Intrinsic dynamic and static natures of ^APn--X⁺--^BPn σ(3c–4e) type interactions (^APn = ^BPn = N, P, As and Sb; X = H, F, Cl, Br and I) in bicyclo[3.3.3] and bicyclo[4.4.4] systems and their behaviour, elucidated with QTAIM dual functional analysis

Abstract

The intrinsic dynamic and static natures of ^APn--X⁺--^BPn (^APn = ^BPn: N, P, As and Sb; X = H, F, Cl, Br and I) in 1a⁺–8c⁺ were elucidated with the quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA). Species 1a⁺–8c⁺ were formed by incorporating X⁺ between ^APn and ^BPn of ^APn(CH₂CH₂CH₂)₃^BPn (1–4) and ^APn(CH₂CH₂CH₂CH₂)₃^BPn (5–8). The relative stabilities between the symmetric and nonsymmetric structures along with their transition states were investigated. Various natures from typical hydrogen bonds (t-HB) to classical covalent bonds were predicted for the ^APn–X/^BPn–X interactions in ^APn--X⁺--^BPn with QTAIM-DFA. The secondary interactions of H–H and X–C were also detected. The vdW to molecular complexes through charge transfer natures were predicted for them. Natural bond orbital analysis clarified that the CT terms were caused by not only n(^APn)→ σ*(X–^BPn) but also σ(^APn–C)→σ*(X–^BPn), σ(^APn–C/^BPn–C)→n_p(X⁺) and n(X)→n_s(Pn⁺). The direction and magnitude of the p-character of n(^APn) were the factors that determined the types of donor–acceptor interactions. Estimating the order of the interaction strengths was attempted. The σ(3c–4e) characters of ^APn--X⁺--^BPn were also examined by analysing the charge distributions on ^APn--X⁺--^BPn. These results would provide fundamentally important insight into designing molecules with high functionality containing X⁺ in symmetric and nonsymmetric structures.

---

Introduction

Chemical bonds and interactions are the most important concepts in chemical and biological sciences. Intrinsic dynamic and static natures of various types of chemical bonds and interactions have been elucidated with the quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA).^1–4 Hydrogen bonds⁵ and chalcogen bonds⁶ are the typical examples of such interactions. What are the natures of the noncovalent interactions, that play a highly important role in the biological systems, such as enzymes? The next target of our project is to elucidate the natures of such interactions, that play a highly important role in the biological species, by applying QTAIM-DFA. It is well known that conformations of the reaction sites, in highly efficient enzymatic reactions, for example, are highly controlled. It is necessary to accumulate fundamental knowledge on the behavior of such interactions under the conditions, to achieve the above goal. The natures of the interactions in cage compounds, called atranes, are investigated, as a first stage of our investigations.

Many cage compounds have been prepared, so far, and they play an important role in chemistry.^7,8 Bicyclo[3.3.3]undecane (I) and bicyclo[4.4.4]tetradecane (II) provide typical types of the frameworks. The bridgehead carbon distances in I and II are considered to be the medium size. The original interactions between the methine groups in I and II would be negligibly small. However, if the methine groups are replaced by A and A′ of the main group atoms or the derivatives in the neutral and charged forms, the A⋯A′ interactions become much stronger. The A⋯A′ interactions make it possible to investigate the natures of the interactions in more detail. The interactions would consist of the repulsive and/or attractive terms. The A⋯A′ interactions change greatly the energy profiles around the interactions relative to the original cases. The changes have profound effects not only on the fine details of the structures but also on the natures of the interactions, which are to be elucidated.

Among a lot of such species,^9–13 the research group of Kawashima demonstrated that 1-hydro-5-carbaphosphatrane and 1-hydro-6-carbaphosphatrane could form an “anti-apicophilic” arrangement.^14–17 Thus, the skeletons of I and II could supply the diverse electronic structures and have unique functionalities, such as superbase,^18–20 catalysis^21–23 and biological activity.^24–27 Alder and coworkers synthesized various derivatives of I and II and related species.^9,11,28–31 1,5-Diazabicyclo[3.3.3]undecane (1)²⁸ and 1,6-diazabicyclo[4.4.4]tetradecane (5)²⁹ are the typical examples. They reported protonation in cages of various sizes.^32–36 Much attention has been paid to prepare the species with the interactions of the so-called proton sponges and to clarify the properties. While 5 could contain a proton in the cage to form a [N--H--N]⁺ bond, N is protonated from outside the cage in 1. However, Alder mentioned that 1 containing a proton in the cage could potentially be kinetically persistent if once formed.³⁴

Recently, molecular designs to form [N⋯F–N]⁺ were theoretically proposed by using the skeletons of 1 and 5,³⁷ in addition to [C--H--C]⁻ using the bicyclo[3.3.3]undecane backborn.^38,39 Much effort has been made experimentally and theoretically to clarify properties of the symmetric (sym) and nonsymmetric (nsym) interactions of the [N--X--N]⁺ (X = F, Cl, Br and I) form.^40–44 However, few reports seem to contain the systematic research on the compressed [N--X--N]⁺ interactions (X = F, Cl, Br and I) in cage structures of the medium size.

Our investigations were focused to elucidate the natures and the properties of the interactions in [N--X--N]⁺ (X = H, F, Cl, Br and I) and the pnictogen derivatives, to achieve our purpose. Chart 1 shows the structures of 1a⁺–8e⁺ and the definition of the symmetric (sym) and nonsymmetric (nsym) shapes. The [^APn--X--^BPn]⁺ interaction will form through the incorporation of X⁺ into ^APn⋯^BPn of 1–8. Herein, we report the results of the investigations on [^APn--X--^BPn]⁺ in 1a⁺–8e⁺, using the original frameworks of I and II, where ^APn = ^BPn = N (1x⁺/5x⁺), P (2x⁺/6x⁺), As (3x⁺/7x⁺) and Sb (4x⁺/8x⁺); X = H (x = a), F (b), Cl (c), Br (d) and I (e) (see Chart 1). The nature of [^APn--X--^BPn]⁺ in 1a⁺–8e⁺ was elucidated after clarifying the structural features and energetic behaviour of the species. The subscribe ⁺ as in [^APn--X--^BPn]⁺ will be often omitted from the interaction as in ^APn--X--^BPn, in this paper. It is the second issue of this study to clarify the stabilities and the properties of 1–8 and 1a⁺–8e⁺ with the larger pnictogen atoms, in place of the CH groups, at the bridgehead positions of I and II. The results of this study will provide useful insight into the ^APn--X--^BPn σ(3c–4e) interactions such as those in 1a⁺–8e⁺.

Chart 1 Species 1a⁺–8e⁺ and the definition of the symmetric (sym) and nonsymmetric (nsym) shapes.

Survey of quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA)

The natures of the ^APn--X--^BPn σ(3c–4e) type interactions are calculated with QTAIM-DFA,^1–4 which we proposed. QTAIM-DFA is formulated based on the QTAIM approach, introduced by Bader.^45,46 QTAIM-DFA has elucidated the (intrinsic) dynamic and static natures of various interactions. The bond critical point (BCP, *) is an important concept in the QTAIM approach.^45,46 The charge density (ρ(r)) at the BCP is described by ρ_b(r_c), as are other QTAIM functions, such as the total electron energy densities H_b(r_c), potential energy densities V_b(r_c) and kinetic energy densities G_b(r_c) at BCPs. A chemical bond or interaction between atoms A and B is denoted by A–B, which corresponds to the bond path (BP) in QTAIM. We will use A-*-B for the BP, in which the asterisk emphasizes the existence of a BCP in A–B. In a nonsymmetric shape, the interaction between ^APn and X is defined to be longer than for ^BPn and X, therefore, they are often described as ^APn--*--X and ^BPn-*-X, respectively, which means ^APn⋯X and ^BPn–X, respectively. The long distances between A and B also denoted by A⋯B, irrespective of above definition.

In the QTAIM approach, H_b(r_c) is plotted versus H_b(r_c) − V_b(r_c)/2 at the BCPs on the interactions in question. The plots are called QTAIM-DFA plots.^1,2 Data from the perturbed structures around the fully optimized structures are plotted in QTAIM-DFA, in addition to the data from the fully optimized ones, in our treatment. Each plot for an interaction is analysed by the polar coordinate (R, θ) parameters with the (θ_p, κ_p) ones. The R and θ parameters are calculated for the data point of the interaction in the fully optimized structure at the origin. While R corresponds to the energy for the interaction in question in the plot, θ, measured from the y-axis, classifies the interaction. The (θ_p, κ_p) parameters are measured based on the plot at the data point for the interaction in the fully optimized structure. θ_p is measured from the y-direction, which corresponds to the tangent line of the plot, and κ_p is the curvature.¹ While θ_p characterizes the interaction, κ_p does not play an important role in the analysis, although it may contain deep meaning.

The (R, θ) parameters are calculated based on the data from the fully optimized structures; therefore, they correspond to the static nature of interactions. The dynamic nature of interactions is proposed based on (θ_p, κ_p), which are obtained by analysing the plots. The plots contain the data from the fully optimized and perturbed structures around fully optimized ones. The perturbed structures are generated using the coordinates (C_i) derived from the diagonal elements of the compliance constants (C_ii) for the internal vibrations, in CIV.⁴⁷ The values were calculated using Compliance 3.0.2 program.^48,49 The dynamic nature based on the perturbed structures with CIV is described as the “intrinsic dynamic nature” since the coordinates are invariant to the choice of coordinate system and thus the perturbed structures are highly reliable.⁴⁷ The standard criteria that distinguish the interactions in question from others are obtained by applying QTAIM-DFA to standard interactions.^1–4 The criteria and QTAIM-DFA with the basic concept of the QTAIM approach are shown by Scheme SA1–SA3, Fig. SA1, SA2, Table SA1 and eqn (SA1)–(SA7) of the ESI.†

Methodological details in calculations

Gaussian 09 program was used for the calculations.⁵⁰ Various basis set systems (BSSs) were examined to determine suitable BSSs. Table 1 shows typical BSSs, examined. The BSS-A employs the basis sets of the Sapporo-TZP with diffuse functions of the 1s1p type (Sapporo-TZP + 1s1p) for all atoms, as implemented from the Sapporo Basis Set Factory.⁵¹ BSS-B employs Sapporo-TZP + 1s1p for ^APn, ^BPn and (X = H, F, Cl, Br and I), together with Sapporo-DZP + 1s1p for C and H. For BSS-C, Sapporo-TZP + 1s1p is applied to ^APn, ^BPn and X, whereas Sapporo-DZP is applied to C and H. The Møller–Plesset second order energy correlation (MP2) level⁵² was employed for the calculations (MP2/BSS-A, MP2/BSS-B and MP2/BSS-C). The optimized structures were confirmed by frequency analysis. The results of frequency analysis were used to derive the coordinates C_i and the constants C_ii. Natural bond orbital (NBO) analysis was performed by the NBO6 program⁵³ under M06-2X/BSS-B//MP2/BSS-B. The relativistic effects were not considered, which would be the next them of the study, especially on the element of 5th period of Sb and I.

Table 1 Basis set systems (BSSs) employed for the calculations

BSSs	C and H	^APn, ^BPn and X (= H, F, Cl, Br and I)
BSS-A	Sapporo-TZP + 1s1p	Sapporo-TZP + 1s1p
BSS-B	Sapporo-DZP + 1s1p	Sapporo-TZP + 1s1p
BSS-C	Sapporo-DZP	Sapporo-TZP + 1s1p

---

Eqn (1) explains the process to generate the perturbed structures with CIV.⁴⁷ The kth perturbed structure in question (S_kw) was generated by adding the coordinates of C_i to the standard orientation of a fully optimized structure (S_o) in the matrix representation. The coefficient g_kw controls the structural difference between S_kw and S_o. g_kw was determined to satisfy r in eqn (2), where r and r_o are the interaction distances in question in perturbed structures and a fully optimized structure, respectively, and a_o is the Bohr radius (0.52918 Å). The C_i values of six digits were used to calculate the S_kw.

