Introduction

Studies of chemical equilibria have provided important foundations of chemistry. The law of mass action allows us to characterize equilibria with equilibrium constants (K)1,2,3, and decomposition analyses of Gibbs energy changes (ΔG) allow us to quantitatively analyze chemical interaction energetics including ΔH and ΔS4,5. Similarly, the chemistries of weak interactions embedded in elaborate molecular structures are being deepened and developed with studies of the equilibria of supramolecular complexes6,7. Although various interesting supramolecular systems are emerging, the weak interactions and complex structures often make it difficult to decide one of the most important characteristics of the equilibrium, i.e., the stoichiometry of the components involved in the equilibrium. Thus, traditionally, the stoichiometries of supramolecular complexes have been determined by the method of continuous variation using so-called Job plots of titration data, which allows us to identify a single equilibrium model for further analyses (Fig. 1a)6,8,9,10. However, since the credibility of Job plots was questioned in 201611,12, the development of alternative protocols to characterize equilibria has become an important subject particularly for supramolecular systems. An interesting proposal has recently been made for determining stoichiometries from fitting analyses (Fig. 1a). In this case, the titration data are fitted with multiple possible models with different stoichiometries (e.g. 1:1 vs. 1:2 in Fig. 1b), and the fitting data are then compared to assess credibility. A statistical method for the F-test using the P-value measure is useful to afford quantitative evaluations for the model comparison analyses, which has also been suggested by Hibbert/Thordarson among various measures12,13,14,15. More recently, we introduced an information-theoretic method using Akaike’s information criterion (AIC) with the Akaike weight (wi) measure as a more versatile method for comparison16,17,18,19. In the present investigation of CH-π hydrogen bonding, we noticed that the protocol needs further development and found that the van ’t Hoff validation step should be used to examine the validity of the model comparison step. The stoichiometries of supramolecular complexes should be carefully examined by adopting multiple measures of different origins.

Fig. 1: Characterization of chemical equilibria.
figure 1

a Protocols for characterization. b An example of the fitting equations used with NMR titration experiments. GOF goodness-of-fit.

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Results

Synthesis of a hydrocarbon cage host

A problem was found in stoichiometry determinations during our studies of a cage-shaped hydrocarbon molecule. The hydrocarbon cage phenine polluxene (1) was recently synthesized as a phenine version of a minimal diamond twin cage20,21,22,23, and host-guest complexation was noticed in this study of its structural diversification. The synthesis of an additional congener (1a, R = H; Fig. 2a) is described first. Following the preceding synthetic route for the D3-symmetric phenine polluxene (1b, R = t-Bu), we started the synthesis from a decagonal macrocycle (2) composed of ten phenine units. After the introduction of two biphenyl arms via Suzuki-Miyaura coupling with 3, the precursor was subjected to Ni-mediated Yamamoto coupling, which afforded the target polluxene (1a) in 49% yield. As shown in Fig. 2b, we determined the crystal structure of 1a. The crystal structure showed a cage-shaped structure for the molecule, which revealed the presence of multiple chloroform molecules trapped inside the cage24,25,26. Some chloroform molecules were found at nearly identical positions in the case of the D3-symmetric congener (1b)20 (Supplementary Fig. 2). For instance, one chloroform molecule was found at the center of the cage, which indicated the presence of CH-π hydrogen bonding between them. Based on our interest in weak hydrogen bonds with CH-π contacts27,28,29,30,31,32,33, we then investigated solution-phase assembly of the phenine polluxene and chloroform complex.

Fig. 2: Phenine polluxene 1.
figure 2

a Synthesis of 1a. b Crystal structure of 1a. Three chloroform molecules were identified, and highly disordered residues were removed by the SQUEEZE protocols (Supplementary Fig. 2). The chloroform molecule at the center had two up and down orientations, and the up-oriented molecule is shown for a representative molecule with 50% occupancy. See Supplementary Fig. 2 for a detailed comparison with 1b. dppf 1,1’-bis(diphenylphosphino)ferrocene, THF tetrahydrofuran, Bu butyl, COD 1,5-cyclooctadiene, DMF N,N-dimethylformamide, Me methyl.

