Dirac many-particle theory

The Dirac and Breit equations do not account for several subtle effects. They ace predicted by QED, a many-particle theory. 

Of course, Dirac was well aware of the many difficulties arising from the electron-positron interpretation — from the mass-dissymmetry if the negative-energy electron was a proton to the fact that he has actually created a many-particle theory which, as a true many-particle theory, would have been difficult to study. We shall have a closer look into these difficulties below. 

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

Awkward questions about the electromagnetic and gravitational fields of infinitely many particles in the vacuum remain unanswered. Also, the Dirac theory, amended by the hole proposition is certainly not a one-particle theory, and hence not a relativistic generalization of Schrodinger s equation. 

The history of quantum chemistry is very closely tied to the history of computation, and in order to place Carl Ballhausen s work in context, it is relevant to review the enormously rapid development of computing during the twentieth century. The fundamental equations governing the physical properties of matter, while deceptively simple to write down, are notoriously difficult to solve. Only the simplest problems, for example the harmonic oscillator and the problem of a single electron moving in the field of a fixed nucleus, can be solved exactly. However, no solutions to the wave equations for interacting many-particle systems such as atoms or molecules are known, and it is quite possible that no simple solutions exist. In 1929, P.A.M. Dirac summarized the position since the discovery of quantum theory with his famous remark ... 

From a very general point of view every ion-atom collision system has to be treated as a correlated many-body time-dependent quantum system. To solve this from an ab initio point of view is still impossible. So, one has to rely on various approximations. Nowadays the best method which can be applied to realistic collision systems (which we discuss here) is on the level of the non-selfconsistent time-dependent Hartree-Fock-Slater or, in the relativistic case, the Dirac-Fock-Slater method. Up-to-now no correlation beyond this approximation can be taken into account in the case of 3 or more electrons. (This is in accordance with the definition of correlation given by Lowdin [1] in 1956) In addition no QED contributions, i.e. no correction to the 1/r Coulomb interaction between the electrons, ever have been taken into account, although in very heavy collision systems this effect may become important. This will be discussed in section 5. A short survey of the theory used is followed by our results on impact parameter dependent electron transfer and excitation calculations of ion-atom and ion-solid collisions as well as first results of an ab initio calculation of MO X-rays in such complicated many particle scattering systems. 

The following approach will show how the already proved quite reliable approach of path integrals is naturally needed within the Dirac formalism of quantum mechanics applied on many-particle systems, specific to chemical structures formed by many-electrons in valence state, by means of the celebrated density matrix formalism - from where there is just a step to the observable density functional theory of many-body systems. 

Furthermore, starting from the Dirac equation, the assumptions of hole theory lead automatically to a formulation for an infinite number of particles occupying the Dirac sea, and the actual number of electrons and positrons cannot be deduced from hole theory. This is a most disturbing aspect of Dirac s hole theory, namely that it is actually a flni/-particle theory. So far we have considered only a single fermion in the universe and set up an equation to describe its motion. Now, we face a conceptual generalization, which in turn requires a description of the motion of infinitely many fermions. This leads to the inclusion of additional interaction operators in the Dirac equation, and it is this ineraction of electrons which makes life difficult for molecular scientists, as we shall discuss in parts III and IV of this book. 

In the previous chapter we derived the Dirac equation for an electron moving in an arbitrary electromagnetic potential. We next proceed step-wise toward a many-electron theory which incorporates any kind of electromagnetic interactions. The most simple example is the hydrogen atom consisting of two interacting particles the electron and the proton. 

In approximate Cl methods, the set of many-particle basis functions is restricted and not infinitely large, i.e., it is not complete. Then, the many-particle basis is usually constructed systematically from a given reference basis function. (such as the Slater determinant, which is constructed to approximate the ground state of a many-electron system in (Dirac-)Hartree-Fock theory). 

In the preceding chapter on isolated, spherically symmetric atoms, we chose to derive Cl and SCF equations from the most general expression of the total electronic energy in which the total state is expanded in a CSF many-particle basis set. Here, we could proceed in the same way but start instead with the simple Dirac-Hartree-Fock theory for the sake of convenience. Its extension we discuss later in this chapter. 

To apply the Dirac theory to the many-particle system the one-particle Dirac operator (8.36) is augmented by the Coulomb or Coulomb-Breit operator as the two-particle term, to produce the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from the quantum electrodynamics [496 98] ... 

Another subtlety is that the assumption nuclei behave as Dirac particles, amounts to assuming that all nuclei have spin 1/2. However, it is not uncommon to have nuclei with spin as high as 9/2 worse nuclei with integer spins are bosons and do not obey Fermi-Dirac statistics. The only justification to use equation (75) for such a case is that the resulting theory agrees with experiment. Under the assumption, we are in a position to extend our many-fermion Hamiltonian to molecules assuming that the nuclei are Dirac particles with anomalous spin. The molecular Hamiltonian may then be written as... 

We will describe, in some detail, one such modification, an effective Dirac equation (EDE) which was derived in a number of papers [7, 8, 9, 10]. This new equation is more convenient in many applications than the original BS equation, and we will derive some general formulae connected with this equation. The physical idea behind this approach is that in the case of a loosely bound system of two particles of different masses, the heavy particle spends almost all its life not far from its own mass shell. In such case some kind of Dirac equation for the light particle in an external Coulomb field should be an excellent starting point for the perturbation theory expansion. Then it is convenient to choose the free two-particle propagator in the form of the product of the heavy particle mass shell projector A and the free electron propagator... 

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