Occupation number notation

In general whether we are discussing symmetrical or antisymmetrical states, the numbers nx are usually termed the occupation numbers, and if we are given a complete spectrum of one-particle states, indicated by A, then the set of occupation numbers assigned to the values of A, specifies the state of the system of JY identical particles just as well as the assignment of a A-value to each particle. Thus we may use the notation just as well as A to specify the state, and so write... 

The wave function of a many-electron system can be written as a Slater determinant (see App. B). However, a more convenient notation is provided by using the occupation number representation, whereby the A-electron de-terminantal function takes the form (March et al. 1967, Raimes 1972)... 

The resolvent in eq. (1.208) is called the one-electron Green s function and the notation for it reads G (z). The integration contour may be set in such a way that it encloses all the poles of the resolvent corresponding to the occupied MOs giving by this the required total projection operator. In the spin-orbital occupation number and the second quantization representations related to each other, one can write the operator projecting to the occupied (spin)-MO as an operator of the number of particles in it. Indeed, the expression... 

In the present case, there are 2 such states (i.e. the various occupancies of the N sites of the lattice by A and B atoms), each of which has a probability P cri,02, , on). This notation refers to the probability that site 1 has occupation number site 2 has occupation number 02 and so on. We introduce the alternative notation P oi]), which again signifies the probability of a given state characterized by the set of occupation numbers ai. In light of this more explicit notation, we can rewrite eqn (6.31) as... 

This la.st equation may be given a neater appearance by noting that, in our definition of the matrices hg, the shell occupation number appears as a simple factor. If we extract this factor and quote it explicitly without changing the notation for the hq we have ... 

Cvi k) such that an appropriate fc integration must be performed. To make the notation more compact, it is customary to merge the occupation number fi and the above mixing coefficients (eigenvectors) and convert them into the entries of a fc-dependent density matrix P iv k) by writing... 

Let us take a closer look at the C2H4 + C2H4 C4H8 reaction. Why do the orbitals change their occupation numbers Let us introduce the following notations ... 

The antisymmetry of electronic fermion wave functions can lead to a cumbersome notation, especially when describing many-electron systems. In the method of second-quantization, only the occupied single particle levels are specified together with their occupation numbers. This is the only information which is compatible with the indistinguishability of electrons. 

A complete specification of how an atom s electrons are distributed in its orbitals is called an electron configuration. There are three common ways to represent electron configurations. One is a complete specification of quantum numbers. The second is a shorthand notation from which the quantum numbers can be inferred. The third is a diagrammatic representation of orbital energy levels and their occupancy. 

In this book we are particularly interested in simple descriptions of structures that are easily visualized and providing information of the chemical environment of the ions and atoms involved. For metals, there is an obvious pattern of structures in the periodic table. The number of valence electrons and orbitals are important. These factors determine electron densities and compressibilities, and are essential for theoretical band calculations, etc. The first part of this book covers classical descriptions and notation for crystals, close packing, the PTOT system, and the structures of the elements. The latter and larger part of the book treats the structures of many crystals organized by the patterns of occupancies of close-packed layers in the PTOT system. 

The objective is to use simple, clear, and informative notation for designating the close-packing type and layers occupied, including partial occupancy. From this information and knowledge of close-packed structures we can determine coordination numbers and local symmetries. For these purposes there are two parts of the notation the index, and the layers occupied (L) ... 

Electrosorption is a replacement reaction. We have already discussed the role of the solvent in the interphase, in the context of its effect on the double-layer capacitance. It is most important for our present discussion to know that the electrode is always solvated and that the solvent molecules are held to the surface both by electrostatic and by chemical bonds. Adsorption of a molecule on such a surface requires the removal of the appropriate number of solvent molecules, to make place for the new occupant, so to speak. This is electrosorption. In this chapter we shall restrict our discussion to the electrosorption of neutral organic molecules from aqueous solutions, without charge transfer. Using the notation RH for an unspecified organic molecule, we can then represent electrosorption in general by the reaction... 

We have dealt with the structure of the hydrogen atom—it has one electron in the Is orbital. Helium completes the building up of the Is orbital with a structure Is2—we commonly use this notation for convenience. The figure 1 indicates the value of the principal quantum number. The letter (s) indicates the type of orbital, and the index (2) indicates the degree of occupation, i.e. there are two electrons in this orbital. 

Element symbol, atom identifier for this atom, oxidation state, number of positions, Wyckoff notation, atomic coordinates xyz, isotropic or anisotropic displacement factors, site occupation, all values with s.u. (standard uncertainty) Conditions of measurement (by defined acronyms) ... 

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