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.