Fermi-Fock space

In Section 2 we discuss the Lie algebra structure of Fermi-Fock space, which is the direct sum of all of the state spaces W, in Section 3 we will describe the possible decompositions of the Lie algebra u 2s) of U 2s), the one-particle group associated with, and thus the factorizations of U (25) that lead to manifolds of CSs in that can be based on U 2s). In Section 4 we study in detail the forms of AGP CSs. We also show how the CS construction applied to the unitary group of A-electron space can reproduce the familiar configurational interaction expansion. We conclude with a summary and discussion in Section 5. 

The most convenient representation of the generators of the Lie algebra of U (25 ) for fermionic quantum mechanics is by second quantized operators acting in Fermi-Fock space, Tfp, which is defined to be the direct sum... 

Let us fix some partitioning of all energy levels to two equal parts and choose some one-to-one correspondence between them. Say we can consider /cth level down from Fermi bound f and agree that it corresponds to kth level up from ep. We shall denote jth level down from Fermi bound by ordinary letter and fh level up from Fermi bound by j. Call the first level jth lower level and the second one yth upper level. Fock space T can be represented as T 0 G. .. 0 kFi.. where each Tj corresponds to jth pair of the corresponding energy levels. Consider a subspace I1) in Tj which... 

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. 

We turn now to the interaction energy e2/r12 between electrons and consider first its effect on the Fermi surface. The theory outlined until this point has been based on the Hartree-Fock approximation in which each electron moves in the average field of all the other electrons. A striking feature of this theory is that all states are full up to a limiting value of the energy denoted by F and called the Fermi energy. This is true for non-crystalline as well as for crystalline solids for the latter, in addition, occupied states in fc-space are separated from unoccupied states by the "Fermi surface . Both of these features of the simple model, in which the interaction between electrons is neglected, are exact properties of the many-electron wave function the Fermi surface is a real physical quantity, which can be determined experimentally in several ways. 

I. Flamant et al. successfully applied the FGSO basis set in Fourier Space Restricted Hartree Fock (FS-RHF) in a study of identification of conformational signatures in valence band of polyethylene. In 1998, they used a distributed basis set of s-type Gaussian function (DSGF) in FS-RHF. The method briefly is to use RHF-Bloch states (p (k,r), which are doubly occupied up to the Fermi energy Ep and orthonormalized. k and n the wave number and the band index, respectively. 

Actually, a Hartree-Fock wave function does have some instantaneous electron correlation. A Hartree-Fock function satisfies the antisymmetry requironent. Therefore [Eq. (10.20)], it vanishes when two electrons with the same spin have the same spatial coordinates. For a Hartree-Fock function, there is little probability of finding electrons of the same spin in the same region of space, so a Hartree-Fock function has some correlation of the motions of electrons with the same spin. This makes the Hartree-Fock energy lower than the Hartree energy. One sometimes refers to a Fermi hole around each electron in a Hartree-Fock wave function, thereby indicating a region in which the probability of finding another electron with the same spin is small. 

In a study in 1982, Luken and Culberson analyzed the change of the Fermi hole shape with respect to the position of reference electron to gain information about the spatial localizatirai of electrons [36], The Fermi hole density is derived from the same-spin pair density and describes the probability density to find an electron at given position, when another same-spin electron is localized at the reference position with all the other electros located somewhere in the space. Like in Sect. 2.2, it shows how the electronic motion of electrons creating a same-spin pair is correlated. For a closed-shell Hartree-Fock wave function, the so-called Fermi hole mobility function F(r) ... 

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