Figure 2. Symmetry-unique key valence
functions (and occupation numbers where appropriate) from analysis of
the SCGVB(6) description of the D 3h[BeH3Be]+ cation. Also shown are
QTAIM bond paths.
We turn now to outcomes of DAFH analysis for theD 3h[BeH3Be]+ SCGVB(6) wavefunction.
In addition to the one-electron density and QTAIM analysis as input,
full DAFH analysis for non-exhaustive (combinations of) individual QTAIM
domains requires in this case the use of the two-electron density, which
was readily available to us in the case of SCGVB and CASSCF
wavefunctions. Such DAFH analysis for the domain comprising the union of
the two beryllium atom QTAIM domains produces three symmetry-equivalent
valence functions, each populated by 0.19 electrons (see the first image
in the second row of Figure 2). Such an occupation number of 0.19 can be
understood as the joint contribution from the two metal atoms to a
shared electron pair of a particular 3c‑2e Be−H−Be linkage. The
corresponding DAFH analysis for the domain consisting of the union of
the three hydrogen atom QTAIM domains also produces three
symmetry-equivalent valence functions (see the second image in the
second row of Figure 2). Each of these functions has an occupation
number of 1.80, which can be understood as the complementary
contribution to the shared electron pair of a particular 3c‑2e Be−H−Be
linkage. The considerable difference between the occupation numbers from
complementary BeBe and HHH domains indicates that the electron pair in
each Be−H−Be linkage is shared very unevenly, implying high polarity
across the three centres. The occupation numbers of the other non-core
functions produced by the DAFH analysis were all sufficiently small that
we did not examine them in any detail.
The existence of the three 3c‑2e Be−H−Be bonding units is also
straightforwardly corroborated by the forms of the localized natural
orbitals (LNOs) that result from simpler DAFH analysis performed for the
domain involving the whole molecule. Such analysis still requires the
one-electron density and QTAIM analysis as input, but not the
two-electron density. We find that it yields in this case three
degenerate valence LNOs with occupancies close to two. The forms of
these LNOs (see the third image in the top row of Figure 2) are again
strongly suggestive of three equivalent highly polar 3c‑2e Be−H−Be
bonding units. The occupation numbers of all of the remaining non-core
LNOs were sufficiently small that we did not examine them in any detail.
As can be seen from Figures S2 and S3 in the Supporting Information,
there were no significant differences in the corresponding valence LNOs
and DAFH functions for the D 3h[NgBeH3BeNg]+ SCGVB(6)
wavefunctions. At most we observed fairly small changes in occupation
numbers, on the order of 0.01. We have also examined the corresponding
valence LNOs and DAFH functions for D 3h[BeH3Be]+ and
[NgBeH3BeNg]+ cations using
different levels of theory, including CASSCF(6,11), CCSD and B3LYP. We
found that there were slightly larger (but still very small) changes to
occupation numbers but negligible changes to the forms of the various
functions. (Note that for the DAFH analysis for BeBe and HHH domains at
the CCSD and B3LYP levels of theory we used a reliable one-electron
approximation[32] based on natural orbital
occupation numbers.[38]) As a representative
demonstration of the high similarity between these different sets of
analyses we display in Figures S4 and S5 in the Supporting Information
the dominant symmetry-unique valence LNOs that were obtained at
different levels of theory.
We now examine the QTAIM bond paths for theD 3h[BeH3Be]+ cation, calculated using
the SCGVB(6) wavefunction. These are depicted in the third image in the
second row of Figure 2. (Note that in addition to the bond critical
points, which are shown, there are corresponding ring and cage critical
points, which have not been displayed.) Clearly there are curved bond
paths linking Be and H atoms. There are also corresponding bond paths
linking the H atoms, but no BeBe bond path. Indeed, the critical point
at the centre of this cation turns out to be a ring critical point,
sandwiched between two cage critical points along the BeBe axis. We
checked that the pattern of critical points and of bond paths was
unchanged when we switched from SCGVB(6) to CASSCF(6,11), CCSD or B3LYP
descriptions. (Additionally, as can be seen from the third image in the
second row of Figure S2 in the Supporting Information, the basic pattern
in the BeH3Be moiety was unchanged upon capping the
‘bare’ cation with He atoms.)
Additional corroboration for the absence of any significant direct
bonding between the beryllium centres is provided by various indicators
of the metal-metal bond order, for which we report here the values of
two somewhat different quantities. The first of these is the
shared-electron distribution index (SEDI),[39]
also known as a delocalization index,[40] which
provides a measure of the distribution of the electrons that are shared
between two atomic domains. The other, denoted here asWAB , is based on an improved definition of
two-centre bond orders for correlated singlet
systems,[41] but re-expressed in AIM-generalized
form.[11, 42] The values of SEDI(Be,Be) and ofW BeBe for the SCGVB(6) description of theD 3h[BeH3Be]+ cation turn out to be
just 0.033 and 0.027, respectively, providing further confirmation of
the absence of any significant direct metal-metal bonding. (Analogous
small values were found for the D 3h[NgMH3MNg]+ cations and we checked
that changing the level of theory to CASSCF(6,11), CCSD or B3LYP, using
the same cc‑pVQZ basis sets, did not lead to significant changes.)
Clearly there is no evidence in the forms of any of the valence LNOs,
DAFH functions or SCGVB orbitals, or in bond indices and QTAIM analysis,
for any significant direct BeBe bonding in theD 3h[BeH3Be]+ and
[NgBeH3BeNg]+ cations
(Ng = He, Ne) at their all-electron CCSD(T)/cc‑pVQZ geometries. Instead
the two positively-charged beryllium centres are held together with a
very short BeBe distance by the three negatively-charged hydrogen
centres, with a stabilizing contribution from three equivalent sets of
highly polar 3c‑2e Be−H−Be bonding character.