Breit-Pauli theory

Therefore, we may here start from the Breit interaction to derive the pseudo-relativistic Hamiltonians instead of following the somewhat meandering historical path from 1926 to about 1932, which was mainly based on classical considerations. As usual, a quantum-electrodynamical derivation is also possible and has been presented by Itoh [678], but the sound basis of our semi-classical theory, which we pursue throughout this book, is necessarily the Breit equation. Needless to say, the rigorous transformation approach to the Dirac-Coulomb-Breit Hamiltonian yields results identical to those from the QED-based derivation. 

The resulting Hamiltonian is the Breit-Pauli Hamiltonian and is still in current use in perturbation theory calculations of especially spin-orbit effects with sophisticated Cl-type wave functions (see also section 14.1). 

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The first term is the non-relativistic result derived from Breit-Pauli theory, and is the Lande -factor, gji, while the second is the first-order relativistic correction, Ajk, which is proportional to the small-component electron density. We compare, in Table 2, the values which we have calculated directly from Eq. 

In chapter 5 we mentioned the Lamb shift in passing and the difficulty of calculating it due to the renormalizations required. In fact, there is a well-developed perturbation theory of the Lamb shift in the same framework as Breit-Pauli theory. We do not propose to derive the expressions here, for which the reader is referred to Bethe and Salpeter (1957). Instead, we report the results for the lowest-order terms, which turn out to be expressible as corrections to the Darwin and spin-orbit one-electron operators. The combined operators may be written... 

The calculation of properties using direct perturbation theory follows exactly the same lines as we used for Breit-Pauli theory. As we noted above, stationary direct perturbation theory leads to precisely the same equations we would have obtained by simply expanding the perturbed wave functions in the set of eigenfunctions of the zeroth-order Hamiltonian, and on this basis we proceed with the development of multiple direct perturbation theory for properties. 

The two perturbations will have a different form for direct and Breit-Pauli perturbation theory. The wave function also will be different in form two components in Breit-Pauli theory and four components in direct perturbation theory. In addition, the metric must be expanded in the direet scheme,... 

The conclusion is that there are real differences between Pauli and direct perturbation theory for the lowest-order relativistic corrections to both electric and magnetic properties, which do not vanish even for exact wave functions. Direct perturbation theory is in general to be preferred because it is convergent and therefore can be used to higher order, in contrast to Breit-Pauli theory. 

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

The spin-orbit mean field (SOMF) operator (56-58) is used to approximate the Breit—Pauli two-electron SOC operator as an effective one-electron operator. Using second-order perturbation theory (59), one can end up with the working equations ... 

The Breit-Pauli spin-orbit operator has one major drawback. It implicitly contains terms coupling electronic states (with positive energy) and posi-tronic states (in the negative energy continuum) and is thus unbounded from below. It can be employed safely only in first-order perturbation theory. 

For many types of electron spectroscopies there are still comparatively few studies of SOC effects in molecules in contrast to atoms, see, e.g., [1, 2, 3, 4, 5, 6, 7] and references therein. This can probably be referred to complexities in the molecular analysis due to the extra vibrational and rotational degrees of freedom, increased role of many-body interaction, interference and break-down effects in the spectra, but can also be referred to the more difficult nature of the spin-orbit coupling itself in polyatomic species. Modern ab initio formulations, as, e.g., spin-orbit response theory [8] reviewed here, have made such investigations possible using the full Breit-Pauli spin-orbit operator. 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

The analysis of the relativistic Renner coupling has been extended to 11 states, including the two-electron part of the Breit-Pauli operator, thus generalizing previous result of Hougen [45,46]. Other extensions of the theory are S — 11 coupling in the doublet manifold [47] and SO coupling in a half-filled n shell, as found, for example, in carbenes [48]. 

Breit-Pauli effective Hamiltonian [63, 39], [64, Appendix 4] which is regularly used to describe relativistic corrections to the familiar nonrelativistic theory of atoms and molecules. It is important to realize that this widely used perturbation expansion contains less physics than the simple relativistic interactions used here. 

The Breit—Pauli spin-orbit Hamiltonian is very useful for organic molecules when the matrix elements of are computed from the nonrelativistic wave functions using perturbation theory or response theory, but it often overestimates the magnitude of spin-orbit splitting. It also suffers from... 

Degenerate perturbation theory is implemented in the computer programs MELD and DALTON. Both use the Breit-Pauli spin-spin Hamiltonian with ROHF and MCSCF wave functions MELD also allows the use of Cl wave functions. 

This version of perturbation theory without exphcit consideration of the excited state wave functions was implemented by Yarkony et al. using the full Breit-Pauli spin-orbit and spin-spin Hamiltomians smd... 

The computer program DALTON implements MCSCF response theory within the full Breit-Pauli The coupled-cluster (CC) response theory is implemented in the program ACES II. It calls DALTON if the full Breit-Pauli is to be used or the program AMFI for the mean-field approximation. 

Quasi-degenerate perturbation theory is implemented in several program packages. The program GAMESS utilizes MCSCF wave functions and the full Breit—Pauli Hamiltonian... 

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