S_kw = S_o + g_kw·C_i (1)

r = r_o + wa_o (w = (0), ±0.05 and ±0.1; a_o = 0.52918 Å) (2)

y = c_o + c₁x + c₂x² + c₃x³ (R_c² = square of correlation coefficient) (3)

QTAIM functions were analysed by the AIM2000 ⁵⁴ and AIMAll⁵⁵ programs. Data from five points of w = 0, ±0.05 and ±0.1 in eqn (2) were used in a QTAIM-DFA plot, unless otherwise noted. Each plot was analysed by using the regression curve of the cubic function, shown in eqn (3), where (x, y) = (H_b(r_c) − V_b(r_c)/2, H_b(r_c)), with an R_c² generally greater than 0.99999.⁴

Results and discussion

Exploring a suitable method for calculations

The structures of 2, 5 and 5a⁺, optimized with MP2/BSS-A, MP2/BSS-B and MP2/BSS-C, were carefully compared with the structures, determined by X-ray crystallography.^30,35 Scheme 1 illustrates the structural parameters of θ (θ_A and θ_B), d (d_A and d_B) and the least-square planes σ (σ_A and σ_B) for the three C atoms bonding to Pn (^APn or ^BPn) of 1–4 and 5–8. The d values are defined to be positive and negative, respectively, if Pn are located outside and inside the σ plane with respect to the molecular centre. The “out” and “in” notations⁵⁶ correspond to the positive and negative d values, respectively.

Scheme 1 Selected structural parameters of θ (θ_A and θ_B) and d (d_A and d_B) for 1–4 (n = 1) and 5–8 (n = 2) with the least-square planes σ (σ_A and σ_B) for the three C atoms bonded to Pn (^APn and ^BPn). The d values are defined to be positive and negative, respectively, if Pn are located outside and inside the σ plane with respect to the molecular centre. The “out” and “in” notations correspond to the positive and negative d values, respectively.

Table S1 of the ESI† lists the selected observed and calculated structural parameters for the interaction lengths between ^APn and ^BPn (r(^APn–^BPn)), θ_A, θ_B, d_A and d_B. The differences between the calculated and observed r(^APn–^BPn) values (Δr(^APn–^BPn) = r_calc(^APn–^BPn) − r_obsd(^APn–^BPn)) are also provided in Table S1.† The Δr(^APn–^BPn) values were 0.027 Å, 0.041 Å and 0.022 Å for 2, respectively, when calculated with MP2/BSS-A, MP2/BSS-B and MP2/BSS-C. The values were −0.035 Å, −0.024 Å and −0.054 Å for 5, respectively, if the three methods were applied. The Δr(^APn–^BPn) values were 0.003–0.006 Å for 5a⁺ by the three methods. The coincidence between the calculated and observed values are excellent for 5a⁺.

The MP2/BSS-A method provided excellent quality results; however, the cost performance in the calculations were poorer if the numbers of primitive Gaussian functions necessary for the calculations became larger in 1–8 and 1a⁺–8e⁺. Both MP2/BSS-B and MP2/BSS-C were (very) good with some advantages, whereas disadvantages were for our purpose. We employed MP2/BSS-B since the results with MP2/BSS-B were closer to those with MP2/BSS-A relative to those with MP2/BSS-C. One imaginary frequency was predicted for ^AN--H--^BN in 1a⁺_sym with MP2/BSS-A and MP2/BSS-B, but all real frequencies were predicted with MP2/BSS-C. All real frequencies were predicted for 1a⁺_nsym with MP2/BSS-B. The results led us to select MP2/BSS-B for the calculations. The diffuse functions appeared to play an important role in predicting an imaginary frequency around the bridgehead symmetric interactions in 1a⁺.

Optimizations and structural features of 1–8 and 1a⁺–8c⁺

Optimizations were started assuming the C_3h symmetry for 1a⁺–4e⁺ and the D₃ symmetry for 5a⁺–8e⁺,⁵⁷ although the symmetries were often broken during the optimizations. The minimal structures were optimized with MP2/BSS-B for 1a⁺–1c⁺ (X = H, F and Cl), 2a⁺–4b⁺ (X = H and F), 5a⁺–6e⁺ (X = H, F, Cl, Br and I), 7a⁺–7d⁺ (X = H, F, Cl and Br) and 8a⁺–8c⁺ (X = H, F and Cl). The optimized minimal structures are collectively written as 1a⁺–8c⁺. The optimized minimal structures were only the symmetric shape for 1c⁺, 4b⁺, 5c⁺–5e⁺, and 6e⁺ and only the nonsymmetric shape for 1a⁺, 5a⁺ and 7a⁺, while they were both for 1b⁺, 2a⁺, 2b⁺, 3a⁺, 3b⁺, 4a⁺, 5a⁺, 5b⁺, 6a⁺–6d⁺, 7b⁺–7d⁺ and 8a⁺–8c⁺. The difference between ^ASb--H (1.8823 Å) and ^BSb--H (1.8815 Å) in the optimized structure of 4a⁺ in the C₃ symmetry was negligibly small in magnitude (0.0008 Å); therefore, it could be recognized to have the symmetric shape. The optimizations were not successfully performed for 1e⁺–4e⁺ (X = I). The reason would be the too large size of I⁺ to incorporate into ^APn⋯^BPn to give the stable species of 1e⁺–4e⁺. Moreover, the optimizations were successful for those other than above species, but multiple imaginary frequencies were predicted after the frequency analysis. Efforts were made to search the minimal structures for the species. However, the trials were unsuccessful under the calculation conditions. The structures of the symmetric and nonsymmetric shapes are conveniently denoted by mx⁺_sym and mx⁺_nsym (mx = 1a–8e), respectively.

Two types of the transition states (TSs) were optimized. One type appears in the site exchange process between the topological isomers of the nsym shape, where the longer and shorter distances in ^A/BPn--X--^B/APn are changed. Another type appears between the sym and nsym shapes. While the TSs will be mx⁺_tp:TS if the optimized minimal structures are only mx⁺_nsym, they will be mx⁺_s–ns:TS if both mx⁺_sym and mx⁺_nsym are optimized for a species. No TSs will be optimized for a species if the optimized structure is only mx⁺_sym. As a result, mx⁺_tp:TS will be optimized for 1a⁺, 5a⁺ and 7a⁺, while mx⁺_s–ns:TS will appear for 1b⁺, 2a⁺, 2b⁺, 3a⁺, 3b⁺, 4a⁺, 5b⁺, 6a⁺–6d⁺, 7b⁺–7d⁺ and 8a⁺–8c⁺, under the calculation conditions. Only one imaginary frequency was predicted for each TS. The vibrational motions of TSs corresponding to the imaginary frequencies are shown in Fig. S1 of the ESI,† which confirms the intervention of the TSs between the corresponding two structures. Fig. 1 explains the processes via mx⁺_tp:TS (a) and mx⁺_s–ns:TS (b), exemplified by 5a⁺_tp:TS and 6b⁺_s–ns:TS, respectively. Fig. 1a illustrates the exchange the longer and shorter distances of ^A/BPn--X--^B/APn in 5a⁺_nsym via 5a⁺_tp:TS, where 5a⁺_nsym is (very) close to 5a⁺_tp:TS, since 5a⁺_nsym is almost symmetric. Fig. 1b shows the clear exchange of the geometries around ^AP and ^BP between 6b⁺_sym and 6b⁺_nsym via 6b⁺_s–ns:TS.

Fig. 1 Processes via the TSs of mx⁺_tp:TS (mx = 5a) (a) and of mx⁺_s–ns:TS (mx = 6b) (b).

The selected structural parameters optimized with MP2/BSS-B are listed in Table S2 of the ESI;† this table contains the lengths between ^APn and X (r(^APn–X)) and between ^BPn and X (r(^BPn–X)), θ_A, θ_B, d_A, d_B and the types of ^APn and ^BPn, defined in Scheme 1. The in/in type was predicted for 1a⁺_tp:TS, 1a⁺_nsym, 5a⁺_tp:TS, 5a⁺_nsym, 5b⁺_sym, 5c⁺_sym, 5d⁺_sym, 6a⁺_sym and 7a⁺_tp:TS, whereas the out/in type was for the nonsymmetric shape of 2a⁺, 5b⁺, 6a⁺–6d⁺, 7a⁺–7c⁺, 8a⁺ and 8b⁺. The out/out type was optimized for those other than the one listed above. The optimized structures are not shown in the figures, but they can be found in molecular graphs of 1–8 and 1a⁺–8c⁺, drawn on the optimized structures with MP2/BSS-B.

The Δr_Cov(^A/BPn–X) values for r(^A/BPn–X) of 1a⁺–8c⁺ in the optimized structures were calculated from the sum of the covalent radii⁵⁸ (r_Cov(^A/BPn–X)). The Δr_vdW(^A/BPn–X) values were similarly calculated from the sum of the van der Waals radii^59,60 (r_vdW(^A/BPn–X)). The values are listed in Table S2 of the ESI.† Each r(^A/BPn–X) was smaller than the corresponding r_vdW(^A/BPn–X), respectively. The Δr_Cov(^A/BPn–X) values were plotted versus the species numbers, which was shown in Fig. S2 of the ESI.† Each r(^A/BPn–X) of the symmetric shape was larger than the corresponding r_Cov(^A/BPn–X) (Δr_Cov(^A/BPn–X) > 0), respectively, except for r(^A/BSb–F) of 4b⁺_sym. While r(^APn⋯X) and r(^BPn–H) in each of mx⁺_nsym was longer and shorter than the corresponding r_Cov(Pn–X), respectively, with the exception of r(^BN–H) in 1a⁺_nsym and 5a⁺_nsym and r(^BN–F) in 1b⁺_nsym and 5b⁺_nsym. The overall order of the lengths was r(^APn⋯X)_nsym > r(^APn–X)_sym > r(^BPn–X)_nsym. While r(^APn⋯X)_nsym and r(^APn–X)_sym seem noncovalent with the exception of r(^ASb–F) in 4b⁺_sym, r(^BPn–X)_nsym became covalent with the exception of r(^BN–H) of 1a⁺_nsym and 5a⁺_nsym and r(^BN–F) of 1b⁺_nsym and 5b⁺_nsym.

Stability of symmetric and nonsymmetric shapes and TSs

The energy values (E: E_ES and E_ZP) for mx⁺_sym, mx⁺_nsym, mx⁺_s–ns:TS and/or mx⁺_tp:TS (mx = 1a–8e) are collected in Table S3 of the ESI,† where E_ES and E_ZP are E on the energy surface and those corrected with the zero-point energy, respectively. The relative energies (ΔE: ΔE_ES and ΔE_ZP) are also shown in the table for the species from the global minimum in each group. Each group contains one mx⁺_sym, one mx⁺_nsym, one mx⁺_s–ns:TS and/or one mx⁺_tp:TS. A very good correlation was obtained in the plot of ΔE_ZP versus ΔE_ES (y = −0.73 + 0.93x, R_c² = 0.991), as shown in Fig. S3 of the ESI.† As a result, the energy profiles can be discussed using the ΔE_ES values.

What are the energies of mx⁺_sym, mx⁺_nsym, mx⁺_s–ns:TS and/or mx⁺_tp:TS (mx = 1a–8c), originally? It is instructive to consider the energies of (mx⁺_sym and mx⁺_nsym) from (1–8 and X⁺). The energy differences correspond to those for the formation of (mx⁺_sym and mx⁺_nsym) from (1–8 and X⁺); therefore, they are denoted by ΔE_f (ΔE_f:ES and ΔE_f:ZP) (= E(mx⁺) − E(mx + X⁺)), here. ΔE_f:ES must strongly reflect the binding energies around the Pn–X–Pn moieties. The ΔE_f values are calculated for mx⁺_sym, mx⁺_nsym, mx⁺_s–ns:TS and/or mx⁺_tp:TS, which are collected in Table S3 of the ESI.†

Fig. 2 illustrates the ΔE_f:ES values for the global minimum in each group, mx⁺_sym or mx⁺_nsym. The energy levels and arrows are shown in black if mx⁺_sym are the global minima whereas they are in red if mx⁺_nsym is the global minima. Fig. 2 clearly tells us that 1–4 react with H and F to give the stable ma⁺ and mb⁺ (m = 1–4), respectively. The structures are all mx⁺_sym for X = F, while they are all mx⁺_nsym for X = H, except for 4a⁺_sym. The cavity size of the bicyclo[3.3.3] system would not be enough large to give the stable moieties of [Pn--X--Pn]⁺ with X⁺ (X = Cl and Br). Indeed, Cl⁺ reacts with 1 to give 1c⁺_sym, but it is less stable than the components by 307 kJ mol⁻¹. The high energy barrier inside the cavity in 1c⁺_sym prevents Cl⁺ to escape from the cavity, irrespective of the less stable property of 1c⁺_sym in energy. It is of interest that H⁺ tends to give mx⁺_nsym, whereas F⁺ to mx⁺_sym.

