Operators kinetic energy, Dirac

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

The Dirac operator incorporates relativistic effects for the kinetic energy. In order to describe atomic and molecular systems, the potential energy operator must also be modified. In non-relativistic theory the potential energy is given by the Coulomb operator. 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

By inserting the equations defining the kinetic energy operators and the pairwise interaction operators into Eq. (8) we obtain the Dirac-Coulomb-Breit Hamiltonian, which is in chemistry usually considered the fully relativistic reference Hamiltonian. 

For materials containing atoms with large atomic number Z, accelerating the electrons to relativistic velocities, one must include relativistic effects by solving Dirac s equation or an approximation to it. In this case the kinetic energy operator takes a different form. 

The operator H is intended to describe the energy of a particle in a given external field. As in nonrelativistic quantum mechanics, the influence of an external field is described by a potential-energy V (x) that is added to the kinetic energy Hq. Hence the Dirac operator with an external field reads... 

In nonrelativistic quantum mechanics, the angular momentum barrier prevents the collaps to the center. The angular momentum barrier is an effective potential of the form + l)/r that appears if one writes the kinetic energy in polar coordinates. For the Dirac equation the role of the angular momentum barrier is obviously played by the term /c/r in the radial Dirac operator. This term is effectively repulsive for both signs of k, because it appears only off-diagonal. The point is, that the repulsive angular momentum barrier k jr cannot balance the attractive Coulomb potential 7/r for r —> 0, as soon as I7I > k. ... 

The superscript (4) indicates the Dirac four-component picture of operators and wave functions. is the relativistic kinetic energy functional of the Dirac-Kohn-Sham (DKS) reference system of non-interacting electrons with ground state density yO [45] ... 

Relativistic corrections would take into account the mass increase of the electrons (plus indirect relativistic effects) occurring in heavier atoms (see Section 2.11.1), to be dealt with properly by Dirac s equation, a relativistic theory. But let us concentrate on the above nonrelativistic Hamilton operator and analyze its ingredients. Obviously, H is composed of five parts the first two represent the kinetic energies of the N electrons with masses m and the M nuclei with masses Ma (remember that neither electrons nor nuclei can be expected to be standing still) the differential operator... 

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