General information1H 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), CD3NO2 (δ 4.33). Electrospray ionization time-of-flight (ESI-TOF) mass spectra were obtained using a Waters Xevo G2-S Tof mass spectrometer and Bruker micrOTOF II-KE02. CSI FT-ICR Mass spectra were obtained using Bruker scimaX. Single-crystal X-ray analyses were conducted by Rigaku VariMax and SuperNova. Ligands 159,60, 261,62, and 463 were prepared according to the literature. Molecular mechanics calculations were performed using Materials Studio software (BIOVIA) with the universal force field.MaterialsUnless 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.Synthesis of 2,4,6-tri(4-pyridyl)-1,3,5-triazine (1)18-Crown-6 (200 mg, 0.76 mmol) and KOH (45 mg, 0.8 mmol) were added to EtOH (4 mL), stirring at room temperature for 20 min. Then, 4-cyanopyrine (2.0 g, 19 mmol) was added to the reaction mixture, heating at 200 °C for 6 h. After cooling down to room temperature, pyridine (24 mL) was added to the reaction mixture, stirring for 5 min. The precipitation was obtained after filtration, washing with pyridine (10 mL) and toluene (10 mL). The solid was dissolved to 2 M. HCl aq. (24 mL) and collected into the filtrate. After the addition of NH3 aq. to the solution, until precipitation was generated. After filtration, tritopic ligand 1 was obtained as an off-white powder (0.51 g, 1.6 mmol, 25%). 1H NMR (500 MHz, CD3NO2/CDCl3/CD3OD = 7/3/2 (v/v/v), 298 K): δ 8.91 (dd, J = 1.5 and 4.3 Hz, 6 H), 8.71 (dd, J = 1.5 and 4.3 Hz, 6 H).Synthesis of 1,3,5,-tri(4-pyridyl)-benzene (2)In a 100-mL sealed tube flask, 1,3,5-tribromobenzene (0.3 g, 0.95 mmol, 1 eq.), 4-pyridyl-boronic acid (0.47 g, 3.8 mmol, 4 eq.), Na2CO3 (3.0 g, 29 mmol, 30 eq.), Pd(PPh3)4 (0.17 g, 0.14 mmol, 15 mol%) were mixed under inert atmosphere in dioxane (30 mL) and H2O (20 mL). The resulting mixture was stirred at 110 °C for 2 days. After evaporation of the mixture, the residue was added to H2O (50 mL), and extraction was performed using CHCl3 (40 mL, three times). The organic layer was dried over anhydrous MgSO4, and the solvent was removed by evaporation to obtain the crude product. After purification by silica gel column chromatography (Eluent: 5% CH3OH containing CHCl3 solution), tritopic ligand 2 was obtained as a colorless solid (0.22 g, 0.70 mmol, 73%). 1H NMR (500 MHz, CDCl3, 298 K): δ 8.76 (dd, J = 1.5 and 4.4 Hz, 6 H), 7.92 (s, 3 H), 7.61 (dd, J = 1.5 and 4.4 Hz, 6 H).Synthesis of 4,4’,4”-(2,4,6-trimethylbenzene-1,3,5-triyl)tripyridine (4)In a 100-mL sealed tube flask, 1,3,5-tribromo-2,4,6-trimethylbenzene (357 mg, 1.0 mmol, 1 eq.), 4-pyridyl-boronic acid (1.23 g, 10 mmol, 10 eq.), K3PO4 (2.55 g, 12 mmol, 12 eq.), Pd2(dba)3 (20.1 mg, 22 µmol, 0.022 eq.), and SPhos (18.1 mg, 44 µmol, 0.044 eq.) were mixed under inert atmosphere in n-butanol (40 mL). The resulting mixture was stirred at 80 °C for 3 days. After cooling to room temperature, the reaction mixture was filtered, and the filtrate was evaporated to remove the solvent. The residue was added to H2O (10 mL), and extraction was performed using CHCl3 (20 mL, three times). The organic layer was dried over anhydrous MgSO4, and the solvent was removed by evaporation to yield a yellow solid. After purification by silica gel column chromatography (Eluent: AcOEt), tritopic ligand 4 was obtained as a colorless solid (0.15 g, 0.42 mmol, 42%). 1H NMR (500 MHz, CD2Cl2, 298 K): δ 8.65 (dd, J = 1.5 and 4.3 Hz, 6 H), 7.14 (dd, J = 1.5 and 4.3 Hz, 4 H), 1.67 (s, 9 H).Self-assembly of the [Pd
1228]24+ open icosahedronIn a vial, [Pd(CH3CN)2](BF4)2 (12 mg, 24.2 µmol) was added to a solution of tritopic ligand 2 (5.0 mg, 16 µmol) in CH3NO2 in the presence of n-Bu4N·NO3 (2.5 mg, 8.1 µmol). The solution was heated at 363 K for 5 days. After cooling to room temperature, the solvent was removed by evaporation, and the residue was washed three times with CHCl3 (1.0 mL). The pale brown solid was dried in vacuo to obtain [Pd1228]24+ (15 mg, quant.). 1H NMR (500 MHz, CD3NO2, 353 K) δ 9.39 (d, J = 6.0 Hz, 48 H), 8.13 (s, 24 H), 7.94 (d, J = 6.0 Hz, 48 H), 3.28 (m, 48 H), 2.98 (s, 72 H), 2.84 (s, 72 H). 13C NMR (125 MHz, CD3NO2, 353 K): δ 153.2, 153.1, 140.2, 130.6, 127.4, 52.1, 51.9.Self-assembly of the [Pd
