A comparison of non-covalent interactions in the crystal structures of two σ-alkane complexes of Rh exhibiting contrasting stabilities in the solid state

Non-covalent interactions surrounding the cationic Rh s -alkane complexes within the crystal structures of [(Cy 2 PCH 2 CH 2 PCy 2 )Rh(NBA)][BAr F4 ], [1-NBA][BAr F4 ] (NBA = norbornane, C 7 H 12 ; Ar F = 3,5-(CF 3 ) 2 C 6 H 3 ), and [1-propane][BAr F4 ] are analysed using Quantum Theory of Atoms in Molecules (QTAIM) and Independent Gradient Model approaches, the latter under a Hirshfeld partitioning scheme (IGMH). In both structures the cations reside in an octahedral array of [BAr F4 ] − anions within which the [1-NBA] + cation system exhibits a greater number of C – H / F contacts to the anions. QTAIM and IGMH analyses indicate these include the strongest individual atom – atom non-covalent interactions between the cation and the anion in these systems. The IGMH approach highlights the directionality of these C – H / F contacts that contrasts with the more di ﬀ use C – H / p interactions. The accumulative e ﬀ ects of the latter lead to a more signi ﬁ cant stabilizing contribution. IGMH % d G atom plots provide a particularly useful visual tool to identify key interactions and highlight the importance of a – {C 3 H 6 } – propylene moiety that is present within both the propane and NBA ligands (the latter as a truncated – {C 3 H 4 } – unit) and the cyclohexyl rings of the phosphine substituents. The potential for this to act as a privileged motif that confers stability on the crystal structures of s -alkane complexes in the solid-state is discussed. The greater number of C – H / F inter-ion interactions in the [1-NBA][BAr F4 ] system, coupled with more signi ﬁ cant C – H / p interactions are all consistent with greater non-covalent stabilisation around the [1-NBA] + cation. This is also supported by larger computed d G atom indices as a measure of cation – anion non-covalent interaction energy.

Understanding the stability of these s-alkane complexes in the solid-state is a central goal for our research in this area, that aims to exploit SMOM chemistry in catalysis. 1,17In an early study we showed that computed NBA binding energies across a range of [(R 2 P(CH 2 ) n PR 2 )M(alkane)] + molecular cations (i.e.neglecting the effect of the solid-state environment, n = 2; R = Cy, i Bu, i Pr, O i Pr; n = 3, R = i Pr) showed no correlation with the stability of [(R 2 P(CH 2 ) n PR 2 )M(alkane)][BAr F 4 ] salts in the solid state. 9Bistoni and co-workers subsequently discussed the role of intramolecular dispersion in the alkane molecular binding energies. 21However, our subsequent work has shown the importance of including the solid-state environment via periodic DFT calculations to achieve reasonable structures and energetics for these s-alkane complexes. 19In a recent study, normalised lattice energies, alkane incorporation energies and molecular alkane binding energies were computed for a range of [(Cy 2 P(CH 2 ) 2 PCy 2 )M(alkane)][BAr F 4 ] complexes (see Fig. 2 for denitions). 16All three measures were consistent with the greater stability of ; moreover, a signicant, additional stabilization provided by the solid-state environment, DE SS , was identied and was larger in [1-NBA][BAr F 4 ] (14.0 kcal mol −1 cf.8.3 kcal mol −1 ).Previously, non-covalent interactions (NCI) plots 22 have highlighted broad areas of dispersive stabilisation between the alkane ligand and the neighbouring [BAr F 4 ] − anions in these s-alkane complexes. 11,13,14,17,18In addition, more localised disk-like features have suggested the presence of C-H/F contacts and we have commented on the potential role of such interactions in conferring stability in the solid state. 17Here we employ Quantum Theory of Atoms in Molecules (QTAIM) and Independent Gradient Model calculations under the recently proposed Hirshfeld partitioning scheme (IGMH 23 ) to explore further the role of these noncovalent interactions on the stability of s-alkane complexes.Our study will focus on the [1-NBA][BAr F 4 ] and [1-propane][BAr F 4 ] systems as these exhibit each end of the range of stabilities seen to date in crystallographically characterised s-alkane complexes in the solid state.

