Spin coordinates

It is possible to write down a many-body wavefiinction that will reflect the antisynmietric nature of the wavefiinction. In this discussion, the spin coordinate of each electron needs to be explicitly treated. The coordinates of an electron may be specified by rs. where s. represents the spin coordinate. Starting with one-electron orbitals, ( ). (r. s), the following fomi can be invoked ... 

The tliree protons in PH are identical aud indistinguishable. Therefore the molecular Hamiltonian will conmuite with any operation that pemuites them, where such a pemiutation interchanges the space and spin coordinates of the protons. Although this is a rather obvious syimnetry, and a proof is hardly necessary, it can be proved by fomial algebra as done in chapter 6 of [1]. 

We restrict ourselves again to symmetric tetraatomic molecules (ABBA) with linear eqnilibrium geometi7. After integrating over electronic spatial and spin coordinates we obtain for A elecbonic states in the lowest order (quartic) approximation the effective model Hamiltonian H — Hq+ H, which zeroth-order part is given by Eq. (A.4) and the perturbative part of it of the form... 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

The Born interpretation of quantum mechanics tells us that s)dTds gives the chance of finding the electron in the spatial volume element dr and with spin coordinate between s and s + ds. Since probabilities have to sum to 1, we have... 

The integration is over all the space and spin coordinates of electrons 2, 3,..., m. Many of the operators that represent physical properties do not depend on spin, and so we often average-out over the spin variable when dealing with such properties. The chance of finding electron 1 in the differential space element dt] with either spin, and the remaining electrons anywhere and with either spin is... 

Before proving this theorem, we will make some general remarks about the nature of the one-electron functions ipk(x) or spin orbitals. For the two values of the spin coordinate f — 1, such a function y)k(r, f) has two space components... 

Introduction of the half-integral spin of the electrons (values h/2 and —fe/2) alters the above discussion only in that a spin coordinate must now be added to the wavefunctions which would then have both space and spin components. This creates four vectors (three space and one spin component). Application of the Pauli exclusion principle, which states that all wavefunctions must be antisymmetric in space and spin coordinates for all pairs of electrons, again results in the T-state being of lower energy [equations (9) and (10)]. 

In this volume dedicated to Yngve Ohm we feel it is particularly appropriate to extend his ideas and merge them with the powerful practical and conceptual tools of Density Functional Theory (6). We extend the formalism used in the TDVP to mixed states and consider the states to be labeled by the densities of electronic space and spin coordinates. (In the treatment presented here we do not explicitly consider the nuclei but consider them to be fixed. Elsewhere we shall show that it is indeed straightforward to extend our treatment in the same way as Ohm et al. and obtain equations that avoid the Bom-Oppenheimer Approximation.) In this article we obtain a formulation of exact equations for the evolution of electronic space-spin densities, which are equivalent to the Heisenberg equation of motion for the electtons in the system. Using the observation that densities can be expressed as quadratic expansions of functions, we also obtain exact equations for Aese one-particle functions. 

The electron density is the diagonal element of the number density matrix N(r,r ), i.e the first order redueed density matrix after integration over the spin coordinates, ... 

We first label the particle with coordinates qi as particle 1, the one with coordinates q2 as particle 2, and the one with coordinates qs as particle 3. The Hamiltonian operator H(, 2, 3) is dependent on the positions, momentum operators, and perhaps spin coordinates of each of the three particles. For identical particles, this operator must be symmetric with respect to particle interchange... 

Remember from basic quantum mechanics that to completely describe an electron its spin needs to be specified in addition to the spatial coordinates. The spin coordinates can only assume the values Vr, the possible values of the spin functions a(s) and fits) are raO/i) = (K- /i) = 1 and a(-V4) = (K /i) = 0. 

The probability interpretation from equation (1-7) of the wave function leads directly to the central quantity of this book, the electron density p(r). It is defined as the following multiple integral over the spin coordinates of all electrons and over all but one of the spatial variables... 

As a typical example we illustrate in Figure 2-1 the electron density of the water molecule in two different representations. In complete analogy, p(x) extends the electron density to the spin-dependent probability of finding any of the N electrons within the volume element dr, and having a spin defined by the spin coordinate s. 

The wave function for any system is a function of both the spatial coordinates of the electrons and of the spins of the electrons. It is convenient to describe the two possible values of the spin angular momentum of an electron as the two possible values of its spin coordinate,... 

The many-electron wave function (40 of any system is a function of the spatial coordinates of all the electrons and of their spins. The two possible values of the spin angular momentum of an electron—spin up and spin down—are described respectively by two spin functions denoted as a(co) and P(co), where co is a spin degree of freedom or spin coordinate . All electrons are identical and therefore indistinguishable from one another. It follows that the interchange of the positions and the spins (spin coordinates) of any two electrons in a system must leave the observable properties of the system unchanged. In particular, the electron density must remain unchanged. In other words, 4 2 must not be altered... 

A corollary of the Pauli principle is that no two electrons with the same spin can ever simultaneously be at the same point in space. If two electrons with the same spin were at the same point in space simultaneously, then on interchanging these two electrons, the wave function should change sign as required by the Pauli principle (4 —> - 4 ). Since in this case the two electrons have the same space and spin coordinates (i.e.,... 

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