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 : where is the Dirac Hamiltonian for particle i at position ri and φ 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 : where the Dirac matrices for electroni: a = . 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 can be decomposed into the following practical energy operators for electrons in electric and magnetic fields , which have well-defined meanings in the interaction of molecules with magnetic fields : in which the consecutive partial operators are:
is the nonrelativistic Hamiltonian.
is connected to the dependence of mass on velocity:.
is a correction that partly accounts for retardation and can be described as the interaction between the magnetic dipole moments of the particles, which arise from the orbital motion of charges.
is the classical interaction between the orbital magnetic moments and spin magnetic moments. The first term describes the interaction of a particle's spin with its own orbital moment, and the second term between two different particles.
is a nonclassical term characteristic for Dirac theory, sometimes called the Darwin term.
is the magnetic momentspin-spin interaction. The first term is called the contact interaction, because it is nonzero only when the particles are at the same position; the second term is the interaction of the classical dipole-dipole type.
is the interaction between spin and orbital magnetic moments with an external magnetic fieldH.