Methodology
Geometries for [1-NBA][BAr F 4 ] 9 and [1-propane][BAr F 4 ] 16 were taken from the reported crystal structures and partially optimised with periodic-DFT calculations at the PBE-D3 level using the CP2K program (see ESI † for full details).B, C, P and Rh atoms were xed at their experimentally determined positions while H and F atoms were fully relaxed in order to provide reliable H atom positions and to resolve any rotational disorder associated with the CF 3 groups.Unit cell parameters were xed at the experimental values.Intermolecular interactions were analysed between one central Rh cation and the surrounding pseudo-octahedron of [BAr F 4 ] − anions.Short contacts within crystal structures were analysed with CrystalExplorer 24,25 with van der Waals radii taken from Alvarez (H: 1.20 Å; C: 1.77 Å; F: 1.46 Å). 26 In the following a 'short contact' will refer to a distance at or below the sum of the relevant van der Waals radii.Quantum theory of atoms in molecules (QTAIM 27 ) analyses employed the AIMAll program 28 and used the extended wavefunction format.Independent gradient model calculations used Multiwfn 29 and employed the Hirshfeld partitioning scheme (IGMH method). 23Surfaces were visualised with VMD. 30

Geometric analysis of the cation microenvironment
Structures will be analysed in terms of the interaction of the [1-NBA] + and [1propane] + cations with the octahedral array of the six surrounding [BAr F 4 ] −

Faraday Discussions Paper
Faraday Discuss.This journal is © The Royal Society of Chemistry 2023 ) all but one is a short contact, and these are also among the closest to linear (150-170°).These features have been taken as evidence for C-H/F H-bonding [31][32][33] and this could be promoted by the relative acidity of the C-H H-bond donors that are located between two CF 3 substituents on the aryl rings.A more detailed discussion of the role of these inter-anion contacts will be the subject of a separate study.
Comparing the alkane ligands, in [1-propane] + the propane displays four C-H/F contacts below 3.0 Å, although none is below the sum of the van der Waals radii.The NBA in [1-NBA] + has 10 C-H/F contacts, four of which are short contacts.A general trend towards wider C-H/F angles with shorter H/F distances can be seen in the alkane C-H/F data for [1-NBA][BAr F 4 ] and this is similar to the data in Boese, Nangia and Desiraju's early study on uorobenzenes, where C-H/F H-bonding was proposed especially when more acidic C-H bonds are present. 32This is discussed in the context of the present systems aer these C-H/F contacts have been considered in more detail via electronic structure analyses.

Electronic structure analyses
5][36] Such interactions have been ascribed weak H-bonding character that is enhanced by a more acidic C-H bond. 32,33An electron decient cationic Rh(I) centre could enhance such interactions via polarization of both the endo-C-H bonds directly involved in the

Faraday Discussions Paper
Faraday Discuss.This journal is © The Royal Society of Chemistry 2023 s-interaction and, through induction effects, the geminal exo-C-H bonds and beyond. 37To assess this, the molecular graphs in Fig. 5 were recomputed with the {(Cy 2 P(CH 2 ) 2 PCy 2 )Rh} + fragments removed.Computed QTAIM charges conrmed the alkane hydrogen atoms do indeed become more positively charged when the alkane are bound to Rh, however, the same networks of bond paths with virtually unchanged BCP metrics were seen (see Fig. S25 †).This lack of inuence of the Rh fragment implies at best weak H-bonding character in these C-H/F interactions.
9][40][41] Previously, NCI plots have highlighted broad areas of dispersive stabilisation between s-bound alkanes and the adjacent anions. 11,13,14,17,18Here, we have used the Independent Gradient Model (IGM) approach 42,43 to isolate non-covalent interactions between the cation and anion.This method is based on atomic electron densities and uses the descriptor, dG, the difference in the upper limit of the electron density gradient of the non-interacting system vs. the electron density gradient of the interacting system.Here the Hirshfeld partitioning scheme of the molecular electron density proposed by Lu and Chen (and implemented within Multiwfn) is employed to dene the atomic electron densities (IGMH approach). 23he results of these IGMH analyses for the top ion-pairs are displayed in Fig. 6, both as isosurfaces of dG inter and as colour-coded %dG atom scores that highlight the atoms that contribute most signicantly to these inter-ion interactions.These %dG atom scores are the sum of all the interactions of a given atom on the cation with all the atoms of the anion; individual atom-atom interactions can also be quantied as dG atom indices.
The IGMH isosurfaces amplify the features seen in the QTAIM molecular plots.For

