Methylammonium Tetrel Halide Perovskite Ion Pairs and Their Dimers: The Interplay between the Hydrogen-, Pnictogen- and Tetrel-Bonding Interactions

The structural stability of the extensively studied organic–inorganic hybrid methylammonium tetrel halide perovskite semiconductors, MATtX3 (MA = CH3NH3+; Tt = Ge, Sn, Pb; X = Cl, Br, I), arises as a result of non-covalent interactions between an organic cation (CH3NH3+) and an inorganic anion (TtX3−). However, the basic understanding of the underlying chemical bonding interactions in these systems that link the ionic moieties together in complex configurations is still limited. In this study, ion pair models constituting the organic and inorganic ions were regarded as the repeating units of periodic crystal systems and density functional theory simulations were performed to elucidate the nature of the non-covalent interactions between them. It is demonstrated that not only the charge-assisted N–H···X and C–H···X hydrogen bonds but also the C–N···X pnictogen bonds interact to stabilize the ion pairs and to define their geometries in the gas phase. Similar interactions are also responsible for the formation of crystalline MATtX3 in the low-temperature phase, some of which have been delineated in previous studies. In contrast, the Tt···X tetrel bonding interactions, which are hidden as coordinate bonds in the crystals, play a vital role in holding the inorganic anionic moieties (TtX3−) together. We have demonstrated that each Tt in each [CH3NH3+•TtX3−] ion pair has the capacity to donate three tetrel (σ-hole) bonds to the halides of three nearest neighbor TtX3− units, thus causing the emergence of an infinite array of 3D TtX64− octahedra in the crystalline phase. The TtX44− octahedra are corner-shared to form cage-like inorganic frameworks that host the organic cation, leading to the formation of functional tetrel halide perovskite materials that have outstanding optoelectronic properties in the solid state. We harnessed the results using the quantum theory of atoms in molecules, natural bond orbital, molecular electrostatic surface potential and independent gradient models to validate these conclusions.

In organic-inorganic hybrid perovskites [4][5][6], the organic cation couples with the inorganic anion to form an ion pair [20][21][22][23][24]; applying periodic boundary conditions leads The unit cell (left) and 2 × 2 supercell (right) structure of cubic MAPbI 3 in the hightemperature phase (T > 330 K [95]), showing the orientation of the organic cation along the (a) (111) and (b) (011) directions [25]. (c) Two different polyhedral views of the low-temperature orthorhombic phase of fully relaxed (PBEsol) geometry of MAPbI 3, with the titling angle (in degrees), ∠Pb-I-Pb, along two crystallographic directions (see ESI for details), obtained using the VASP code (version 5.4) [96][97][98][99][100]. Each polyhedron represents the PbI 6 4− octahedron. Three types of bonds For a given halogen derivative in [CH3NH3 + •PbX3 − ], the N-H···X hydrogen bond distances are shorter than the C-H···X hydrogen bonds, in agreement with what was observed in the solid state for CH3NH3PbX3 (X = Cl, Br, I) perovskites [22,23]. This means that the charge-assisted N-H···X hydrogen bonds formed by the ammonium end of MA are stronger than the C-H···X hydrogen bonds formed by the methyl end of MA, and their strength increases with the electronegativity of the halogen (F > Cl > Br > I). For both Conf. 1 and Conf. 2, the hydrogen bond distance decreases as the halogen becomes more electron-withdrawing (see Figure 2a,d,g,j or Figure 2b,e,h,k), a feature found for ion pairs when Tt = Sn and Ge, but not when Tt = Si (see Figures S1-S4 of the Electronic For a given halogen derivative in [CH 3 NH 3 + •PbX 3 − ], the N-H···X hydrogen bond distances are shorter than the C-H···X hydrogen bonds, in agreement with what was observed in the solid state for CH 3 NH 3 PbX 3 (X = Cl, Br, I) perovskites [22,23]. This means that the charge-assisted N-H···X hydrogen bonds formed by the ammonium end of MA are stronger than the C-H···X hydrogen bonds formed by the methyl end of MA, and their strength increases with the electronegativity of the halogen (F > Cl > Br > I). For both Conf. 1 and Conf. 2, the hydrogen bond distance decreases as the halogen becomes more electron-withdrawing (see Figure 2a,d,g,j or Figure 2b,e,h,k), a feature found for ion pairs when Tt = Sn and Ge, but not when Tt = Si (see Figures S1-S4 of the Electronic Supplementary  Information, ESI). In all cases, the hydrogen bonds are non-linear (∠N-H···X or ∠C-H···X < 143 • ), regardless of their type, and (based on the bond lengths) the strength of the Pb-X coordinate bonds decreases passing from F through Br to Cl to I in [CH 3 NH 3 + •PbX 3 − ] (see Figure 2a,d,g,h, for example).
Replacement of Pb by Sn and Ge in [CH 3 NH 3 + •PbX 3 − ] did not appreciably change the nature of the coordination between Tt and X in TtX 3 − , although the hydrogen-bonded environment was affected (see Figures S1a-l and S2a-l of the ESI). When Tt = Si in [CH 3 NH 3 + •TtX 3 − ], Conf. 3 was found to be unstable for X = I and Br (see Figure S3a- Conf. 3 was obtained from the initial geometry shown in Figure S4d for each of the three tetrel halide perovskite ion pair series, [CH 3 NH 3 + •PbX 3 − ] (X = Cl, Br, I). However, when this initial configuration was used for relaxation of the [CH 3 NH 3 + •SiF 3 − ] ion pair, the resulting geometry was no longer an ion pair and the initial configuration was not retained after energy minimization. This was not the case when the methyl and ammonium ends of the organic cation were pointed toward the center of the trifluoride triangular face of SiF 3 − . The energy-minimized geometry in Figure S4d shows that there is a significant interaction between the two ionic moieties when in close proximity, with a breaking of the N-H covalent bond and the formation of two neutral species, SiHF 3 and CH 3 NH 2 , due to complete hydrogen transfer from the ammonium end to the anion. (This is addressed later.) Due to this, we then used the optimized geometry of the [PbF 3 − ···NH 3 CH 3 + ] ion pair ( Figure 2l) as an initial configuration, replacing Pb with Si; the final energy-minimized geometry is shown in Figure S4c. Although this was the case for the system with the anion SiF 3 − , a similar attempt did not stabilize the analogous geometry of the ion pair when X = I and Br.
