Relativistic methods 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... 

Needless to say, many-electron atoms and molecules are much more complicated than one-electron atoms, and the realization of the nonrelativistic limit is not easily accomplished in these cases because of the approximations needed for the description of a complicated many-particle system. However, the signature of relativistic effects (see, for example, Chapter 3 in this book) enables us to identify these effects even without calculation from experimental observation. Two mainly experimentally oriented chapters will report astounding examples of relativistic phenomenology, interpreted by means of the methods of relativistic electronic structure theory. These methods for the theoretical treatment of relativistic effects in many-electron atoms and molecules are the subject of most of the chapters in the present volume, and with the help of this theory relativistic effects can be characterized with high precision. 

The model used in these considerations is liighly simplified only one particle is assumed to move, the other constituents of the nucleus not being considered separately, but only as the sources of the force acting on the moving particle. Recently various attempts have been made to treat the nucleus a,s a system of many particles with the help of the methods of non-relativistic quantum mechanics. For this purpose definite assumptions about the forces between the elementary particles (neutrons and protons) are necessary. It then becomes pos-... 

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 technique for dealing with this problem is well known from nonrelativistic calculations on many-electron systems. One-particle basis sets are developed by considering the behavior of the single electron in the mean field of all the other electrons, and while this neglects a smaller part of the interaction energy, the electron correlation, it provides a suitable starting point for further variational or perturbational treatments to recover more of the electron-electron interaction. It is only natural to pursue the same approach for the relativistic case. Thus one may proceed to construct a mean-field method that can be used as a basis for the perturbation theory of QED. In particular, the inclusion of the Breit interaction in the mean-field calculations ensures that the terms of O(a ) are included to infinite order in QED. 

The coupled cluster (CC) approach is the most powerful and accurate of generally applicable electron correlation methods. This has been shown in many benchmark applications of 4-component relativistic CC methods to atoms [11-18] and molecules [19-31]. The CC method is an all-order, size-extensive, and systematic many-body approach. Multireference variants of relativistic 4-component CC methods capable of handling quasidegeneracies, which are important for open-shell heavy atomic and molecular systems, have been developed in recent years [15,17-19,21,31]. In particular, the multireference FSCC scheme [32,33] is applicable to systems with a variable number of particles, and is an ideal candidate for merging with QED theory to create an infinite-order size-extensive covariant many-body method applicable to systems with variable numbers of fermions and bosons [6,7]. 

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... 

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