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Titration experiments and fitting analyses

Titration experiments were performed to study the association equilibrium for polluxene and chloroform. Thus, to a solution of 1a (0.503 mM) in deuterated cyclohexane (C6D12) was gradually added a solution of chloroform in C6D12 (1.01 M) at 298 K, and the 1H NMR spectra were recorded. The resonance for chloroform showed upfield shifts upon mixing with 1a, and the changes (Δδ) were plotted against the host-guest ratios CHCl30/1a0 (Fig. 3a). The titration was performed in triplicate, and fitting analyses were performed with all of the data to obtain association constants (Kn) by using equations for 1:1 and 1:2 (see Fig. 1b and Supplementary Information for detailed procedures). Both equilibrium equations resulted in acceptable goodness-of-fit (GOF) levels with determination coefficients of R2 > 0.99 for the fit. We first compared fitting credibility with F-tests using the P-values12. The F-test compares the credibilities of two models (1:1 vs. 1:2), and a small P-value below 0.05 indicates that the more complex model (1:2) is more credible13. As shown in Fig. 3a for 1a as the host, a P-value on the order of 10–5 was obtained. We thus concluded that for the equilibrium between 1a and chloroform, 1:2 was more credible than 1:1. The fitting credibility was also examined with the information-theoretic measure of AIC16, and high wi values were obtained for 1:2 (0.9165 with 1a and 0.9405 with 1b) with minute supports for 1:1 (wi < 0.1)17. Thus, two different fitting analyses supported the credibility of 1:2 for the 1a⊃(CHCl3)n complex. Similarly, the association equilibrium between 1b and chloroform was examined by titration. As summarized in Fig. 3b, the credibility of 1:2 for the 1b⊃(CHCl3)n complex was also supported by both the F-test and the AIC.

Fig. 3: Titration experiments for solution-phase assembly of 1⊃(CHCl3)n complexes.
figure 3

The triplicate titration data were fitted with two equilibrium models for 1:1 (red curve) and 1:2 (blue curve). The fit quality was evaluated with R2-values indicating the GOF, and the model credibility was examined with two measures: P-values for the F-test and wi values for the AIC. The blue and red lines show fitting curves for 1:2 and 1:1, respectively. a Experiments with 1a and chloroform. b Experiments with 1b and chloroform. GOF goodness-of-fit, AIC Akaike’s information criterion, Bu butyl.

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van ’t Hoff validation of model credibility

During our examinations of the two-stage association constants for the 1⊃(CHCl3)2 complexes, however, we noticed unreasonable discrepancies, which prompted us to investigate model validation further. Thus, for two phenine polluxene complexes with different substituents (1a: R = H and 1b: R = t-Bu), the first-stage association constants (K1) were comparable at 10–2 M–1, but the second-stage association constants (K2) were significantly different for 1a and 1b and showing a difference of nearly two orders of magnitude (10–3 M–1 vs. 10–1 M–1; Fig. 3). Considering the subtle structural differences at the outer surfaces of the cages, we found this discrepancy chemically unreasonable and started further verification of the models selected for stoichiometry determination. After several trials with alternative measures, we found that the introduction of a thermodynamic verification measure provided an additional criterion for model selection. Thus, we performed triplicate titration experiments at 6 different temperatures (283, 288, 298, 308, 318, and 328 K; Supplementary Fig. 3), and the association constants from the fitting analyses were plotted in ln Kn vs. 1/T graphs for the van ’t Hoff analysis2,3,34,35. For both 1a and 1b with the 1:2 stoichiometry, the data were fitted with the van ’t Hoff equation. The fitting equations with statistical factors are ln K1 = –ΔH1•(1/T) + ΔS1/R and ln K2 = –ΔH2•(1/T) + ΔS2/R – ln 2 where R is the gas constant14,36,37. As shown in Fig. 4a, the R2 values for the van ’t Hoff fits were mostly poor, ranging from 0.0005 to 0.8751. These values showed that the thermodynamics determined with the van ’t Hoff analyses disproved 1:2 as the stoichiometry model for the 1⊃(CHCl3)n complex. We then performed the same van ’t Hoff validation for 1:1. As shown in Fig. 4b, we obtained linear fits with high R2 values of 0.9397 and 0.9714 for 1a and 1b, respectively. With this result from the van ’t Hoff validation, we therefore concluded that phenine polluxene and chloroform formed 1:1 complexes in solution. Further support for this conclusion was derived from theoretical analyses of the 1⊃(CHCl3) complex.