Fig. 2 Energy profiles for the formation of mx⁺_sym and/or mx⁺_nsym of the minimal structures from 1–8 and X⁺ (X = H, F, Cl, Br and I), optimized with MP2/BSS-B. The energy of each combination of 1–8 and X⁺ was taken as the standard for each energy producing mx⁺_sym and/or mx⁺_nsym, which are given in kJ mol⁻¹. The energy levels and the values are described in black for mx⁺_sym, whereas they are given in red for mx⁺_nsym in the diagram.

Species of 5–8 essentially react with all X⁺ (X = H, F, Cl, Br and I) to give stable species. While 6 (Pn = P) reacts with all X⁺ to give stable species, the adduct of 5 (Pn = N) with I⁺ was optimized but it is less stable than the components. The adduct of 7 (Pn = As) with I⁺ was not optimized; moreover, those of 8 (Pn = Sb) with Br⁺ and I⁺ were not, either. The larger cavities in 5–8 of the bicyclo[4.4.4] system seem superior to the narrower ones in 1–4 of the bicyclo[3.3.3] system to give the stable species in the reaction with X⁺, especially for X = Br and I. It is also confirmed that the Pn atom affects on the stability of the adducts.

The ΔE_ES values for (1a⁺_tp:TS from 1a⁺_nsym) and (5a⁺_tp:TS from 5a⁺_nsym) were predicted to be very small values of 0.02 and 0.04 kJ mol⁻¹, respectively, in magnitudes. Therefore, it would be difficult to distinguish the two structures clearly. The optimized structures of mx⁺_tp:TS (mx = 1a and 5a) must be very close to the corresponding mx⁺_nsym (and mx⁺_sym) (see Table S3† for 1a and 5a). However, ΔE_ES for 7a⁺_tp:TS from 7a⁺_nsym were predicted to be very large value of 76.2 kJ mol⁻¹. 7a⁺_tp:TS must be substantially different from 7a⁺_nsym. The optimized structure of 7a⁺_tp:TS seems symmetric. The differences between Pn = As in 7a⁺ and Pn = N in 1a⁺ and 5a⁺ would be responsible for the results.

Fig. 3 shows the relative values of ΔE_ES (mx⁺_sym), ΔE_ES (mx⁺_s–ns:TS) and ΔE_ES (mx⁺_nsym) from the global minimum of each group. As a result, all ΔE_ES (mx⁺) are plotted upside of the corresponding global minimum. The energy levels and the model lines for the reaction coordinates are illustrated in black if ΔE_ES (mx⁺_sym) < ΔE_ES (mx⁺_nsym), whereas they are in red when ΔE_ES (mx⁺_sym) > ΔE_ES (mx⁺_nsym). The colors in Fig. 2 and 3 are related to each other. Only one plot in black appeared for each of 1b⁺, 4a⁺ and 5b⁺; therefore, the plots are shown as Fig. 3a. The plots for (2a⁺ and 2b⁺), (3a⁺ and 3b⁺), (6a⁺–6d⁺), (7b⁺–7d⁺) and (8a⁺–8c⁺) are drawn in Fig. 3b–f, respectively.

Fig. 3 Plots of ΔE_ES (mx⁺_sym), ΔE_ES (mx⁺_s–ns:TS) and ΔE_ES (mx⁺_nsym), optimized with MP2/BSS-B. Plots are for 1b⁺, 4a⁺ and 5b⁺ (a), 2a⁺ and 2b⁺ (b), 3a⁺ and 3b⁺ (c), 6a⁺–6d⁺ (d), 7b⁺–7d⁺ (e) and 8a⁺–8c⁺ (f). The energy levels and the model lines for the reaction coordinates are illustrated in black when ΔE_ES (mx⁺_sym) < ΔE_ES (mx⁺_nsym) and in red if ΔE_ES (mx⁺_sym) > ΔE_ES (mx⁺_nsym).

The ΔE_ES values for (1b⁺_nsym and 1b⁺_s–ns:TS from 1b⁺_sym), (4a⁺_nsym and 4a⁺_s–ns:TS from 4a⁺_sym), and (5b⁺_nsym and 5b⁺_s–ns:TS from 5b⁺_sym) were predicted to be (70.7 and 72.7 kJ mol⁻¹), (10.0 and 14.8 kJ mol⁻¹) and (91.9 and 95.3 kJ mol⁻¹), respectively, in magnitudes. The results are illustrated in Fig. 3a with the ΔE_ES values. Similarly, the calculated ΔE_ES values for mx⁺_nsym and mx⁺_s–ns:TS from mx⁺_sym (mx⁺ = 2a⁺–8c⁺) are drawn in Fig. 3b–f, respectively.

Fig. 2 and 3 are illustrated in such a way that all processes can be visualized for the formation of mx⁺_sym, mx⁺_nsym, mx⁺_s–ns:TS and mx⁺_tp:TS (mx = 1a–8c) from the components, 1–8 and X⁺ (X = H, F, Cl, Br and I), by overlapping the energy levels of the common species in the two figures. While the red lines should connect to the red ones, the black lines to the black ones, in this process. The ΔE_f:ES values for all species are obtained by adding the values along the processes via the global minimum in a group. (The memory widths on the vertical axes are different.) The ΔE_ES values are given in the figures, which are also found in Table S3 of the ESI.†

The ΔE_ES values for both mx⁺_nsym and mx⁺_s–ns:TS increase in the order of Pn = P (in 6c⁺) < As (in 7c⁺) < Sb (in 8c⁺). However, the orders for others are unclear, since some data are lacking due to unsuccessful optimizations and some disorders seem to occur. In the case of 6b⁺–6d⁺ and 7b⁺–7d⁺, the ΔE_ES values for mx⁺_nsym increase in the order of X = F < Cl < Br whereas the values for mx⁺_s–ns:TS decreased in the order of X = F > Cl > Br. The opposite trends between mx⁺_nsym and mx⁺_s–ns:TS are of interest. The similar trend was detected in 8b⁺_nsym and 8c⁺_nsym, of which ΔE_ES values increased in the order of X = F < Cl.

The ^APn--X--^BPn and/or ^APn⋯X–^BPn interactions plays an important role in stabilising the symmetric and nonsymmetric shapes of 1a⁺–8c⁺, respectively, together with the TSs, although the contributions seem complex. The secondary interactions also contribute to control the fine details of the structures.

Molecular graphs with contour plots for 1–8 and 1a⁺–8c⁺

Fig. 4 shows the molecular graphs with contour plots drawn on the selected minimal structures of 1a⁺–8c⁺, optimized with MP2/BSS-B. Those of others, containing those for 1–8, are shown in Fig. S4–S6 of the ESI.† The BCPs corresponding to the ^APn⋯^BPn interactions were detected for ^AN-*-^BN in 1 and 5 and ^AP-*-^BP in 6, although such BCPs were not detected for 2–4, 7 and 8, due to the clear out/out structures with the large ^APn⋯^BPn distances for 7 and 8 (see Table S2 of the ESI†). The BCPs were detected for ^APn--X--^BPn in 1a⁺–8c⁺, along with those for the secondary H-*-H and X-*-C interactions in 2–8 and 1b⁺–8c⁺, together with that for H-*-^BSb in 8a⁺_sym.

Fig. 4 Molecular graphs with contour plots for 1b⁺_nsym, 6a⁺_sym, 6a⁺_nsym, 6b⁺_nsym, 6c⁺_sym, 6c⁺_nsym, 6d⁺_sym, 6d⁺_nsym, 6e⁺_sym, 7b⁺_sym, 8b⁺_sym and 8b⁺_nsym ((a)–(l), respectively), calculated with MP2/BSS-B. The BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, CCPs (cage critical points) by green dots and BPs by pink lines. Carbon, hydrogen, nitrogen, phosphorus, arsenic, antimony, fluorine, chlorine, bromine and iodine atoms are shown in black, grey, blue, orange, light purple, purple, light green, green, dark red and dark purple, respectively. Contour plots are drawn on the planes containing at least ^APn, ^BPn and X.

The BPs in ^APn--*--^BPn and ^APn-*-X appeared straight, as shown in Fig. 4 and S4–S6 of the ESI.† What is the linearity of the BPs for ^APn--*--^BPn and Pn-*-X? The lengths of BPs (r_BP) and the straight-line distances (R_SL) corresponding to r_BP of ^APn--*--^BPn in 1, 5 and 6 and Pn-*-X in 1a⁺–8c⁺ are listed in Table S4 of the ESI,† along with the differences between r_BP and R_SL, Δr_BP (= r_BP −R_SL). The ^APn--*--^BPn and Pn-*-X interactions in 1, 5, 6 and 1a⁺–8c⁺ were recognized to be straight, since the Δr_BP values were less than 0.01 Å. However, the BPs for H--*--H and H--*--^BSb in 8a⁺_nsym and some X--*--C were bent compared with those of ^APn--*--^BPn and Pn-*-X, of which Δr_BP values are provided in Table S5 of the ESI.† The Δr_BP values were 0.02–0.23 Å for H--*--H, H--*--^BSb in 8a⁺_nsym, Cl--*--C in 6c⁺_nsym, Br--*--C in 6d⁺_sym and 6d⁺_nsym, I--*--C in 6e⁺_sym, and F--*--C in 7b⁺_nsym, 8b⁺_sym and 8b⁺_nsym. For other X--*--C, the Δr_BP values were less than 0.20 Å; thus, they were approximately straight.

The values of ρ_b(r_c), V_b(r_c), G_b(r_c) and H_b(r_c), which were called the QTAIM functions here, were calculated at the BCPs that appeared in molecular graphs, as shown in Fig. 4, for example. Table 2 lists the values of the QTAIM functions of ρ_b(r_c), H_b(r_c) − V_b(r_c)/2 and H_b(r_c), evaluated with MP2/BSS-B.