Paper Faraday Discussions
This journal is © The Royal Society of Chemistry 2023 Faraday Discuss.
[BAr F 4 ] are associated with the bridging methylene (C 7 , H 71 and H 72 ) whereas H 31 is most prominent in [1-propane][BAr F 4 ].The atoms involved in the exo-C-H/F contacts show little lightening of the blue colour, indicating a minimal contribution.This is conrmed by the %dG atom values: the bridgehead and bridging methylene atoms that make up the -{C 3 H 4 }-"propane" fragment within the NBA ligand account for 77.0% of the total interaction, whereas the exo-hydrogen atoms, H 42 and H 52 , that are involved in H/F short contacts contribute only 4.6% and 3.9% respectively.The dG atom indices, that quantify individual atom-atom contributions, show that the H 42 /F 21 and H 52 /F 11 pairs are ranked 2nd and 4th strongest (1.8% and 1.7%) and are comparable in strength to the H 3 /C 2para and H 6 /C 1para pairs (1.9% and 1.8%, see Fig. S11 †).These data are also consistent with the similar BCP r(r) values computed for the equivalent bond paths.However, H 42 and H 52 engage in relatively few additional atom-atom contacts whereas all atoms in the -{C 3 H 4 }-"propane" moiety have numerous atom-atom contributions that accumulate to make them more signicant in the interfragment interaction.This difference between the directional C-H/F contacts and the more diffuse C-H/C aryl contacts is a recurring theme in this study.For [1-propane][BAr F 4 ] the equivalent -{C 3 H 4 }-moiety accounts for 87.0% of the total interaction, but of this only 3.4% is due to H 11 /F 11 and H 31 /F 22 atom-atom contributions.
Overall, the alkane ligands dominate the cation component to the noncovalent interactions in these ion-pairs: summing %dG atom values from all atoms of the propane ligand accounts for 94.6% of the interaction while for the NBA ligand the equivalent sum gives 99.5%.These values are summarised in Table 2, where comparison with the other ion-pairs studied below is also made.Also included in Table 2 are the sum of the individual dG atom indices.These give an indication of the accumulative strength of the non-covalent interactions ] top ion-pairs where the cations and anions are defined as separate fragments.Sign(l 2 )r-coloured isosurfaces are plotted with dG inter = 0.003 a.u. and atoms are coloured by %dG atom to highlight their relative contributions.The insets show all %dG atom contributions on the cations that are above 1%.