Fig. 4: van ’t Hoff validation of variable-temperature titration data for the 1⊃(CHCl3)n complexes.
figure 4

a Validation of the 1:2 stoichiometry. b Validation of the 1:1 stoichiometry. Thermodynamic parameters from the van ’t Hoff fits are shown.

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Theoretical studies of the 1:1 complex

We performed a theoretical analysis of the 1⊃(CHCl3) complex with density functional theory (DFT) calculations for 1a with Me substituents as models for the t-Bu substituents. As shown in Fig. 5, the structure of 1:1 was obtained from a geometry optimization exhibiting good convergence, whereas the structure of 1:2 with two chloroform molecules did not converge. The association energy of the 1⊃(CHCl3) complex was estimated as ΔE = –5.6 kcal mol–1 at the LC-BLYP/6-311 G(d) level of theory38,39 via counterpoise basis-set superposition error (BSSE) corrections40 in the presence of the polarizable continuum model (PCM)41 for cyclohexane, which reasonably reproduced the experimental association enthalpy of ΔH = –1.5 kcal mol–142,43. As detailed in Supplementary Information, large |δΔHG| values from fitting analyses indicated the presence of unknown origins in experimental values, which might be one of the reasons of the minor difference. The DFT electron density was also subjected to quantum mechanical atoms-in-molecule (AIM) analyses44, which revealed three intermolecular bonding paths for the host and guest (Fig. 5). The bond paths possessed (3,–1) bond critical points (BCPs), which indicated the presence of CH-π hydrogen bonds between the hydrocarbon cage and chloroform27,29,45,46,47. The relative orientations of the chloroform and a phenine panel in the calculated structure were analyzed to show a CH-π distance (Dpln) of 2.34 Å and a CH-π angle (α) of 179.7°27,48. These geometry parameters matched well with those of the crystal structure (see also Fig. 2b). Thus, the theoretical calculations confirmed that the 1:1 stoichiometry provided a reasonable model for the 1⊃(CHCl3)n complex that was uniquely formed by CH-π hydrogen bonding between the hydrocarbon cage and the guest.

Fig. 5: A DFT structure of the 1⊃(CHCl3) complex.
figure 5

The structure was optimized with DFT calculations at the LC-BLYP/6-311 G(d) level of theory with phenine polluxene containing methyl substituents as a model for 1a. The association energy was estimated after counterpoise BSSE corrections in the presence of PCM for cyclohexane. Quantum mechanical AIM analyses revealed the presence of bond paths and (3,–1) BCP. The CH-π distances (Dpln) and CH-π angles (α) are shown for the theoretical model and the crystal structure. BCP: bond critical point, BSSE: basis-set superposition error. PCM: polarizable continuum model.

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Scope: Applicability of the van ’t Hoff validation

Finally, we examined the scope of the van ’t Hoff validation method by assessing its applicability to other systems. Among various host-guest combinations of nanocarbon molecules23,49, we have previously studied a ball-in-bowl assembly between C60 and phenine 5circulene (5) by performing variable-temperature titration experiments with NMR spectroscopy50,51. The triplicate titration data comprising 252 1H NMR spectra were thus subjected to re-analysis for the fitting analyses, and the fitted data for 1:1, 1:2 and 2:1 models were first compared by using the wi values from AIC analyses16. As shown in Fig. 6, although moderate preference was found for a chemically reasonable 1:1 model that was observed in the crystal structure, the wi values did not consistently support one model over the 6 variable-temperature conditions.

Fig. 6: An assessment study with a ball-in-bowl nanocarbon assembly (5⊃C60): AIC.
figure 6

The wi values for three models (1:1, 1:2 and 2:1) showed that all the models could be supported. See Supplementary Fig. 11 for further details of fitting analyses. Bu butyl.