Table 2 QTAIM functions and QTAIM-DFA parameters of A/BPn-*-X, ^APn--*--X and/or ^BPn-*-X for 1a⁺–8c⁺, together with the C_ii values and the predicted natures, evaluated with MP2/BSS-B^a,b

mx^+sym/nsym (symmetry) (^A/BPn-*-X)	ρb(rc) (eao^−3)	c∇^2ρb(rc)^c (au)	Hb(rc) (au)	R^d (au)	θ^e (°)	Cii^f (Å mdyn^−1)	θp^g (°)	κp^h (au^−1)	Predicted nature
a See Table 1 for BSS-B.b Data are given at the BCPs.c c∇^2ρb(rc) = Hb(rc) − Vb(rc)/2, where c = ħ^2/8m.d R = (x^2 + y^2)^1/2, where (x, y) = (Hb(rc) − Vb(rc)/2, Hb(rc)).e θ = 90° − tan^−1(y/x).f Cij = ∂^2E/∂fi∂fj, where i and j refer to internal coordinates, and fi and fj, corresponding to i and j, respectively, are the external force components acting on the system.g θp = 90° − tan^−1(dy/dx).h κp = |d^2y/dx^2|/[1 + (dy/dx)^2]^3/2.i Two sets of very close data are predicted from ^ASb-*-H and ^BSb-*-H. One set of data are shown here.j Data from w = ±0.05, ±0.025 and 0 are used for the plot because a poor correlation is obtained for the data from w = ±0.1, ±0.05 and 0.0.
1a^+nsym (C1) (^AN--*--H)	0.1815	−0.0445	−0.1805	0.1859	193.9	19.19	205.2	0.1	SS/Cov-s
(^BN-*-H)	0.2364	−0.1009	−0.2959	0.3126	198.8	12.80	206.6	0.0	SS/Cov-s
1b^+sym (C3h) (^A/BN-*-F)	0.1518	0.0626	−0.0590	0.0860	133.3	0.155	155.0	7.4	r-CS/CT-MC
1b^+nsym (C3) (^AN--*--F)	0.0860	0.0520	−0.0155	0.0543	106.6	0.506	146.5	20.1	r-CS/t-HBwc
(^BN-*-F)	0.3334	−0.0212	−0.3316	0.3323	183.7	0.302	199.0	0.1	SS/Cov-s
1c^+sym (C3h) (^AN-*-Cl)	0.1790	0.0087	−0.1148	0.1151	175.6	0.547	196.2	1.0	r-CS/CT-TBP
2a^+sym (C3h) (^A/BP-*-H)	0.1088	−0.0180	−0.0709	0.0732	194.2	0.613	198.5	2.2	SS/Cov-w
2a^+nsym (C3) (^AP--*--H)	0.0464	0.0098	−0.0108	0.0146	137.7	0.860	182.7	28.9	r-CS/CT-TBP
(^BP-*-H)	0.1980	−0.0091	−0.2209	0.2211	182.4	0.238	128.3	1.7	SS/Cov-s
2b^+sym (C3h) (^A/BP-*-F)	0.0968	0.0237	−0.0645	0.0687	159.8	0.247	116.8	39.8	r-CS/t-HBwc
2b^+nsym (C3) (^AP--*--F)	0.0455	0.0230	−0.0039	0.0234	99.7	0.480	157.6	79.2	r-CS/CT-MC
(^BP-*-F)	0.1987	0.1707	−0.1335	0.2167	128.0	0.125	106.4	0.1	r-CS/t-HBwc
3a^+sym (C3h) (^A/BAs-*-H)	0.0924	−0.0047	−0.0487	0.0489	185.6	0.800	187.1	4.0	SS/Cov-w
3a^+nsym (C3) (^AAs--*--H)	0.0438	0.0089	−0.0101	0.0134	138.7	0.981	180.8	21.5	r-CS/CT-TBP
(^BAs-*-H)	0.1800	−0.0177	−0.1457	0.1468	186.9	0.279	168.6	2.4	SS/Cov-w
3b^+sym (C3h) (^A/BAs-*-F)	0.0918	0.0404	−0.0321	0.0516	128.5	0.267	123.2	0.7	r-CS/t-HBwc
3b^+nsym (C3) (^AAs--*--F)	0.0431	0.0213	−0.0036	0.0216	99.6	0.584	140.7	22.0	r-CS/t-HBwc
(^BAs-*-F)	0.1848	0.1156	−0.1202	0.1668	136.1	0.154	128.8	1.4	r-CS/t-HBwc
4a^+sym (C3) (^A/BSb-*-H)^i	0.0764	0.0084	−0.0305	0.0316	164.5	0.646	148.9	4.9	r-CS/t-HBwc
4a^+nsym (C3) (^ASb--*--H)	0.0409	0.0076	−0.0097	0.0123	142.0	1.174	169.6	15.4	r-CS/CT-MC
(^BSb-*-H)	0.1308	0.0123	−0.0744	0.0754	170.6	0.435	158.1	1.4	r-CS/CT-MC
4b^+sym (C3h) (^A/BSb-*-F)^j	0.0856	0.0654	−0.0111	0.0663	99.7	0.282	109.3	164.4	r-CS/t-HBwc
5a^+nsym (C3) (^AN--*--H)	0.1491	−0.0306	−0.1291	0.1327	193.3	10.36	204.6	0.2	SS/Cov-w
(^BN-*-H)	0.2108	−0.0895	−0.2519	0.2673	199.6	6.953	206.1	0.0	SS/Cov-s
5b^+sym (D3) (^A/BN-*-F)	0.1116	0.0487	−0.0264	0.0554	118.5	0.305	148.0	9.8	r-CS/t-HBwc
5b^+nsym (C3) (^AN--*--F)	0.0542	0.0330	−0.0015	0.0331	92.5	1.156	128.4	30.8	r-CS/t-HBwc
(^BN-*-F)	0.3091	−0.0186	−0.2833	0.2839	183.8	0.284	197.7	0.0	SS/Cov-s
5c^+sym (C3) (^A/BN-*-Cl)	0.1722	0.0049	−0.1042	0.1043	177.3	0.424	194.3	1.6	r-CS/CT-TBP
5d^+sym (C2) (^AN-*-Br)	0.1508	0.0072	−0.0963	0.0966	175.7	0.405	186.5	2.2	r-CS/CT-TBP
5e^+sym (D3) (^AN-*-I)	0.1262	0.0225	−0.0729	0.0763	162.8	0.403	149.6	14.8	r-CS/t-HBwc
6a^+sym (C2) (^A/BP-*-H)	0.1242	−0.0228	−0.0736	0.0771	197.2	2.470	206.0	0.1	SS/Cov-w
6a^+nsym (C2) (^AP--*--H)	0.0161	0.0048	−0.0003	0.0048	93.9	3.611	134.2	260.5	r-CS/t-HBwc
(^BP-*-H)	0.1857	−0.0217	−0.2041	0.2053	186.1	0.269	129.7	2.6	SS/Cov-s
6b^+sym (D3) (^A/BP-*-F)	0.0738	0.0095	−0.0352	0.0365	164.8	0.594	184.0	10.6	r-CS/CT-TBP
6b^+nsym (C3) (^AP--*--F)	0.0212	0.0116	0.0023	0.0118	78.9	1.017	91.6	56.9	p-CS/t-HBnc
(^BP-*-F)	0.1843	0.1406	−0.1248	0.1880	131.6	0.149	105.3	0.3	r-CS/t-HBwc
6c^+sym (C1) (^A/BP-*-Cl)	0.0895	−0.0016	−0.0488	0.0488	181.8	0.331	192.8	0.1	SS/Cov-w
6c^+nsym (C1) (^AP--*--Cl)	0.0433	0.0123	−0.0074	0.0144	120.8	0.623	170.1	56.4	r-CS/CT-MC
(^BP-*-Cl)	0.1670	0.0120	−0.1657	0.1661	175.9	0.201	121.4	3.4	r-CS/t-HBwc
6d^+sym (C3) (^A/BP-*-Br)	0.0899	−0.0014	−0.0445	0.0445	181.8	0.326	193.8	1.9	SS/Cov-w
6d^+nsym (C3) (^AP--*--Br)	0.0498	0.0112	−0.0109	0.0157	134.3	0.590	181.3	32.2	r-CS/CT-TBP
(^BP-*-Br)	0.1572	−0.0297	−0.1594	0.1621	190.6	0.259	155.2	20.3	SS/Cov-s
6e^+sym (D3) (^A/BP-*-I)	0.0851	−0.0014	−0.0397	0.0397	182.1	0.586	196.9	1.0	SS/Cov-w
7a^+nsym (C1) (^AAs--*--H)	0.0222	0.0054	−0.0014	0.0056	104.8	3.013	153.3	199.5	r-CS/CT-MC
(^BAs-*-H)	0.1717	−0.0211	−0.1337	0.1354	189.0	0.301	168.3	2.9	SS/Cov-w
7b^+sym (D3) (^A/BAs-*-F)	0.0689	0.0226	−0.0186	0.0293	129.4	0.622	133.6	17.3	r-CS/t-HBwc
7b^+nsym (C3) (^AAs--*--F)	0.0200	0.0104	0.0020	0.0106	79.2	1.190	91.0	69.7	p-CS/t-HBnc
(^BAs-*-F)	0.1734	0.0999	−0.1094	0.1481	137.6	0.174	130.0	1.3	r-CS/t-HBwc
7c^+sym (C2) (^A/BAs-*-Cl)	0.0757	0.0083	−0.0301	0.0312	164.6	0.362	168.4	11.2	r-CS/CT-MC
7c^+nsym (C1) (^AAs--*--Cl)	0.0394	0.0113	−0.0061	0.0129	118.3	0.712	166.0	57.7	r-CS/CT-MC
(^BAs-*-Cl)	0.1471	0.0113	−0.0972	0.0979	173.4	0.245	157.5	2.7	r-CS/CT-MC
7d^+sym (C2) (^A/BAs-*-Br)	0.0754	0.0057	−0.0296	0.0302	169.2	0.358	176.5	6.9	r-CS/CT-MC
7d^+nsym (C3) (^AAs--*--Br)	0.0464	0.0101	−0.0098	0.0140	134.2	0.743	198.7	46.2	r-CS/CT-TBP
(^BAs-*-Br)	0.1256	−0.0038	−0.0759	0.0760	182.9	0.504	178.4	14.9	SS/Cov-w
8a^+sym (C3) (^A/BSb-*-H)	0.0689	0.0017	−0.0267	0.0267	176.3	1.108	164.5	25.4	r-CS/CT-MC
8a^+nsym (C1) (^ASb--*--H)	0.0220	0.0047	−0.0019	0.0051	111.8	2.545	162.3	136.5	r-CS/CT-MC
(^BSb-*-H)	0.1320	0.0098	−0.0754	0.0760	172.6	0.357	153.0	1.4	r-CS/CT-MC
8b^+sym (C3) (^A/BSb-*-F)	0.0632	0.0345	−0.0072	0.0353	101.8	0.596	99.8	5.0	r-CS/t-HBwc
8b^+nsym (C3) (^ASb--*--F)	0.0208	0.0096	0.0010	0.0096	84.3	1.821	102.6	66.6	p-CS/t-HBnc
(^BSb-*-F)	0.1353	0.1004	−0.0465	0.1106	114.8	0.200	121.6	1.5	r-CS/t-HBwc
8c^+sym (D3) (^A/BSb-*-Cl)	0.0640	0.0162	−0.0183	0.0244	138.5	0.418	124.0	7.8	r-CS/t-HBwc
8c^+nsym (C3) (^ASb--*--Cl)	0.0336	0.0094	−0.0051	0.0107	118.7	0.908	154.0	4.6	r-CS/CT-MC
(^BSb-*-Cl)	0.1130	0.0314	−0.0488	0.0580	147.3	0.272	134.6	2.0	r-CS/t-HBwc

---

Intrinsic dynamic and static natures of ^APn-*-X and ^BPn-*-X in 1a⁺–8c⁺, elucidated with QTAIM-DFA

The natures of the interactions in question in the optimized minimal species are mainly discussed in this section. Fig. 5 shows the QTAIM-DFA plots of H_b(r_c) versus H_b(r_c) − V_b(r_c)/2, although the data are limited to those for ^A/BPn-*-X in mx⁺_sym (see Table 2 for the data), while those for ^APn--*--X and ^BPn-*-X in mx⁺_nsym are shown in Fig. S7 of the ESI.† Data from the perturbed structures generated with CIV are also employed for the plot. The 17 (data) points appeared in the SS region. While the plots for others than those above appeared in the r-CS region, except for three, which appeared in the p-CS region (see Fig. 5 and S7 of the ESI† for the interactions and the species). The results imply that ^APn-*-X, ^APn--*--X and ^BPn-*-X show various natures of the wide ranges over the p-CS to SS regions.

Fig. 5 QTAIM-DFA plots of ^A/BPn-*-X in mx⁺_sym for 1b⁺, 1c⁺, 2a⁺, 2b⁺, 3a⁺, 3b⁺, 4a⁺, 4b⁺, 5b⁺–5e⁺, 6a⁺–6e⁺, 7b⁺–7d⁺ and 8a⁺–8c⁺. See Table 2 for the values of H_b(r_c) and H_b(r_c) − V_b(r_c)/2.

