Pathway selection in the self-assembly of Rh₄L₄ coordination squares under kinetic control

Abstract

Pathway selection principles in reversible reaction networks such as molecular self-assembly have not been established yet, because achieving kinetic control in reversible reaction networks is more complicated than in irreversible ones. In this study, we discovered that coordination squares consisting of cis-protected dinuclear rhodium(II) corner complexes and linear ditopic ligands are assembled under kinetic control, perfectly preventing the corresponding triangles, by modulating their energy landscapes with a weak monotopic carboxylate ligand (2,6-dichlorobenzoate: dcb^–) as the leaving ligand. Experimental and numerical approaches revealed the self-assembly pathway where the cyclization step to form the triangular complex is blocked by dcb^–. It was also found that one of the molecular squares assembled into a dimeric structure owing to the solvophobic effect, which was characterized by nuclear magnetic resonance spectroscopy and single-crystal X-ray analysis.

Introduction

Pathway-dependent molecular self-assembly has often been seen in the realm of nature, especially in the multicomponent self-assembly of viruses¹ and proteasomes². One of the reasons why such a sequential assembly process is adopted is because the formation of undesired assemblies whose structures are far from the desired one is efficiently prevented in the assembly process. If there exist some other assemblies with similar thermodynamic stabilities, selective assembly of the desired one under thermodynamic control is impossible. In such cases, pathway selection under kinetic control is inevitable, but the general principles of pathway selection in complicated reversible reaction networks have yet to be fully understood³.

Molecular square is one of the most representative molecular architectures in coordination self-assembly^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}. Thanks to the L–M–L bond angle of 90° around transition metal ions with octahedral and square planer coordination spheres, upon complexation with linear ditopic ligands, square complexes are expected to be thermodynamically most stable over other polygonal structures. However, in many cases triangular complexes were coproduced in solution even using rigid linear ditopic ligands as the side of the polygons, although triangles are geometrically largely different from squares^{25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45}. Therefore, the selective formation of molecular squares and the understanding of their origin remain challenging in supramolecular chemistry⁴⁶.

Under the condition where the energy barriers between square and triangular compounds are relatively high, there is a chance to produce the square in a higher yield than that obtained at equilibrium, kinetically preventing the triangle. Such a kinetic approach is quite limited in artificial molecular self-assembly, except for unexpected observations of kinetic traps^{47,48,49,50,51,52,53,54,55}. Our understanding of the energy landscapes of molecular self-assembly is not as thorough as we can rationally design them, and to the best of our knowledge, there is no report on the selective formation of a triangular or square complex under kinetic control.

Here, we report the self-assembly of the Rh₄L₄ squares in solution under kinetic control, perfectly preventing the formation of the Rh₃L₃ triangle using a designed monotopic carboxylate as the leaving ligand (dcb^– in Fig. 1). Experimental and numerical analyses of the self-assembly mechanism indicate that the free energy landscape of the self-assembly was properly modulated so that the pathways to the triangular complex were blocked. Furthermore, the monocarboxylate facilitates ligand exchanges, resulting in the conversion of the kinetically trapped triangles into the squares under mild condition. It was also found that one of the Rh₄L₄ squares aggregates to form a structurally well-defined dimer in solution by the solvophobic effect, owing to the electronically neutral nature of the Rh(II) complexes.

Fig. 1: Self-assembly of the Rh₄L₄ squares in this study.

Schematic representation of kinetically controlled self-assembly of the Rh₄L₄ squares from cis-protected dinuclear Rh(II) complex (Rh²⁺: [Rh₂(DAniF)₂]²⁺) and linear dicarboxylate ligand (L^2–: 1^2– or 2^2–) in solution. When the metal source is [Rh(CH₃CN)₄]²⁺, where CH₃CN is the leaving ligand, Rh₃L₃ triangle and Rh₄L₄ square complexes were produced in a 1:2 ratio. In contrast, when the self-assembly was carried out using Rh(dcb)₂ as the metal source (dcb^–: 2,6-dichlorobenzoate), dcb^– modulates the energy landscape of the self-assembly so that the formation of the Rh₃L₃ triangle is prevented, producing the Rh₄L₄ square only. The axial ligands on the dinuclear Rh(II) centers are omitted for clarity.

Results and discussion

Self-assembly of Rh(II)-triangle and square with CH₃CN as the leaving ligand

Cotton and his co-workers and others reported that molecular squares composed of cis-protected dinuclear Rh(II) complexes (Rh²⁺: [Rh₂(DAniF)₂]²⁺⁵⁶ and [Rh₂(O₂C-R-CO₂)]²⁺⁵⁷) and linear dicarboxylates (L^2–) were selectively obtained in crystalline states^56,57. As Rh(II)–carboxylate coordination bonds in dinuclear Rh(II) complexes are relatively strong, the reversibility of the coordination bond is low⁵⁸, which would allow the self-assembly of the Rh₄L₄ squares to proceed under kinetic control. Thus, we were interested in why the Rh₄L₄ squares were selectively produced in the solid state in the previous research.

We first performed the self-assembly from [Rh(CH₃CN)₄](BF₄)₂ and 1^2– in CH₃CN ([1^2–] = 10 mM) according to the literature⁵⁶. Dark red solid precipitated in 1 day. Its ¹H NMR spectrum in CDCl₃ showed two prominent singlet signals of H^d (aromatic protons of 1^2– as shown in Fig. 2) in a 1:3 integral ratio, suggesting the formation of two highly symmetric cyclic structures (Supplementary Fig. 1). Crystallization of this mixture in CHCl₃ gave single crystals of Rh₄1₄ as was reported in the literature⁵⁶. The major ¹H NMR signals are the same as those of Rh₄1₄ (Supplementary Fig. 2), so the minor signals are expected to correspond to the Rh₃1₃ triangle, which is supported by ¹H DOSY spectroscopy (Supplementary Fig. 3), ESI-TOF mass spectrometry (Supplementary Fig. 4a), and other experiments (details are shown in Supplementary Fig. 2). Therefore, the Rh₄1₄ square and the Rh₃1₃ triangle were produced in CH₃CN.