To help determine the validity of these conclusions we performed an MESP analysis; the results are summarized in Table 1. Analysis of these data suggests that the Si surfaces of [SiX 3 − ···NH 3 CH 3 + ] (X = I and Br) ion pairs are weakly positive but are negative in the remaining eight ion pairs of the same series regardless of the identity of the halogen derivative. The halogen in [SiX 3 − ] has high electron density compared to the surface of the Si atom. There are three equivalent negative σ-holes on Si in each of the two ion pairs for a given halogen except for Conf. 3, and their strength decreases with an increase in the size of the halogen in [SiX 3 − ]. The σ-hole on Si is more electrophilic when the ammonium head faces the triangular X 3 face of [SiX 3 − ] than when the methyl head faces that face. This is reasonable since the -NH 3 group in MA is strongly electrophilic and hence relatively more electron-withdrawing than the methyl head. The first arrangement causes the development of a structure with relatively greater depletion of charge density on the surface of the Si atom along the outermost extensions of the three X-Si bonds. Since the lone pair of the Si lies along the outer extension of the C 3v axis, the V S,min associated with it is increasingly more positive as the halogen becomes more electron-withdrawing, making the entire surface of the Si atom nucleophilic. Table 1. Selected [ωB97X-D/def2-TZVPPD] computed 0.001 a.u. isoelectronic density envelope mapped potential maxima and minima (kcal mol −1 ) on the electrostatic surfaces of ion pairs X 3 Tt − ···MA (X = I, Br, Cl, F; Tt = Pb, Sn, Ge, Si; MA = NH 3 CH 3 + ). The 0.0015 a.u. isoelectronic density envelope mapped potentials are given in parentheses for selected systems.   Figure 3a-c (bottom), respectively. These results indicate that the coordinately bonded Pb atom has three electrophilic σ-holes in these molecular ion pairs, and its surface is entirely electrophilic. An exception was found for [F 3 Pb − ···CH 3 NH 3 + ] when the 0.001 a.u. isoelectronic density envelope was invoked on which to compute the potential (V S,max (F-Pb) = −0.4 kcal mol −1 ), but a larger isoelectronic density envelope of 0.0015 a.u. gave V S,max (F-Pb) = 4.1 kcal mol −1 .

Ion Pairs
Both C and N atoms along the N-C and C-N bond extensions have electrophilic σ-holes in Conf. 1 and Conf. 2, respectively, and their strength in the former was weaker than that in the latter (V S,max (N-C) and V S,max (C-N) 38.2 and 90.4 kcal mol −1 , respectively). Similarly, the lateral portions of the halogen in Conf. 1 or Conf. 2 are equipotential, described by V S,min (X) < 0, although this was not the case in Conf. 3 because of symmetry. For instance, the surface of the halogen that is hydrogen-bonded to the methyl group is less negative than the halogens bonded to the ammonium group in Conf. 3  (X-Pb) > 0) ( Figure 3). They are equivalent in Conf. 1 or 2 but not in Conf. 3. The surface of Pb along the outermost extension of the C3v axis is also positive, represented by a positive local minimum of potential (VS,min (Pb) > 0). For example, values of VS,min (Pb) are 22.2, 9.6 and 18.3 kcal mol −1 for Conf. 1, Conf. 2 and Conf. 3, as shown in Figure 3a-c (bottom), respectively. These results indicate that the coordinately bonded Pb atom has three electrophilic σ-holes in these molecular ion pairs, and its surface is entirely electrophilic. An exception was found for [F3Pb − ···CH3NH3 + ] when the 0.001 a.u. isoelectronic density envelope was invoked on which to compute the potential (VS,max (F-Pb) = −0.4 kcal mol −1 ), but a larger isoelectronic density envelope of 0.0015 a.u. gave VS,max (F-Pb) = 4.1 kcal mol −1 . Both C and N atoms along the N-C and C-N bond extensions have electrophilic σ-holes in Conf. 1 and Conf. 2, respectively, and their strength in the former was weaker than that in the latter (VS,max(N-C) and VS,max(C-N) 38.2 and 90.4 kcal mol −1 , respectively). Similarly, the lateral portions of the halogen in Conf. 1 or Conf. 2 are equipotential, described by VS,min (X) < 0, although this was not the case in Conf. 3 because of symmetry. For instance, the surface of the halogen that is hydrogen-bonded to the methyl group is less negative than the halogens bonded to the ammonium group in Conf  As observed in the case of methylammonium lead halide perovskite ion pairs, the tin surfaces of the three tin-based halide perovskite ion pairs (I3Sn···NH3CH3, I3Sn···H3NCH3 As observed in the case of methylammonium lead halide perovskite ion pairs, the tin surfaces of the three tin-based halide perovskite ion pairs (I 3 Sn···NH 3 CH 3 , I 3 Sn···H 3 NCH 3 and Br 3 Sn···NH 3 CH 3 ) are entirely positive, while the surfaces of the same atom in the remaining nine tin halide perovskite ion pairs are not completely positive; V S,min (Sn) is negative in the first two and negative in the latter nine ion pairs (Table 1). Except for F 3 Sn···CH 3 NH 3 , the three σ-holes on the surface of Sn along the X-Sn bond extensions are electrophilic (V S,max (X-Sn) > 0). They are negative for F 3 Sn···CH 3 NH 3 regardless of the isoelectronic density envelopes used for mapping of the potential (cf. Table 1).
