The Breit Hamiltonian

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

The Breit Hamiltonian for two electrons consists essentially of a Dirac Hamiltonian for each electron, with interaction terms. It may be written 

The first interaction term is, of course, the Coulomb interaction the second interaction term has the same form as the classical expression for the retarded interaction of two particles, derived in section 3.8. However, the Breit Hamiltonian suffers from the defect that the interaction terms are not Lorentz invariant. Detailed investigations using 

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Semi-relativistic Functionals from the Breit Hamiltonian... 

One of the purposes of this work is to make contact between relativistic corrections in quantum mechanics and the weakly relativistic limit of QED for this problem. In particular, we will check how performing plane-wave expectation values of the Breit hamiltonian in the Pauli approximation (only terms depending on c in atomic units) we obtain the proper semi-relativistic functional consistent in order ppl mc ), with the possibility of analyzing the separate contributions of terms with different physical meaning. Also the role of these terms compared to next order ones will be studied. 

Neglecting spin-orbit contributions (smaller than other relativistic corrections for the ground state of atoms, and zero for closed-shell ones), the Breit hamiltonian in the Pauli approximation [25] (weak relativistic systems) can be written for a many electron system as ... 

We have reproduced the c terms by means of the analysis of the different terms of the Breit hamiltonian with the exception of Darwin ones. These terms, denoted by Hp and given by Eq. (24) are not hermitian in general. The adjoint is given by... 

In the Breit Hamiltonian in (3.2) we have omitted all terms which depend on spin variables of the heavy particle. As a result the corrections to the energy levels in (3.4) do not depend on the relative orientation of the spins of the heavy and light particles (in other words they do not describe hyperfine splitting). Moreover, almost all contributions in (3.4) are independent not only of the mutual orientation of spins of the heavy and light particles but also of the magnitude of the spin of the heavy particle. The only exception is the small contribution proportional to the term Sio, called the Darwin-Foldy contribution. This term arises in the matrix element of the Breit Hamiltonian only for the spin one-half nucleus and should be omitted for spinless or spin one nuclei. This contribution combines naturally with the nuclear size correction, and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear size contribution. 

Calculation of the leading recoil corrections of order a Za) becomes now almost trivial. One has to take into account that in our approximation the analogue of the Breit Hamiltonian in (3.3) has the form [20]... 

Now the Breit Hamiltonian for two electrons in the presence of electromagnetic fields is, as we have seen,... 

The Breit Hamiltonian operates on sixteen-component spinor functions which contain fourtypes of function, designated A L. 1- fiu- fu, which represent upper and lower... 

It now remains to expand the operators in (3.235) using the definitions given in (3.230) but before we do so we must draw attention to a difficulty with (3.235). The final term, containing the operator (00)2 is not obtained if a more sophisticated treatment starting from the Bethe Salpeter equation is used. The reader will recall our earlier comment that the interaction term in the Breit Hamiltonian is acceptable provided it is treated by first-order perturbation theory. Rather than launch into quantum electrodynamics at this stage, we shall proceed to develop (3.235) but will omit the (00)2 term without further comment. 

The last stage is now to replace tv, by P, + eA, and to use explicit expressions for the potentials A, and (pi. Our previous expressions for A, and electron interactions have now been derived more naturally by starting with the Breit Hamiltonian and the vector and scalar potentials therefore contain only terms describing external fields or electrostatic interactions involving the nuclear charge. Hence we make the substitutions... 

It is possible to obtain the nuclear spin magnetic interaction terms by starting from the Breit equation. We recall that the Breit Hamiltonian describes the interaction of two electrons of spin 1 /2, each of which may be separately represented by a Dirac Hamiltonian ... 

Any operator can be separated into odd and even contributions. The Dirac operator fi commutes with an even operator and anticommutes with an odd operator. The Breit Hamiltonian can be decomposed into even-even (EE), even-odd (EO), odd-even (OE) and odd-odd (OO) components, depending on whether the part of the operator acting on electrons one and two, respectively, is even or odd... 

The separated upper-upper part of the Breit Hamiltonian is satisfactory to the order of 1 /c2 and is... 

The Breit Hamiltonian operates on sixteen-component spinor functions which contain fourtypes of function, designated V uL. fm, V i,whichrepresentupperandlower components as previously defined, the small letters u, referring to the first particle (1) and the capital letters U, L referring to the second particle (2). Our aim is to find a transformation which gives a Hamiltonian operating only on the components uu in other words, we seek a Hamiltonian which, to order c, contains only terms which are overall even-even in character. 

The Breit Hamiltonian (in our example, for two electrons in an electromagnetic field) can be approximated by the following useful formula. known as the Breit-Pauli Hamiltonian ... 

Finally, the Breit equation has been given. The equation goes beyond the Dirac model by taking into account the retardation effects. The Breit-Pauli expression for the Breit Hamiltonian contains several easily interpretable physical effects. 

To describe the interactions of the spin magnetic moments, this Hamiltonian will soon be supplranented by the relativistic terms from the Breit Hamiltonian (p. 147). 

Let us assume the Bom-Oppenheimer approximation (p. 269). Thus, the nuclei occupy some fixed positions in space, and in the electronic Hamiltonian equation (12.59), we have the electronic charges qj = —e and masses mj = nto = m (we skip the subscript 0 for the rest mass of the electron). Now, let us refine the Hamiltonian by adding the interaction of the particle magnetic moments (of the electrons and nuclei the moments result from the orbital motion of the electrons, as well as from the spin of each particle) with themselves and with the external magnetic field. We have, therefore, a refined Hamiltonian of the system, the particular terms of the Hamiltonian corresponds to the relevant terms of the Breit HamiltonianS (p. 147)... 

The interaction of the spin magnetic moments of the electrons (Hsh) and of the nuclei Hih) with the field H. These terms come from the first part of the term H(, of the Breit Hamiltonian, and represent the simple Zeeman terms ... 

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