Fig. 2: ¹H NMR spectra of the self-assembly of the Rh₄1₄ square.

¹H NMR spectra (500 MHz, CDCl₃, 298 K, aromatic region) of the self-assembly of the Rh₄L₄ squares under various conditions ([1^2–] = [Rh²⁺] = 1 mM). a Self-assembly from [Rh(CH₃CN)₄](BF₄)₂ and 1^2– in CDCl₃ at 298 K, giving the Rh₃1₃ triangle and the Rh₄1₄ square in 23% and 39% yields, respectively. Asterisks indicate the carbon satellite of CHCl₃. b Addition of n-Bu₄N·dcb in a mixture of the Rh₃1₃ triangle and the Rh₄1₄ square obtained from the self-assembly of [Rh(CH₃CN)₄](BF₄)₂ and 1^2– in CDCl₃ after convergence. The Rh₃1₃ triangle was converted into the Rh₄1₄ square at 298 K assisted by dcb^–, although heating at 100 °C for 2 days is necessary without dcb^– (Supplementary Fig. 14). c Self-assembly of the Rh₄1₄ square from Rh(dcb)₂ and 1^2– in CDCl₃ at 298 K to produce the Rh₄1₄ square in a 65% yield without formation of the Rh₃1₃ triangle during self-assembly. The yields were determined based on the internal standard.

To investigate the solution process, self-assembly was performed in CDCl₃ at 298 K ([1^2–] = 1.0 mM), where precipitation did not occur during the self-assembly, resulting in a mixture of Rh₃1₃ and Rh₄1₄ in a 1:2 ratio (Fig. 2a). Similarly, self-assembly of [Rh(CH₃CN)₄](BF₄)₂ and another ditopic ligand (2^2–) in CDCl₃ gave the triangular and square complexes in a 1:2 ratio (Supplementary Figs. 2, 4c, 5, and 6). Therefore, the self-assembly of [Rh(CH₃CN)₄]²⁺ and the linear ditopic ligands (L^2–) at room temperature gave a mixture of a molecular square (major) and a triangle (minor) in solution, and the Rh₄1₄ square was selectively crystallized from the mixture.

Kinetic self-assembly of Rh(II)-squares

We then investigated the kinetic effect of the leaving ligand on Rh-based self-assembly. 2,6-Dichlorobenzoate (dcb^–) was chosen as the leaving ligand (Fig. 1). Because the pK_a value of Hdcb in H₂O (1.82)^59,60 is lower than that of benzoic acid (4.21), the coordination ability of dcb^– is lower than those of the ditopic ligands (L^2–), suggesting that dcb^– can act as a leaving ligand. To test the potential of dcb^– as a leaving ligand, the ligand exchange of dcb^– in Rh(dcb)₂ with p-toluate (tol^–) in CDCl₃ was monitored by ¹H NMR spectroscopy (Supplementary Fig. 7a). The ligand exchange proceeded at 298 K to produce Rh(tol)₂ in 82% yield (Supplementary Fig. 7b), indicating that dcb^– can be used as a leaving ligand in Rh-based self-assembly at room temperature.

The self-assembly of Rh(dcb)₂ and L^2– (1^2– and 2^2–) in CDCl₃ at 298 K was monitored by ¹H NMR spectroscopy (Fig. 2c and Supplementary Fig. 8). Surprisingly, the ¹H NMR signals of the Rh₃L₃ triangle did not appear at all during the self-assembly for 1^2– and 2^2–. These results indicate that the energy landscape of the self-assembly was dramatically altered by dcb^–, kinetically producing only the Rh₄L₄ squares.

Self-assembly mechanism of Rh(II)-squares

We were interested in why the Rh₄L₄ squares were selectively produced with dcb^– as the leaving ligand under kinetic control, so the self-assembly processes were investigated by QASAP (quantitative analysis of self-assembly process)^61,62,63. Basically, it is difficult to obtain information about intermediates in molecular self-assembly, because most of intermediates are unobservable. Indeed, only small minor signals for intermediates were observed by ¹H NMR spectroscopy during the self-assembly of the Rh₄L₄ squares (Fig. 2c and Supplementary Fig. 8). In such cases, QASAP provides valuable insight into the self-assembly pathway from time-development of the average composition of all intermediates even when the intermediates cannot be observed. We have applied QASAP to Pd(II)- and Pt(II)-based coordination self-assemblies so far to determine the major self-assembly pathway(s) and the rate-determining step. QASAP of the Rh₄L₄ squares is the first example of QASAP for metal-organic assemblies with transition metals other than Pd(II) and Pt(II) ions.

The substrates (Rh(dcb)₂ and L^2–) and the products (Rh₄L₄ and dcb^–) were quantified by ¹H NMR spectroscopy (Fig. 3c), and the change in the average composition of all intermediates with time was plotted in a 2D map (n-k map) (Fig. 3b). The species regarding the self-assembly are expressed by Rh_aL_b(dcb)_c (a–c are 0 or positive integer), which is indicated as (a,b,c) for simplicity. The (n, k) values of Rh_aL_b(dcb)_c are defined as n = (2a – c)/b and k = a/b^61,62,63. The n value indicates the average number of Rh²⁺ bound to a single ditopic ligand, L^2– . The k value indicates the ratio of Rh²⁺ against L^2–. The experimentally obtained (n, k) values, which is indicated as (〈n〉, 〈k〉), are calculated for the average composition of all intermediates (Rh_〈a〉L_〈b〉(dcb)_〈c〉), whose time-development was used for the discussion on the self-assembly process (Fig. 3b).