The plots were analysed according to eqn (SA3)–(SA6) of the ESI.† (See the appendix in the ESI for eqn (SA3)–(SA6) and Fig. SA1†). The θ values of 180° < θ < 206.6° correspond to the shared-shell (SS) interactions of H_b(r_c) − V_b(r_c)/2 < 0, and the values less than 180° (θ < 180°) do to the closed-shell (CS) interactions of H_b(r_c) − V_b(r_c)/2 > 0. The CS interactions are subdivided into regular CS (r-CS) interactions by 90° < θ < 180° (H_b(r_c) < 0 and H_b(r_c) − V_b(r_c)/2 > 0) and pure CS (p-CS) interactions by 45° < θ < 90° (H_b(r_c) > 0 and H_b(r_c) − V_b(r_c)/2 > 0). The classical covalent interactions (Cov) of SS are subdivided by R into strong (Cov-s: R > 0.15 au) and weak (Cov-w: R < 0.15 au).

While the typical HB (t-HB) nature with covalency (t-HB_wc), molecular complex formation through CT (CT-MC), and TBP adduct formation through CT (CT-TBP) are contained in r-CS, the t-HB nature with no covalency (t-HB_nc) and van der Waals interactions (vdW) belong to p-CS. The nature of each Pn-*-X interaction in 1a⁺–8c⁺ was classified and characterized with the QTAIM-DFA parameters of (R, θ, θ_p), employing the criteria as guidance. Table 2 lists the analysed QTAIM-DFA parameters of (R, θ) and (θ_p, κ_p), along with the C_ii values, corresponding to generate the perturbed structures, evaluated with MP2/BSS-B. Table 2 also provides the intrinsic dynamic and static natures predicted for the interactions.

After careful examination of (θ, θ_p) for standard interactions, the borderlines between the CS interactions were determined as follows: The (θ, θ_p) values of (75°, 90°), (90°, 125°), (115°, 150°), (150°, 180°) and (180°, 190°) correspond to the borderlines between vdW/t-HB_nc, t-HB_nc/t-HB_wc, t-HB_wc/CT-MC, CT-MC/CT-TBP and CT-TBP/Cov-w, respectively. The parameters in bold are superior to those in plain font in the prediction of the natures, while those in plain font are provided as tentative ones.

While θ classifies the interactions, θ_p characterizes them. The (R, θ, θ_p) values are (0.162–0.332 au, 182.4–199.6°, 128.3–206.6°), for ^AN--*--H and ^BN-*-H (1a⁺_nsym), ^BN-*-F (1b⁺_nsym and 5b⁺_nsym), ^BP-*-H (2a⁺_nsym and 6a⁺_nsym), ^BN-*-H (5a⁺_nsym) and ^BP-*-Br (6d⁺_nsym); therefore, the interactions are predicted to have the SS/Cov-s nature. Similarly, the (R, θ, θ_p) values were (0.040–0.147 au, 181.8–197.2°, 168.3–206.0°) for ^A/BP-*-H (2a⁺_sym and 6a⁺_nsym), ^A/BAs-*-H (3a⁺_sym), ^BAs-*-H (3a⁺_nsym and 7a⁺_nsym), ^AN--*--H (5a⁺_nsym), ^A/BP-*-Cl (6c⁺_sym), ^A/BP-*-Br (6d⁺_sym), ^A/BP-*-I (6e⁺_sym) and ^BAs-*-Br (7d⁺_nsym); thus, these were predicted to be the SS/Cov-w nature. In the case of CS interactions, the r-CS/CT-TBP nature was predicted for ^A/BN-*-Cl (1c⁺_sym and 5c⁺_sym), ^AP--*--H (2a⁺_nsym), ^AAs--*--H (3a⁺_nsym), ^A/BN-*-Br (5d⁺_sym), ^A/BP-*-F (6b⁺_sym), ^AP--*--Br (6d⁺_nsym) and ^AAs--*--Br (7d⁺_nsym), of which the (θ, θ_p) values were (134.2–177.3°, 180.8–198.7°). The r-CS/CT-MC nature was predicted for ^A/BN-*-F (1b⁺_sym), ^AP--*--F (2b⁺_nsym), ^ASb--*--H/^BSb-*-H (4a⁺_nsym and 8a⁺_nsym), ^AP--*--Cl (6c⁺_nsym), ^AAs--*--H (7a⁺_nsym), ^A/BAs-*-Cl (7c⁺_sym), ^AAs--*--Cl/^BAs-*-Cl (7c⁺_nsym), ^A/BAs-*-Br (7d⁺_sym), ^A/BSb-*-H (8a⁺_sym) and ^ASb--*--Cl (8c⁺_nsym), due to the (θ, θ_p) values of (99.7–176.3°, 153.0–176.5°).

The r-CS/t-HB_wc nature was predicted for ^AN--*--F (1b⁺_nsym and 5b⁺_nsym), ^A/BP-*-F (2b⁺_sym), ^BP-*-F (2b⁺_nsym and 6b⁺_nsym), ^A/BAs-*-F (3b⁺_sym and 7b⁺_sym), ^AAs--*--F (3b⁺_nsym), ^BAs-*-F (3b⁺_nsym and 7b⁺_nsym), ^A/BSb-*-H (4a⁺_sym), ^A/BSb-*-F (4b⁺_sym and 8b⁺_sym), ^A/BN-*-F (5b⁺_sym), ^A/BN-*-I (5b⁺_sym), ^AP--*--H (6a⁺_nsym), ^BP-*-Cl (6c⁺_nsym), ^BSb-*-F (8b⁺_nsym), ^A/BSb-*-Cl (8c⁺_sym) and ^BSb-*-Cl (8c⁺_nsym), of which (θ, θ_p) values were (92.5–175.9°, 99.8–149.6°). The (θ, θ_p) values for ^AP--*--F (6b⁺_nsym), ^AAs--*--F (7b⁺_nsym) and ^ASb--*--F (8b⁺_nsym) were (78.9–84.3°, 91.0–102.6°); thus, they were predicted to be the p-CS/t-HB_nc nature.

The differences between θ and θ_p (Δθ_p = θ_p − θ) are positive for usual interactions, which show normal behaviour. However, the Δθ_p values are sometimes negative, which show inverse behaviour. The inverse versus normal behaviour of interactions in relation to the Δθ_p values, which we proposed recently, would correspond to some basic properties of interactions.⁶¹ The inverse behaviour is often observed when the interaction occurs between the atoms, of which atomic numbers are (very) large. Such cases can be easily found in Table 2. The abnormal character of G_b(r_c) at the BCP for the interactions seems responsible for the inverse behaviour.

While the negative ∇²ρ_b(r_c) values (= 8m/ħ²(H_b(r_c) − V_b(r_c)/2)) were predicted for some interactions, irrespective of the negative Δr_Cov(^A/BPn–X) values, which were examined in more detail through the negative Laplacian maps. Fig. 6 shows the maps for 5a⁺_tp:TS, 5a⁺_nsym, 6b⁺_nsym, 6c⁺_nsym, 7b⁺_nsym and 8a⁺_nsym, where the maps for 5a⁺_tp:TS and 5a⁺_nsym were very close to those previously reported.⁶² Similar maps for those other than those above are shown in Fig. S8 and S9 of the ESI.† The negative ∇²ρ(r) area near the electronegative atoms around the ^BPn–X interactions was more widely extended relative to that near the electropositive atoms for 2b⁺_nsym, 3b⁺_nsym, 4a⁺_nsym, 6b⁺_nsym, 6c⁺_nsym, 7b⁺_nsym, 7c⁺_nsym, 8a⁺_nsym, 8b⁺_nsym and 8c⁺_nsym, along with ^A/BPn–X one in 4b⁺_sym.

Fig. 6 Negative Laplacian maps of 5a⁺_tp:TS, 5a⁺_nsym, 6b⁺_nsym, 6c⁺_nsym, 7b⁺_nsym and 8a⁺_nsym ((a)–(f), respectively), evaluated with MP2/BSS-B. The BCPs are denoted by red dots, RCPs (ring critical points) by yellow dots, CCPs (cage critical points) by green dots and BPs by pink lines. Carbon, hydrogen, phosphorus, arsenic, antimony, fluorine and chlorine atoms are shown in black, grey, orange, light purple, purple, light green and green, respectively. Contour plots are drawn on the planes containing at least ^APn, ^BPn and X. The red and blue lines correspond to the negative and positive areas of ∇²ρ(r), respectively.

Cremer and Kraka reported that the homonuclear covalent bonds resided in the negative ∇²ρ_b(r_c) area, except for F₂, whereas the positive ∇²ρ_b(r_c) values sometimes appeared around the heteronuclear bonds.⁶³ They also demonstrated that distances from the hydrogen atoms to the BCPs changed depending on the differences in the electronegativities of the interacting atoms. As a result, the BCPs moved towards the electropositive atoms rather than the electronegative ones since the electronegative atoms were more negatively charged than the electropositive ones. Similar conditions were observed for ^BP-*-F (2b⁺_nsym and 6b⁺_nsym), ^BAs-*-F (3b⁺_nsym and 7b⁺_nsym), ^BSb-*-H (4a⁺_nsym and 8a⁺_nsym), ^A/BSb-*-F (4b⁺_sym), ^BP-*-Cl (6c⁺_nsym), ^BAs-*-Cl (7c⁺_nsym), ^BSb-*-F (8b⁺_nsym) and ^BSb-*-Cl (8c⁺_nsym).

H_b(r_c) is a better indicator than ∇²ρ_b(r_c) to elucidate the nature of the CS interactions.⁶³ The H_b(r_c) values for all Pn⋯X⋯Pn bonds in question collected in Table 2 are arranged increasing order, which is shown in eqn (S1) of ESI.† Eqn (4)–(6) show the partial orders from eqn (S1) of ESI† for X = H, X = F, and X = Cl, Br and I, respectively. Eqn (4) shows that (1) H_b(r_c) for Pn-*-H increases in the order of ^BPn-*-H (ma⁺_nsym) < ^A/BPn-*-H (ma⁺_sym) < ^APn--*--H (ma⁺_nsym); (2) regarding m, H_b(r_c) increases in the order of m = (1 and 5: Pn = N) < (2 and 6: Pn = P) < (3 and 7: Pn = As) < (4 and 8: Pn = Sb); (3) the order in (2) is clearly observed for the strong interactions in ^BPn-*-H and ^A/BPn-*-H whereas the order becomes vague for the weak ^APn--*--H. In the case of eqn (5), the trends for X = F are very close to that observed X = H in eqn (4), although some discrepancies are observed in the order, such as ^AN--*--F (1b⁺_nsym: −0.016 and 5b⁺_nsym: −0.0015) and ^A/BP-*-F (2b⁺_sym: −0.065 and 6b⁺_sym: −0.035). The order in eqn (6) implies that (1) H_b(r_c) of Pn-*-X increases in the order of X = Cl < Br < I; (2) the same order for H_b(r_c) of ^BPn-*-X (mx⁺_nsym) < ^A/BPn-*-X (mx⁺_sym) < ^APn--*--X (mx⁺_nsym) was also observed for X = Cl, Br, and I. However, the order becomes unclear between ^BPn-*-X (mx⁺_nsym) and ^A/BPn-*-X (mx⁺_sym); (3) the difference in the reactivity between the bicyclo[4.4.4] system versus the bicyclo[3.3.3] system becomes much larger for X = Cl, Br, and I, if compared with the case of X = H and F.

