The Breit-Pauli Hamiltonian

The terms in the first square bracket come from the renormalization operator. The terms in the second square bracket are not renormalized because they are already of order 1/c. The renormalization terms cancel with the terms in the second square bracket. 

Since the operators are symmetric in i and j we may reindex and combine the terms, and substitute Vf l/rij) = -47tS(rij) and Vi(l/r,y) = -rijfrfj to give the operator 

The spin-orbit term involves the interaction of the spin of the electron with its own orbital angular momentum around the other electrons, and is often called the spin-own-orbit interaction or spin-same-orbit interaction. 

We now turn to the Gaunt interaction, and use the terms from the modified Dirac representation in (15.54) to derive the Breit-Pauli operators. These terms need no renormalization, because they are all of order 1/c. The three classes of operators defined in (15.54) are considered in turn. 

The first class contains the spin-free operators. 

Finally, the full Breit-Pauli Hamiltonian contains the one-electron terms discussed in the beginning of this chapter as well as the magnetic interactions of the electrons that can be rewritten as interacting (coupled) angular momenta. Starting from the full Dirac-Breit equation for two electrons, Eq. (8.19), we obtain the purely two-component external-field-free Breit-Pauli Hamiltonian [72, p. 377], 

In the Breit-Pauli Hamiltonian, the one-electron Pauli Hamiltonians and the two-dimensional Breit operator can be clearly identified. The factor eh/ 2m.eC) in these operators represents the Bohr magneton ji-q, which we have already encountered in section 5.4.3, and the gradient of the nuclear potential is the electric field strength 

However, remember the caveats made with respect to the use of the one-electron terms above. Witness also that the one-electron spin-orbit terms are hidden in the vector product of the electric field with the momentum operator in the H3-operator above. The H4-operator produces the Laplacian of the potential, i.e., the one-electron Darwin term as derived in the beginning of this chapter. However, the potential due to the other electron(s) also produces terms of the Darwin form 

In the Breit-Pauli Hamiltonian, we identify iso -coupling terms, namely the orbit-orbit — also called orbit-other-orbit — and spin-spin — also called 

---

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

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 Breit-Pauli Hamiltonian with an external field contains all standard one- and two-electron contributions as well as the magnetic interaction of the electrons and their interactions with an external electromagnetic field. We may group the various contributions in the Breit-Pauli Hamiltonian according to one-and two-electron terms,... 

In order to readily compare these terms to their expressions in the Breit-Pauli Hamiltonian given for example in Refs. (21,54), we insert the definition of the electric field strength E, and the... 

The nuclear spins give rise to additional terms in the Breit-Pauli Hamiltonian due to the interaction of the electrons with the magnetic moment of the nuclei and the electrostatic interaction with the electric quadrupole interaction of the nuclei. The magnetic interaction term of the spins with the nuclei is of the same type as the spin-spin interaction and following Abragam and Pryce (61) can be written as... 

In ESR spectroscopy the terms in the effective Hamiltonian are typically expressed by virtue of effective coupling matrices or tensors, whereas in this review we shall relate them to their corresponding terms in the Breit-Pauli Hamiltonian. The effective coupling matrices parametrize the electronic structure of the molecule under study and can be calculated from the Breit-Pauli Hamiltonian by employing a suitable representation of the molecular wave function. 

Though the true electron spin operators were employed here as well as in the Breit-Pauli Hamiltonian, the phenomenological Spin Hamiltonian, in which the spin coupling is an exchange effect, is in sharp contrast to the Breit-Pauli Hamiltonian, that is including the (magnetic) spin-spin interactions. Since the exchange effect is an effect introduced by the Pauli principle imposed on the wave function, we may write the electron-electron interaction as an expectation value,... 

The spin operators Sx, and Sg which occur in the Breit-Pauli Hamiltonian form a basis for the Lie algebra of SU(2). The concept of the electron as a spinning particle has arisen through the isomorphism between SU(2) and the angular momentum operators. This analogy is unnecessary and often undesirable. 

Many electronic systems are well described by the Breit-Pauli Hamiltonian... 

Here r is the radius vector of the single unpaired electron, k = 1,2,3, denote the positions of the nuclei, and q is the effective charge of the three identical nuclei. The Breit-Pauli Hamiltonian of this system is [17]... 

The Breit-Pauli Hamiltonian for a single unpaired electron in the field of four identical atomic centers reads... 

Calculations of total energies and widths of doubly excited states using the Breit-Pauli Hamiltonian and comparison with results from the use of Hylleraas-type basis functions [55]... 

Most applications to molecules that do not contain heavy atoms (beyond Br in the periodic table) have been based on the Breit-Pauli Hamiltonian which consists of six terms and operates on ordinary nonrelativistic wave functions. In the Breit-Pauli approximation, the total Hamiltonian is defined as... 

Table 3.1 Effective Nuclear Charge (Scaling Parameter) Z ff for Approximate Spin-Orbit Interaction Calculations Using the One-Electron Term in the Breit-Pauli Hamiltonian (Equation 3.8, developed by Koseki et...

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

Abstract Variational methods can determine a wide range of atomic properties for bound states of simple as well as complex atomic systems. Even for relatively light atoms, relativistic effects may be important. In this chapter we review systematic, large-scale variational procedures that include relativistic effects through either the Breit-Pauli Hamiltonian or the Dirac-Coulomb-Breit Hamiltonian but where correlation is the main source of uncertainty. Correlation is included in a series of calculations of increasing size for which results can be monitored and accuracy estimated. Examples are presented and further developments mentioned. 

---