Perturbative Treatment of the Lamb Shift

In chapter 5 we mentioned the Lamb shift in passing and the difficulty of calculating it due to the renormalizations required. In fact, there is a well-developed perturbation theory of the Lamb shift in the same framework as Breit-Pauli theory. We do not propose to derive the expressions here, for which the reader is referred to Bethe and Salpeter (1957). Instead, we report the results for the lowest-order terms, which turn out to be expressible as corrections to the Darwin and spin-orbit one-electron operators. The combined operators may be written 

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 

The multiplying factor is clearly related to the expectation of the Dirac delta function or of the spin-orbit operator, both of which scale as Z jr , and hence we can derive a correction factor for the Darwin operator of 

A more elaborate treatment of the Lamb shift for general polyatomic molecules is not at present available. 

---