Relativistic two-electron Hamiltonian terms

The derivation of the electron-electron interaction needs deep analysis originating in the special theory of relativity. Let us consider two coordinate systems C(x,y,z, r) and C x, y, z, t ) with the time-related coordinate r = ict. Let the inertial frame C be moving along the x-direction with a constant velocity v relative to C. The systems of coordinates are interrelated by the Lorenz transformation and its inverse according to Table 4.4, where the dimensionless time dilation factor 

Any four-vector V Vi(x), X2(y), Xfz), X4(t) obeys a similar transformation. The electromagnetic potential four-vector is S = Ax, Ay, Az, i4 /c and its transformation becomes 

HA5a) 4 0 8 Ge)( ext) rGe) XNe ext)] spin-orbit Zeeman gauge correction 

Hlb(2a) Jl(2 p BarlGe) orbital Zeeman-kinetic energy correction 

Lorenz transformation of Inverse transformation of Lorenz transformation of a 

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Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

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. 

Visscher et have applied this formalism at the RPA level to calculate the J coupling tensor in hydrogen halides using the DIRAC code. Only the Coulomb interaction in the two-electron part of the Hamiltonian was considered, disregarding the Breit interaction byki, Eq. (21). Relativistic values were compared with those obtained by applying the Breit scaling factors 5(n,Z) to the FC term ... 

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