Fermi functions

Effects of projection weights on total energy and kinetic energy (in hartree) for H-(-C=C-) H. The projection weights are generated from eqs.(18) and (42)-(49). 

A common Fermi energy across the whole molecule is physically important, particularly when polar and ionic groups are involved. It controls the charge transfers among subsystems. Polar bonds can be broken in the divide step. This has been done for the tetraglycine molecule. Both the Harris and the self-consistent results are similar to what is shown here for the nonpolar molecules H-(-C=C-)-H [57,58]. 

For finite temperature DFT the Fermi function in eqs.(l 1) and (12) is exact [43], The divide-and-conquer method gives the same results as the conventional KS method does except the orbitals. The parameter /3 now gets a physical meaning. It is the inverse of the temperature. 

Analysis of electron distributions among orbitals for H-(-C=C-)t,H after the self-consistency is achieved. The number of electrons in this molecule is 38. Projection weights are used in all cases. K — number of orbitals picked without Fermi fraction occupation numbers. 

The performance of basis sets also depends on the projection weights used. Using fewer buffer atoms requires faster decaying g ( r ) s. When many buffer atoms are used the sharpness of the g ( r ) s is not expected to be that important. After all, in the limiting case all basis sets include contributions from all atoms in the molecule. The conventional KS results will emerge no matter what form is used for g ( r ) s. 

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The occupation of the energy levels of the conduction band in metals is described by the Fermi function... 

Particles that obey Fermi statistics are called Fermi particles or fermions. The probability density of Fermi particles in their energy levels is thus represented by the Fermi function, fiz), that gives the probability of fermion occupation in an energy level, e, as shown in Eqn. 1-1 ... 

Fig. 1-1. Probability density functions of partiele energy distribution (a) Fermi function, (b) Bose-Einstein function, e = particle energy f(i) - probability density function cp = Fermi level sb - Bose-Einstein condensation level.

For high density electron ensembles such as free valence electrons in solid metals where electrons are in the state of degeneracy, the distribution of electron energy follows the Fermi function of Eqn. 1-1. According to quantum statistical dynamics [Davidson, 1962], the electrochemical potential, P., of electrons is represented by the Fermi level, ep, as shown in Eqn. 1-10 ... 

Electrons thermally excited from the valence band (VB) occupy successively the levels in the conduction band (CB) in accordance with the Fermi distribution function. Since the concentration of thermally excited electrons (10 to 10 cm" ) is much smaller than the state density of electrons (10 cm ) in the conduction band, the Fermi function may be approximated by the Boltzmann distribution function. The concentration of electrons in the conduction band is, then, given by the following integral [Blakemore, 1985 Sato, 1993] ... 

The concentration of electrons, n, in the conduction band of n-type semiconductors and the hole concentration, p, in the valence band of p-type semiconductors are given by Eqn. 2-7 and Eqn. 2-10, respectively. The concentration of ionized donors, IVd-, and the concentration of ionized acceptors, iVx-, are derived by using the Fermi function approximated by the Boltzmann function as shown in Eqns. 2-18 and 2-19, respectively ... 

The concentration of electrons that occupy a part of the total concentration, N, of the energy states available for electrons is obtained by integrating the product of the state density. Die), and the Fermi function, fie), as shown in Eqn. 2-30 ... 

Fig. 2-20. Electron state density and ranges of Fermi energy where electron occupation probability in the conduction band of an electron ensemble of low electron density (e.g., semiconductor) follows Boltzmann function (Y i)or Fermi function (y > 1) y = electron activity coeffident ET =transition level from Y 4= 1 to Y > 1 0(t) = electron energy state density CB = conduction band. [From Rosenberg, I960.]...

In cases in which the surface state density is high Nc/i,Nm, Ny/i,Nm - 1), electron distribution in the siuface state conforms to the Fermi function (the state of degeneracy) and the Fermi level is pinned at the surface state level. This is what is called the Fermi level pinning at the surface state. 

The potential i sc of the space charge layer can also be derived as a fixnction of the surface state charge Ou (the surface state density multiplied by the Fermi function). The relationship between of a. and M>sc thus derived can be compared with the relationship between and R (Eqn. 5-67) to obtain, to a first approximation, Eqn. 5-68 for the distribution of the electrode potential in the space charge layer and in the compact layer [Myamlin-Pleskov, 1967 Sato, 1993] ... 

Note that the exchange term is of the form / y(r,r ) h(r )dr instead of the y (r) (r) type. Equation (1.12), known as the Hartree-Fock equation, is intractable except for the free-electron gas case. Hence the interest in sticking to the conceptually simple free-electron case as the basis for solving the more realistic case of electrons in periodic potentials. The question is how far can this approximation be driven. Landau s approach, known as the Fermi liquid theory, establishes that the electron-electron interactions do not appear to invalidate the one-electron picture, even when such interactions are strong, provided that the levels involved are located within kBT of Ep. For metals, electrons are distributed close to Ep according to the Fermi function f E) ... 

For the Boltzmann formulation we introduce, for each point on the Fermi surface, the relaxation time t, defined so that any disturbance 5/ of the equilibrium Fermi function / decays according to the law... 

The density of states function given here follows from the assumption of a parabolic energy band and includes spin degeneracy (Bube, 1974, p. 172). (Note that the Fermi function effectively cuts off the integrand at a few kT above eF, so that for kT the upper limit may be extended to infinity.) Also we have set c n = 0. 

In conclusion, let us say that even though the method becomes rather complicated here because the factor in polynomial form is replaced by a Fermi function, we consider that the method nevertheless presents a certain interest because at the same time it gives a solid base for comparison between the LCAO model and the metallic model. 

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