In the case of germanium halide perovskite ion pairs, V S,min (Ge) is found to be negative for all 12 ion pairs, evidence that the lone pair of Ge is active along the outer extension of the C 3v axis. The σ-holes on the surface of Ge in I 3 Ge···CH 3 NH 3 (X = I, Br, Cl) are not all nucleophilic, although they are for the germanium fluoride perovskite ion pairs (F 3 Ge···NH 3 CH , F 3 Ge···CH 3 NH 3 and F 3 Ge···H 3 NCH 3 ). This suggests that the formation of an MAGeF 3 perovskite is unlikely in the solid state, as inferred for the MASiX 3 systems (vide supra). The synthesis of the chloro-and bromo-analogues of MAGeF 3 may be feasible, although the interaction between the ion pairs will be weaker and thus it is expected that the solid-state compounds would be readily degraded.
Among the ion pairs investigated, the lead atom in Br 3 Pb···NH 3 CH 3 provides the strongest σ-holes, suggesting that they are likely to form stronger oligomers, or clusters, when a number of such ion pairs are in close proximity. The phenomenon is likely to persist in the crystalline phase and rationalizes why MAPbBr 3 perovskites were observed to be significantly more stable under ambient conditions than MAPbI 3 perovskite materials [59,101,102]. A number of suggestions have been provided to explain the enhanced stability of MAPbBr 3 compared to MAPbI 3 , including bromide migration, and the reduced activation energy, diffusion coefficient and concentration for halide ions in MAPbBr 3 compared to MAPbI 3 [59].

QTAIM, IGM-δg inter and NBO Analysis of Ion Pairs
The molecular graphs of all twelve ion pairs associated with the three conformers QTAIM misses the bond path topologies of the C-H···X hydrogen bonds in the molecular graphs shown in Figure 4c,f,i,l; this is not unexpected since the intermolecular distances associated with these weakly bonded close contacts (cf. Figure 2c,f,i,l) are longer than the sum of the van der Waals radii of X and H (r vdW (H) = 1.20 Å; r vdW (F) = 1.46 Å; r vdW (Cl) = 1.82 Å; r vdW (Br) = 1.86 Å; r vdW (I) = 2.04 Å) [103]. Very similar results have been reported recently in a study that focused on the physical chemistry of anion-molecule systems driven by tetrel bonds [82]. QTAIM identifies a C···X tetrel bond between the methyl C of MA and an X site on PbX 3 − in Conf. 3 (cf. Figure 4c,f,i,l), instead of C-H···X hydrogen bonds; the emergence of such an interaction is not surprising since the outer electrostatic surface of the C atom along the H-C bond extensions is positive.
The topological charge density characteristics of the intermolecular interactions identified by QTAIM are different from those of the Pb-X coordinate bonds. Most of the former, but not all, are characterized by ∇ 2 ρ b > 0 and H b > 0, whereas the latter are characterized by ∇ 2 ρ b > 0 but H b < 0 and a relatively large ρ b , where ∇ 2 ρ b and H b are the Laplacian of the charge density and the total energy density, respectively. These results suggest that most of the intermolecular (interionic) interactions are of the closed-shell type [104], whereas the coordinate bonds in the ion pairs are of mixed character [105]. For each series with a given halogen derivative, the value of ρ b is the largest for the N-H···X hydrogen bond (see Conf. 1 (Figure 4j)). The three equivalent N-H···X hydrogen bonds in Conf. 1 of each series possess some covalency character since Hb < 0 at the bcps [105] (see Figure 4a,d,g,j).   That QTAIM missed the C-H···X hydrogen bonds in Conf. 3 is confirmed by our IGM-δg inter analysis, presented in Figure 5. The results demonstrate that the IGM-δg inter isosurface volumes (circular or flat), colored bluish green (or green), spread within the area between the interacting atomic basins that are non-covalently engaged with each other, a color code that appears only when there is an attractive interaction between the interacting basins. These isosurfaces are representatives of ρ × sign (λ 2 ) < 0, where λ 2 is the second eigenvalue of the Hessian of the charge density matrix. When ρ × sign (λ 2 ) > 0, one would expect repulsion between interacting basins, and these are generally described by red isosurfaces. Van der Waals interactions correspond to ρ × sign (λ 2 ) ≈ 0 [93,94]. Clearly, as inferred from geometries alone (see Figure 2, for example), Conf. 1 and Conf. 2 are stabilized by N-H···I and C-H···I hydrogen bonds, respectively, with the former relatively stronger than the latter. Similarly, Conf. 3 is stabilized by N-H···I, C···I and C-H···I close contacts. The latter are very weak and appear only when an isovalue < 0.01 a.u. is used. The flat capsule type isosurface volume between the methyl group in MA and I in the inorganic anion in Figure 5d suggests that the attraction between the interacting units is weakly dispersive and includes weak C-H···I hydrogen bonds and the C···I tetrel bond. The presence of the former was confirmed when the interacting atomic basins (H and I) in the ion pair were used in the analysis of IGM-δg inter for which a smaller isovalue of 0.005 a.u. was invoked. This is shown in Figure 5d (right), and the IGM-δg inter isosurface volume between I and H representing C-H···I is more localized and weaker. These conclusions may hold for ion pairs formed with any other halogen derivative. That QTAIM missed the