Fermi function smearing

Eq. 26 has two mathematical properties important for the adiabatic motion. Firstly, the dot charge in units e has, for sure, to be between zero and one. This circumstance helps one find an approximate solution in the form of a converging power series over Q, provided u = Uc/ ksT < 1. On the other hand, the decomposition over exponent power cu(i can be employed for u > 1. Secondly, if the thermal smearing is marginal, the dot population at the point x, p of the phase space switches quickly between the equilibrium states 0 and 1. The solution is therefore expected to be expressed in terms of the step-like Fermi functions or their derivatives. 

For closed-shell and open-shell molecules, spin-restricted Kohn-Sham (RKS) and spin-unrestricted Kohn-Sham (UKS) density functional calculations were employed, respectively. Except for the calculations of excited states and the cases where pure states are sought, we have employed an approximation in which electron density is smeared among the closely spaced orbitals near the Fermi levels. In this procedure, fractional occupations are allowed for those frontier orbitals with energy difference within 0.01 hartree to avoid the violation of the Aufbau principle (46). 

The line shapes observed in Figure 3.2.2.29a were reproduced nicely in calculations of spectral functions A(k,e) using simple parameterizations for Ee(lc, E) and Edef(k, E) and a many-body treatment for Eph(k, E) [101]. Important parameters such as the electron-phonon coupling constant X could be extracted from this comparison. For measurements at higher temperatures, the two-peak structure becomes blurred because both, the broadening of the Fermi edge and the thermal excitation of phonons lead to a smearing out of the spectral function. 

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