Electron, noninteger number

Zhang, Y., Yang, W., 1998, A Challenge for Density Functionals Self-Interaction Error Increases for Systems With a Noninteger Number of Electrons , J. Chem. Phys., 109, 2604. 

Density Functional Theory for Noninteger Number of Electrons.8... 

DENSITY FUNCTIONAL THEORY FOR NONINTEGER NUMBER OF ELECTRONS... 

Since the definitions given by Equations 2.5 and 2.6 require the energy to be well defined for noninteger number of electrons and to be differentiable with respect to N, let us present first the extension of DFT to open systems that can exchange electrons with its environment, developed by Perdew, Parr, Levy, and Balduz (PPLB) [23]. 

By differentiating with respect to the number of electrons, the preceding equations presuppose that one can compute the energy (Equation 18.5) and the density (Equation 18.7) for systems with noninteger numbers of electrons. But how Every real and finite system has an integer number of electrons. 

Y. Zhang and W. Yang, A Challenge for density functionals Selfinteraction error increases for systems with a noninteger number of electrons, J. Chem. Phys. 109, 2604-2408 1998. 

Finally, there is the matter of fractional numbers of electrons in Eqs. (30) and (31) and throughout Sec. 4, we found it convenient to consider systems with noninteger numbers of electrons. Since no such systems exist in nature, the properties of these systems must be defined in an appropriate way. Several different arguments converge on the same result the properties of a system with N+s electrons (0 < s < 1) should be taken as the appropriate weighted average of the properties of the systems with integer numbers of electrons [5,90] ... 

So the highest occupied Kohn-Sham orbital has a fractional occupation number to. The fact that is uniquely defined by follows directly from the Hohenberg-Kohn theorem applied to the non-interacting system. The proof of the Hohenberg-Kohn theorem for systems with noninteger number of electrons proceeds along the same lines as for systems with integer particle number (alternatively it can be obtained from the zero temperature limit of temperature dependent density functional theory [82]). We further split up v, as... 

Since j = x, this approximation for /a reduces to the MuUiken definition for the electronegativity [16]. Within the ensemble approach, the hardness 17 as defined in Equation 34.3 would be zero for noninteger N and undefined for integer values of the electron number. One obtains quantitative values for this concept for integer N, using a finite difference approximation of the chemical potentials yu,1 and /a, i.e.,... 

D. (a) Find the number of rings plus double bonds in a molecule with the composition C 4H,2 and draw one plausible structure, (b) For an ion or radical, the rings + double bonds formula gives noninteger answers because the formula is based on valences in neutral molecules with all electrons paired. How many rings plus double bonds are predicted for C4Hl0NO+ Draw one structure for C4H 0NO+. 

In 1924, Wolfgang Pauh (1900-58) added to the earlier work of Bohr and Arnold Sommerfeld (1868-1951) and identified the fourth quantum number, spin or s, confined to two (noninteger) values +5 and -Vi. This was followed by the Pauli exclusion principle no two electrons in the same atom can have four identical quantum numbers n, 1, m, and s, and these are subject to the following rules n= 1,2, 3,4,... 

COMMENTS The number of electrons transferred in the elementary charge transfer step at the cathode, c, can be a noninteger valne if derived experimentally since there can be more than one charge transfer reaction in parallel. The kinetic polarization voltages in this example are typical relative to one another. That is, the ORR losses usually dominate activation losses when pure hydrogen is nsed as the fnel. 

---