Fig. 3: Self-assembly mechanism of the Rh₄2₄ square under kinetic control.

a Three types of possible chain intermediates in the self-assembly of M₄L₄ square. Type I: M_mL_m+1, Type II: M_m+1L_m+1(dcb), Type III: M_m+2L_m+1(dcb)₂. b Plots of the existence ratios of the substrates and products in the self-assembly of the Rh₄2₄ square from Rh(dcb)₂ and 2^2– in CDCl₃ at 298 K. [Rh]₀ = [2^2–]₀ = 0.86 mM. c Plots of the (〈n〉, 〈k〉) values in the n-k map of the Rh₄2₄ square (red filled circles). Green crosshairs indicate the (n, k) values of the chain intermediates. The three types of chain intermediates, Types I, II, and III, are plotted on each straight line. The brown number indicates m in each type of oligomer in a. The definitions of 〈n〉 and 〈k〉 values are shown in the main text (Eqs. 1 and 2). The data in (b) and (c) are shown as the average of the three runs of QASAP with standard errors. d Comparison of the existence ratio of the substrates and the products between QASAP (red) and NASAP (blue). e Comparison of the n-k plot between QASAP (red) and NASAP (blue). Both (d) and (e) indicate that the numerical simulation results reproduce the experimental counterparts well. f Dominant self-assembly pathways of the Rh₄2₄ square. Red arrows indicate the major self-assembly pathway. (a,b,c) indicates Rh_a2_b(dcb)_c. The numbers above the reaction arrows indicate the elementary reactions with high net reaction frequency listed in Supplementary Table 4. Longer Type II oligomers than (4,4,1), such as (5,5,1) and (6,6,1), are produced during the self-assembly, although they are not involved in the major self-assembly pathway (Supplementary Fig. 13). g Two possible pathways to produce triangle (3,3,0) from Type II intermediate (3,3,1) (left) and from Type II oligomers with more than three Rh²⁺ units (4+x, 4+x, 1) (x ≥ 0) (right). The green sphere indicates the leaving ligand(s) (dcb^– or two molecules of CH₃CN). h A plausible key structure in the triangular formation through associative ligand exchange process. Steric repulsion caused by ArCO₂^– (in green) in the cyclic intermediate and transition state of the triangular formation process would prevent the cyclization.

There are three types of chain intermediates classified according to the difference in the terminals (Types I–III) (Fig. 3a)^64,65. The (n, k) values of the three types of oligomers are distinctly plotted on different straight lines in the n-k map (Fig. 3b). With an increase in the degree of oligomerization (m in Fig. 3a), the (n, k) values become close to (2, 1), which is the (n, k) value of the cyclic structures (triangle and square). Thus, the (n, k) plot enables us to discuss which type of oligomers was mainly produced and how large the chain-like oligomers grew during self-assembly.

With regard to QASAP for the Rh₄1₄ square, the 〈n〉 value was smaller than 1 at an early stage (Supplementary Fig. 9 and Supplementary Table 1), which suggests that Rh²⁺ units in the intermediates have more than two carboxylate ligands (1^2– and/or dcb^–), so further analysis could not be performed (for detailed discussion, see Supplementary Fig. 9). Such a strange behavior was not found for the self-assembly of the Rh₄2₄ square (Fig. 3b and Supplementary Fig. 10 and Supplementary Table 2). The 〈n〉 value increased with an almost constant 〈k〉 value of 1, indicating the formation of Type II oligomers as main intermediates, whose m value finally reached 3 or 4, corresponding to 4 or 5 Rh²⁺ units in the oligomers.

Numerical analysis of self-assembly pathway

The major self-assembly pathway was determined by numerical analysis (NASAP: numerical analysis of self-assembly process)⁶⁶. In NASAP, the QASAP data were numerically analyzed based on the reaction network model where possible intermediates and their elementary reactions were considered. We prepared a reaction network model consisting of 34 species with a maximum number of Rh²⁺ of 10 and 1694 elementary reactions between them (Supplementary Fig. 11). The elementary reactions in the network were classified into 10 groups according to the reaction type (Supplementary Fig. 12). Then, the forward and backward rate constants for the elementary reactions were used as variable parameters to fit the QASAP data. No good dataset was found unless the rate constants of the triangular formations (k₈ and k₉) were set to almost zero, which is consistent with the experimental result that the ¹H NMR signals of Rh₃2₃ did not appear during the self-assembly (Supplementary Fig. 8). A set of rate constants that reproduced the experimental results well (Fig. 3d, e) was found (Supplementary Table 3).

With the dataset of the rate constants thus obtained, the self-assembly of Rh₄2₄ was numerically simulated in the reaction network, which enabled us to determine the major self-assembly pathway. The top 10 elementary reactions with high net frequencies are listed in Supplementary Table 4. Net frequency represents the difference between the occurrence in the forward direction and that in the backward direction for an elementary reaction; thus, the net frequency indicates the actual degree of progress of the elementary reaction⁶⁷. Connecting the elementary reactions with the highest net reaction frequency from Rh₄2₄ to the substrates resulted in the dominant self-assembly pathways (Fig. 3f). As expected from the n-k analysis, Rh₄2₄ was assembled, producing Type II species as the major intermediates, and the largest intermediate in the major pathway is Rh₄2₄(dcb)₁, which is indicated as (4,4,1) (Fig. 3f).

The change in the existence ratios of the intermediates in the major self-assembly pathway shows that the existence of (4,4,1) is much smaller than that of the other intermediates (Supplementary Fig. 13), indicating that the cyclization of (4,4,1) occurs quickly and that the formation of (4,4,1) from (2,2,1) and from (3,3,1) is the rate-determining steps (Fig. 3f)⁶⁸.

