Relativistic theory applications

Methods for treating relativistic effects in molecular quantum mechanics have always seemed to me, if I may say so without appearing too impertinent to those who work in the field, a complete dog s breakfast. The difficulty is to know to what question they are supposed to be the answer, in the circumstances in which we find ourselves. We do not know what a relativistically invariant theory applicable to molecular behaviour might look like. As was pointed out to us at the last meeting, the Dirac equation certainly will not do to describe interacting electrons and even at the single particle level, where it seems to work, there is an inconsistency in interpreting its solutions in terms... 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

The metric geometry of equilibrium thermodynamics provides an unusual prototype in the rich spectrum of possibilities of differential geometry. Just as Einstein s general relativistic theory of gravitation enriched the classical Riemann theory of curved spaces, so does its thermodynamic manifestation suggest further extensions of powerful Riemannian concepts. Theorems and tools of the differential geometer may be sharpened or extended by application to the unique Riemannian features of equilibrium chemical and phase thermodynamics. 

I.P. Grant, H.M. Quiney, Application of relativistic theories and quantum electro-dynamics to chemical problems, Int.. Quant. Chem. 80 (2000) 283-297. 

The mathematical problem associated with the Dirac Hamiltonian, i.e. the starting point of the relativistic theory of atoms, can be phrased in simple terms. The electron-positron field can have states of arbitrarily negative energy. As a general feature of the Dirac spectrum this instability occurs even in the case of extended nuclei and even in the absence of any nucleus (free Dirac spectrum), the energy is not bounded from below. This gives rise to the necessity of renormalization and well-established renormalization schemes have been around for many decades. Despite their successful applications in physics, we may ask instead whether there exist states that allow for positivity of the energy. 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

A detailed presentation of relativistic effects on magnetic properties is found in Ref. [60], especially for the H-atom in a homogeneous magnetic field in Ref. [61] Application of DPT to first-order magnetic properties were published by Hennum, et al. [62]. An earlier, more intuitive formulation, especially for NMR chemical shifts was given by Nomura et al. [63]. The fully relativistic theory has been studied by Pyykko [64] and Pyper [65]. 

Debashis Mukherjee is a Professor of Physical Chemistry and the Director of the Indian Association for the Cultivation of Science, Calcutta, India. He has been one of the earliest developers of a class of multi-reference coupled cluster theories and also of the coupled cluster based linear response theory. Other contributions by him are in the resolution of the size-extensivity problem for multi-reference theories using an incomplete model space and in the size-extensive intermediate Hamiltonian formalism. His research interests focus on the development and applications of non-relativistic and relativistic theories of many-body molecular electronic structure and theoretical spectroscopy, quantum many-body dynamics and statistical held theory of many-body systems. He is a member of the International Academy of the Quantum Molecular Science, a Fellow of the Third World Academy of Science, the Indian National Science Academy and the Indian Academy of Sciences. He is the recipient of the Shantiswarup Bhatnagar Prize of the Council of Scientihc and Industrial Research of the Government of India. 

This value cannot be properly derived from the non-relativistic theory because the application of the classical expression for the magnetic moment of a charged particle associated with some angular momentum /... 

Today, the role relativistic effects play for NMR and EPR parameters has been appreciated to very different extents for different properties and by different communities of experimentalists and theoreticians. For example, it has been known early on in the EPR community that the electronic g-tensors of EPR spectroscopy are basically dominated by spin-orbit coupling and are thus intrinsically relativistic [2]. On the other hand, in spite of much early work on relativistic theories of NMR chemical shifts, and much associated recent cori5)utational developments and applications [3,4,5,6,7], most users of NMR spectroscopy still seem largely unaware of the important role of relativistic effects. This holds in particular for the role of spin-orbit effects, in what is often simply called heavy-atom effects on NMR chemical shifts. This can be seen easily when inspecting most NMR textbooks and much of the research literature. 

Kutzelnigg, W. Fundamentals of Nonrelativistic and Relativistic Theory of NMR and EPR Parameters. In Calculation of NMR and EPR Parameters Theory and Applications-, Kaupp, M., Biihl, M., Malkin, V. G., Eds. Wiley-VCH VerlagGmbH Weinheim, 2004 Chapter 5, pp 43-82. 