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We then subjected the fitted data to the van ’t Hoff validations by plotting the K values in the 1/T-ln K graphs. As shown in Fig. 7, the 1:1 model showed a linear relationship with a high R2 value of 0.8968, whereas other models (1:2 and 2:1) failed to show consistent linear relationships. Considering the observation of the 1:1 crystal structure of 5⊃C60 (Fig. 6), we concluded that the van ’t Hoff validation method was applicable to this ball-in-bowl system that was tightly assembled with a K value of (3.9 ± 0.1) × 104 M–1 (298 K). The results showed that the van ’t Hoff validation method can be versatile to be applied to many other supramolecular systems.

Fig. 7: An assessment study with a ball-in-bowl nanocarbon assembly (5⊃C60): van ’t Hoff validation.
figure 7

The Kn values from the fitting analyses with three models (1:1, 1:2 and 2:1) were respectively plotted in the 1/T-ln K graph.

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Discussion

In summary, we investigated measures and methods for determining the stoichiometries of supramolecular complexes. The results obtained from careful analyses of 954 1H NMR spectra showed that drawing conclusions about the stoichiometry from a single measure is risky, and we propose the following procedure for stoichiometry analyses. 1. Fit the titration data with multiple models. 2. The GOF for the fit should be evaluated with the R2 value. 3. The credibilities of model stoichiometries should be examined with quantitative measures such as P-values and/or wi values. 4. The Ka values derived from the step 1 should be examined ideally by comparing congeners and structural models. 5. Model comparisons should be validated by performing van ’t Hoff analyses. When the van ’t Hoff validations indicate non-linear relationships in the 1/T-ln K graph, one may also need to check the existence of temperature-dependent enthalpy, which should give rise to new subjects to be investigated2,3,34,52. 6. By examining the results from the necessary measures and steps, and probably with the aid of theoretical calculations, a chemically sound stoichiometry should be concluded. In the present study, we found that a phenine polluxene hydrocarbon cage served as an interesting host for a small molecule such as chloroform via formation of CH-π hydrogen bonds. Further structural diversifications are currently under investigation, and this study showed that host-guest complexations should also be considered for the development of unique functions.

Methods

Syntheses

The syntheses of the precursors are described in Supplementary Information, and the final cyclization step for 1a is described here. A mixture of 2,2’-bipyridine (848 mg, 5.43 mmol), 1,5-cyclooctadiene (668 μL, 5.43 mmol) and Ni(cod)2 (1.49 g, 5.43 mmol) was stirred in DMF (11.3 mL) at 80 °C for 30 min. A solution of 4 (115 mg, 67.8 µmol) in DMF (22.6 mL) was then added dropwise over 120 min, and the mixture was stirred at 80 °C for 2 h. After the mixture was cooled down to ambient temperature, 2 M aq. HCl (30 mL) was added. The organic layers was separated, and the aqueous layer was extracted with chloroform (20 mL × 3). The combined organic layer was washed with saturated aq. NaHCO3 (50 mL), dried over Na2SO4 and concentrated in vacuo. The crude material was purified by gel permeation chromatography (GPC) to afford 1a in 49% yield (54.0 mg, 33.2 µmol). 1H NMR (600 MHz, CDCl3) δ 7.96 (s, 2H), 7.93 (s, 2H), 7.91 (s, 4H), 7.79 (s, 2H), 7.78 (s, 4H), 7.76-7.74 (m, 10H), 7.65-7.63 (m, 6H), 7.60-7.58 (m, 6H), 7.57 (s, 4H), 7.56 (s, 4H), 1.49 (s, 36H), 1.37 (s, 36H), 1.36 (s, 18H); 13C NMR (151 MHz, CDCl3) δ 152.4, 152.3, 152.2, 143.3, 143.1, 143.1, 142.9, 142.8, 142.3, 141.3, 141.1, 141.1, 140.9, 129.4 (CH), 127.7 (CH), 126.2 (CH), 125.8 (CH), 125.8 (CH), 125.2 (CH), 124.8 (CH), 124.6 (CH), 124.5 (CH), 124.3 (CH), 124.0 (CH), 123.8 (CH), 123.7 (CH), 123.4 (CH), 123.1 (CH), 35.2, 35.1, 35.1, 31.6 (2 overlapping CH3), 31.6 (CH3); HRMS (MALDI-TOF) (m/z): M+ calcd. for C124H134 1623.0480, found 1623.0497. Chromatograms and spectra are included in Supplementary Information.