Faraday Discussions Paper
Faraday Discuss.  1 show all four equatorial ionpairs feature more C-H/F bond paths than around the alkanes in the top ionpairs', and that these are greater in number around [1-NBA] + than around [1propane] + .Both alkanes show one C-H/F bond path to each of the Eq-1 and Eq-3 anions that arise from terminal hydrogen atoms within the "propane" moieties (i.e.H 11 and H 31 in [1-propane] + and H 3 and H 6 in [1-NBA] + ), although these have r(r) values below 3.2 × 10 −3 a.u.The H 41 and H 51 exo-hydrogens in [1-NBA] + contribute an additional three bond paths, two of which equate to short contacts (2.60 Å, r(r) = 5.8 × 10 −3 a.u.; 2.65 Å, r(r) = 4.9 × 10 −3 a.u.).All the remaining C-H/F bond paths involve cyclohexyl hydrogens.These equatorial ion-pairs also feature several C-H/C aryl bond paths and C-H/H aryl bond paths, the latter always to ortho ring H atoms.
The molecular graph for the ion-pair with Eq-1 in [1-propane][BAr F 4 ] is shown in Fig. 7(a).This displays a pattern, common to all these ion-pairs, in which one cyclohexyl substituent (Cy inner , here Cy 1 ) sits within a conical pocket dened by three of the anion aryl groups (Ar F eq11 , Ar F eq12 and Ar F eq13 ) while a second cyclohexyl substituent (Cy outer , here Cy 2 ) sits outside this pocket.The former displays a range of C-H/F, C-H/C aryl and C-H/H aryl bond paths while the latter only exhibits C-H/F bond paths.In the Eq-1 ion-pair Cy 1 lies approximately parallel to the aromatic ring of Ar F eq11 and shows ve C-H/F bond paths that are distributed across all three Ar F groups of the pocket.In contrast the C-H/C aryl bond paths are all directed to carbons on Ar F eq11 and the shortest of these involves the C3-axial hydrogen and an ortho ring-carbon (H Cy131 /C eq112 = 3.06 Å; r(r) = 4.2 × 10 −3 a.u.).All these bond paths have low r(r) values below 4.2 × 10 −3 a.u.In contrast, the C3-equatorial hydrogen, H Cy132 , is directed away from Ar F eq11 and engages in two C-H/F bond paths (r(r) = 2.0 × 10 −3 a.u. and 3.4 × 10 −3 a.u.) and one C-H/H aryl bond path to H eq121 (r(r) = 3.5 × 10 −3 a.u.).Cy 2 exhibits two C-H/F bond paths, one of which has a very short contact (H Cy222 / F eq112 = 2.22 Å) and the highest r(r) value of 14.1 × 10 −3 a.u.within this ion-pair and indeed across both systems.This corresponds to the intense red C-H/F short contact noted on the Hirshfeld surface in Fig. 3 beside Eq-1.
These different interactions are reected in the IGMH isosurfaces in Fig. 7(b).The C-H/F interactions from Cy 2 can be clearly seen with the darker green region equating to the short H Cy222 /F eq112 contact noted above.The orange colour of F eq112 in the %dG atom colouring conrms a signicant contribution from that atom: the H Cy222 /F eq112 pair also has the single largest atom-atom dG atom index at 4.1%.More signicant, however, is the green isosurface between Cy 1 and Ar F eq11 .In this case the %dG atom colouring highlights the C3-axial hydrogen, H Cy131 , in red, indicating the largest single atomic contribution (15.0%).The C3equatorial hydrogen contributes 9.5%somewhat lower as it is engaged in more directional C-H/F interactions.More generally, the cation interaction is dominated by the -{C 3 H 6 }-propylene motif centred on the C Cy12 , C Cy13 and C Cy14 carbons and ringed in Fig. 7(c).Together these atoms contribute 62.9% of the interaction from the cation.

Faraday Discussions Paper
Faraday Discuss.This journal is © The Royal Society of Chemistry 2023 Within the other equatorial ion-pairs a similar pattern of bond paths is seen, with BCP r(r) values generally between 1.0 and 6.0 × 10 −3 a.u.Any bond paths with higher BCP r(r) values all correspond to short contacts below the sum of the relevant van der Waals radii: there are four such C-H/F, two C-H/C aryl and one C-H/C aryl bond paths, within which the highest r(r) value is 8.2 × 10 −3 a.u.Of the three remaining equatorial ion-pairs Eq-3 behaves like Eq-1 with one Cy group lying side-on within the conical pocket and the equivalent -{C 3 H 6 }propylene motif centred on the C3 methylene group and contributing 62.0% of the cation interaction.With Eq-2 and Eq-4 the inner cyclohexyl groups (Cy 2 and Cy 4 , respectively) show a more end-on geometry within the pocket (see Fig. 8 for the ion-pair with Eq-2).As a result, both the equatorial and axial hydrogens of the C4 atom (H Cy241 and H Cy242 ) engage in C-H/C aryl bond paths and this is most clearly seen when visualised as %dG atom in the IGMH plot.This methylene group alone accounts for 49.6% of the cation interaction with Eq-2 and this increases to 76.9% when expanded to include the neighbouring CH 2 groups (circled in Fig. 8(c)).
The four equatorial ion-pairs in the

Paper Faraday Discussions
This journal is © The Royal Society of Chemistry 2023 Faraday Discuss.