Reason for the selective formation of Rh(II)-squares under kinetic control

Next, we discuss why the pathways to the Rh₃L₃ triangles were blocked with dcb^– as the leaving ligand. There are two types of reactions that produce Rh₃L₃ by the cyclization of Type II intermediates: (i) the cyclization of (3,3,1) to form (3,3,0) by breaking the coordination bond between Rh(II) and the leaving ligand (CH₃CN or dcb^–) (Fig. 3g, left) and (ii) the cyclization of oligomers longer than (3,3,1), (4+x, 4+x, 1) (x ≥ 0), by breaking a Rh–L coordination bond (Fig. 3g, right). Note that the latter reaction is not affected by the leaving ligand. Therefore, the formation of Rh₃L₃ triangles from (4+x, 4+x, 1) (x ≥ 0) should be prevented either using Rh(dcb)₂ or [Rh(CH₃CN)₄]²⁺. In addition, when dcb^– is the leaving ligand, the cyclization of (3,3,1) to form (3,3,0) should be prevented. In other words, the cyclization of Rh₃L₃(CH₃CN)₂ occurs, while that of Rh₃L₃(dcb) is kinetically blocked.

Considering the associative ligand exchange mechanism of Rh(II)–carboxylate bonds^69,70, triangular formation should take place by the attack of the terminal non-coordinated carboxylate group of L to the axial site of a Rh(II) center in Type II intermediates (Fig. 3h). The instability of a resultant cyclic intermediate or a transition state with dcb^– would be the reason for the suppression of the cyclization of Rh₃L₃(dcb).

Conversion of Rh(II)-triangles into Rh(II)-squares assisted by dcb^–

The conversion between the triangular and square complexes was very slow due to the inertness of Rh–carboxylate bonds. Heating a 1:2 mixture of Rh₃1₃ and Rh₄1₄ in CDCl₃ at 60 °C did not show any change of their ratio (Supplementary Fig. 14a). The conversion of Rh₃1₃ into Rh₄1₄ was realized by heating the mixture in dimethylacetamide (DMA) at 100 °C for 2 days (Supplementary Fig. 14b).

Considering the associative ligand exchange mechanism of Rh(II)-carboxylate bonds, the ligand exchange is expected to be facilitated by monotopic carboxylate with weak coordination ability such as dcb^–. Thus, n-Bu₄N·dcb was added to a mixture of Rh₃1₃ and Rh₄1₄ in CDCl₃, and the reaction at 298 K was monitored by ¹H NMR spectroscopy (Fig. 2b and Supplementary Table 5). The signals assigned to Rh₃1₃ slowly decreased with time and almost disappeared after 18 h. The conversion of Rh₃1₃ into Rh₄1₄ was also confirmed by ESI-TOF mass spectrometry (Supplementary Fig. 4b). Likewise, the conversion of Rh₃2₃ into Rh₄2₄ also occurred at 298 K in 13 h with dcb^– (Supplementary Figs. 4d and 15 and Supplementary Table 6). These results indicate that dcb^– greatly decreased the energy barriers of the conversion of Rh₃L₃ triangles to Rh₄L₄ squares as a catalyst. However, the time scale of the conversion of the triangle into the square using the catalyst is much slower than that of the assembly of the square from the substrates (Rh₂(dcb)₂ and the ditopic ligand). The idea that the selective self-assembly of the Rh₄L₄ squares took place by the conversion of the Rh₃L₃ triangles into the Rh₄L₄ squares by the catalytic effect of dcb^– during the self-assembly is ruled out by the fact that the signals of the Rh₃L₃ triangles were not observed during the self-assembly (Fig. 2c and Supplementary Fig. 8).

Supramolecular dimerization of Rh(II)-square by solvophobic effect

The dinuclear Rh(II)-based assemblies are neutral molecules, which is largely different from other positively or negatively charged coordination assemblies with counter ions. One of the advantages of neutral coordination assemblies is that they can be assembled into higher order structure without electrostatic repulsion between themselves, leading to a closely packed structure in the crystalline state and in solution. A new set of signals appeared when Rh₄1₄ was dissolved in CD₃NO₂/CDCl₃ (9:1 (v/v)) (Fig. 4c). A similar spectral change was also observed in acetone-d₆ and CD₃CN with 10 volume% of CDCl₃ (Supplementary Fig. 16). A significant up-field shift of aromatic protons (H^a and H^b) suggests the aggregation of the square (Fig. 4a). All signals except for H^c were observed as two sets with a 1:1 integral ratio, which is consistent with the crystal structure of the [Rh₄1₄(dmso-S)₄]₂ dimer (Fig. 4b) obtained from acetone/DMSO.

Fig. 4: Dimerization of the Rh₄1₄ square in solution.

a Supramolecular dimer formation of the Rh₄1₄ square by the solvophobic effect. b Crystal structure of [Rh₄1₄(dmso-S)₄]₂. Two Rh₄1₄ squares engaged each other are shown in red and blue. DMSO molecules axially coordinating to the Rh(II) centers are colored in green. c ¹H NMR spectra (500 MHz, 298 K, aromatic region) of (Rh₄1₄)₂ in CD₃NO₂/CDCl₃ (9:1 (v/v)) and Rh₄1₄ in CDCl₃. d ¹H DOSY spectrum of (Rh₄1₄)₂ in CD₃NO₂/CDCl₃ (9:1 (v/v)).

