Fermi hole definition

Equation (4.49) indicates that for this wave function the classical Coulomb repulsion between the electron clouds in orbitals a and b is reduced by Kab, where the definition of this integral may be inferred from comparing the third equality to the fourth. This fascinating consequence of the Pauli principle reflects the reduced probability of finding two electrons of the same spin close to one another - a so-called Fermi hole is said to surround each electron. 

We see that the Fermi hole for electrons of the same spin is an approximate substitute for the Coulomb hole and that the correlation problem for electrons of the same spin is, at least partially, eliminated. The situation is far more dramatic for electrons of different spin — and two electrons paired in the same molecular orbital enter this category — for which there is no Fermi hole. Two such electrons have a definite probability of being in the same volume element and the correlation problem here is particularly acute. 

The quasi-Fermi level is often interpreted as a thermodynamic driving force. Whether or not this is appropriate is a matter of some debate. While it may provide useful insights, the concept has been derived from kinetic arguments, and can at best provide a quasi-thermodynamic description. Whereas true equilibrium thermodynamics are universally valid, the predictive value of the quasi-Fermi level as a thermodynamic driving force for, e.g., chemical reactions may depend on the reaction mechanism. One particular aspect to be noted in this respect is that the quasi-Fermi level definitions in (2.62) and (2.63) only consider electrons and holes in the conduction and valence bands. They do not account for any changes in the... 

As tools for measuring electron delocalization, we opt for the visual inspection of the occupied orbitals and especially the so-called domain-averaged Fermi-hole (DAFH) analysis and MCI. Both of these require availability of atomic overlap matrices, where the overlap between two molecular orbitals is obtained in atom condensed form. This requires the definition of an atom in the molecule and in this study we chose the Ftirshfeld-I method. Finally, we introduce ring current maps in the ipso-centric approach. 

This is called the domain-averaged Fermi hole (DAFH), although in the general definition above, domain-averaged correlation hole might be more appropriate. It expresses how the presence of electrons in the chosen domain has an impact on the probability of finding an electron at some coordinate r,. As a one electron function. 

In addition to the numerical information provided by the values of various bond indices, the family of tools for the description of molecular structure was complemented some time ago by the new approach based on the analysis of the so-called domain averaged Fermi holes (DAFH). The most straightforward definition of these holes is via the restricted integration of the exchange part of the pair density [cf. eqn (2))... 

The detection of sharp plasmon absorption signifies the onset of metallic character. This phenomenon occurs in the presence of a conduction band intersected by the Fermi level, which enables electron-hole pairs of all energies, no matter how small, to be excited. A metal, of course, conducts current electrically and its resistivity has a positive temperature coefficient. On the basis of these definitions, aqueous 5-10 nm colloidal silver particles, in the millimolar concentration range, can be considered to be metallic. Smaller particles in the 100-A > D > 20-A size domain, which exhibit absorption spectra blue-shifted from the plasmon band (Fig. 80), have been suggested to be quasi-metallic [513] these particles are size-quantized [8-11]. Still smaller particles, having distinct absorption bands in the ultraviolet region, are non-metallic silver clusters. 

It is presently hard to give a definitive explanation of the unexpected appearance of the spectrum of slightly deuterated NMA, although one is tempted to suspect a Fermi resonance interaction giving rise to an Evans hole centered around 2300 cm-1. 

From this definition it is evident that application of Yj to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in 14>0 >. The effect of YA on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in I 0>. The effect of YA" on the Fermi vacuum is the creation of a particle in the virtual spin-orbitals and finally, the effect of YA" is the annihilation of a particle in virtual spin-orbitals. Thus e.g., a singly excited Slater determinant I ) can be described as... 

The sign of the thermopower is negative for electrons and positive for holes and its measurement provides a definitive identification of the type of conduction. When conduction is by a single type of carrier (either electrons or holes) far from the Fermi energy in a homogeneous material, the thermopower is... 

This definition ensures that together with the Kohn-Sham theory Fermi plP(r, F), and the quantum-mechanical Fermi-Coulomb p (r, F) holes, the Coulomb hole pP(r, F) too corresponds to the system density p(r). The correlation energy EP[p] is then... 

FIGURE 10.1 Classification of the electron orbitals. Each column in this figure can be interpreted as (the occupation of) a determinant model space. While it is convenient to distinguish between four classes of orbitals for open-shell atoms and molecules (left side), the definition of the Fermi level itself just separates the particlefrom the hole states (right side). 

A special limit of the Lindhard response function is known as the Thomas-Fermi screening. This is based on the limit of the Lindhard function for k kp. Consistent with our earlier remarks, this corresponds to electron-hole excitations with small total momentum relative to the Fermi momentum. In this case, expanding the occupation numbers n(k k/2) through their definition in Eq. (2.106) about k = 0, we obtain to first order in k... 

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