Order of H_b(r_c) for Pn-*-X (X = H):

^BN-*-H (1a⁺_nsym: H_b(r_c)/au = −0.296) < ^BN-*-H (5a⁺_nsym: −0.252) < ^BP-*-H (2a⁺_nsym: −0.221) < ^BP-*-H (6a⁺_nsym: −0.204) < ^AN--*--H (1a⁺_nsym: −0.181) < ^BAs-*-H (3a⁺_nsym: −0.146) < ^BAs-*-H (7a⁺_nsym: −0.134) < ^AN--*--H (5a⁺_nsym: −0.129) < ^BSb-*-H (8a⁺_nsym: −0.075) ≤ ^BSb-*-H (4a⁺_nsym: −0.0744) ≤ ^A/BP-*-H (6a⁺_sym: −0.0736) < ^A/BP-*-H (2a⁺_sym: −0.071) < ^A/BAs--*--H (3a⁺_sym: −0.049) < ^A/BSb-*-H (4a⁺_sym: −0.030) < ^A/BSb-*-H (8a⁺_sym: −0.027) < ^AP--*--H (2a⁺_nsym: −0.0108) ≤ ^AAs--*--H (3a⁺_nsym: −0.0101) < ^ASb--*--H (4a⁺_nsym: −0.0097) < ^ASb--*--H (8a⁺_nsym: −0.0019) ≤ ^AAs--*--H (7a⁺_nsym: −0.0014) ≤ ^AP--*--H (6a⁺_nsym: −0.0003) (4)

Order of H_b(r_c) for Pn-*-X (X = F):

^BN-*-F (1b⁺_nsym: H_b(r_c)/au = −0.332) < ^BN-*-F (5b⁺_nsym: −0.283) < ^BP-*-F (2b⁺_nsym: −0.134) < ^BP-*-F (6b⁺_nsym: −0.125) < ^BAs-*-F (3b⁺_nsym: −0.120) < ^BAs-*-F (7b⁺_nsym: −0.109) < ^A/BP-*-F (2b⁺_sym: −0.065) < ^A/BN-*-F (1b⁺_sym: −0.059) < ^BSb-*-F (8b⁺_nsym: −0.047) < ^A/BP-*-F (6b⁺_sym: −0.035) < ^A/BAs-*-F (3b⁺_sym: −0.032) < ^A/BN-*-F (5b⁺_sym: −0.0264) < ^A/BAs-*-F (7b⁺_sym: −0.019) < ^AN--*--F (1b⁺_nsym: −0.016) < ^A/BSb-*-F (4b⁺_sym: −0.0111) < ^A/BSb-*-F (8b⁺_sym: −0.0072) < ^AP--*--F (2b⁺_nsym: −0.0039) ≤ ^AAs--*--F (3b⁺_nsym: −0.0036)) < ^AN--*--F (5b⁺_nsym: −0.0015) ≤ ^ASb--*--F (8b⁺_nsym: 0.0010) ≤ ^AAs--*--F (7b⁺_nsym: 0.0020) ≤ ^AP--*--F (6b⁺_nsym: 0.0023) (5)

Order of H_b(r_c) for Pn-*-X (X = Cl, Br, and I):

^BP-*-Cl (6c⁺_nsym: H_b(r_c)/au = −0166) < ^BP-*-Br (6d⁺_nsym: −0.159) > ^A/BN-*-Cl (1c⁺_sym: −0.115) < ^A/BN-*-Cl (5c⁺_sym: −0.104) < ^BAs-*-Cl (7c⁺_nsym: −0.097) ≤ ^A/BN-*-Br (5d⁺_sym: −0.096) < ^BAs-*-Br (7d⁺_nsym: −0.076) < ^A/BN-*-I (5e⁺_sym: −0.0729) < ^A/BP-*-Cl (6c⁺_sym: −0.04881) ≤ ^BSb-*-Cl (8c⁺_nsym: −0.04880) < ^A/BP-*-Br (6d⁺_sym: −0.045) < ^A/BP-*-I (6e⁺_sym: −0.040) < ^A/BAs-*-Cl (7c⁺_sym: −0.0301) ≤ ^A/BAs-*-Br (7d⁺_sym: −0.0296) < ^A/BSb-*-Cl (8c⁺_sym: −0.018) < ^AP--*--Br (6d⁺_nsym: −0.0109) ≤ ^AAs--*--Br (7d⁺_nsym: −0.0098) < ^AP--*--Cl (6c⁺_nsym: −0.0074) ≤ ^AAs--*--Cl (7c⁺_nsym: −0.0061) ≤ ^ASb--*--Cl (8c⁺_nsym: −0.0051) (6)

The strengths of interactions become weakened inversely proportional to H_b(r_c) being increased. Therefore, the orders in eqn (4) and (5) show the rough orders of the strengths of the interactions being weakened. However, some deviations were observed.

What is the behaviour of H_b(r_c) in 1a⁺–8c⁺? Fig. 7 shows the plot of H_b(r_c) versus ρ_b(r_c) for the sym and nsym shapes of 1a⁺–8c⁺ (see Table 2 for the data). The plot was analysed, assuming the linear correlations, by the four groups of Group A (G(A)), G(B), G(C) and G(D). The correlations are very good for G(A) (y = 0.090–1.57x: R_c² = 0.992), G(B) (y = 0.099–1.35x: R_c² = 0.987) and G(C) (y = 0.136–1.38x: R_c² = 0.994) and fairly good for G(D) (y = 0.007–0.35x: R_c² = 0.78). The plot for G(A) appears most downside of the four and that for G(C) does most upside, while that for G(B) exists between the two. The data for G(D) correspond to those from the much weaker bonds of ^APn--*--X in the nsym shapes; therefore, the (data) points are located near the origin of the plots. The correlation constant for G(D) is about one-fourth to one-fifth of those for G(A), G(B) and G(C). Most of the data are contained in G(A), which consists of the data from 14 sym shapes and seven nsym shapes. G(B) contains those from five sym shapes and eight nsym shapes. While G(C) contains four sym shapes seven nsym shapes G(D) does those for 16 nsym shapes, except for four (see Fig. 7 for the interactions and the species).

Fig. 7 Plots of H_b(r_c) versus ρ_b(r_c) for Pn-*-X in 1a⁺–8c⁺, evaluated with MP2/BSS-B.

The trend in the correlations can be variously explained, but our explanation is as follows: Almost common mechanisms are operating to give H_b(r_c) and ρ_b(r_c) for both sym and nsym shapes of 1a⁺–8c⁺. The mechanisms to give H_b(r_c) and/or ρ_b(r_c) would be very similar for most species but somewhat different for some species. The species with the negligibly small differences in the mechanisms forms a group of G(A), G(B), G(C) or G(D); however, there are some differences among the groups.

The natures of the secondary interactions of X--*--C and H--*--H for 1–8 and 1a⁺–8c⁺ were also investigated with MP2/BSS-B, along with ^APn--*--^BPn of 1, 5 and 6. The QTAIM functions and QTAIM-DFA parameters are provided in Table S6 of the ESI,† along with the C_ii values, corresponding to generate the perturbed structures, and the predicted natures. Various natures of p-CS/vdW, p-CS/t-HB_nc, r-CS/t-HB_wc and r-CS/CT-MC were predicted for X--*--C and H--*--H, together with ^APn--*--^BPn. These interactions contributed to stabilizing the system, in addition to ^A/BPn-*-X, ^APn--*--X and ^BPn-*-X.

The natures were also elucidated for the interactions in question in the transition states. The analysed results by QTAIM-DFA are collected in Table S7 of the ESI,† along with molecular graphs and negative Laplacian maps shown in Fig. S10 and S11 of the ESI,† respectively.

Second-perturbation energies E(2)

The contributions from the CT terms of NBO(i)→NBO(j) to 1a⁺–8c⁺ were estimated through the second-perturbation energies (E(2)), calculated using the NBO analysis with M06-2X/BSS-B//MP2/BSS-B. Table 3 lists the E(2) values for 1a⁺–8c⁺. Several types of the CT terms contribute to stabilize the species. The first is the n(^APn)→σ*(X–^BPn) type, which operates in 17 species. The E(2) values are shown in parenthesis after the compound: The values larger than 30 kcal mol⁻¹ are 1a⁺_nsym (E(2) = 78 kcal mol⁻¹), 5a⁺_nsym (80 kcal mol⁻¹), 1b⁺_sym (344 kcal mol⁻¹), 1c⁺_sym (230 kcal mol⁻¹), 2b⁺_sym (565 kcal mol⁻¹), 5b⁺_sym (243 kcal mol⁻¹), 5c⁺_sym (191 kcal mol⁻¹), 5d⁺_sym (171 kcal mol⁻¹), 5e⁺_sym (132 kcal mol⁻¹), 6a⁺_sym (288 kcal mol⁻¹), 6c⁺_sym (357 kcal mol⁻¹), 7c⁺_sym (506 kcal mol⁻¹), 8a⁺_sym (349 kcal mol⁻¹), 8b⁺_sym (429 kcal mol⁻¹), and 8c⁺_sym (762 kcal mol⁻¹). The E(2) values for the nsym shapes are small to moderate, whereas the values are very large for the sym shapes. The shorter interaction distances in the sym shapes would be responsible for the results.

Table 3 Contributions from the CT terms of the NBO(i)→NBO(j) form to the interactions in 1a⁺–8c⁺, estimated through the second-perturbation energies (E(2)), evaluated with NBO analysis under the M06-2X/BSS-B//MP2/BSS-B conditions.^a

Species (^APn (^BPn), X)	NBO(i)→NBO(j)	E(2)^b (kcal mol^−1)	F(i,j)^c (au)	Species (^APn (^BPn), X)	NBO(i)→NBO(j)	E(2)^b (kcal mol^−1)	F (i,j)^c (au)
a See Table 1 for BSS-B.b Second-perturbation energy.c The off-diagonal NBO Fock matrix elements.d Detected as σ(3c–4e).e Six CT contributions are detected.f Total values are given in parenthesis.g Three CT contributions are detected.h No CT contributions in question are detected.
1a^+nsym (N, H)	n(^AN)→σ*(H–^BN)	77.6	0.212	6a^+sym (P, H)	n(^A/BP)→σ*(H–^B/AP)	288.1^d	0.256
1b^+sym (N, F)	n(^A/BN)→σ*(F–^B/AN)	344.2^d	0.198	6a^+nsym (P, H)	σ(^AP–C)→σ*(H–^BP)	4.2^g (12.6)^f	0.052
1b^+nsym (N, F)	n(^AN) → σ*(F–^BN)	11.1	0.064	6b^+sym (P, F)	n(F)→σ*(^AP–^BP)	101.8	0.171
1c^+sym (N, Cl)	n(^A/BN)→σ*(Cl–^B/AN)	229.6^d	0.277	6b^+nsym (P, F)	^h	^h	^h
2a^+sym (P, H)	n(^AP/^BP)→s(H^+)	31.9	0.060	6c^+sym (P, Cl)	n(^A/BP)→σ*(Cl–^B/AP)	356.9^d	0.171
	σ(^AP–C/^BP–C)→s(H^+)	55.8^e (334.8)^f	0.134	6c^+nsym (P, Cl)	σ(^AP–C)→σ*(Cl–^BP)	1.4^g (4.1)^f	0.031
2a^+nsym (P, H)	σ(^AP–C)→σ*(H–^BP)	10.1^g (30.3)^f	0.083	6d^+sym (P, Br)	σ(^AP–C/^BP–C)→n(Br^+)	61.5^e (369.2)^f	0.116
2b^+sym (P, F)	n(^AP)→σ*(F–^BP)	565.0	0.142	6d^+nsym (P, Br)	σ(^AP–C)→σ*(Br–^BP)	8.3^g (24.8)^f	0.066
2b^+nsym (P, F)	σ(^AP–C)→σ*(F–^BP)	1.2^g (3.6)^f	0.027	6e^+sym (P, I)	σ(^AP–C/^BP–C)→n(I^+)	58.9^e (353.2)^f	0.122
3a^+sym (As, H)	σ(^AAs–C/^BAs–C)→s(H^+)	74.6^e (447.6)^f	0.138	7a^+nsym (As, H)	σ(^AAs–C)→σ*(H–^BAs)	7.0^g (21.0)^f	0.065
3a^+nsym (As, H)	σ(^AAs–C)→σ*(H–^BAs)	13.5^g (40.5)^f	0.091	7b^+sym (As, F)	n(F)→σ*(^AAs–^BAs)	80.0	0.147
3b^+sym (As, F)	σ(^AAs–C)→σ*(F–^BAs)	3.3^g (9.8)^f	0.037	7b^+nsym (As, F)	^h	^h	^h
3b^+nsym (As, F)	σ(^AAs–C)→σ*(F–^BAs)	1.2^g (3.6)^f	0.025	7c^+sym (As, Cl)	n(^A/BAs)→σ*(Cl–^B/AAs)	505.8^d	0.151
4a^+sym (Sb, H)	σ(^ASb–C/^BSb–C)→s(H^+)	94.3^e (565.7)^f	0.123	7c^+nsym (As, Cl)	σ(^AAs–C)→σ*(Cl–^BAs)	4.3^g (12.8)^f	0.046
4a^+nsym (Sb, H)	σ(^ASb–C)→σ*(H–^BSb)	14.3^g (43.0)^f	0.086	7d^+sym (As, Br)	σ(^AAs–C/^BAs–C)→n(Br^+)	76.6^e (459.5)^f	0.111
4b^+sym (Sb, F)	n(F)→n(Sb^+)	66.8	0.066	7d^+nsym (As, Br)	σ(^AAs–C)→σ*(Br–^BAs)	9.9^g (29.7)^f	0.067
5a^+nsym (N, H)	n(^A/BN)→σ*(H–^B/AN)	80.4^d	0.220	8a^+sym (Sb, H)	n(^A/BSb)→σ*(H–^B/ASb)	348.6^d	0.184
5b^+sym (N, F)	n(^A/BN)→σ*(F–^B/AN)	242.7^d	0.162	8a^+nsym (Sb, H)	σ(^ASb–C)→σ*(H–^BSb)	6.7^g (20.0)^f	0.059
5b^+nsym (N, F)	n(^AN)→σ*(F–^BN)	12.0	0.058	8b^+sym (Sb, F)	n(^ASb)→σ*(F–^BSb)	429.2	0.086
5c^+sym (N, Cl)	n(^A/BN)→σ*(Cl–^B/AN)	191.4^d	0.259	8b^+nsym (Sb, F)	n(F)→σ*(^ASb–C)	1.7^g (5.1)^f	0.042
5d^+sym (N, Br)	n(^A/BN)→σ*(Br–^B/AN)	170.5^d	0.251	8c^+sym (Sb, Cl)	n(^A/BSb)→σ*(Cl–^B/ASb)	762.1^d	0.130
5e^+sym (N, I)	n(^A/BN)→σ*(I–^B/AN)	131.8^d	0.231	8c^+nsym (Sb, Cl)	σ(^ASb–C)→σ*(H–^BSb)	2.9^g (8.6)^f	0.035