The (Rh₄1₄)₂ dimer in CD₃NO₂/CDCl₃ (9:1 (v/v)) was characterized by ¹H DOSY, (H, H)-COSY, and (H, H)-NOESY spectroscopy (Fig. 4d and Supplementary Figs. 17 and 18). A cross peak was found between H^d(out) and H^d(in) protons in the NOESY spectrum (Supplementary Fig. 18), indicating the chemical exchange of the two proton signals caused by the rotation of the 1,4-phenylene ring in 1^2–, which would be the reason for the broadening of the two H^d signals (Fig. 4c). The equilibrium of the monomer (Rh₄1₄) and the dimer ((Rh₄1₄)₂) shifted toward the dimer with an increase in the composition ratio of CD₃NO₂/CDCl₃ (Supplementary Fig. 19), indicating the dimerization of the Rh₄1₄ square due to the solvophobic effect. The ¹H NMR signals of the monomer did not appear even at a low concentration of [Rh₄1₄]₀ = 4.6 μM (Supplementary Fig. 20). Further dilution caused the NMR spectrum to significantly broaden with a low S/N ratio. According to these results, the dimerization constant for (Rh₄1₄)₂ is estimated to be higher than 10⁷M^–1 under the assumption that 10% monomer exists (Supplementary Fig. 20).

Conclusions

In conclusion, the self-assembly of the Rh₄L₄ squares was kinetically controlled to produce Rh₄L₄ squares selectively assisted by monotopic carboxylate ligand (dcb^–), which prevents the cyclization of the Rh₃L₃(dcb) chain intermediate. This is the first example of the selective formation of coordination squares under kinetic control in solution. The metastable yet kinetically highly stabilized Rh₃L₃ triangles can be transformed into the Rh₄L₄ squares with dcb^– under very mild conditions. These results indicate that the energy landscape of the dinuclear Rh(II)-based coordination self-assembly can be tuned by the leaving ligand. This basic knowledge can be applied to a wide range of Rh(II)-based metal-organic polyhedra (MOPs)^71,72,73, whose self-assembly is limited compared with MOPs composed of other transition metal ions owing to the relatively inert equatorial Rh(II)–carboxylate bonds. Because Rh(II)-based MOPs possess high thermal and chemical stabilities and catalytic activity, they are expected as a platform for new materials, so exploring the novel structures of Rh(II)-based MOPs and their efficient self-assembly by modulating the energy landscapes will make a great progress in this field.

Methods

General information

¹H NMR spectra were recorded using a Bruker AV-500 (500 MHz) spectrometer. All ¹H NMR spectra were referenced using a residual solvent peak, CDCl₃ (δ 7.26), acetone-d₆ (δ 2.05), CD₃NO₂ (δ 4.33). Electrospray ionization time-of-flight (ESI-TOF) mass spectra were obtained using a Waters Xevo G2-S Tof mass spectrometer.

Materials

Unless otherwise noted, all solvents and reagents were obtained from commercial suppliers (TCI Co., Ltd., WAKO Pure Chemical Industries Ltd., KANTO Chemical Co., Inc., and Sigma-Aldrich Co.) and were used as received. Deuterated solvents were used after dehydration with Molecular Sieves 4 Å. [Rh(CH₃CN)₄](BF₄)₂ (Rh: Rh₂(DAniF)₂, DAniF: N,N′-bis(4-methoxyphenyl)formamidinate)⁷⁴ was prepared according to the literature. The ¹H and ¹³C{¹H} NMR spectra of all substrates are available in Supplementary Data 1. Rh(dcb)₂ was prepared by mixing [Rh(CH₃CN)₄](BF₄)₂ and n-Bu₄N·dcb in a 1:2 ration in NMR tube at 298 K and were used for the self-assembly after the quantification based on the internal standard ([2.2]paracyclophane). See more detail in section “Monitoring of the self-assembly of the Rh₄L₄ squares”.

QASAP of the Rh ₄L₄ squares

Monitoring of the self-assembly of the Rh₄L₄ squares

A 2.4 mM solution of [2.2]paracyclophane in CHCl₃ (125 μL), which was used as an internal standard, was added to two NMR tubes (tubes I and II) and the solvent was removed in vacuo. A CDCl₃ solution of Rh(dcb)₂ was prepared as solution A (10 mM). Solution A (50 μL) and CDCl₃ (400 μL) were added to tube I. The exact concentration of Rh(dcb)₂ in solution A was determined through the comparison of the signal intensity with [2.2]paracyclophane by ¹H NMR. A solution of ditopic ligand (n-Bu₄N)₂·L (L = 1^2– or 2^2–; 2.5 mM) in CDCl₃ (200 μL) and CDCl₃ (250 μL) were added to tube II and the exact amount of L^2– in tube II was determined through the comparison of the signal intensity with [2.2]paracyclophane by ¹H NMR. Then, 1.0 eq. (against the amount of ligand L^2– in tube II) of solution A (ca. 50 μL; the exact amount was determined based on the exact concentrations of solution A and of L^2– in tube II) was added to tube II. The self-assembly of the Rh₄L₄ square was monitored at 298 K by ¹H NMR spectroscopy. The selected ¹H NMR spectra measure during the self-assembly are shown in Fig. 2c and Supplementary Fig. 8 ([1^2–]₀ = 1.0 mM and [2^2–]₀ = 0.86 mM). The amounts and the existence ratios of L^2–, Rh(dcb)₂, the Rh₄L₄ square, and dcb^– were quantified by the integral value of each ¹H NMR signal against the signal of the internal standard ([2.2]paracyclophane) according to the determination methodology described in the section “Determination of the existence ratios of each species”. The data, the average values of the existence ratios, and the (〈n〉, 〈k〉) values with standard errors are listed in Supplementary Tables 1 and 2 for the self-assembly of the Rh₄1₄ and Rh₄2₄ squares, respectively.