If relativistic wave functions are used, second-order perturbation theory is enough. The ultimate goal, of course, remains a fully relativistic theory of both J and o. The analogues to Ramsey s theories for them using the Dirac equation were formulated in Ref. [25] and in Refs. [26-28], respectively. The first numerical applications are now starting to appear, see, for example, the two conference proceedings [29, 30]. 

Relativity adds a new dimension to quantum chemistry, which is the choice of the Hamiltonian operator. While the Hamiltonian of a molecule is exactly known in nonrelativistic quantum mechanics (if one focuses on the dominating electrostatic monopole interactions to be considered as being transmitted instantaneously), this is no longer the case for the relativistic formulation. Numerical results obtained by many researchers over the past decades have shown how Hamiltonians which capture most of the (numerical) effect of relativity on physical observables can be derived. Relativistic quantum chemistry therefore comes in various flavors, which are more or less well rooted in fundamental physical theory and whose relation to one another will be described in detail in this book. The new dimension of relativistic Hamiltonians makes the presentation of the relativistic many-electron theory very complicated, and the degree of complexity is far greater than for nonrelativistic quantum chemistry. However, the relativistic theory provides the consistent approach toward the description of nature molecular structures containing heavy atoms can only be treated correctly within a relativistic framework. Prominent examples known to everyone are the color of gold and the liquid state of mercury at room temperature. Moreover, it must be understood that relativistic quantum chemistry provides universal theoretical means that are applicable to any element from the periodic table or to any molecule — not only to heavy-element compounds. 

As well as the chemical shift, indirect nuclear spin-spin coupling constant (SSCC) is one of the most important molecular properties measured routinely in NMR experiments. Much effort has been devorted until now to the development of theoretical tools to compute SSCCs from first-principles theory. For SSCCs in heavy element compounds, a relativistic theory is needed. Moreover, SSCCs are sensive to electron correlation. Therefore, in the computation of SSCCs for heavy-atom included systems, density functional theory (DFT) employing two-component relativistic methods plays a major role because of its applicability to relatively large molecules. 

I. Lindgren, A. Rosen Relativistic self-consistent field calculations with application to hyperfine interaction. Pt.I Relativistic self-consistent fields, Pt.II Relativistic theory of atomic hyperfine interaction. Case Stud. Atom. Phys. 4, 93 (1974), Pt.IlI Comparison between theoretical and experimental hyperfine structure results. Case Stud. Atom. Phys. 4, 197 (1974)... 

Relativistic Quantum Chemistry Four Components Good, Two Components Bad (b) I. P. Grant and H. M. Quiney, Int. ]. Quantum Chem., 80,283 (2000). Application of Relativistic Theories and Quantum Electrodynamics to Chemical Problems. 

The Electronic Structure of Organoactinide Complexes via Relativistic Density Functional Theory Applications to the Actinocene Complexes An(rj -C8H8)2 (An = Th-Am)... 

While it is the accepted wisdom that relativistic calculations (see Relativistic Theory and Applications) are not worthwhile for molecules containing only first- and perhaps second-row... 

Field (CASSCF) Second-order Perturbation Theory (CAS-PT2) Configuration Interaction Core-Valence Correlation Effects Coupled-cluster Theory Experimental Data Evaluation and Quality Control G2 Theory Heats of Formation Isoelectronic Isogyric Reactions M0ller-Plesset Perturbation Theory Numerical Hartree-Fock Methods for Molecules r 12-Dependent Wavefunctions Relativistic Theory and Applications Spectroscopy Computational Methods Spin Contamination Transition Metals Applications,... 

Configuration Interaction Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Density Functional Theory Applications to Transition Metal Problems Metal Complexes Relativistic Effective Core Potential Techniques for Molecules Containing Very Heavy Atoms Relativistic Effects of the Superheavy Elements Relativistic Theory and Applications Transition Metal Chemistry Transition Metals Applications. 

Relativistic Effects of the Superheavy Elements Relativistic Theory and Applications. 

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