Concept of the Fermi Vacuum

Show that the cyclic permutation of indices p, v, p or that of X, a, x in Problem 5.2 does not affect the value of the matrix element. 

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. 

To begin with, consider the matrix element of the operator string aj aj between a Hartree-Fock wave function, i.e. determinant, built up from orbitals i 2 Vn- Assume i, k g 1,2. N. We have then  

To determine all possible pairings, or to move the strings to the left seems to be rather complicated. However, the annihilation operator a, 1 g 1,2.N, can always be moved just before aj if 1 i and 1 k by an even number of successive 

This latter expression is easy to evaluate the result is zero (notice, for example, that two electrons are being created on orbital which is impossible). Thus one has  

---

In what follows the concept of the Fermi vacuum will be utilized in many derivations and the reader will see how powerful it is. 

In this chapter we introduce and discuss a number of concepts that are commonly used in the electrochemical literature and in the remainder of this book. In particular we will illuminate the relation of electrochemical concepts to those used in related disciplines. Electrochemistry has much in common with surface science, which is the study of solid surfaces in contact with a gas phase or, more commonly, with ultra-high vacuum (uhv). A number of surface science techniques has been applied to electrochemical interfaces with great success. Conversely, surface scientists have become attracted to electrochemistry because the electrode charge (or equivalently the potential) is a useful variable which cannot be well controlled for surfaces in uhv. This has led to a laudable attempt to use similar terminologies for these two related sciences, and to introduce the concepts of the absolute scale of electrochemical potentials and the Fermi level of a redox reaction into electrochemistry. Unfortunately, there is some confusion of these terms in the literature, even though they are quite simple. 

For a metal, the negative of the work function gives the position of the Fermi level with respect to the vacuum outside the metal. Similarly, the negative of the work function of an electrochemical reaction is referred to as the Fermi level Ep (redox) of this reaction, measured with respect to the vacuum in this context Fermi level is used as a synonym for electrochemical potential. If the same reference point is used for the metal s,nd the redox couple, the equilibrium condition for the redox reaction is simply Ep (metal)= Ep(redox). So the notion of a Fermi level for a redox couple is a convenient concept however, this terminology does not imply that there are free electrons in the solution which obey Fermi-Dirac statistics, a misconception sometimes found in the literature. 

THE VACUUM SCALE OF ELECTRODE POTENTIAL AND THE CONCEPT OF THE SOLUTION FERMI LEVEL... 

The concept of the solution Fermi level is very useful in photoelectrochemistry, althongh it strikes some as strange at first, because the term Fermi level was first introduced and defined for an electronically conducting phase, such as a metal or semiconductor, which contains free electrons. In this context, the Fermi level is defined as the energy, measured with respect any convenient reference level, for which the probability that an electronic energy level is occupied is one-half. The most convenient reference level to use in photoelectrochemistry is the local vacuum level of the solution. The Fermi level of any conducting phase a is then synonymous with the electrochemical potential of an electron in that phase, i.e. E = fi . 

It is well established that the Auger parameter is a very useful concept, which is not affected by the reference level used in the analysis of the data (the Fermi or the vacuum level) [75]. Moreover, the Auger parameter, being a difference between two peaks recorded on the same energy scale, does not depend on surface charging. It appears that Auger parameter measurements are very useful... 

The selection of a reference potential at the Ohmic contact is arbitrary and was chosen to emphasize the degree of band bending and straightening in the semiconductor. The development of Mott-Schottky theory in Section 12.3.2 employs a potential referenced to the Ohmic contact. A difference in sign will be seen if the potential is referenced instead to a reference electrode located in the electrolyte. The potential of the electrolyte has been fovmd to be independent of current and illumination intensity when referenced to an external quantity such as the Fermi energy of an electron in vacuum. This concept has proved useful for predicting the interaction between semiconductors and a variety of redox couples. The lUPAC standard for photoelectrochemical systems, in fact, is that the potential is referred to a reference electrode in the electrolyte. ... 

Two-photon photoemission can be viewed as regular (one-photon) photoemission from a state after excitation of the surface by another photon. All the well-known concepts of photoemission can be apphed to two-photon photoemission energy, spin, and (parallel) momentum conservation for the emitted electron as well as dipole selection rales for optical transitions. The main realm of two-photon photoemission is the spectroscopy of excited intermediate states with energies above the Fermi level, which are normally unoccupied. This energy range, in particular, the part below the vacuum level, is otherwise accessible only by inverse photoemission. 

---