Determination of the existence ratio of each species

The relative integral value of each ¹H NMR signal against the internal standard [2.2]paracyclophane is used as the integral value in this description. We define the integral values of the signal for the substrates and the products at each time t as follows:

QASAP for the Rh₄1₄ square

I_L(t): 1/4 of the integral value of the d proton in free ligand 1^2–

I_M(t): 1/2 of the integral value of the e proton of dcb^– in Rh(dcb)₂

I₄₄₀(t): 1/4 of the integral value of the d proton in the Rh₄1₄ square

I_dcb(t): the integral value of the f proton of free dcb^– (before 1 h) or 1/2 of the integral value of the e proton of free dcb^– (after 1 h)

QASAP for the Rh₄2₄ square

I_L(t): 1/4 of the integral value of the e proton in free ligand 2^2–

I_M(t): 1/2 of the integral value of the f proton of dcb^– in Rh(dcb)₂

I₄₄₀(t): 1/4 of the integral value of the d proton in the Rh₄2₄ square

I_dcb(t): the integral value of the g proton of free dcb^–

I_M(0) was determined based on the exact concentration of solution A determined by ¹H NMR and the exact volume of solution A added into tube II.

I_L(0) was determined by ¹H NMR measurement before the addition of solution A into tube II.

Existence ratio of Rh(dcb)₂

As the total (initial) amount of Rh(dcb)₂ corresponds to I_M(0), the existence ratio of Rh(dcb)₂ at t is expressed by I_M(t)/I_M(0).

Existence ratio of L^2–

As the total amount of free ligand L^2– corresponds to I_L(0), the existence ratio of L^2– at t is expressed by I_L(t)/I_L(0).

Existence ratio of the Rh ₄L₄ square

As the total amount of the Rh₄L₄ square is quantified based on L^2–, the existence ratio of the Rh₄L₄ square at t is expressed by I₄₄₀(t)/I_L(0).

Existence ratio of dcb^–

As the total amount of dcb^– corresponds to I_M(0), the existence ratio of dcb^– at t is expressed by I_dcb(t)/I_M(0).

Existence ratio of the total intermediates not observed by ¹H NMR (Int)

The existence ratio of the total intermediates not observed by ¹H NMR (Int) is determined based on the amount of L^2– in Int. Thus, the existence ratio of Int is calculated by subtracting the other species containing L^2– (free L^2–, and the Rh₄L₄ square) from the total amount of L^2– (I_L(0)). The existence ratio of Int at t is expressed by (I_L(0) – I_L(t) – I₄₄₀(t))/I_L(0).

〈a〉

The initial amount of Rh centers corresponds to I_M(0)/2.

The amount of Rh centers in Rh(dcb)₂ at t corresponds to I_M(t)/2.

The amount of Rh centers in the Rh₄L₄ square at t corresponds to I₄₄₀(t).

The amount of Rh centers in Int at t is thus expressed by I_M(0)/2 – I_M(t)/2 – I₄₄₀(t).

〈b〉

The initial amount of L^2– corresponds to I_L(0).

The amount of free L^2– at t corresponds to I_L(t).

The amount of L^2– in the Rh₄L₄ square at t corresponds to I₄₄₀(t).

The amount of L^2– in Int at t is thus expressed by I_L(0) – I_L(t) – I₄₄₀(t).

〈c〉

The total amount of dcb^– corresponds to I_M(0).

The amount of free dcb^– at t corresponds to I_dcb(t).

The amount of dcb^– in Rh(dcb)₂ at t corresponds to I_M(t).

The amount of dcb^– in Int at t is thus expressed by I_M(0) – I_dcb(t) – I_M(t).

The 〈n〉 and 〈k〉 values are determined with these 〈a〉, 〈b〉, and 〈c〉 values by Eqs. (1) and (2).

$$\left\langle n\right\rangle =\frac{2\left\langle a\right\rangle -\left\langle c\right\rangle }{\left\langle b\right\rangle }$$

(1)

$$\left\langle k\right\rangle =\frac{\left\langle a\right\rangle }{\left\langle b\right\rangle }$$

(2)

NASAP of the Rh ₄2₄ square

The reaction network model and the classification of the elementary reactions

For the numerical analysis of self-assembly process (NASAP), a reaction network for the self-assembly of the Rh₄L₄ square from RhX₂ (X: the leaving ligand) and L^2– (2^2–) is constructed as follows. Starting from the substrates, that is, RhX₂ and L^2–, the reaction path is traced to form the possible intermediate chemical species with up to ten ditopic ligands (L^2–). In this network, the Rh₃L₃ triangle is also considered as intermediate species. With these procedures taken, it is found that the total of 34 molecular species (including both the substrates and the product themselves) construct a reaction network composed of 1,698 elementary reactions. It should be noted that some of those reactions have the forward and backward processes (see the classification of the reaction type given below). All the intermediates considered in this network model and a simplified reaction network are shown in Supplementary Fig. 11.

This reaction network turns out to be so large that it is impossible to assign an individual rate constant to each reaction and to search for the parameter in such a vast parameter space to fit the experimental results best. Therefore, we divided the whole elementary reactions into ten classes possessing similar characteristics and defined rate constants as follows:

i. Incorporation of free L^2– by RhX₂ with releasing a leaving ligand X.

k₁ [min^–1 M^–1] and k_–1 [min^–1 M^–1] for forward and backward reactions, respectively.

ii. Reaction of RhX₂ and the second site of L^2–, whose opposite site is already bonded with another Rh.

k₂ [min^–1 M^–1] and k_–2 [min^–1 M^–1].

iii. Reaction of free L^2– and the second site of Rh, which is already coordinated by another L^2–.

k₃ [min^–1 M^–1] and k_–3 [min^–1 M^–1].

iv. Reaction at the second sites of L^2– and Rh, both of whose other sites are already bonded with Rh and L^2–, respectively.

k₄ [min^–1 M^–1] and k_–4 [min^–1 M^–1].

v. L-L exchange reaction. This reaction can lead to both the growth and the decomposition of oligomers, or produce no net change in the reaction system.

k₅ [min^–1 M^–1].

vi. Cyclization to make a square via L-X exchange.

k₆ [min^–1] and k_–6 [min^–1 M^–1].

vii. Cyclization to make a square via L-L exchange.

k₇ [min^–1] and k_–7 [min^–1 M^–1].

viii. Cyclization to make a triangle via L-X exchange.

k₈ [min^–1] and k_–8 [min^–1 M^–1].

ix. Cyclization to make a triangle via L-L exchange.

k₉ [min^–1] and k_–9 [min^–1 M^–1].

x. X-X exchange, which practically produces no change in the reaction system.

k₁₀ [min^–1 M^–1].

