Third virial coefficient for hard spheres with embedded dipoles

The doublet stability of the SCF solutions for the allyl radical is re-examined for various ab-initio model hamiltonians, based on the minimum, double-zeta and double-zeta plus polarization atomic orbital basis sets in the LCAO scheme. The calculations are carried out for a wide range of the C-C bond lengths (1.3 Å to 2·0 Å). In all cases the critical bond length, separating regions of stability and instability, is found to be shorter than the corresponding equilibrium C-C bond length (1·36 Å versus 1·44 Å, respectively, for the minimum basis set). In contrast to the results of McKelvey and Berthier we find that the doublet-stability problem yields at most one negative eigenvalue in the whole range of the C-C bond lengths studied, this implying only one possible (other than spin) symmetry breaking, in complete agreement with the semi-empirical PPP model results for this system. The basis dependence of the doublet instability is also re-examined and the generality of this phenomenon is firmly established for the system studied : while the actual values of the energy lowering due to the symmetry breakdown and the critical value of the C-C bond length for the onset of instability will of course depend on the basis and approximations used, there is no doubt about the general occurrence of doublet instability in this system for large enough C-C internuclear separations, and about the implied non-analytic behaviour of the potential energy hypersurface in this region, for any basis. Finally, a simple physically and chemically plausible source of the doublet instability of allyl RHF solutions is pointed out.