Interaction Breit

The Breit interaction [29, p. 170] is that part of the interaction between electrons mediated by exchange of transverse photons. The lowest-order energy shift associated with the exchange of a transverse photon between two electrons in states a and b is 

In the direct matrix element babab, where fco = 0, the frequency-dependent Breit interaction reduces to its limiting static form  

Differences between the frequency-dependent Brcit interaction and its static limit given in Eq. (86) are of relative order a Z, and therefore important primarily for highly-charged ions. 

The correction to the many-electron Hamiltonian of Eq. (74) from the Breit interaction 

The corresponding first-order Breit correction to the energy of a closed-shell atom is 

---

Note that the subscript on the a matrices refers to the particle, and a here includes all of the tlx, tty and components in eq. (8.4). The first correction term in the square brackets is called the Gaunt interaction, and the whole term in the square brackets is the Breit interaction. The Dirac matiices appear since they represent the velocity operators in a relativistic description. The Gaunt term is a magnetic interaction (spin) while the other term represents a retardation effect. Eq. (8.27) is more often written in the form... 

Gorcek, O. and Indelicato, P. (1988) Effect of the complete Breit interaction on two-electron ion energy levels. Physical Review A, 37, 1087-1094. 

Lindroth, E. and Martensson-Pendrill, A.-M. (1989) Eurther analysis of the complete Breit interaction. Physical Review A, 39, 3794-3802. 

Perdew, J.P. and Cole, L.A. (1982) On the local density approximation for Breit interaction. Joumoi of Physics C, 15, L905-L908. 

In a rigorous treatment, one replaces the one-electron operator h by the four-component Dirac-operator hjj and perhaps supplement the two-electron operator by the Breit interaction term [15]. Great progress has been made in such four-component ab initio and DPT methods over the past decade. However, they are not yet used (or are not yet usable) in a routine way for larger molecules. 

The plan of this paper is as follows - In section 2, the basic experimental data required in the re-evaluation of the empirical correlation energies of the N2 CO, BF and NO molecules are collected. The essential theoretical ingredients of our re-determination are given in section 3 including new fully relativistic calculations including the frequency independent Breit interaction and electron correlation effects described by second order diagrammatic perturbation theory for the Be-like ions B", C, O" ... 

On the other hand, the estimate of the Breit interaction energy is, in all cases, quite satisfactory in any atomic or ionic superposition model. The reason for this has already been discussed the Breit interaction energy arises due to electron current density in the neighbourhood of the nuclei, which is dominated by the core electrons and is apparently insensitive to the valence electron environment. 

Table 7 Estimates of total relativistic correction, E , and the first-order Breit energy correction,, obtained by combining the atomic or ionic contributions indicated by the second column. They may be compared with the values of the total relativistic correction, Er. and the first-order Breit interaction, Es, obtained directlyfrom matrix Dirac-Elartree-Fock and Elartree-Fock calculations of the molecular structure using BERTEIA [12], Only the results of the 13s7p2d atom-centred basis sets for Er and Eb are quoted. All energies in atomic units.

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

The molecular orbitals in the nonrelativistic and one-component calculations and the large component in the Dirac-Fock functions were spanned in the Cd s Ap9d)l[9slp6d basis of [63] and the H (5s 2p)/[35 l/>] set [61]. Contraction coefficients were taken from corresponding atomic SCF calculations. The basis for the small components in the Dirac-Fock calculations is derived by the MOLFDIR program from the large-component basis. The basis set superposition error is corrected by the counterpoise method [64]. The Breit interaction was found to have a very small effect and is therefore not included in the results. 

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 matrices J, K and B are direct, exchange and Breit interaction matrices, of which only the first is block diagonal. Their matrix elements are linear combinations of interaction integrals over G-spinors. 

The Breit interaction matrix can be treated in a similar way. The off-diagonal blocks can be written in terms of the magnetic fields using... 

It is known that the Breit interaction can give contributions in excess of one thousand wave numbers even to energies of transitions between lowest-lying states of very heavy elements (see, e.g., tables 7 and 8). It is also clear that the point nuclear model becomes less appropriate when the nuclear charge is increased. Therefore, the RECPs designed for accurate calculations of actinide and SHE compounds should allow one to take into account the Breit interaction and the finite size of nuclei. The most economic way is to incorporate the corresponding contributions into the RECP operator. 

ACCOUNTING FOR THE BREIT INTERACTION BETWEEN DIFFERENT SHELLS... 

Let us analyze contributions of the Breit interaction between electrons from different shells to the energy of a heavy atom [27]. We will use the estimate (e.g., see [32])... 

As a result, the Breit interaction between the one-electron states P and P can be estimated as... 

---