Typical examples of each type of elementary reaction are shown in Supplementary Fig. 12. Note that reactions in classes (i)–(v) and (x) are the intermolecular ligand exchanges and those in classes (iv)–(ix) are intramolecular ones. It should be noted here that each rate constant is defined per reaction site, based on the above modeling procedure. Therefore, the actual reaction rate for each reaction is estimated as the defined constant multiplied by the total number of available combinations. For example, for the reaction between RhX₂ and L^2– to produce [RhLX]^– and X, the rate constant is given as k₁ times 2 (the number of Rh–X bonds in RhX₂) times 2 (the number of coordination sites in L^2–), i.e.,

$${{{{{\boldsymbol{Rh}}}}}}{{{{{{\rm{X}}}}}}}_{2}+{{{{{{\rm{L}}}}}}}^{2{{{-}}}}\mathop{\to }\limits^{4{k}_{1}}{\left[{{{{{\boldsymbol{Rh}}}}}}{{{{{\rm{LX}}}}}}\right]}^{{{{-}}}}+{{{{{{\rm{X}}}}}}}^{{{{-}}}}$$

It was adopted this setting to explicitly distinguish the structural difference among the species with the same composition.

Numerical fitting of the rate constants

In order to numerically track the time evolution of the existence ratios for both reactants and products and the (〈n〉, 〈k〉) values, we have adopted a stochastic approach based on the chemical master equation, the so-called Gillespie algorithm. In this algorithm, for all the possible N chemical reactions including molecular species S_ai, S_bi, S_ci, …,

$${S}_{{ai}}+{S}_{{bi}}+{{\cdot \cdot \cdot }}\to {S}_{{ci}}+{{{{{\rm{\cdot \cdot \cdot }}}}}}\left(i=1,\ldots ,N\right)$$

the total reaction rate R_tot is calculated as

$${R}_{{{{{{\rm{tot}}}}}}}={r}_{1}+{r}_{2}+{{\cdot \cdot \cdot }}+{r}_{N}\left(i=1,\ldots ,N\right)$$

$${r}_{i}={k}_{i}\left[{S}_{{ai}}\right]\left[{S}_{{bi}}\right]{{\cdot \cdot \cdot }}$$

Starting from the initial time t = 0, at each instant t, which one of the reactions to occur is determined with the uniform random number s₁ ∈ (0,1) as

if ${s}_{1}\le {r}_{1}/{R}_{{{{{{\rm{tot}}}}}}},$ then reaction 1 occurs,

if ${r}_{1}/{R}_{{{{{{\rm{tot}}}}}}} < \, {s}_{1}\le ({r}_{1}+{r}_{2})/{R}_{{{{{{\rm{tot}}}}}}}$, then reaction 2 occurs,

if $({r}_{1}+...+{r}_{N-1})/{R}_{{{{{{\rm{tot}}}}}}} < \, {s}_{1}\le 1$, then reaction N occurs.

Another uniform random number s₂ ∈ (0,1) is independently given to fix the time incremental dt as

$${dt}={{{{\mathrm{ln}}}}}(1/{s}_{2})/{R}_{{{{{{\rm{tot}}}}}}}$$

Time is updated as t = t + dt, together with the update of the numbers of corresponding molecular species, i.e., 〈S_ai〉 → 〈S_ai〉 – 1, 〈S_bi〉 → 〈S_bi〉 – 1, 〈S_ci〉 → 〈S_ci〉 + 1, …. The reason why this approach traces the chemical reactions and actually works well is given in the literature in detail, along with the practical way to implement it^75,76,77,78.

The program for fixing the rate constant values is handmade for only this purpose. In order to obtain one of the best parameter sets, the following practical procedures are taken:

First, the rate constant sets are defined in the form of k_i = 10 ^mi. Starting from the initial guess value for each m_i, time evolutions of the global self-assembly event are traced with giving each m_i a stepwise increment or decrement, searching for smaller values of the residual sum of squares (RSS) for the experimental data at hand.

Initial guess sets are given in several different ways for exploring as broad parameter space as possible. An example is the uniform one like (m₁, m₂, m₃, …) = (0, 0, 0, …). In another guess, rate constants for the oligomerization are given 10 to 100 times larger values than others by intention, for the reason that those reactions (especially, the inclusion of the free multitopic ligand L) generally occur very fast at the initial stage of the global self-assembly.

As a result of the global search, most of k_i’s settle into almost definite values, with others floating within relatively wide numerical ranges. From our numerical experiences, the latter constants do not largely affect the global reproduction of the experimental self-assembly process and the dominant reaction pathways. Therefore, for those rate constant parameters, representative values are picked up within the scope of our chemical intuition.

We have to admit that at the final point an arbitrariness occurs. And as the parameter search is performed in this procedure for giving a good fit to the experimental counterpart, we cannot calculate the amounts of statistics such as the standard deviation for the obtained rate constant values.

With the initial conditions (numbers of species), 〈RhX₂〉₀ = 12,900, 〈2^2–〉₀ = 12,900, 〈others〉₀ = 0, rate constant search was performed in an eighteen-dimensional parameter space (k₁, k_–1, k₂, k_–2, k₃, k_–3, k₄, k_–4, k₅, k₆, k_–6, k₇, k_–7, k₈, k_–8, k₉, k_–9, k₁₀). The Avogadro number and the volume of the simulation box were set to be N_A = 6.0 × 10²³ and V = 2.5 × 10^–17 L, respectively, which give the same concentration as the experiments were carried out under [2^2–]₀ = [Rh]₀ = 0.86 mM.

