Figure 1. Schematic representation of the geometries of D 3h[NgMH3MNg]+ cations.

The spin-coupled generalized valence bond (SCGVB) wavefunctions considered here for the six valence electrons of the MH3M moieties take the form

in which the \(\phi_{i}\) are the six singly-occupied nonorthogonal active orbitals and the total active space spin function\(\Theta_{0,0}^{6}\) is a linear combination of all five linearly independent ways that the spins of these six electrons can be coupled to yield a state with S  =  0 andMS  = 0.[22] All of the closed-shell and active orbitals were fully optimized as expansions in the full molecular basis set, simultaneously with the expansion coefficients of \(\Theta_{0,0}^{6}\). All of these SCGVB(6) calculations were carried out using the CASVB module[23-27] in MOLPRO.[20-21]

We also calculated the corresponding ‘6 electron in 6 orbitals’ complete active space self-consistent field, i.e. CASSCF(6,6), wavefunctions. Additionally, full-valence CASSCF calculations, with all allowed distributions of six electrons in eleven orbitals, were carried out for the ‘bare’ D 3h[MH3M]+ cations (M = Be, Mg); such CASSCF(6,11) calculations were also carried out for the corresponding systems capped by Ng (Ng = He, Ne) atoms. All of these various CASSCF wavefunctions were generated using MOLPRO.[20-21]

In addition to examining the highly-visual SCGVB(6) descriptions of these various systems, we also performed domain-averaged Fermi hole (DAFH) analysis.[28-34] It proved especially informative to consider ‘holes’ that are averaged over multi-atom domains, each formed as the union of individual quantum theory of atoms in molecules (QTAIM) domains.[35] Such DAFH analysis not only provides information about any electron pairs that remain intact within each of the chosen fragments but also about the broken or dangling valences that are created by the (formal) bond splitting that would be required to isolate that fragment from the rest of the molecule. With this in mind, we took one domain to be the union of the two M atom QTAIM domains and another to be the complementary domain formed from the union of the three H atom QTAIM domains. In this way, we can detect whether there is any direct MM bonding and we can identify the nature of the MHM interactions.

In the special case that the domain is taken instead to be the whole molecule, i.e. exhausting the complete space, DAFH analysis corresponds to the generation of localized natural orbitals (LNOs). All of the DAFH and LNO analysis described in the present work used our own codes, with the required QTAIM analysis[35] carried out with AIMAll.[36] Using the same isovalue throughout, pictorial depictions of SCGVB orbitals, LNOs and DAFH functions were produced using Virtual Reality Markup Language (VRML) files generated with Molden.[37]