X-ray crystallographic analysis

A single crystal was grown from a solution of 1a in chloroform by slowly diffusing methanol vapor at 25 °C. The crystal was mounted on a thin polymer tip with cryoprotectant oil and frozen via flash cooling. The diffraction experiment was carried out at 95 K with a synchrotron X-ray source at the BL17A beamline equipped with a Dectris EIGER X 16 M PAD detector (KEK Photon Factory), and the diffraction data were processed with XDS53. The structure was solved by direct methods with SHELXT54 and refined by full-matrix least-squares on F2 using SHELXL-2018/355 running with Yadokari-XG 200956. In the refinements, the t-Bu groups and solvent molecules were restrained by SIMU, DELU, ISOR, DFIX and DANG. The nonhydrogen atoms were analyzed anisotropically, and hydrogen atoms were input at the calculated positions and refined with a riding model. Some solvent molecules could not be modeled due to severe disorder, and the residual density was eliminated by using the PLATON/SQUEEZE protocol57,58. Crystal and structure refinement data are included in Supplementary Information.

Titration experiments and fitting analyses

A solution of 1a (5.63 mg, 3.47 µmol/Precision balance XPE205V, Mettler Toledo) was prepared in C6D12 (0.690 mL; 5.02 mM/Gastight syringe1001, Hamilton), and a solution of chloroform (242 mg, 2.02 mmol/XPE205V) was prepared in C6D12 (2.00 mL; 1.01 M/Gastight syringe1001). To the solution of 1a in an NMR tube (0.500 mL/Gastight syringe1001) was added 1.01 M solution of chloroform (+1.00 µL/Gastight syringe1701), and 1H NMR spectra were respectively recorded at 283 K, 288 K, 298 K, 308 K, 318 K and 328 K. When the temperature was adjusted, the sample was maintained at the temperature for approximately 15 min before recording the spectrum. The spectra were subjected to fourfold zero filling to secure a digital resolution of 0.013 Hz (6.6 kHz range/512k points). The NMR titration experiments were performed by adding additional chloroform solution (+3.00 µL, +4.00 µL, +4.00 µL, +6.00 µL, +7.00 µL/Gastight syringe1701, +25.0 µL, +50.0 µL/Microsyringe MSGFN50, Ito, +100 µL, +100 µL, +100 µL, +200 µL and +200 µL/Gastight syringe1725, and these were repeated three times to generate triplicate titration data. The spectra were recorded on JEOL RESONANCE JNM-ECA II 600 spectrometer equipped with a temperature-controlled UltraCOOL probe (error: ±0.1 °C). A solution of 1b (8.21 mg, 4.73 µmol) was also prepared in C6D12 (0.940 mL; 5.03 mM), which was subjected to identical titration experiments with a solution of chloroform (238 mg, 1.99 mmol) in C6D12 (2.00 mL; 1.00 M). As a third entry of the cage host, 1c was subjected to the titration experiment by using 1c (5.91 mg, 3.51 µmol) in C6D12 (7.00 mL; 5.01 mM) and chloroform (240 mg, 2.01 mmol) in C6D12 (2.00 mL; 1.01 M). The chemical shift of chloroform was measured by using a singlet peak from C6H12 at 1.38000 ppm as a reference, and the changes in the chemical shifts (Δδ) were derived by using a chemical shift of unbound free chloroform (7.11590 ppm) as a standard. In total, 702 1H NMR spectra were recorded and used for the analyses. All the source data for the fitting analyses were provided in Supplementary Information (excel files). The data were plotted against CHCl30/10 (Fig. 3), where CHCl30 and 10 are the total concentrations of the guest and host. To derive the association constants (Kn), the titration data were fitted with the following two equations6:

