Dirac equation many-particle systems

Relativistic Methods 204 8.1 Connection Between the Dirac and Schrodinger Equations 207 8.2 Many-particle Systems 210 8.3 Four-component Calculations 213 11.4.1 Ab Initio Methods 272 11.4.2 DFT Methods 273 11.5 Bond Dissociation Curve 274 11.5.1 Basis Set Effect at the HF Level 274 11.5.2 Performance of Different Types of Wave Function 276... 

The mathematical basis of the relativistic quantum mechanical description of many-electron atoms and molecules is much less firm than that of the nonrelativistic counterpart, which is well understood. As we do not know of a covariant quantum mechanical equation of motion for a many-particle system (nuclei plus electrons), we rely on the Dirac equation for the quantum mechanical characterization of a free electron (positron) (Darwin 1928 Dirac 1928,1929 Dolbeault etal. 2000b Thaller 1992)... 

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 ... 

It has often been claimed that the Dirac equation allows no generalization to a many-particle system. In a sense, that is not true. The Dirac equation can be generalized to several particles as well as the Schrodinger equation, but there are some technical problems with the perturbation-theoretical treatment. We refer to later sections in this book for more details. In order to treat many particle problems, one considers antisymmetric tensor products of spinor-valued wave functions similar to the many-particle wave functions in nonrelativistic quan-... 

A fully relativistic treatment of more than one particle would have to start from a full QED treatment of the system (Chapter 1), and perform a perturbation expansion in terms of the radiation frequency. There is no universally accepted way of doing this, and a full relativistic many-body equation has not yet been developed. For many-particle systems it is assumed that each electron can be described by a Dirac operator (ca n -I- P me 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 Hamiltonian operator in non-relativistic theory. Since this approach gives results that agree with experiments, the assumptions appear justified. 

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... 

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

An assembly of nuclei and electrons could be described very accurately within QED. There would still just be a cluster of particles, our molecule, and any structure would have to arise out of the dynamics of the system. For reasons pointed out earlier, QED—if viable at all— would be a very expensive path to calculation of the electronic stracture and chemical properties of molecules. For electrons, we circumvented this problem by going to a many-particle treatment based on the Dirac equation, as discussed in chapter 5, and we could presumably do the same here for our cluster of electrons and nuclei. In doing this, we choose a Hamiltonian description of the system, but alternative approaches based on a Lagrangian formalism are also possible. In this process we draw a formal distinction between the molecule and the electromagnetic field, which leaves us with the normal Coulomb interactions between the particles in the molecule and the radiation field as an entity external to the molecule. 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

The equations of quantum statistical mechanics for a system of non-identical particles, for which all solutions of the wave equations are accepted, are closely analogous to the equations of classical statistical mechanics (Boltzmann statistics). The quantum statistics resulting from the acceptance of only antisymmetric wave functions is considerably different. This statistics, called Fermi-Dirac statistics, applies to many problems, such as the Pauli-Sommerfeld treatment of metallic electrons and the Thomas-Fermi treatment of many-electron atoms. The statistics corresponding to the acceptance of only the completely symmetric wave functions is called the Bose-Einstein statistics. These statistics will be briefly discussed in Section 49. 

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