Fig. 2
Fig. 2 Computed normalised lattice energies (DE lattice /Z, where Z is the number of formula units in the unit cell), alkane incorporation energies (DE incorp ) and molecular alkane binding energies (DE molec ) for [1-NBA][BAr F 4 ] and [1-propane][BAr F 4 ].The solidstate stabilization energy (DE SS ) is the difference between DE incorp and DE molec .All values are based on relative SCF energies in kcal mol −1 computed with periodic-DFT.

Fig. 3
Fig. 3 Hirshfeld surfaces mapped over d norm for [1-NBA] + and [1-propane] + within the octahedral arrays of neighbouring [BAr F 4 ] − anions with the anion labelling system indicated.Views are taken from above the 'top' anion' looking down the B top /Rh/B bottom axis; the 'bottom' anion is obscured by the cation surface.

Fig. 4
Fig. 4 Scatterplots of H/F distance (Å) vs. C-H/F angle (degrees) for the [1-NBA] + and [1-propane] + cations within the octahedral arrays of neighbouring [BAr F 4 ] − anions.The dashed vertical line indicates the sum of the van der Waals radii for H and F. a Data points correspond to two equivalent H/F contacts.
[1-NBA][BArF  4  ] and [1-propane][BArF   4 ] 'octahedra'i.e.where the central cation is common to all six ion-pairs and is treated in turn with each adjacent anion.The range of C-H/F, C-H/C aryl and C-H/H aryl bond paths characterised are summarised in Table1.Individual bond critical point (BCP) metrics are listed in the ESI (see Fig.S5-S16 †).The C-H/F bond paths are most numerous, there being almost twice as many of these as the total of C-H/ C aryl and C-H/H aryl bond paths.Overall the [1-NBA] + cation exhibits signicantly more bond paths to the surrounding anions (88 cf.72 around [1-propane] + ) and this is primarily due to a greater number of C-H/F bond paths (61 cf.46).2.3.1.The top ion-pairs.The propane ligand in [1-propane][BAr F 4 ] exhibits 10 bond paths in total (four C-H/F and six C-H/C aryl ) compared to 14 around the NBA in [1-NBA][BAr F 4 ] (nine C-H/F and ve C-H/C aryl ).In both cases all but two of these involve the top anion and molecular graphs for these ion-pairs are compared in Fig. 5.For the [1-propane][BAr F 4 ] top ion-pair the C 1 -H 11 /F 11 and C 1 -H 11 /F 12 bond paths correspond to two of the four C-H/F contacts noted above (the remaining two contacts are to equatorial anions, see below).In contrast, the equivalent C 3 -H 31 /F 21/22 distances exceed 3.30 Å and no bond path is seen.H 11 and H 31 also exhibit bond paths and short contacts to the paracarbons on the adjacent Ar F groups (C 1para and C 2para ): these have the largest bond critical point BCP electron densities in this ion-pair (C 1 -H 11 /C 1para = 2.52 Å, r(r) = 10.3 × 10 −3 a.u.; C 3 -H 31 /C 2para = 2.67 Å, r(r) = 7.8 × 10 −3 a.u.) and correspond to the red feature beneath the top anion noted previously in the Hirshfeld surfaces.The four C-H/C aryl bond paths involving H 21 and H 22 on the central methylene correspond to long contacts (>3.25 Å; r(r) < 3.2 × 10 −3 a.u.).Within the [1-NBA][BArF  4 ]  ion-pair the -{C 3 H 4 }-moiety comprising the bridgehead (C 3 -H 3 and C 6 -H 6 ) and central bridging methylene (C 7 -H 71 /C 7 -H 72 ) bonds behaves as a "propane" fragment and forms two C-H/F and ve C-H/ C aryl bond paths that are equivalent to similar bond paths in [1-propane][BArF  4 ].Consistent with this, the highest r(r) values within this set are for the C 3 -H 3 / C 2para and C 6 -H 6 /C 1para bond paths (r(r) = 5.9 × 10 −3 and 7.7 × 10 −3 a.u.).The NBA system has additional bond paths from both pairs of exo-C-H bonds.Of these the C 4 -H 42 /F 21 and C 5 -H 52 /F 11 bond paths correspond to the two shortest C-H/F contacts (2.47 Å and 2.53 Å) and have higher r(r) values (7.7 × 10 −3 a.u. and 6.7 × 10 −3 a.u.).The C 1 -H 12 /F 12 and C 2 -H 22 /F 22 bond paths are weaker, with longer contacts (2.77 Å and 2.94 Å) and lower r(r) (3.6 × 10 −3 a.u. and 2.4 × 10 −3 a.u.).The nal additional feature of the NBA system is a weak C 7 -H 72 / F 22 bond path (3.02 Å; r(r) = 2.6 × 10 −3 a.u.).
[1-NBA][BAr F 4 ] distinct disk-like features consistent with exo-C-H/F contacts are seen and these are much more pronounced than in [1-propane][BAr F 4 ].The peripheral alkane C-H/C para (anion) contacts are also common to both structures, but these are more extensive in [1-NBA][BAr F 4 ].This is conrmed by the % dG atom values, where the relative contribution of each atom is represented on a blue-green-red scale (blue: zero; red: the maximum contribution, see also inset for % atomic contributions in the cations).The strongest contributions in[1-NBA]

