Higher order coulombic mechanisms

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

In our first simple example the electrostatic potential set up by CsCl is almost but not quite a minimal surface [10]. The reason is that the Coulomb electrostatic energy is only a part of the whole electromagnetic field. Two body, three and higher order, non-additive van der Waals interactions contribute to the complete field, distributed within the crystal. This leads one to expect that the condition that the stress tensor of the field is zero, as for soap films, yields the condition for equilibrium of the crystal. Precisely that condition is that for the existence of a minimal surface. Strictly speaking the minimal surface might be defined by the condition that the electromagnetic stress tensor is zero. But in any event, we see in this manner that the occurrence of minimal surfaces, should be a consequence of equilibrium (cf. Chapter 3,3.2.4). Indeed a statement of equilibrium may well be equivalent to quantum statistical mechanics. 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

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