Many-electron systems spin part

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

We have described the spin of a single electron by the two spin functions a(this section we will discuss spin in more detail and consider the spin states of many-electron systems. We will describe restricted Slater determinants that are formed from spin orbitals whose spatial parts are restricted to be the same for a and p spins (i.e., xi = il iP ), Restricted determinants, except in special cases, are not eigenfunctions of the total electron spin operator. However, by taking appropriate linear combinations of such determinants we can form spin-adapted configurations which are proper eigenfunctions. Finally, we will describe unrestricted determinants, which are formed from spin orbitals that have different spatial parts for different spins (i.e., fjS ). 

Precise evaluations of the atomic energy for many-electron systems require the sort of explicit analysis we were just looking at, in which the wavefunctions are written out and symmetrized. In a properly symmetrized many-electron wave-function, reversing the labels of any two electrons in the function must change the sign of the function. This leads to complicated wavefunctions for many-electron atoms, particularly because in most cases the spin and spatial parts of the wave-function can no longer be separated. Commonly, matrix algebra is used to determine these wavefunctions in a task we usually leave to computers. 

This is not the place to expose the parts of quantum mechanics and group theory used in M.O. theory, since many excellent textbooks exist.However, many new examples of how to use this mathematical technique in the description of actual compounds will be given. We may recall that the wave function T for q electrons in a many-electron system is a function of Aq variables three space-variables, and one spin-variable of each electron. Since observables in electron systems do not seem to occur which depend on the relative position of more than two electrons at a time (while three-particle forces perhaps occur in nuclei), the wave function T for three or more electrons is actually more complicated than is needed for the calculation of any interesting property. 

We have demonstrated that by replacing it with a-it we can indeed introduce spin into the nonrelativistic Schrodinger equation. In this form, spin appears explicitly in the wave function through the Pauli spinors (or products of these for many-electron systems), and its interaction with magnetic fields appears naturally in the Hamiltonian and need not be grafted on ad hoc when required. However, apart from the fact that it yields a convenient form of the Schrodinger equation, it is not immediately evident why the operator a it should be used. And we still have the problem that the free-electron part of the Hamiltonian is not Lorentz invariant. So we must look for an alternative route to a relativistic quantum theory for the electron, one which preferably also accounts for spin. Our experiences from the derivations in this section show us that this route may lead to multicomponent wave functions. 

The partition of the total system into a spin part and a lattice part is, in principle, fuzzy. Other nuclear or electronic spins can either be included in the spin space or into the lattice. In a perturbation treatment, it depends on the interaction strength and on the time-scales of different processes. If nuclear spins are interacting over long periods (e.g. within a small molecule), the different spins cannot be considered separately. However, electron spins can in many cases be put in the lattice space since the difference in time scales for the electron spin and nuclear spin dynamics is prohibitive an effective coupling. 

For an overall view of transition metal systems one has to confront a number of problems besides correlation, or sometimes as a part of it many electrons in open shells producing large number of close-lying electronic states, different spin multiplicities, magnetism, metal-metal bonds, core or " semicore electrons that occupy the same part of the space as the valence shell, fast electrons requiring relativistic corrections etc. In addition, many of transition metal systems of practical interest are highly complex and often incompletely characterized, so that careful modeling is to be added to the list of difficulties. 

Because the one-electron operators are identical in form to the one-electron operator in hydrogen-like systems, we use for the independent particle model of Eq. (8.96) for the basis of the many-electron wave function a product consisting of N such hydrogen-like spinors. This ansatz allows us to treat the nonradial part analytically. The radial functions remain unknown. In principle, they may be expanded into a set of known basis functions, but we focus in this chapter on numerical methods, which can be conveniently employed for the one-dimensional radial problem that arises after integration of all angular and spin degrees of freedom. 

Before looking in more detail to the Af-electron wave functions, we will first shortly summarize the most important aspects of the spin part of a one-electron wave function. We will follow the common practice to use lower case symbols when dealing with one-particle wave functions and uppercase for many-particle systems. 

The electron spin resonance spectra of many aromatic univalent ions have been studied. They prove unambiguously that the added electron is not localized on a particular carbon atom, but is distributed over the whole molecule and forms an integral part of the delocalized ir-electron system. 

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