---

The second is the σ(^APn–C/^BPn–C)→n_p(X⁺) type, which operates in 6 species of 2a⁺_sym (335 kcal mol⁻¹), 3a⁺_sym (448 kcal mol⁻¹), 4a⁺_sym (566 kcal mol⁻¹), 6d⁺_sym (369 kcal mol⁻¹), 6e⁺_sym (353 kcal mol⁻¹) and 7d⁺_sym (460 kcal mol⁻¹). Total six equivalent σ(Pn–C) bonds from three σ(^APn–C) and three σ(^BPn–C) bonds contribute to this type; therefore, the total contribution is six times larger than that of each contribution. The total values are also very large. The third case is the σ(^APn–C)→σ*(X–^BPn) form, which operates in 14 species. The total contribution is three times larger than that of each contribution. The total contributions larger than 30 kcal mol⁻¹ are 3a⁺_nsym (41 kcal mol⁻¹), 4a⁺_nsym (43 kcal mol⁻¹). The fourth is the n_p(F)→σ*(^APn–^BPn), which operated in 6b⁺_sym (102 kcal mol⁻¹) and 7b⁺_sym (80 kcal mol⁻¹). Other cases of which contributions larger than 10 kcal mol⁻¹ are n(^AP/^BP)→σ(H⁺) (32 kcal mol⁻¹) in 2a⁺_sym and n(F)→n(Sb⁺) (67 kcal mol⁻¹) in 4b⁺_sym.

The high contributions from the CT terms mainly arise from the n(^APn)→σ*(X–^BPn) types. The σ(^APn–C/^BPn–C)→n_p(X⁺) type interactions also have the substantially high contributions. The reason for the contributions was intriguing at first glance but it would be difficult to explain. The reason could be interpreted by considering the through space and the through bond mechanisms. The contributions increased rapidly as the interaction distances became shorter. The contributions in question could be understood based on the same reason. A similar mechanism would also operate in the CT interactions in σ(^APn–C/^BPn–C)→n_p(X⁺).

The very high contributions from the of n(^APn)→σ*(X–^BPn) type, accompanied by the very large E(2) values, must correlate deeply to the (very) high p-characters in n(^APn). The occupancy ratios between s- and p-characters in n(^APn) were calculated under the same conditions. The results are shown in Table S8 of the ESI.† The planarity around ^APn will affect much to the ratio of the p-character, together with the energy difference between atomic s- and p-orbitals of ^APn. High ratios of p-characters over 70% were predicted for n(^APn) in 1a⁺–1c⁺ (86.0–98.7%), 2b⁺ (97.6%), 3b⁺ (97.4%), 5a⁺–5e⁺ (84.4–99.6%), 6a⁺ (87.9%), 6c⁺ (96.2%), 7c⁺ (96.5%), 8a⁺ (92.6%), 8b⁺(97.7%) and 8c⁺ (96.3%). However, the predicted rations were 32.1–62.4% for the others. Indeed, high ratios of the p-characters are expected to enlarge E(2) for n(^APn)→σ*(X–^BPn), but the correlation between the ratio and E(2) seems unclear, as shown in Table 3. Other factors, such as the atomic types of ^APn, X and ^BPn, must also operate to control the E(2) values.

The NBO analysis demonstrated the large contributions from the ^APn--X--^BPn and ^APn⋯X–^BPn interactions and clarified the reasons for the high contributions from n(^APn)→σ*(X–^BPn). While the σ(^APn–C/^BPn–C)→n_p(X⁺) interactions also contribute much to stabilise the system, the contributions from σ(^APn–C)→σ*(X–^BPn) seem smaller. The contribution from n(F)→n_s(Sb⁺) was also of interest. The n(F)→n_s(Sb⁺) interaction, in place of n_p(Sb)→σ*(F–Sb), contributed to 4b⁺_sym. A heavy CT occurred from Sb to F⁺ due to the very large electronegativity of F, relative to Sb, resulting in the formation of F and Sb⁺, which led to the above interaction, as listed in Table 3.

Characteristic nature of [Pn--X--Pn]⁺ with the behaviour of charge

What is the character of Pn--X⁺--Pn in 1a⁺–8c⁺? The ψ₃, ψ₂ and ψ₁ orbitals of σ(3c–4e) are looked for among MOs in 1a⁺–8c⁺. Fig. 8 shows HOMO-5, HOMO and LUMO of 6c⁺_sym, for example, which correspond to the orbitals, respectively. The results clearly demonstrate the σ(3c–4e) nature for Pn--X⁺--Pn.

Fig. 8 HOMO–5, HOMO and LUMO in 6c⁺_sym (Pn = P, X = Cl), which correspond to ψ₁, ψ₂ and ψ₃ of σ(3c–4e), respectively.

How are the behaviour of Qn, in relation to σ(3c–4e)? The sym and nsym shapes are optimized for all 6a⁺–6e⁺, except for 6e⁺_sym; therefore, it is convenient to discuss the overall trends in the changes of Qn, so are 5⁺, 7⁺ and 8⁺. Fig. 9 shows the plot of Qn(^AP), Qn(X), Qn(^BP) and Qn(Total) (= Qn(T) = Qn(^AP) + Qn(X) + Qn(^BP)) for 6a⁺_sym–6e⁺_sym and 6a⁺_nsym–6d⁺_nsym versus X. The values of Qn(^AP), Qn(X), Qn(^BP) and Qn(T) for 1–8 and 1a⁺–8c⁺ are collected in Table S8 of the ESI.†

Fig. 9 Plots of Qn(^APn), Qn(X), Qn(^BPn) and Qn(T) versus X for 6a⁺–6e⁺.

The Qn(X) values in sym and nsym shapes of 6a⁺–6e⁺ become larger in the order of X = F < Cl < Br < I, as expected, where Qn(H) seems close to Qn(Cl) or Qn(Br). Qn(P) behaves the exact opposite of Qn(X). The exact opposite behaviour between Qn(P) (or Qn(^BPn)) and Qn(X) are confirmed by examining Qn(T), which are almost constant for both shapes. It is noteworthy that Qn(^AP) are almost constant in the nsym shapes, irrespective of the large changes in Qn(X) and Qn(^BPn). The results may support the formation of the (very) strong ^BP–X bond with the (very) weak ^APn⋯X in the nsym shapes. The plots for 5⁺, 7⁺ and 8⁺, drawn in Fig. S12 of the ESI,† are very similar to that in Fig. 9.

It seems curious that the Qn(T) values of 6a⁺_sym–6e⁺_sym are close to 2 (e⁺), which is much larger than 1 (e⁺), originated from X⁺, at first glance. The results could be very different from those generally expected for the formation of σ(3c–4e). Fig. 10 explains the changes of Qn(^A/BP) and Qn(X) in the formation of 6c⁺_sym, starting from 6_nsym + Cl⁺, via [6c⁺_nsym]_CT=0. Our explanation for the change in Qn is as follows. The mechanism for the change is explained by the three processes, starting from the neutral 6_nsym + Cl⁺. (1) The Qn(^APn/^BPn) values in 6_nsym are 0.78e⁺ and 0.82e⁺, respectively. (2) When Cl⁺ is incorporated into P⋯P of 6_nsym, the double donation occurs from the lone pair electrons of P to the vacant p-orbital of Cl⁺. The Qn(Pn) and Qn(Cl) values would be 1.8e⁺ and −1.0e⁺, respectively, where the donated electrons are assumed to share equally between P and Cl. The primitive structure with no CT is denoted by [6c⁺_nsym]_CT=0. (3) The charges in P–Cl–P of [6c⁺_nsym]_CT=0 redistribute to form 6c⁺_sym.

Fig. 10 Changes in Qn(^A/BPn) and Qn(X) in the formation of 6c⁺_sym via [6c⁺_nsym]_CT=0.

The CT occurs from [6c⁺_nsym]_CT=0 to 6c⁺_sym, contrary to the electronegativities of P and Cl, in this process, resulting in Qn(Pn) = 1.02e⁺ and Qn(Cl) = −0.31e⁺. The σ(3c–4e) character of ^APn--X⁺--^BPn is demonstrated again, through the examination of the changes of the charges on the atoms. Indeed, some Pn--X⁺--Pn interactions would not be typical σ(3c–4e), depending on the combination of Pn and X, but the bonds, analysed in this work, are recognized to be the σ(3c–4e) type, for the unified understanding of the bonds.

The σ-hole on a halogen atom must correlate deeply in the behavior of ^APn--X⁺--^BPn and/or Pn⋯X⁺–Pn. Molecular electrostatic potentials (MEPs) will present a fine picture for the visualization. The MEPs are shown in Fig. S13 of the ESI,† exemplified by 6d⁺_nsym, [H–C(CH₂CH₂CH₂CH₂)₃P–Br]⁺ (9d⁺) and [H–C(CH₂CH₂CH₂CH₂)₃P–I]⁺ (9e⁺).

Conclusions

The possibility of large atoms to incorporate in stable cage compounds of medium rings was examined with the atoms X of H (a), F (b), Cl (c), Br (d) and I (e). The skeletons of bicyclo[3.3.3]undecane and bicyclo[4.4.4]tetradecane were employed for the purpose, in which the bridgehead atoms were substituted by pnictogens (^APn = ^BPn = N, P, As and Sb: 1x⁺–8x⁺). In this system, X acted as X⁺ to form the linear ^APn--X⁺--^BPn σ(3c–4e) type interactions, which were fixed by the bicyclo systems. Moreover, 1⁺ (^APn = ^BPn = N) was successfully optimized for X of each H, F and Cl, and 2x⁺ (^APn = ^BPn = P), 3x⁺ (^APn = ^BPn = As) and 4x⁺ (^APn = ^BPn = Sb) were carefully optimized when X = H and F. In the case of 5x⁺ (^APn = ^BPn = N) and 6x⁺ (^APn = ^BPn = P) were successfully optimized for X of H, F, Cl, Br and I, 7x⁺ (^APn = ^BPn = As) was optimized for X of H, F, Cl and Br, and 8x⁺ (^APn = ^BPn = Sb) was optimized for X of H, F and Cl. The structures of the symmetric and nonsymmetric shapes were optimized for 1a⁺–8c⁺, with their relative stabilities, and their transition states. The symmetric shapes appeared to be more stabilized as ^APn, ^BPn and X became larger, with the exception of ^APn = ^BPn = N.