$${\Delta }\delta {=}\frac{{\delta }_{{\Delta }{{{{{\rm{H}}}}}}{{{{{\rm{G}}}}}}}{{K}}_{1}[{{{{{\rm{H}}}}}}][{{{{{\rm{G}}}}}}]}{{[{{{{{\rm{G}}}}}}]}_{0}}$$
(1)
$${\Delta }\delta {=}\frac{{\delta }_{{\Delta }{{{{{\rm{H}}}}}}{{{{{\rm{G}}}}}}}{{K}}_{1}[{{{{{\rm{G}}}}}}]+2{\delta }_{{\Delta }{{{{{{\rm{H}}}}}}{{{{{\rm{G}}}}}}}_{2}}{K}_{1}{K}_{2}{[{{{{{\rm{G}}}}}}]}^{2}}{1+{K}_{1}[{{{{{\rm{G}}}}}}]+{K}_{1}{K}_{2}{[{{{{{\rm{G}}}}}}]}^{2}}\cdot \frac{{[{{{{{\rm{H}}}}}}]}_{0}}{{[{{{{{\rm{G}}}}}}]}_{0}}$$
(2)

where H is the host (1) and G is the guest (chloroform) in the 1:1 and 1:2 equilibrium models, respectively, and OriginPro 2023 (OriginLab Corporation) was used for fitting. The experimental 1H NMR spectra with resonances derived from the fitting analyses are shown in Supplementary Fig. 4 (1:1) and Supplementary Fig. 5 (1:2), which showed a higher consistency between 1a and 1b for the 1:1 model. The fitted data were analyzed with GOF/R2 values, F-test/P-values and AIC/wi values, and the fitting data as well as Kn values are included in Supplementary Information along with the equations and codes. To derive the determination coefficients, the temperature-dependent Kn values were plotted in 1/T-ln Kn graphs and fitted with the van ’t Hoff equations with statistical factors considered for one-site hosts: ln K1 = –ΔH1•(1/T) + ΔS1/R and ln K2 = –ΔH2•(1/T) + ΔS2/R – ln 214,37. To confirm the repeatability of the titration experiments, independent titration data at 298 K were separately fitted to derive averaged Kn values with standard deviations (SD) of three titrations as follows (Kn ± SD). 1a⊃(CHCl3)1: K1 = (3.3 ± 0.2) × 10–2 M–1; 1a⊃(CHCl3)2: K1 = (3.8 ± 4.4) × 10–2 M–1, K2 = (8.4 ± 6.1) × 10–3 M–1; 1b⊃(CHCl3)1: K1 = (3.0 ± 0.2) × 10–2 M–1; 1a⊃(CHCl3)2: K1 = (1.1 ± 0.4) × 10–2 M–1, K2 = 9.6 ± 9.8 M–1. The SD values are reasonably low particularly for 1:1 models. We also analyzed the fitted chemical shifts for 1:1 and 1:2 both with 1a and 1b as the host (see Supplementary Information). The results with 1c were summarized in Supplementary Figs. 6–9.

Theoretical studies with DFT calculations

DFT calculations at the LC-BLYP/6-311 G(d) level38 of theory were performed with Gaussian 1659. For the polluxene hosts, the t-Bu substituents were replaced by Me groups in the model. Geometry optimizations were performed for 1, chloroform and the 1⊃(CHCl3) complex in the presence of cyclohexane PCM41, and the association energies were estimated with BSSE corrections40. Quantum mechanical AIM analyses were performed with Multiwfn60. The results are summarized in Fig. 5 and Supplementary Fig. 12, and the cartesian coordinates are provided as Source Data.

Assessment study of the van ’t Hoff validation method via re-analyses

The titration data from the previous report of 5⊃C60 were used for the re-analyses51. For the titration experiment, 5 and C60 were mixed in CDCl3 at 14 different ratios for triplicate titrations under 6 different temperature conditions to record 14 × 3 × 6 = 252 spectra (Supplementary Fig. 10). A 1H resonance of 5 at the most downfield region of 9.008 ppm (δunbound) was used as a reference, and the Δδ values were respectively fitted with three different models (1:1, 1:2 and 2:1 models) (Supplementary Fig. 11). At the stage 3, AIC analyses were performed to obtain wi values (Fig. 6), which failed to support one single model. The temperature-dependent K values were then plotted in 1/T-ln K graphs for the van ’t Hoff validations (Fig. 7). The van ’t Hoff validation supported the 1:1 model by showing a linear relationship in the graph, which matched well with the structure observed in the crystal (see Fig. 6). In the previous study, an averaged aromatic resonance values were used for the fitted analyses, and minor differences were found for the K values and thermodynamic parameterers51.