C-H···X hydrogen bonds in Conf. 3 is confirmed by our IGMδg inter analysis, presented in Figure 5. The results demonstrate that the IGM-δg inter isosurface volumes (circular or flat), colored bluish green (or green), spread within the area between the interacting atomic basins that are non-covalently engaged with each other, a color code that appears only when there is an attractive interaction between the interacting basins. These isosurfaces are representatives of ρ × sign (λ2) < 0, where λ2 is the second eigenvalue of the Hessian of the charge density matrix. When ρ × sign (λ2) > 0, one would expect repulsion between interacting basins, and these are generally described by red isosurfaces. Van der Waals interactions correspond to ρ × sign (λ2) ≈ 0 [93,94]. Clearly, as inferred from geometries alone (see Figure 2, for example), Conf. 1 and Conf. 2 are stabilized by N-H···I and C-H···I hydrogen bonds, respectively, with the former relatively stronger than the latter. Similarly, Conf. 3 is stabilized by N-H···I, C···I and C-H···I close contacts. The latter are very weak and appear only when an isovalue < 0.01 a.u. is used. The flat capsule type isosurface volume between the methyl group in MA and I in the inorganic anion in Figure 5d suggests that the attraction between the interacting units is weakly dispersive and includes weak C-H···I hydrogen bonds and the C···I tetrel bond. The presence of the former was confirmed when the interacting atomic basins (H and I) in the ion pair were used in the analysis of IGM-δg inter for which a smaller isovalue of 0.005 a.u. was invoked. This is shown in Figure 5d (right), and the IGM-δg inter isosurface volume between I and H representing C-H···I is more localized and weaker. These conclusions may hold for ion pairs formed with any other halogen derivative.  We carried out a second-order natural bond orbital analysis [106,107] to provide some insight into the nature of the charge transfer (hyperconjugative) interactions that occur between the "filled" (donor) Lewis-type NBOs and "empty" (acceptor) non-Lewis NBOs, but only for the three confirmations of [CH 3 NH 3 + •PbI 3 − ] ( Figure 5). The results suggest that the charge transfer interactions between the acceptor and donor NBOs responsible for either of the two N-H···I hydrogen bonds in Conf. 3 ([PbI 3 − ···H 3 NCH 3 + ]) can be described by n(3)I → σ*(N-H) (E (2) = 6.71 kcal mol −1 ) and σ(Pb-I) → σ*(N-H) (E (2) = 2.0 kcal mol −1 ), whereas the C-H···I hydrogen bond and H-C···I tetrel bond are described by n(3)I → σ*(N-C) (E (2) = 0.17 kcal mol −1 ) and n(3)I → σ*(N-C) (E (2) = 0.16 kcal mol −1 ), respectively, where n(3) refers the third lone pair and E (2) is the second-order stabilization energy [106,108] , indicating that the σ* anti-bonding orbital of the C-N bond of the organic cation has the ability to act as an acceptor of electronic charge density via a charge transfer interaction with the π-type lone pair orbital of iodine in the inorganic anion and corresponds to an I-C···I tetrel bond. Since the geometric topology is very similar for all ion pairs investigated, it may be assumed that similar charge transfer interactions occur between the interacting ions that lead to the formation of the other ion pairs investigated.

Energy Stability of the Ion Pairs
The interaction energies ∆E of all the ion pairs investigated are summarized in Table 2; the total electronic energies obtained from the geometries of the ion pairs and the individual ions in the energy-minimized geometries of the same ion pairs were used. The counterpoise procedure of Boys and Bernardi [109] was followed, as implemented in Gaussian 16 [110] (see Section 4 for details). Among the three conformers examined for each tetrel derivative for a given halide, the hydrogen-bonded ion pair that has the ammonium head pointing towards the X 3 triangular face of the anion was found to be the most stable. For example, the relative stability of the three conformers of [CH 3 NH 3 + •PbI 3 − ] follows the order: Conf. 1 (I 3 Pb···NH 3 CH 3 ) > Conf. 3 (I 3 Pb···H 3 NCH 3 ) > Conf. 2 (I 3 Pb···CH 3 NH 3 ) (Figure 2a-c), which is similar to that of their corresponding interaction energies, summarized in Table 1. The most stable ion pair was F 3 Pb···NH 3 CH 3 , with an interaction energy of −138.54 kcal mol −1 ( Table 2).
The interaction energy listed in Table 2 for each ion pair is not the consequence of a single hydrogen bond. For example, this means that the ∆E(BSSE) of −104.85 kcal mol −1 for I 3 Pb···NH 3 CH 3 arises from three equivalent charge-assisted I···H(N) hydrogen bonds; that of −76.33 kcal mol −1 for I 3 Pb···CH 3 NH 3 is due to three equivalent charge-assisted I···H(C) hydrogen bonds; and that of −97.53 mol −1 for I 3 Pb···H 3 NCH 3 is the result of a pair of equivalent charge-assisted I···H(N) hydrogen bonds and a pair of charge-assisted I···H(C) hydrogen bonds.
For ion pairs formed with a given tetrel derivative and a variable halide, i.e., [X 3 Pb···NH 3 CH 3 ] where X = F, Cl, Br, I, the interaction energy increases (F > Cl > Br > I) as the polarizability of the halide decreases (I > Br > Cl > F). This is intuitively obvious since the electronegativity of the halides plays a significant role in determining the strength of an intermolecular charge-assisted hydrogen bonds.
