Two-Electron Darwin Term

One recognizes the two-electron Darwin term and the two-electron spin-(same)-orbit term (see also sections 2.12 and 4.6). 

The Bethe logarithm. In Kq, is a number that is independent of Z but depends on n, the principal quantum number of the atomic electron. This would be awkward for quantum chemical calculations, but since the most important differential effects for chemistry would come from the valence shell, a good approximation is to use the value for the valence shell (Pyykko et al. 2001). Bethe logarithms have been tabulated by Drake and Swainson (1990). The correction to the spin-orbit interaction is small, being less than 0.25%, but the correction to the Darwin term varies from nearly 10% for hydrogen down to 1% for heavier elements. There is also a perturbative correction to the two-electron Darwin term, which we write as... 

This is the correction to the Hartree-Fock energy. Obviously, only the terms that are diagonal in the spin and are spatially totally symmetric will contribute. These terms are the scalar relativistic corrections the mass-velocity term, the one- and two-electron Darwin terms, and the orbit-orbit term. Other terms such as the spin-spin term and the z component of the spin-orbit interaction contribute for open-shell systems, where the spin is nonzero. 

The third term of Hn is identical with the second one except for the interchange of electrons, so that the individual contributions should be counted twice. These two terms can be rewritten as a sum of three contributions the first is the electron Darwin term... 

The contributions from the (2S.2.10) integrals to (2S.2.8) vanish for the same reason as for the two-electron Darwin operator. The second-quantization representation of the two-electron spin-spin contact term then becomes... 

Semiempirical spin-orbit operators play an important role in all-electron and in REP calculations based on Co wen- Griffin pseudoorbitals. These operators are based on rather severe approximations, but have been shown to give good results in many cases. An alternative is to employ the complete microscopic Breit-Pauli spin-orbit operator, which adds considerably to the complexity of the problem because of the necessity to include two-electron terms. However, it is also inappropriate in heavy-element molecules unless used in the presence of mass-velocity and Darwin terms. 

After the abovementioned cancellation (of and Darwin terms), the retardation becomes one of the most important relativistic effects. As seen from Table 3.1, die effect is about 100 times larger (both for the ionization energy and the polarizability) for the electron-electron retardation than for that of the nucleus-electron. This is quite understandable because the nucleus represents a massive rock" (it is about 7000 times heavier in comparison to an electron). it moves slowly, and in the nucleus-electron interaction, only the electron contributes to the retardation effect. Two electrons make the retardation much more serious. 

We have already discussed in chapters 12 and 13 that low-order scalar-relativistic operators such as DKH2 or ZORA provide very efficient variational schemes, which comprise all effects for which the (non-variational) Pauli Hamiltonian could account for (as is clear from the derivations in chapters 11 and 13). It is for this reason that historically important scalar relativistic corrections which can only be considered perturbatively (such as the mass-velocity and Darwin terms in the Pauli approximation in section 13.1), are no longer needed and their significance fades away. There is also no further need to develop new pseudo-relativistic one- and two-electron operators. This is very beneficial in view of the desired comparability of computational studies. In other words, if there were very many pseudo-relativistic Hamiltonians available, computational studies with different operators of this sort on similar molecular systems would hardly be comparable. 

Second, if only the one-electron mass-velocity and Darwin terms are included in the relativistic perturbation—a common choice since they are relatively easy to evaluate— the relativistic first-order wave function (which is 0 c )) must consist only of the Brillouin single excitations. But these have a zero matrix element of the correlation perturbation with the ground state, and hence the correction is zero. It would then be necessary to include the two-electron terms to get a correction at this order to the ground state. This is true for both closed-shell and open-shell systems. 

In order (Z a)", one finds a p /m mass correction term, the Darwin term that shifts s states, and the spin-orbit interaction -(e/2m r) (dU/dr)s lT. If we look at the special hydrogenic case, U=Ze/r, we can get a rough idea of what these lower order rela-tivisic terms do to the wavefunction of an atom or ion. We find that the spin-orbit interaction causes j-dependent shifts in the one-electron energy, and that the mass correction term causes an 1-dependent shift. Adding the Darwin term to these two corrections, one finds that the energy is given approximately by... 

The first two terms, the mass-velocity and the Darwin operators, are called scalar relativistic terms since they do not involve the electron spin. They are given by... 

Because of the importance of Darwin s expression for the classical electromagnetic interaction of two moving charges (section 3.5), we are particularly interested in the frequency-independent radial form of the Breit operator. This represents the consistent interaction term to approximately include the retarded electromagnetic interaction of the electrons in our semi-classic formalism that describes only the elementary particles (electrons) quantum mechanically. In this long-wavelength limit, m —> 0, the radial operator Vv l,2) in Eq. (9.16) becomes D (l, 2) — already known from the Coulomb case in Eq. (9.9)... 

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