Electron occupation number

In view of the result just found, it is interesting to contrast exact and approximate behaviour of the density n(x F) [eqs. p.5) and (2.17), respectively]. Some insight into the nature of the approximations contained in our treatment is gained through the inspection of Table 1, which collects Fermi-level energy values calculated for several electron occupation numbers and two different electric field amplitudes. The entries have... 

AB and CD denoted the center positions of the bonds formed by atoms A and B and C and D respectively. La and Ly represent the lone pair positions. As we will see, this formulation takes into account bonds and lone pairs hybridation, each one of the term depending of an overlap functional. N0CC(AB) and Nocc(CD) are the electron occupation numbers of the AB and BC bonds. Therefore, Nocc is equal to 2 for usual bonds and lone pairs. R, i .ci) is the distance between the barycenters of the AB and CD bonds. 

Ligands and molecules bind to organometallic and coordination complexes with energies comparable to the adsorption energies found for surfaces. As on surfaces, the chemical bond formed in such complexes also contains significant s or p character. A major difference with metal surfaces is the importance of discrete valence electron occupation numbers on complex stability. A prototype coordination compound is Ni(CO)4. 

The parameter Ae should not be confused with the actual difference in electronic occupation numbers of the two CT states. When the eigenfunctions (])+ corresponding to the eigenstates F (Y ) are represented... 

Number of atomic pv orbitals Number of jt electrons Occupation number of the it MO /... 

Orbital Energies and Optimum it Electron Occupation Numbers... 

Fig. 9. Specific HMO 7r-bond energies E /N for idealized types of cyclic (pd)n systems in dependence on the ring size N for optimum 7t-electron occupation number...

The modified additivity rule introduced by Deutsch et al. (1997) attempted to account for the effects of molecular bonding by introducing empirically determined weighting factors that depend on the atomic orbital radii and the electron occupation numbers of the various atomic orbitals. A detailed comparison with existing molecular ionization cross section data for molecules of the form AB suggested the following explicit form of the ionization cross section of... 

As in general all the y-coefficients do not vanish one has to assume a more general reference state than the single determinant SCF state. This is the rather well-known problem of finding the consistent reference state for the Random Phase Approximation (RPA). It also means that the field operator basis can be enlarged and can for instance include the iV-electron occupation number operators (in this discussion, electron field operators and their adjoints are used referring to a basis of spin orbitals that are the natural spin orbitals of the reference state, as will be discussed below, i.e., the spin orbitals that diagonalize the one-matrix)... 

It is observed that at an electron occupation number of one electron per atom and for an adsorbate orbital of cr symmetry ... 

Molecular orbital Molecular orbitals n center two electron Occupation number... 

The extended Falicov-Kimball model with hybridization ( Rare Earth Elements C 7, 1983, p. 306) is used by Kanda etal. [26] to calculate the electron occupation number of the conduction band as a function of the localized level. The Falicov-Kimball model is also used to describe the intermediate valence state and the phase transition by Schweitzer [27], Singh et al. [28]. 

From Section 5.2, we recall that it is impossible to tell from its occupation number alone whether the orbital decreases or increases the probability of having the two electrons close to the same nucleus. Indeed, this information can be obtained only from an inspection of the two-electron density matrix or the wave function itself. The one-electron occupation numbers are nevertheless useful for analysing the electronic state, providing information about the importance of the individual otbitals for the one-electron density. 

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