While the interaction energies of the fluorinated ion pairs (viz. F 3 Pb···H 3 NCH 3 and F 3 Sn···H 3 NCH 3 ) are very large, and are of the ultra-strong type, these ion pairs hinder the formation of their corresponding perovskite structures in the solid state since the volume of the cage formed by the inorganic moiety is unlikely to be large enough to accommodate the organic cation.
On the other hand, the interaction between the two neutral molecules SiHF 3 and CH 3 NH 2 in F 3 HSi···NH 2 CH 3 ( Figure S4d) is not weak. The monomers in the complex are bonded to each other through a Si···N tetrel bond. The formation of this bond is perhaps obvious since the lone pair on the ammonium N is directly engaged with the positive σ-hole on the Si atom in SiHF 3 , which appears along the F-Si bond extension. The uncorrected (and BSSE-corrected) interaction energy of this bond is -22.80 (-22.23) kcal mol −1 , indicative of a reasonably strong non-covalent interaction [81]. The fully relaxed geometries of [CH 3 NH 3 + •PbI 3 − ] 2 (X = F, Cl, Br,I) dimers are shown in Figure 6. The connectivity between the ion pairs is driven by three types of intermolecular interactions: the N-H···X hydrogen bonds, the C-N···X pnictogen bonds and the X-Pb···X tetrel bonds. A tetrel bond occurs when there is evidence of a net attractive interaction between an electrophilic region associated with a covalently or coordinately bonded tetrel atom (or atoms) in a molecular entity and a nucleophilic region in another, or the same, molecular entity [81]. The same underlying definition applies to a pnictogen bond, with the terms "tetrel bond" and "tetrel atom" replaced by the terms "pnictogen bond" and "tetrel atom", respectively. Both the pnictogen and tetrel bonds are linked, with a bent ∠Pb-X···Pb angle between the two ion pairs forming the [CH 3 NH 3 + •PbI 3 − ] 2 dimer. Although these were obtained from gas phase calculations, a very similar geometric feature was found in the low-temperature orthorhombic structure of the CH 3 NH 3 PbX 3 perovskites (X = Cl, Br) (Figure 6e-g); however, the X-Pb bonds in the latter are nearly equivalent as a result of crystal packing forces.
A question that arises is why the [PbI 3 − ] anions are bonded to each other in the crystal, forming an infinite array of [PbX 6 4− ] ∞ octahedra that form cage-like structures to host the organic cations, even though the anions are expected to coulombically repel each other. An immediate answer to this is probably the presence of CH 3 NH 3 + that locally polarizes the potential on the electrostatic surface of the tetrel atom in [PbI 3 − ] when in close proximity, driven by an extensive number of double-charge-assisted N-H···X and C-H···X hydrogen bonds that appear between them. As the surface of the tetrel atom in the ion pair [CH 3 NH 3 + •TtX 3 − ] is highly electrophilic, featuring three positive σ-holes on Tt (Tt = Pb and Sn), it is capable of accepting electrons simultaneously from three halogens of three interacting ion pairs [CH 3 NH 3 + •TtI 3 − ] at equilibrium, forming the [TtX 6 4− ] octahedra in the crystalline phase. QTAIM calculations suggest there is an appreciable transfer of charge between the organic and inorganic ions, facilitating the formation of the ion pair, and this is true in all three conformations investigated for [CH 3 NH 3 + •TtX 3 − ], a feature that also occurs between the ion pairs responsible for the dimers. When the same process of assembly continues with [CH 3 NH 3 + •PbX 3 − ] ion pairs, the formation of CH 3 NH 3 TtX 3 perovskite in the solid state is the likely consequence, a result of interplay between the σ-hole-centered tetrel bonds and other non-covalent interactions. The MESP plots that provide evidence of the formation of X-Tt···X tetrel bonds (dotted lines) between two ion pairs, leading to the formation of [CH 3 NH 3 + •PbX 3 − ] 2 dimers, are shown in Figure 7a-d (left). They show that the positive σ-hole on Pb in an ion pair is in coulombic attraction with the negative halide of the anion with which it interacts.
As shown in Figure 6a-d, the organic cation in the [CH 3 NH 3 + •PbX 3 − ] ion pair on the left connects with another ion pair on the right through N-H···X hydrogen bonds and X-Tt···X tetrel bonds. The former are equivalent in a given ion pair, and become shorter as the size of the halogen decreases from I through to F. Based on the bond distances, it is clear that the hydrogen bonds are stronger in [CH 3 (Figure 6d); this may indicate that the formation of the perovskite CH 3 NH 3 PbF 3 is unlikely in the solid state. The result also suggests that the formation of a different type of ion pair, methylammonium fluoride ([CH 3 NH 3 + •F − ]), is a likely consequence when more than two units of the [CH 3 NH 3 + •PbF 3 − ] ion pair are in close proximity. The geometric arrangement between them would hinder the formation CH 3 NH 3 PbF 3 perovskite in the crystalline phase and might explain why CH 3 NH 3 PbF 3 is unknown in the solid state.  [ωB97X-D/def2-TZVPPD] level fully relaxed gas phase geometries of (a) [CH 3    The X-Tt···X tetrel bonds formed between the ion pairs in the dimers at equilibrium are all less than the sum of the van der Waals (vdW) radii of Tt and X; for instance, the tetrel bond distances of 3.582, 3.351, 3.163 and 2.530 Å in [CH 3 [81].
