Relativistic Hamiltonian, derivation

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

When a proper relativistic derivation of the spin-orbit-interaction term Hs.o. the atomic Hamiltonian is carried out, one finds that for a one-electron atom (see Bethe and Jackiw, Chapters 8 and 23)... 

We have developed a new relativistic treatment in the QMC technique using the ZORA Hamiltonian. We derived a novel relativistic local energy using the ZORA Hamiltonian and tested its availability in the VMC calculation. In addition, we proposed a relativistic electron-nucleus cusp correction scheme for the relativistic ZORA-QMC method. The correction scheme was a relativistic extension of the MO correction method where the 1 s MO was replaced by a correction function satisfying the cusp condition. The cusp condition for the ZORA wave function in electron-nucleus collisions was derived by the expansion of the ZORA local energy and the condition required the weak divergence of the orbital itself. The proposed relativistic correction function is the same as the NR correction function of Ma et al. in the NR... 

In recent years, there has been an increasing interest in the inclusion of relativistic effects for molecules containing heavy atoms. One of the most practical yet reliable methods is to use relativistically derived effective core potentials. Major relativistic effects such as the Darwin and mass-velocity effects are easily taken into account in the form of a spin-free (SF) one-electron operator. The spin-orbit (SO) interaction is in general too strong to be considered as a small perturbation, and therefore should be treated explicitly as a part of the total Hamiltonian. 

Although the Pauli Hamiltonian was derived for use in perturbational computations as corrections to nonrelativistic Hamiltonian, these terms have been used in variational self-consistent field (SCF) calculations by Hay and Wadt, These authors have incorporated the relativistic terms into the ECPs, except that the spin-orbit effects are not included in their ECPs, and, thus, in the molecular calculations based on this scheme. 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

We would like to point out some steps of derivation of the nonrelativistic limit Hamiltonians by means of the Foldy-Wouthuyisen transformation (Bjorken and Drell, 1964). The method is based on the transformation of a relativistic equation of motion to the Schrodinger equation form. 

Apparently, a large number of successful relativistic configuration-interaction (RCI) and multi-reference Dirac-Hartree-Fock (MRDHF) calculations [27] reported over the last two decades are supposedly based on the DBC Hamiltonian. This apparent success seems to contradict the earlier claims of the CD. As shown by Sucher [18,28], in fact the RCI and MRDHF calculations are not based on the DBC Hamiltonian, but on an approximation to a more fundamental Hamiltonian based on QED which does not suffer from the CD. At this point, let us defer further discussion until we review the many-fermion Hamiltonians derived from QED. 

The first rigorous derivation of such a relativistic Hamiltonian for a two-fermion system that makes use of Feynman [13,14] formalism of QED was due to Bethe and Salpeter [30,31]. Recently, Broyles has extended it to many-eleetron atoms and molecules [32]. A detailed account of Broyle s derivation ean be found elsewhere [32,33] and will not be repeated here. Following Broyles, the stationary state many-fermion Hamiltonian based on QED ean be written as... 

The electron coupled interaction of nuclear magnetic moments with themselves and also with an external magnetic field is responsible for NMR spectroscopy. Since the focus of this study is calculation of NMR spectra within the non-relativistic framework, we will take a closer look at the Hamiltonian derived from equation (76) to describe NMR processes. In this regard, we retain all the terms, which depend on nuclear magnetic moments of nuclei in the molecule and the external magnetic field through its vector potential in addition to the usual non-relativistic Hamiltonian. The result is... 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

The beauty of the above results is that, apart firom the use of a non-relativistic Bom-Oppenheun r Hamiltonian, no apprommationa have been made-, the density functions are all rigorously derivable, in principle, firom an exact wavefiinction containing no orbital approximations and remain valid for any system at any level of approximation. 

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

The older, Pauli-Hamiltonian based QR method has recently been compared to the more modem ZORA approach (10). The comparison has been done based on methane derivatives. We have summarized the results in Table II. Both the ZORA and QR methods agree well for molecules containing atoms no heavier than Cl. This shouldn t be surprising since relativistic effects are still small in such molecules. However, at least for this sample of molecules, the ZORA method is clearly superior for molecules containing heavy nuclei like Br or I. This is reflected in the mean absolute deviation between theory and experiment of 9.2 ppm (ZORA) and 15.6 ppm (QR), respectively (10). Note that some of the same systems have also been studied by other authors (35-37). 

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

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