Spin-coordinate space

Electrons in molecules and crystals repel each other according to Coulomb s law, with the repulsion energy depending on the interelectron distance as This interaction creates a correlation hole around any electron, i.e. the probabihty to find any pair of electrons at the same point of spin-coordinate space is zero. Prom this point of view only the Hartree product of molecular or crystalline spin-orbitals... 

In general, the spin of each electron is specified in advance of a calculation. For instance, with H2O, electrons 1-5 are spin up and electrons 6-10 are spin down. This places the system in one region of spin-coordinate space. Other regions of 5-up/5-down would give the same result. For electrons of the same spin, antisymmetry in the space coordinates is imposed and produces the nodal structure in the space of those electrons. One can devise an overall space-spin wavefunction that is antisymmetric by combining with spin functions. 

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]. 

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. 

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.,... 

When the system is made up of identical particles (e.g. electrons in a molecule) the Hamiltonian must be symmetrical with respect to any interchange of the space and spin coordinates of the particles. Thus an interchange operator P that permutes the variables qi and (denoting space and spin coordinates) of particles i and j commutes with the Hamiltonian, [.Pij, H] = 0. Since two successive interchanges of and qj return the particles to the initial configuration, it follows that P = /, and the eigenvalues of are e = 1. The wave functions corresponding to e = 1 are such that... 

In the first term, Uc, usually called the Coulombic term, the initially excited electron on D returns to the ground state orbital while an electron on A is simultaneously promoted to the excited state. In the second term, called the exchange term, Liex, there is an exchange of two electrons on D and A. The exchange interaction is a quantum-mechanical effect arising from the symmetry properties of the wavefunctions with respect to exchange of spin and space coordinates of two electrons. 

Employing the abbreviated notation j = xj for the composite space/spin coordinates of the jth electron, let... 

The wave function is a function of the 3n space coordinates and n spin coordinates of the n electrons. The one-electron density p r) is obtained from the wave function by integration over all spin coordinates and the space coordinates of all but one electron ... 

The wave function P contains all information of the joint probability distribution of the electrons. For example, the two-electron density is obtained from the wave function by integration over the spin and space coordinates of all but two electrons. It describes the joint probability of finding electron 1 at r, and electron 2 at r2. The two-electron density cannot be obtained from elastic Bragg scattering. 

Suppose that (xi,X2,..., xn) is the position-space representation of the N-electron wavefunction. It is a function of the space-spin coordinates Xk = (Ofcj ctjfc) in which is the position vector of the kth electron and Gk is its spin coordinate. The position-space wavefunction is obtained by solving the usual position- or r-space Schrodinger equation by one of the many well-developed approximate methods [32-34]. 

The counterpart wavefunction in momentum-space, 4>(yi,y2 is a function of momentum-spin coordinates % = (jpk, k) in which pk is the linear momentum of the feth electron. There are three approaches to obtaining the momentum-space wavefunction, two direct and one indirect. The wavefunction can be obtained directly by solving either a differential or an integral equation in momentum- or p space. It can also be obtained indirectly by transformation of the position-space wavefunction. 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

In the above equation, is the antisymmetrizer working on both the space—and spin coordinates, gXrj )are primitive Cartesian Gaussian functions Eq. (23) and... 

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