The geometries of the isolated ion pair, [CH 3 NH 3 + •TtX 3 − ], and its dimer, [CH 3 NH 3 + •TtX 3 − ] 2 , and the intermolecular interactions in them are not identical. A major rearrangement of the monomers occurred on dimer formation. In particular, the organic cation shared between the two inorganic anions rearranged in a manner so as to maximize its non-covalent interactions with the halides of the two inorganic moieties (Figure 6), while the organic cation on the right retains its shape as observed in the isolated ion pair itself (cf. Figure 2). The physical arrangement between the ions in the former (left) part of the dimer resembles the geometry of crystalline tetrel halide perovskites (Figure 6e-g) in the low-temperature orthorhombic phase. The corner-shared [TtX 6 4− ] octahedra in these structures are tilted along the crystallographic a-c-axes. The tilting can be gauged from the ∠Pb-X···Pb angle between a pair of octahedra that are located at corners of a significantly distorted cube.
The tilt angle is smallest in the [CH 3 NH 3 + •PbI It has been shown that to improve the efficiency of perovskite solar cells, it is necessary to tune the degree of octahedral tilting of the halide framework, which affects the optical band gap and the effective mass of the charge carriers [75,114]. The steric effects dominate the magnitude of the tilt in inorganic halides, while hydrogen bonding between the organic cation and the halide frame plays an important role in hybrids, and tuning the degree of hydrogen bonding can be used as an additional control parameter to optimize the photoelectric conversion properties of the perovskites [75]. In the absence of hydrogen bonding, the octahedra in the tetrel halide perovskites do not tilt at all, a view which is in disagreement with our results discussed above and elsewhere [22]. In the case of allinorganic alkali tetrel halide perovskites, the in-phase tilting provides a better arrangement of the larger bromide and iodide anions, which minimizes the electrostatic interactions, improves the bond valence of the A-site cations and enhances the covalency between the A-site metal and Br − or I − ions [115].
The chemical bonding shown in the left portion of the [CH 3 NH 3 + •PbX 3 − ] 2 dimer (Figure 6a-c), which includes N-H···X hydrogen bonds, a C-N···X pnictogen bond and an X-Pb···X tetrel bond, is very similar to that found in crystals of tetrel halide perovskites. A major difference between them is that the gas phase structures do not have hydrogen bonds formed by the methyl group of the organic cation; these can evolve if the [CH 3 NH 3 + •PbX 3 − ] 2 dimer is surrounded with additional ion pairs in all three directions. Likewise, the C-N···X pnictogen bond apparent in the left portion is absent on the right of the [CH 3 NH 3 + •PbX 3 − ] 2 dimer. An extended array of [CH 3 NH 3 + •PbX 3 − ] ion pairs to the right of the dimer will enable the terminal organic cation to rearrange in a manner as on the left so the terminal hydrogen H atom of the ammonium group of the organic cation on the right can engage in the formation of N-H···X bonding interactions with the halogen atoms of a third ion pair; the N site of the same cation can also engage with the tetrel-bonded X site to form a C-N···X pnictogen bond. These intermolecular interactions were confirmed by a QTAIM analysis, see Figure 7 (right). The analysis shows that there is variability in the character of the N-H···X hydrogen bonding interactions in the gas phase geometries, characterized by the topological properties of the charge density.
The QTAIM results suggest that the strength of the C-N···X pnictogen bond increases with an increase in the size of the halogen in [CH 3 NH 3 The relatively longer pnictogen bonds in the [CH 3 NH 3 + •SiX 3 − ] 2 dimers are a result of increasing repulsion between the interacting atomic basins (Si and X) when two ion pairs are in close proximity. The stability of these Si-based dimers is governed by N-H···X hydrogen bonds formed by the organic cation common to both the interacting ion pairs. Details of the topological charge density properties associated with the pnictogen bond are given in Table 3, and the molecular graphs of the dimers with Tt = Sn, Ge and Si are shown in Figures S9a-d, 10a-d and 11a-d of the ESI, respectively. Formation of a dimer is accompanied by an induction of weakly electrophilic σ-holes on the surface of the coordinately bound Si when X = I, Cl and Br but not when X = F, explaining why there is a weak Si···X tetrel bond in the first three dimers but not in [CH 3 NH 3 + •SiF 3 − ] 2 (see Figures S8d and S11d of the ESI). In the case of the latter, the development of electrophilic sites on Si is seen only in the equilibrium structure of the system; induction of the electrophilic sites does not occur during the course of the interaction between the two ion pairs because of coulombic repulsion between entirely negative Si and F sites in the two interacting ion pairs. The feature is shown in Figure S12 of the ESI for [CH 3 NH 3 + •SiX 3 − ] 2 (X = F, Cl, Br, I).  The properties include the charge density (ρ b ), the Laplacian of the charge density (∇ 2 ρ b ) and the total energy density (H b ). b A hydrogen bond (see Figure S11d).
In the case of the [CH 3 NH 3 + •SiF 3 − ] 2 dimer, the Si center in the [CH 3 NH 3 + •SiF 3 − ] ion pair on the left of Figure S8d of the ESI acts as a hydrogen bond acceptor for an ammonium H of the organic cation in the second ion pair on the right when they are in close proximity. The interaction energy of the Si···H hydrogen bond was calculated to be -14.35 kcal mol −1 and this does not involve any secondary interactions between the two ion pairs. Each of the C-N···X (X = F, Cl, Br, I) pnictogen bonds found in the four other [CH 3 NH 3 + •SiX 3 − ] 2 dimers was stronger than the Si···X tetrel bonds found in [CH 3 NH 3 + •SiX 3 − ] 2 (X = I, Br, Cl), which can be inferred from the ρ b values at the bcps shown in Figure S11a-d of the ESI and in Table 3. These are characterized by a positive value of ∇ 2 ρ b and H b , indicative of closed-shell interactions; these are also present in the orthorhombic crystals of MAPbX 3 (see Figure S13 of the ESI for MAPbI 3 and MAPbBr 3 ) and are discussed elsewhere [22,23]. The Si···H hydrogen bond in [CH 3 NH 3 + •SiF 3 − ] 2 has a covalent character since ∇ 2 ρ b > 0 and H b < 0) (see Figure S11d and Table 3 for values).
