# Breit equation

The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and retardation effects to the order of 1/c2. When other quantum electrodynamic effects are negligible, this equation has been shown to give results in good agreement with experiment. It was originally derived from the Darwin Lagrangian but later vindicated by the Wheeler–Feynman absorber theory and eventually quantum electrodynamics.

## Introduction

The Breit equation is not only an approximation in terms of quantum mechanics, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation. Just as does the Dirac equation, it treats nuclei as point sources of an external field for the particles it describes. For N particles, the Breit equation has the form (rij is the distance between particle i and j):

where

${\hat {H}}_{D}(i)=\left[q_{i}\phi ({\mathbf {r} }_{i})+c\sum _{s=x,y,z}\alpha _{s}(i)\pi _{s}(I)+\alpha _{0}(I)m_{0}c^{2}\right]$ is the Dirac hamiltonian (see Dirac equation) for particle i at position ri and φ(ri) is the scalar potential at that position; qi is the charge of the particle, thus for electrons qi = −e. The one-electron Dirac hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/rij, form the Dirac-Coulomb operator. To this, Breit added the operator (now known as the (frequency-independent) Breit operator):

${\hat {B}}_{ij}=-{\frac {1}{2r_{ij}}}\left[{\mathbf {a} }(i)\cdot {\mathbf {a} }(j)+{\frac {\left({\mathbf {a} }(i)\cdot {\mathbf {r} }_{ij}\right)\left({\mathbf {a} }(j)\cdot {\mathbf {r} }_{ij}\right)}{r_{ij}^{2}}}\right]$ ,

where the Dirac matrices for electron i: a(i) = [αx(i),αy(i),αz(i)]. The two terms in the Breit operator account for retardation effects to the first order. The wave function Ψ in the Breit equation is a spinor with 4N elements, since each electron is described by a Dirac bispinor with 4 elements as in the Dirac equation, and the total wave function is the tensor product of these.

## Breit hamiltonians

The total hamiltonian of the Breit equation, sometimes called the Dirac-Coulomb-Breit hamiltonian (HDCB) can be decomposed into the following practical energy operators for electrons in electric and magnetic fields (also called the Breit-Pauli hamiltonian) , which have well-defined meanings in the interaction of molecules with magnetic fields (for instance for nuclear magnetic resonance):

${\hat {B}}_{ij}={\hat {H}}_{0}+{\hat {H}}_{1}+...+{\hat {H}}_{6}$ ,

in which the consecutive partial operators are: