which can be represented in the RVB form [15, 6, 36, 13, 31]. It is natural to try to find electronic models with an exact ground state at half-filling formed by SB functions in the same manner as for above mentioned spin models. The electronic models of these types include the correlated hopping of electrons as well as the spin interactions and pair hopping terms. 

As the first example we consider the ID electronic model with the two-fold degenerate ground state in the form of the simple product of SB dimers, similarly to the ground state of the well-known spin-5 Majumdar-Ghosh model [6]. For the half-filling case the proposed ground state wave functions are ... 

The origin of these effects has been debated. One possibility is the Peierls instability [57], which is discussed elsewhere in this book In a one-dimensional system with a half-filled band and electron-photon coupling, the total energy is decreased by relaxing the atomic positions so that the unit cell is doubled and a gap opens in the conduction band at the Brillouin zone boundary. However, this is again within an independent electron approximation, and electron correlations should not be neglected. They certainly are important in polyenes, and the fact that the lowest-lying excited state in polyenes is a totally symmetric (Ag) state instead of an antisymmetric (Bu) state, as expected from independent electron models, is a consequence... 

Semi-empirical methods (Section 3.9) sometimes employ the so-called half-electron (HE) method for describing open-shell systems, such as doublets and triplets. In this model a doublet state is described by putting two half-electrons in the same orbitals with opposite spins, i.e. constmeting an RHF type wave function where all electron spins arc... 

According to a hospital press release, although the high-tech faucets cut daily water consumption by well over half, Johns Hopkins researchers identified Legionella growing in 50 percent of cultured water samples from 20 electronic-eye faucets in or near patient rooms on three different inpatient units, but in only 1 5 percent of water cultures from 20 traditional, manual faucets in the same patient care areas. Weekly water culture results also showed half the amount of bacterial growth of any kind in the manual faucets than in the electronic models. 

A description of covalent bond formation in terms of atomic orbital overlap is called the valence bond method. The creation of a covalent bond in the valence bond method is normally based on the overlap of half-filled orbitals, but sometimes such an overlap involves a filled orbital on one atom and an empty orbital on another. The valence bond method gives a localized electron model of bonding Core electrons and lone-pair valence electrons retain the same orbital locations as in the separated atoms, but the bonding electrons do not. Instead, they are described by an electron probability density that includes the region of orbital overlap and both nuclei. 

For nearly half a century, Mendeleev s periodic table remained an empirical compilation of the relationship of the elements. Only after the first atomic model was developed by the physicists of the early twentieth century, which took form in Bohr s model, was it possible to reconcile the involved general concepts with the specificity of the chemical elements. Bohr indeed expanded Rutherford s model of the atom, which tried to connect the chemical specificity of the elements grouped in Mendeleev s table with the behavior of electrons spinning around the nucleus. Bohr hit upon the idea that Mendeleev s periodicity could... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

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