Because ¹H NMR signals of the Rh₃L₃ triangle were not observed during the self-assembly (Supplementary Figs. 8 and 10) and because the conversion of the Rh₃L₃ triangle into the Rh₄L₄ square was slow (Supplementary Fig. 12), the rate constant values concerning the trimerization (k₈ and k₉) should be zero. Indeed, any good parameter set of the rate constants with large k₈ and k₉ was not found.

After the rate constant search was finished, refined simulations were performed for some rate parameter sets that give existence ratios and the (〈n〉, 〈k〉) plot in good agreement with the experimental counterparts. The adequacy of the fitting to the experimental data was evaluated from the residual sum of squares (RSS) with the average of the experimental data, obtained from three runs. For all the time steps t_i at which the experimental data of existence ratios R_ex^S and parameters n_ave and k_ave are available, RSS’s are calculated with the numerically obtained values R_nu^S as (note that S = RhX₂, 2^2–, Rh₄2₄, or X),

$${{{{{\rm{RS}}}}}}{{{{{{\rm{S}}}}}}}_{1}=\mathop{\sum}\limits _{{t}_{i}} \mathop{\sum}\limits _{S}{\left({R}_{{nu},{t}_{i}}^{S}-{R}_{{ex},{t}_{i}}^{S}\right)}^{2}$$

$${{{{{\rm{RS}}}}}}{{{{{{\rm{S}}}}}}}_{2}=\mathop{\sum}\limits _{{t}_{i}}{\left({\left\langle n\right\rangle }_{{nu},{t}_{i}}-{\left\langle n\right\rangle }_{{ex},{t}_{i}}\right)}^{2}+\mathop{\sum}\limits _{{t}_{i}}{\left({\left\langle k\right\rangle }_{{nu},{t}_{i}}-{\left\langle k\right\rangle }_{{ex},{t}_{i}}\right)}^{2}$$

A representative numerical result and the corresponding rate constant set are shown in Figs. 3d and 3e and Supplementary Table 3, respectively. In the simulation to obtain Fig. 3d, e, the initial particles and the volume of the simulation box were set to be a thousand times larger than the rough parameter search, i.e., 〈RhX₂〉₀ = 1,290,000, 〈2^2–〉₀ = 1,290,000, and V = 2.5 × 10^–15 L. Although the numerical results of a single run are shown in Fig. 3d, e, a similar behavior was confirmed with several runs for the particle numbers given above.

Determination of the major self-assembly pathway

In the framework of the stochastic algorithm employed in this study, it is possible to count the number of occurrences (we call it as “frequency”) of each elementary reaction in the course of the self-assembly process. Therefore, the dominant reaction pathway as shown in Fig. 3f was determined by tracing back the reaction network from the objective product, Rh₄2₄ in this case, to the substrates along the elementary reactions whose frequencies are larger than the others. For the elementary reaction with forward and backward processes, which direction to proceed is determined by the number difference between the forward and the backward reaction frequencies. Top 10 elementary reactions with high net frequency in the self-assembly of the Rh₄2₄ square are listed in Supplementary Table 4.

The advantage of using the reaction frequency is that the main intermediates participating in the dominant self-assembly pathway are not missed with the reason of the existence ratio being small due to high reaction rate values associated with that species. For example, the existence ratio of (4,4,1) is very low compared with other Type II intermediates (Supplementary Fig. 13, so if the major self-assembly pathway is determined by connecting the species with large existence ratio, (4,4,1) is not involved in the major pathway. On the other hand, the net reaction frequency of the cyclization of (4,4,1) to form (4,4,0) is higher than those of the other elementary reactions to produce (4,4,0) (Supplementary Table 4), indicating that the square (4,4,0) is mainly produced by the cyclization of (4,4,1). The reason for the low existence ratio of (4,4,1) in the self-assembly is because of fast consumption of (4,4,1) by the cyclization. As this example shows, the determination of the major self-assembly pathway based on the existence ratios of the species leads to a wrong conclusion.

X-ray crystallographic structural analysis of the [Rh₄1₄(dmso-S)₄]₂ dimer

A single crystal was immersed in and coated with perfluoropolyalkylether (viscosity 80 cSt; abcr GmbH), and then mounted on a MicroMount^TM (MiteGen LLC). Diffraction data of the single crystal were collected on a VariMax Dual single crystal X-ray diffractometer with PILATUS 200 K detector (Rigaku Corporation) at 93(2) K, using Mo Kα (λ = 0.71073 Å) radiation monochromated by multilayer mirror optics. Bragg spots were integrated using the CrysAlisPro program package (Rigaku Corporation). An empirical absorption correction based on the multi-scan method using spherical harmonics was implemented in the SCALE3 ABSPACK scaling algorithm. The structure was solved by an intrinsic phasing method on the SHELXT program⁷⁹ and refined by a full-matrix least-squares minimization on F² executed by the SHELXL program⁸⁰, using Olex2 software package (OlexSys Ltd)⁸¹. Thermal displacement parameters were refined anisotropically for all non-hydrogen atoms. All the hydrogen atoms except for those on C70, C100, C137, C145 of terminal methoxy groups were located at calculated positions and the parameters were refined with a riding model. The data were corrected for disordered electron density of crystal solvents in void spaces by using the PLATON SQUEEZE method⁸². The crystal structure is shown in Fig. 4b. Crystallographic data of [Rh₄1₄(dmso-S)₄]₂ are summarized in Supplementary Table 7 and Supplementary Data 2.