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
\(\begin{equation}\Psi_{\mathrm{SCGVB(6)}}\mathcal{=A}\left\{\left(\mathrm{closed-shell\ orbitals}\right)\left(\prod_{\mu=1}^{6}\phi_{i}\right)\mathrm{\Theta}_{0,0}^{6}\right\} \end{equation}\)
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]