The uncorrected and BSSE-corrected interaction energies, ∆E and ∆E(BSSE), respectively, of all the [CH 3 NH 3 + •TtX 3 − ] 2 (Tt = Pb, Sn, Ge, Si; X = I, Br, Cl, I) dimers examined are given in Table 4. The interaction energy holding the two ion pairs together in the dimers decreases for a given type of halogen derivative in [CH 3 2 , respectively. The decrease in the interaction energy in the series is also attributed to the weakening of the tetrel bond between the ion pairs (see Figure S7 of  ). It should be noted that the interaction energy is not due solely to the tetrel bond but also due to several hydrogen bonds that are the key forces holding the ion pairs together in the equilibrium geometry of each of the dimers.  2 is clearly a result of strong hydrogen bonding between the ion pairs that play a vital role in stabilizing the tetrel bonds. It is worth mentioning that the trend in the decrease in the interaction energy correlates with the decreasing strength of the σ-hole on Ge in [X 3 Ge···NH 3 CH 3 ] for [CH 3 NH 3 + •GeX 3 − ] 2 series, although this was not so for the [CH 3 NH 3 + •SnX 3 − ] 2 series. We have not performed similar calculations with other theoretical methods, but our MP2 level calculations for the X 3 Pb − ···MA (X = I, Br, Cl, F, and MA = NH 3 CH 3 + ) ion pair series, in conjunction with the def2-TZVPPD and aug-cc-pVTZ basis set, have produced a similar trend (Br > I > Cl > F) for the strength of the σ-hole on Pb along the X-Pb bond extensions (see Tables S1 and S2).

Discussion
The application of the current state-of-the-art theoretical methods has revealed various types of intermolecular (interionic) interactions between organic and inorganic ions, leading to the formation of molecular methylammonium tetrahalide perovskite ion pairs in the gas phase. Of the ion pairs investigated, for any given halogen and tetrel derivative, the conformer with the amine group facing towards the triangular face of the inorganic anion was found to be the most stable (i.e., Conf. 1). This is similar to the geometry observed in the solid state with the organic cation oriented along the (111) direction. The other two conformers (Conf. 2 and 3) known (and extracted) from lead halide perovskite crystal in the cubic phase were found to be the second-and first-order saddle-point structures, respectively.
The conclusion that emerged from combined second-order hyperconjugative charge transfer delocalization and charge-density based IGM-δg inter analyses is that the most stable ion pair for each Tt series [CH 3 NH 3 + •TtX 3 − ] (Tt = Pb, Sn, Ge, Si; X = I, Br, Cl, F), with the organic cation along the (111) direction as observed in the solid state, is not entirely stabilized by N-H···X hydrogen bonds but also partially by a C-N···I pnictogen bond. The latter interaction cannot be readily identifiable by just looking at the geometry of the ion pair yet can be revealed at a very low isovalue around 0.008 a.u. using an IGM-δg inter analysis. This is not unexpected given that anti-bonding σ*(N-C) is an electron-accepting orbital and hence capable of accepting electrons from the lone pair orbital of coordinate I atoms of the inorganic anion when they are in close proximity.
The MESP analysis of the ion pairs led to the identification and subsequent characterization of electrophilic and/or nucleophilic σ-holes on the surface of the tetrel derivative in the ion pairs examined. The strength of the σ-hole was shown to vary with the size of the halogen derivative for any given tetrel atom in the ion pair. There were three such equivalent σ-holes on the electrostatic surface of Tt in each ion pair when the methyl or ammonium end of the organic cation faced towards the triangular face of the inorganic cation. They were inequivalent when the organic cation was in a sitting orientation (viz. (011) orientation in the cubic lattice in the solid state) (Conf. 3). Our calculations suggest that the effect of electrostatic polarization (and/or charge transfer) of the organic cation plays a crucial role in transforming the negative potential to positive on the surface of the tetrel derivative in TtX 3 − in the majority of the ion pairs, and hence three electrophilic σ-holes appeared, especially when Tt = Pb, Sn (except [F 3 Sn···CH 3 NH 3 ]).
Except for the three conformations of F 3 Ge···MA and X 3 Ge···CH 3 NH 3 (X = I, Br, Cl), the surfaces of Ge in the remaining ion pairs of the same series were electrophilic, which suggests that the methyl end is relatively less effective in polarizing the GeX 3 − anion compared to when the ammonium head faces the triangular halide face of the anion ion in the ion pair. Except for a few cases (viz. I 3 Si···NH 3 CH 3 and Br 3 Si···NH 3 CH 3 ), the organic cation was unable to polarize the electron charge density on the surface of Si in [X 3 Si···MA]), thus nucleophilic σ-holes developed on its electrostatic surface.
