Introduction

The results in the preceding chapters on stability of the second kind were mostly of the form E0 > −C(N + M), where E0 denotes the ground state energy of the system, and N and M are the number of electrons and nuclei, respectively. It is obvious, however, that if N is very large or very small compared to M the excess number of particles, positive or negative as the case may be, will float away to infinity. In other words it ought to be possible to reformulate the previous results as E0 > −C′ min{N, M} for a suitable C′ that depends only on the nuclear-electron charge ratio Z.

In the relativistic case discussed in Chapter 8, the energy is actually nonnegative for suitable α and Z, independently of N and M. From this we conclude that also the non-relativistic energy can be bounded below by E0 > −CN independent of M, as discussed in Remark 8.6. (For an alternative method, see.) Also the results in Chapters 9 and 10 yielded bounds of this form, since they rely in an essential way on the non-negativity of the relativistic energy in Chapter 8. This answers half the problem, namely it gives a bound on the energy of the correct form if M is larger than N.