Fig. 6
Fig. 6 IGMH plots for the [1-NBA][BAr F 4 ] and [1-propane][BAr F 4 ] top ion-pairs where the cations and anions are defined as separate fragments.Sign(l 2 )r-coloured isosurfaces are plotted with dG inter = 0.003 a.u. and atoms are coloured by %dG atom to highlight their relative contributions.The insets show all %dG atom contributions on the cations that are above 1%.

Fig. 7
Fig.7(a) QTAIM molecular graph for the Eq-1 ion-pair in [1-propane][BArF  4 ]  with BCPs in green and showing selected atom labels and BCP electron densities (a.u.× 10 −3 ).Bond paths with r(r) < 0.001 a.u., intramolecular bond paths and ring critical points (RCP) are omitted for clarity.(b) IGMH plot where the cation and anion are defined as separate fragments.Sign(l 2 )r-coloured isosurfaces are plotted with dG inter = 0.003 a.u. and atoms are coloured by %dG atom .(c) %dG atom contributions on the cation that are above 1%, with the propylene motif highlighted (see text for details).
[1-NBA][BAr F 4 ] system exhibit the same general geometric pattern as in [1-propane][BAr F 4 ] with each ion-pair having one inner cyclohexyl group, Cy inner , within the [BAr F 4 ] − conical pocket and one outer

Fig. 8
Fig.8(a) QTAIM molecular graph with Eq-2 in [1-propane][BArF  4 ]  with BCPs in green and showing selected atoms and BCP electron densities (a.u.× 10 −3 ).Bond paths with r(r) < 0.001 a.u., intramolecular bond paths and ring critical points (RCP) are omitted for clarity.(b) IGMH plot where the cation and anion are defined as separate fragments.Sign(l 2 )rcoloured isosurfaces are plotted with dG inter = 0.003 a.u. and atoms are coloured by dG atom .(c) %dG atom contributions on the cation that are above 1%, with the propylene motif highlighted (see text for details).

Table 1
Summary of bond paths around the cations in [1-NBA][BAr F a Values in parentheses indicate the number of C-H/F bond paths involving the cyclohexyl substituents within the conical pocket of anion aryl groups (Cy inner ), the cyclohexyl outside that pocket (Cy outer ) and, where appropriate, the alkane ligand, respectively (see text for details).b One C/F bond path is also present; see ESI for details.c Includes C-H/F bond paths from the -C 2 H 4backbone of the diphosphine ligand.
This journal is © The Royal Society of Chemistry 2023 between the cation and anion and show higher values for [1-NBA][BAr F 4 ] over [1propane][BAr F 4 ] (for the top anion 1.39 vs. 0.95).2.3.2.'Equatorial' ion-pairs.The data in Table

Table 2 %
Contributions for the cations fragments indicated within the six ion-pairs in [1propane][BAr F 4 ] and [1-NBA][BAr F 4 ].dG atom indices (a.u.) are summed for all atom pairs across each ion-pair This journal is © The Royal Society of Chemistry 2023Faraday Discuss.