Electromagnetic Interaction Energy Operator

Finally, we may note that the Dirac Eq. (5.121) can be re-arranged to yield an expression for the interaction energy operator V, 

This is exactly the expression for the Dirac velocity operator which we derived in section 5.3.3 from the Heisenberg equation of motion. In section 8.1, this 

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First of all, the theory presented is based on a few assumptions, which, while valid for the molecular systems considered in the literature so far, need to be care-fidly examined in every specific case. As mentioned in Section 8.3, we assume that the effects of external fields on the kinetic energy operator for the relative motion are negligible and that the interactions with electromagnetic fields are independent of the relative separation of the colliding particles. In addition, we ignore the nonadiabatic interactions that may be induced by external fields and that, at present, cannot be rigorously accounted for in the coupled channel calculations. 

II being the crystal energy operator when only the Coulomb interaction is accounted for, H2 is the energy operator of transverse photons of the electromagnetic field in vacuum, and //mti is the operator of the interaction between all carriers with the field of the transverse photons. 

We have already seen in section 8.1 that (i) a Dirac electron with electromagnetic potentials created by all other electrons [cf. Eq. (8.2)] cannot be solved analytically, which is the reason why the total wave function as given in Eq. (8.4) cannot be calculated, and also that (ii) the electromagnetic interactions may be conveniently expressed through the 4-currents of the electrons as given in Eq. (8.31) for the two-electron case. Now, we seek a one-electron Dirac equation, which can be solved exactly so that a Hartree-type product becomes the exact wave function of this system. Such a separation, in order to be exact (after what has been said in section 8.5), requires a Hamiltonian, which is a sum of strictly local operators. The local interaction terms may be extracted from a 4-current based interaction energy such as that in Eq. (8.31). Of course, we need to take into account Pauli exchange effects that were omitted in section 8.1.4, and we also need to take account of electron correlation effects. This leads us to the Kohn-Sham (KS) model of DFT. 

One can see that the full Hamiltonian consists of three terms, two which describe separately the parts for the atom and the field, and one which represents a coupling between the field (vector potential A) and terms from the atom (operator V,-). Obviously, it is this mixed term which is responsible for the photon-atom interaction. Provided perturbation theory can be applied, this term then acts as a transition operator between undisturbed initial and final states of the atom. Following this approach, one has to verify whether the disturbance caused by the electromagnetic field in the atom is small enough such that perturbation theory is applicable. Hence, one has to compare the terms which contain the vector potential A with an energy ch that is characteristic for the atomic Hamiltonian ... 

This variation of AAm/Bm with separation comes from the "relativistic screening function" R (l), which is subsequently elaborated. This factor becomes important at large distances when we must be concerned with the finite velocity of the electromagnetic wave. At short distances, Rn(l) = 1 the energy of interaction between two flat surfaces varies with the square of separation. At large distances any effective power-law variation of the interaction depends on the particular separation of materials and wavelengths of the operative electromagnetic waves between them. 

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