Fractional electron number

An open system with a fluctuating number of particles is described by an ensemble or statistical mixture of pure states and the fractional electron number may arise as a time average. Thus, let No be an integer electron number, and N be an average in the interval Ai> < N < Nt) f 1. In this case Y is an ensemble or statistical mixture of the No -electron pure state with wave function T V 1 and probability 1 — t, and the (A 0 I l)-electron pure state with wave function Tv, i and probability t, where 0 < t < 1. 

This issue was resolved in 1982 by Perdew et al. [2,9], who showed how to formulate DFT for systems with fractional electron number by adapting the zero-temperature grand-canonical ensemble construction of Gyftopoulos and Hatsopoulos [10]. Their analysis reveals that the energy and electron density should be linearly interpolated between integer values ... 

In fact, there is another approach to DFT that allows fractional electron numbers, namely the extended Kohn-Sham (EKS) scheme [23,26,27]. ft allows the use of fractional occupation numbers fi 0 < fi < 1, hence... 

These are fundamental results in DFT. The density functional extended for a fractional number of electrons based on the ensemble approach gives the correct description for fractional electron number systems in the dissociation limit of Hj. A great challenge remains to construct an energy functional E[p] that would have the correct behavior for fractional electron number systems, as described in Eq. (3). 

The resonating-valence-bond theory of metals discussed in this paper differs from the older theory in making use of all nine stable outer orbitals of the transition metals, for occupancy by unshared electrons and for use in bond formation the number of valency electrons is consequently considered to be much larger for these metals than has been hitherto accepted. The metallic orbital, an extra orbital necessary for unsynchronized resonance of valence bonds, is considered to be the characteristic structural feature of a metal. It has been found possible to develop a system of metallic radii that permits a detailed discussion to be given of the observed interatomic distances of a metal in terms of its electronic structure. Some peculiar metallic structures can be understood by use of the postulate that the most simple fractional bond orders correspond to the most stable modes of resonance of bonds. The existence of Brillouin zones is compatible with the resonating-valence-bond theory, and the new metallic valencies for metals and alloys with filled-zone properties can be correlated with the electron numbers for important Brillouin polyhedra. 

At least for the case of a non-degenerate ground state of a closed shell system, it is possible to delineate the standard Kohn-Sham procedure quite sharply. (The caveat is directed toward issues of degeneracy at the Fermi level, fractional occupation, continuous non-integer electron number, and the like. In many but of course not all works, these aspects of the theory seem to be... 

Here i/j. ng l 2mK is the fractional occupation number of the 2ma- sp/n-orbitals of Shell K and is a suitably averaged electron interaction matrix (cf. the usual... 

More insight into these processes is obtained by studying the particle number dependent properties of density functionals. This of course requires a suitable definition of these density functionals for fractional particle number. The most natural one is to consider an ensemble of states with different particle number (such an ensemble is for instance obtained by taking a zero temperature limit of temperature dependent density functional theory [84]). We consider a system of N + co electrons where N is an integer and 0 < m < 1. For the corresponding electron density we then have... 

First we discuss and construct monodisperse two-dimensional arrangements of impenetrable cylinders in terms of radial distance correlation functions, the lateral packing fraction and number density. In the second step, these hard cylinders are covered by the mean electronic density functions of the RISA chain segment ensemble. Last of all, the Fourier transformation and final averaging is... 

To do the calculation in more detail, for the baryons (nuclei) and electrons, we define the ionization fraction X (t), the ratio of the density of ions to neutrals if we assume overall charge-neutrality, this is equal to the ratio of the free electron number density to that of neutrals. We also assume that the number density of any massive species (i) is large enough that it can be described by a Boltzmann distribution,... 

Fig. 2. Electronic dipolar nature of the peptide unit. The numbers adjacent to each atom give the approximate fractional electronic charge attributed to each atom (in units of fundamental electronic charge). The magnitude of the dipole moment is 0.72 ek = 3.46 D.

Because 7 = 3<- ll<- 12<- 13, increased-valence structure 7 involves fractional kx(CN) and 7iy(CN) electron-pair bonds [2,4,10-14,16]. Therefore its C-N bond-number, or bond-order is less than 3. A thin bond line is used to represent a fractional electron-pair bond [2-4]. An N-O double bond, which consists of an electron-pair a bond, and one-electron nx and ny bonds, is also present in this VB structure. With these bond properties, the N-0 and C-N bond-lengths that are implied by increased-valence structure 7 are in accord with the following observations with regard to its bond-lengths [171 ... 

With such a definition in mind, one envisions that an electron will be transferred as a unit and thus reaches the conclusion that the resultant charge must be an integer. Alternately, Shriver, Atkins and Langford [12] seem to have no problem with fractional oxidation numbers and define the term as ... 

MCSCF. Alternatively, a UHF (when different from the RHF) type wave function may also be used. The total UHF density, which is the sum of the a and /3 density matrices, will also provide fractional occupation numbers since UHF includes some electron correlation. The procedure may still fail. If the underlying RHF wave function is poor, the MP2 correction may also give poor results, and selecting the active MCSCF orbitals based on such MP2 occupation numbers may again lead to erroneous results. In practice, however, selecting active orbitals based on for example MP2 occupation numbers appears to be quite efficient, and better than... 

The idea of fractional occupation numbers was introduced by Slater [71], already in 1969. This approach is not limited for the JT systems, e.g. it was explored by Dunlap and Mei [32] for molecules, by Filatov and Shaik [36] for diradicals, and is also used for calculations of solids and metal clusters [8], It rests on a Arm basis in cases when the ground state density has to be represented by a weighted sum of single determinant densities [53,79], One should remember that molecular orbitals (MO) themselves have no special meaning. Thus, using partial occupation is just a way of obtainning electron density of a proper symmetry (HS). 

Convention 1 is fundamental because it guarantees charge conservation The total number of electrons must remain constant in chemical reactions. This rule also makes the oxidation numbers of the neutral atoms of all elements zero. Conventions 2 to 5 are based on the principle that in ionic compounds the oxidation number should equal the charge on the ion. Note that fractional oxidation numbers, although uncommon, are allowed and, in fact, are necessary to be consistent with this set of conventions. 

Here we formulate a fractional electron method (FEM) which allows for non-integer numbers of electrons in a QM system. The approach relies on the use of a pseudo-closed-shell expression for the electronic energy, where fractional occupation numbers for the MO s are assumed at the outset. If M is the number of atomic orbitals (AO), n... 

This results in one oxygen having two lone pairs, one homoatomic bond and one unpaired electron, which means that it has a share in the six electrons in the valence shell, exactly as it would have had in the elemental state, and so its oxidation number is zero. The other oxygen has three lone pairs and one homoatomic bond. Thus, it has a share in seven electrons, which is one more than the elemental state, and so it has an oxidation state of-1. This analysis resolves the problem of fractional oxidation numbers that was encountered above. 

Thirdly, and on a slightly different note, one can hypothesise that there is a reorganisation of the electrons in the n molecular orbitals to give another canonical structure, where the central nitrogen still has an oxidation number of+1, but now each of the terminal nitrogen atoms has an oxidation number of-1. Again, this analysis resolves the problem of fractional oxidation numbers encountered above. 

Oxidation number is a formal concept adopted for our convenience. The numbers are determined by relying on rules. These rules can result in a fractional oxidation number, as shown here. This does not mean that electronic charges are spht. 

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