The intrinsic dynamic and static natures of the Pn-*-X interactions were elucidated for 1a⁺–8c⁺ with QTAIM-DFA. Various natures from vdW to Cov-s were predicted. The ^BP-*-F (2b⁺_nsym and 6b⁺_nsym), ^BAs-*-F (3b⁺_nsym and 7b⁺_nsym), ^BSb-*-H (4a⁺_nsym and 8a⁺_nsym), ^A/BSb-*-F (4b⁺_sym), ^BP-*-Cl (6c⁺_nsym), ^BAs-*-Cl (7c⁺_nsym), ^BSb-*-F (8b⁺_nsym) and ^BSb-*-Cl (8c⁺_nsym) interactions were predicted to be r-CS/t-HB_wc or r-CS/CT-MC, respectively, although their r(^A/BPn–X) were shorter than the r_Cov(^A/BPn–X). In contrast, the SS/Cov-w or SS/Cov-s natures were predicted for ^BPn-*-X interactions than the above such as ^BN-*-H in 1a⁺_nsym and 5a⁺_nsym (with 1a⁺_tp:TS and 5a⁺_tp:TS). The secondary interactions of H-*-H and X-*-C were also detected, of which the predicted natures were p-CS/vdW, p-CS/t-HB_nc, r-CS/t-HB_wc and r-CS/CT-MC. They also contributed to stabilising the systems, along with the main interactions. The order of the interaction strengths was estimated using the H_b(r_c) values.

The NBO analysis was applied to the interactions and evaluated the contributions from the CT terms to the interactions in question. Several types of NBO(i) to NBO(j) interactions were detected in this case. Among the several types, the n_p(^APn)→σ*(X–^BPn) interactions highly contributed, together with σ(^APn–C)→σ*(X–^BPn) and σ(^APn–C/^BPn–C)→n_p(X⁺). In the case of 4b⁺, the CT interaction was described by n_p(F)→n_s(Sb⁺), not by n_p(Sb)→σ*(F–Sb). The very large electronegativity of F was likely responsible for the change. The charge distributions on the atoms of ^APn--X⁺--^BPn were confirmed to show the σ(3c–4e) type behaviour.

Our results provide important insights into the nature of the interactions, especially the hypervalent 3c–4e interactions of the symmetric and nonsymmetric shapes, where the interactions are fixed linearly by the bicyclo systems. These interactions can also be considered as the extension of the hydrogen and halogen bonds. These results are useful to design materials containing X⁺, as in 1–8.

Author contributions

S. H. formulated the project. T. N. contributed to conceptualization, investigation and writing – original draft. W. N. and S. H. contributed to supervision, data curation, resources and writing – review & editing. All authors have read and agreed to the published version of the manuscript.

Conflicts of interest

The authors declare no conflicts of interest.

Acknowledgements

The computations were partially performed at the Research Centre for Computational Science, Okazaki, Japan.

Notes and references

1. W. Nakanishi, S. Hayashi and K. Narahara, J. Phys. Chem. A, 2009, 113, 10050–10057.
2. W. Nakanishi and S. Hayashi, Curr. Org. Chem., 2010, 14, 181–197.
3. W. Nakanishi and S. Hayashi, J. Phys. Chem. A, 2010, 114, 7423–7430.
4. W. Nakanishi, S. Hayashi, K. Matsuiwa and M. Kitamoto, Bull. Chem. Soc. Jpn., 2012, 85, 1293–1305.
5. S. Hayashi, K. Matsuiwa, M. Kitamoto and W. Nakanishi, J. Phys. Chem. A, 2013, 117, 1804–1816.
6. S. Hayashi, T. Nishide, W. Nakanishi, L. Sancineto and C. Santi, RSC Adv., 2019, 9, 39435–39446.
7. G. Montà-González, F. Sancenón, R. Martínez-Máñez and V. Martí-Centelles, Chem. Rev., 2022, 122, 13636–13708.
8. X. Yang, Z. Ullah, J. F. Stoddart and C. T. Yavuz, Chem. Rev., 2023, 123, 4602–4634.
9. R. W. Alder, Tetrahedron, 1990, 46, 683–713.
10. J. G. Verkade, Coord. Chem. Rev., 1994, 137, 233–295.
11. R. W. Alder and D. Read, Coord. Chem. Rev., 1998, 176, 113–133.
12. J. L. Vennerstrom, J. Med. Chem., 1989, 32, 64–67.
13. R. Gleiter, K. Hövermann, J. Ritter and B. Nuber, Angew. Chem., Int. Ed., 1995, 34, 789–791.
14. J. Kobayashi, K. Goto, T. Kawashima, M. W. Schmidt and S. Nagase, J. Am. Chem. Soc., 2002, 124, 3703–3712.
15. S. Nakafuji, J. Kobayashi, T. Kawashima and M. W. Schmidt, Inorg. Chem., 2005, 44, 6500–6502.
16. J. Kobayashi, K. Goto, T. Kawashima, M. W. Schmidt and S. Nagase, Chem.–Eur. J., 2006, 12, 3811–3820.
17. J. Kobayashi and T. Kawashima, C. R. Chim., 2010, 13, 1249–1259.
18. J. Tang, J. Dopke and J. G. Verkade, J. Am. Chem. Soc., 1993, 115, 5015–5020.
19. P. B. Kisanga, J. G. Verkade and R. Schwesinger, J. Org. Chem., 2000, 65, 5431–5432.
20. J. G. Verkade, New Aspects in Phosphorus Chemistry II, ed. J. P. Majoral, Springer, Berlin, Heidelberg, 2003, pp. 1–44.
21. M.-Y. Xu, W.-T. Jiang, Y. Li, Q.-H. Xu, Q.-L. Zhou, S. Yang and B. Xiao, J. Am. Chem. Soc., 2019, 141, 7582–7588.
22. D. Tanaka, A. Konishi and M. Yasuda, Chem.–Asian J., 2021, 16, 3118–3123.
23. M.-Y. Xu and B. Xiao, Chem. Commun., 2021, 57, 11764–11775.
24. J. K. Puri, R. Singh and V. K. Chahal, Chem. Soc. Rev., 2011, 40, 1791–1840.
25. S. N. Adamovich, Appl. Organomet. Chem., 2019, 33, e4940.
26. V. V. Istratov, V. A. Vasnev and G. D. Markova, Molecules, 2021, 26, 1893.
27. S. N. Adamovich, I. A. Ushakov, E. N. Oborina, S. V. Lukyanova and V. Y. Komarov, Int. J. Mol. Sci., 2023, 24, 9965.
28. R. W. Alder, R. B. Sessions, J. M. Mellor and M. F. Rawlins, J. Chem. Soc., Chem. Commun., 1977, 747–748.
29. R. W. Alder and R. B. Sessions, J. Am. Chem. Soc., 1979, 101, 3651–3652.
30. R. W. Alder, D. D. Ellis, R. Gleiter, C. J. Harris, H. Lange, A. G. Orpen, D. Read and P. N. Taylor, J. Chem. Soc., Perkin Trans., 1998, 1, 1657–1668.
31. R. W. Alder, C. P. Butts, A. G. Orpen and D. Read, J. Chem. Soc., Perkin Trans., 2001, 2, 288–295.
32. R. W. Alder, A. Casson and R. B. Sessions, J. Am. Chem. Soc., 1979, 101, 3652–3653.
33. R. W. Alder, R. J. Arrowsmith, A. Casson, R. B. Sessions, E. Heilbronner, B. Kovac, H. Huber and M. Taagepera, J. Am. Chem. Soc., 1981, 103, 6137–6142.
34. R. W. Alder, R. E. Moss and R. B. Sessions, J. Chem. Soc. Chem. Commun., 1983, 997–998.
35. R. W. Alder, A. G. Orpen and R. B. Sessions, J. Chem. Soc. Chem. Commun., 1983, 999–1000.
36. R. W. Alder, R. E. Moss and R. B. Sessions, J. Chem. Soc. Chem. Commun., 1983, 1000–1002.
37. S. A. Harry, S. Vemulapalli, T. Dudding and T. Lectka, J. Org. Chem., 2022, 87, 8413–8419.
38. Y. Wang and Z.-X. Yu, J. Org. Chem., 2020, 85, 397–402.
39. Cage compounds with compressed CX–π interactions, called “iron maiden,” were also studied.^64,65.
40. A. Karim, M. Reitti, A.-C. C. Carlsson, J. Gräfenstein and M. Erdélyi, Chem. Sci., 2014, 5, 3226–3233.
41. M. Bedin, A. Karim, M. Reitti, A.-C. C. Carlsson, F. Topić, M. Cetina, F. Pan, V. Havel, F. Al-Ameri, V. Sindelar, K. Rissanen, J. Gräfenstein and M. Erdélyi, Chem. Sci., 2015, 6, 3746–3756.
42. S. Lindblad, K. Mehmeti, A. X. Veiga, B. Nekoueishahraki, J. Gräfenstein and M. Erdélyi, J. Am. Chem. Soc., 2018, 140, 13503–13513.
43. L. Turunen and M. Erdélyi, Chem. Soc. Rev., 2020, 49, 2688–2700.
44. A. C. Reiersølmoen, S. Battaglia, S. Øien-Ødegaard, A. K. Gupta, A. Fiksdahl, R. Lindh and M. Erdélyi, Chem. Sci., 2020, 11, 7979–7990.
45. Atoms in Molecules. A Quantum Theory, ed. R. F. W. Bader, Oxford University Press, Oxford, UK, 1990.
46. C. F. Matta and R. J. Boyd, An Introduction to the Quantum Theory of Atoms in Molecules, in The Quantum Theory of Atoms in Molecules: from Solid State to DNA and Drug Design, ed. C. F. Matta and R. J. Boyd, WILEY-VCH, Weinheim, Germany, 2007, ch. 1.
47. W. Nakanishi and S. Hayashi, Int. J. Quantum Chem., 2018, 118, e25590.
48. K. Brandhorst and J. Grunenberg, Chem. Soc. Rev., 2008, 37, 1558–1567.
49. K. Brandhorst and J. Grunenberg, J. Chem. Phys., 2010, 132, 184101–184107.
50. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian 09, Revision D.01 and E.01, Gaussian, Inc., Wallingford CT, 2009.
51. T. Noro, M. Sekiya and T. Koga, Theor. Chem. Acc., 2012, 131, 1–8.
52. C. Møller and M. S. Plesset, Phys. Rev., 1934, 46, 618–622.
53. E. D. Glendening, C. R. Landis and F. Weinhold, J. Comput. Chem., 2013, 34, 1429–1437.
54. F. Biegler-König, J. Comput. Chem., 2000, 21, 1040–1048.
55. T. A. Keith, AIMAll (Version 17.11.14), TK Gristmill Software, Overland Park, KS, 2017, https://aim.tkgristmill.com/.
56. R. W. Alder and S. P. East, Chem. Rev., 1996, 96, 2097–2112.
57. The postulated symmetry was often broken during the optimizations, but not significantly altered.
58. P. Pyykkö and M. Atsumi, Chem.–Eur. J., 2009, 15, 186–197.
59. A. Bondi, J. Phys. Chem., 1964, 68, 441–451.
60. M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, J. Phys. Chem. A, 2009, 113, 5806–5812.
61. W. Nakanishi, S. Hayashi, R. Imanaka, T. Nishide, E. Tanaka and H. Matsuoka, Int. J. Mol. Sci., 2023, 24, 2798.
62. D. B. DuPré, J. Phys. Chem. A, 2003, 107, 10142–10148.
63. D. Cremer and E. Kraka, Croat. Chem. Acta, 1984, 57, 1259–1281.
64. R. A. Pascal, Eur. J. Org Chem., 2004, 3763–3771.
65. M. Jabłoński, J. Comput. Chem., 2022, 43, 1206–1220.

---

Footnote

† Electronic supplementary information (ESI) available: Additional tables and figures, explanation of QTAIM-DFA, and the fully optimized and transition structures given by Cartesian coordinates, together with total energies of 1a⁺–8c⁺. See DOI: https://doi.org/10.1039/d3ra08926f

---