Gas phase exploration of the dimers of each of the ion pairs investigated has enabled us to reveal why the inorganic anion is capable of forming corner-shared TtX 6 4− octahedra in the solid state, especially when X = I, Br and Cl and Tt = Pb, Sn and Ge. In particular, our results led to the conclusion that the coulombic attraction between the monovalent anions TtX 3 − leading to the TtX 6 4− octahedra in an infinite array in the crystalline material is driven by (X 3 − )Tt···X(TtX 2 − ) tetrel bonds in the presence of the organic cation MA. Three such tetrel bonds can be formed by each Tt center in each TtX 3 − anion when each ion pair is surrounded by three identical ion pairs, forming the corner-shared TtX 6 4− octahedra, in which each octahedron occupies a corner of a regular cube in the high-temperature phase of the system. Although our calculation was limited to the gas phase and dimers of some representative ion pairs, our results have provided evidence of possible tilting of the TtX 6 4− octahedra observed in the low-temperature orthorhombic phase of MATtX 3 (X = I, Br, Cl; Tt = Pb, Sn), driven by different types of intermolecular interactions. In particular, the electronic structures of the dimer models, in combination with the QTAIM results, suggested that C-N···X pnictogen bonds play a critical role, in addition to the MA(X 3 − )Tt···X(TtX 2 − )MA tetrel bonds and N-H···X and C-H···X hydrogen bonds, which interact in determining the octahedral tilting. The formation of (C)N···X pnictogen bonds is an inherent feature of the organic-inorganic tetrel halide perovskites and is present in all the 16 dimers investigated, whether or not the geometry of the dimer mimics a part of the tetrel halide perovskite structure in the crystalline phase. Therefore, it would be misleading to attribute the octahedral tilting feature in methylammonium tetrel halide perovskites to just the N-H···X hydrogen bonds. It should be borne in mind that the intermolecular interactions between the ions leading to the formation of an ion pair are charge-assisted, whereas those responsible for the interaction between ion pairs are not, given that each ion pair is a neutral system and the assembly between the ion pairs in dimers or extended systems is driven by ordinary medium-to-strong non-covalent interactions.
Our investigation has also shed light on why MATtX 3 (X = F, Br, Cl; Tt = Ge, Si) perovskites are not known in the crystalline phase. The failure to synthesize these systems was speculated on when exploring the MESP of the ion pairs of the corresponding systems. The σ-holes on the Tt sites in these systems were weakly positive (or even nucleophilic) and are unlikely to be able to engage appreciably in attractive (coulombic) intermolecular interactions with the halides of a neighboring ion pair. While [CH 3 NH 3 + •TtX 3 − ] 2 (X = Cl, Br, I; Tt = Si) perovskite dimers are stable in the gas phase as result of N-H···X strong hydrogen bonding, the MASiX 3 perovskites are not likely to be formed in the crystalline phase because the Si···X tetrel bonds between the ion pairs are very weak; they are not only secondary interactions but also driven by N-H···X hydrogen bonds formed between the ion pairs when in close proximity.

Materials and Methods
The DFT-ωB97X-D [87] calculations, in combination with the basis set def2-TZVPPD, were performed to fully relax the geometry of 46 ion pairs and their binary complexes; a binary complex refers to the dimer of an ion pair. The density functional uses a version of Grimme's D2 dispersion model [116], as implemented in Gaussian 16 [110]; the basis set is available in the EMSL basis set exchange library [117,118]. Default convergence criteria (tight SCF and ultrafine integration grid) were used. A harmonic frequency calculation was performed in each case. The identified local minima and saddle points are discussed in the Results section. Second-order Møller-Plesset perturbation theory (MP2) [119,120] calculations were also performed on a few systems to demonstrate the reliability of the results obtained with DFT. Although the theoretical methods discussed above were employed on the chemical systems in the gas phase, the results shown in Figure 1 were obtained using periodic boundary calculations, in which the popular PBEsol [121] functional implemented in VASP 5.4 [96][97][98][99][100] was utilized. This was carried out to illustrate the geometric aspects of analogues of the ion pairs responsible for the crystalline phase.
The MESP model [89][90][91][92] generates two physical descriptors that are the local minima and maxima of the potential (V S,min and V S,max , respectively [71,90,[122][123][124][125][126][127][128]) when mapped on an isoelectronic density envelope of a molecule. The positive/negative signs of V S,min and V S,max ([V S,min > 0 and V S,max > 0]/[V S,min < 0 and V S,max < 0]) represent the electrophilicity/nucleophilicity of a particular region on the electrostatic surface of the molecular domain. The magnitude of V S,min or V S,max determines the strength of the potential and correlates linearly with the strength of the intermolecular interactions. The usefulness of the model has been demonstrated in a number of studies. Following a prior recommendation [129,130], the 0.001 a.u. isoelectronic density was used to map the potential for all cases. However, we have also shown that the choice of this envelope can be misleading in systems where the anion contains a more electronegative (and hence less polarizable) tetrel derivative and the less acidic portion of the organic cation faces the anion to form an ion pair. In this case, the use of a higher-value isodensity envelope is necessary [125,[131][132][133][134][135][136]. A σ-hole on the surface of Tt along the outermost extension of the R-Tt covalent/coordinate σ-bond in a molecule was identified when the most local potential associated with it is positive (V S,max > 0); R is the remaining part of the molecule [81].
QTAIM [88] relies on the zero-flux boundary condition and the bond path topology recovers the connectivity between covalently bonded atoms that make up the ion pair or dimers examined. Analysis of the reduced charge density-based isosurfaces was performed using the actual density computed within the framework of IGM-δg inter [93,94]. Software such as AIMAll [137] and Multiwfn [138,139], together with VMD [140], were used for QTAIM, MESP and IGM-δg inter analyses and drawing of MESP and IGM-δg inter graphs.
The uncorrected and Basis Set Superposition Error (BSSE)-corrected interaction energies (∆E and ∆E(BSSE), respectively) were calculated using Equations (1) and (2). E T in Equation (1) and E(BSSE) in Equation (2) are the electronic total energy of respective species and the error in total electronic energy due to the effect of the basis set superposition accounted for by the counterpoise procedure of Boys and Bernardi [109], respectively.
The total electronic energy of the individual ion (or ion pair) was calculated using the energy-minimized geometry of the ion pair (or binary complex), which was used for the calculation of the interaction energy.