Fermi vacuum expectation value

Note that the first and second terms on the right-hand side of this equation are simply the spin-orbital Fock operator (in normal-ordered form), and the last two terms are the Hartree-Fock energy (i.e., the Fermi vacuum expectation value of the Hamiltonian). Thus, we may write... 

Now we are ready to formulate the time-independent Wick s theorem. Wick s theorem provides a formal mechanism for the simplification of matrix elements and, in particular, of Fermi vacuum expectation values. The time-independent Wick s theorem may be stated as follows ... 

From the above mentioned relations it is easy to see that the vacuum expectation value of the electronic Hamiltonian (3.4) is zero. The particle-hole formalism implies a redefinition of the vacuum state. Since correlation energy is defined with respect to the Hartree-Fock energy, we redefine the vacuum state as being the occupation-number vector corresponding to the converged HF determinant, the Fermi vacuum. This leads to a redefinition of creation... 

If the coupling constant Y is comparable to the coupling constant g, then the Fermi expectation energies of the Fermions occur at the mean value for the Higgs field (4>0)- In this case the vacuum expectation of the vacuum is proportional to the identity matrix. This means that the masses acquired by the right chiral plus left chiral gauge bosons A + /iM are zero, while the left chiral minus right chiral... 

In this section we shall introduce the very useful concept of the Fermi vacuum, which makes the evaluation of certain types of matrix elements much easier. As a matter of fact, many (if not most) quantum-chemical considerations and methods are based on the Hartree-Fock single determinantal wave function which serves also as a zeroth-order wave function ( reference state ) in guessing more accurate wave functions as well. For this reason, one is often interested in evaluating expectation values with respect to Hartree-Fock-type wave functions. The evaluation of such expressions will be analyzed below in some detail. 

Derivations of the above type can be made much simpler by introducing the Fermi vacuum. Consider the expectation value of some operator A with a Hartree-Fock-type wave function ... 

The appearance of the occupation numbers is the only difference one has to keep in mind when evaluating the expectation value of an operator string with respect to the Fermi vacuum. Checking the occupancies may be too time-consuming however, by introducing a simple trick the whole process can be... 

The expectation value of the operator string sandwiched by the Fermi vacuum can be evaluated in several manner. The simplest way is to realize that l k must annihilate the virtuals s r, and i j must re-create q p". Collecting all possibile pairings we get ... 

In the literatme, the work function of a metal, p (in eV), is often used to estimate the degree of charge transfer at semiconductor/metal junctions. The work function of a metal is defined as the minimum potential experienced by an electron as it is removed from the metal into a vacuum. The work function ip is often nsed in lieu of the electrochemical potential of a metal, because the electrochemical potential of a metal is difficult to determine experimentally, whereas tp is readily accessible from vacuum photoemission data. Additionally, the original model of semiconductor/metal contacts, advanced by Schottky, utilized differences in work functions, as opposed to differences in electrochemical potentials, to describe the electrical properties of semiconductor/metal interfaces. A more positive work function for a metal (or more rigorously, a more positive Fermi level for a metal) would therefore be expected to produce a greater amount of charge transfer for an n-type semiconductor/metal contact. Therefore, use of metals with a range of tp (or fip.m) values should, in principle, allow control over the electrical properties of semiconductor/metal contacts. 

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