926]18+ open triaugmented triangular prismIn an NMR tube with [2.2]paracyclophane as an internal standard, [Pd(CH3CN)2](BF4)2 (0.86 mg, 1.8 µmol), tritopic ligand 2 (0.37 mg, 1.2 µmol), and CH3NO2 (480 µL) were added. The solution was then heated at 343 K for 2 days to obtain [Pd926]18+ in 96% NMR yield. 1H NMR (500 MHz, CD3NO2, 298 K): δ 9.21 (d, J = 6.0 Hz, 12 H), 9.15 (d, J = 6.0 Hz, 12 H), 9.10 (d, J = 6.0 Hz, 12 H), 8.08 (d, J = 1.5 Hz, 12 H), 7.95 (d, J = 6.5 Hz, 24 H), 7.66 (dd, J = 2.0, 6.0 Hz, 12 H), 7.58 (s, 6 H), 3.28 (m, 36 H), 3.03 (s, 36 H), 2.97 (s, 36 H), 2.73 (s, 36 H). 13C NMR (125 MHz, CD3NO2, 298 K): δ 153.0, 152.7, 152.5, 139.8, 139.6, 132.7, 128.8, 127.4, 127.0, 126.8, 105.5, 105.1, 51.9, 51.8, 51.5.Self-assembly of the [Pd
232]4+ ringIn an NMR tube with [2.2]paracyclophane as an internal standard, 1 equiv. of [Pd(CH3CN)2](BF4)2 (1.2 µmol) was added to a solution of 3 (1.2 µmol) in CD3NO2 (490 µL). After heating the solution at 343 K for 1 h, the [Pd232]4+ ring was obtained in 92% yield. 1H NMR (500 MHz, CD3NO2, 298 K): δ 9.14 (dd, J = 1.5, 5.5 Hz, 8 H), 7.40 (dd, J = 1.5 and 5.5 Hz, 8 H), 7.14 (s, 2 H), 3.20 (s, 8 H), 2.85 (s, 12 H), 2.06 (s, 12 H), 0.64 (s, 6 H). 13C NMR (125 MHz, CD3NO2, 298 K): δ 155.9, 152.6, 136.7, 136.4, 133.3, 130.3, 130.2, 51.7, 20.8, 20.3, 13.9.Self-assembly of the [Pd
242]4+ ringIn an NMR tube with [2.2]paracyclophane as an internal standard, 1 equiv. of [Pd(CH3CN)2](BF4)2 was added to a solution of 4 in CD3NO2 (500 µL). After heating at 343 K for 4 h, the [Pd242]4+ ring was obtained quantitatively. 1H NMR (500 MHz, CD3NO2, 298 K): δ 9.17 (dd, J = 1.5 and 5.5 Hz, 8 H), 8.65 (dd, J = 1.5 and 4.5 Hz, 4 H), 7.46 (d, J = 1.5 and 5.0 Hz, 8 H), 7.21 (d, J = 1.5 and 4.5 Hz,4 H), 3.18 (s, 8 H), 2.83 (s, 24 H), 1.77 (s, 12 H), 0.71 (s, 6 H). 13C NMR (125 MHz, CD3NO2, 298 K): δ 155.9, 152.8, 151.4, 139.3, 137.0, 134.1, 133.1, 130.2, 126.1, 79.2, 51.7, 21.0, 19.3, 13.9.Self-assembly of the [Pd’
1248]24+ square sheetIn an NMR tube with [2,2]paracyclophane as an internal standard, [Pd’(CH3CN)2](BF4)2 (0.76 mg, 1.8 μmol) was added to a solution of 4 (0.42 mg, 1.2 µmol) in CD3NO2 (500 µL) by titration at 343 K. After the convergence of the self-assembly, the [Pd’1248]24+ square sheet was obtained in 90% NMR yield. 1H NMR (500 MHz, CD3NO2, 298 K): δ 8.99 (d, J = 6.5 Hz, 16 H), 8.94 (d, J = 6.5 Hz, 32 H), 7.44 (d, J = 6.5 Hz, 16 H), 7.28 (d, J = 6.5 Hz, 32 H), 3.10 (br, 48 H), 1.62 (s, 48 H), 0.65 (s, 24 H). 13C NMR (125 MHz, CD3NO2, 298 K): δ 153.2, 137.4, 133.6, 132.6, 129.4, 129.2, 48.6, 47.8, 21.0, 19.1.Single-crystal X-ray analysesSingle crystals of the [Pd1228]24+ open icosahedron suitable for X-ray crystallographic analysis were obtained by vapor diffusion of CH3OH in a CH3NO2 solution of [Pd1228]24+ at room temperature after 15 days. A crystal was immersed in and coated with FluorolubeⓇ (SIGMA-ALDRICH Corp.), and then mounted on a MicroMountTM (MiteGen LLC). The diffraction data of the single crystal were collected on a Rigaku VariMax with a Hybrid Photon Counting Detector at 93 K, using Mo Kα (λ = 0.71075 Å).Single crystals of [Pd232]4+ were obtained at the interface between the two phases with CH3NO2 (100 µL) as a buffer solvent placed between them at room temperature for 15 days. One phase was a solution of 3 (5 mg, 18.2 µmol) and n-Bu4N·NO3 (2.8 mg, 9.1 µmol) in CH3NO2 (200 µL), and the other phase was a solution of [Pd(CH3CN)2](BF4)2 (8.7 mg, 18.2 µmol) in CH3NO2 (200 µL). A crystal was immersed in and coated with immersion oil (Cargille Corp.), and then mounted on a SuperNova single-crystal X-ray diffractometer with an Eos CCD detector (Rigaku Oxford Diffraction) at 180 K, using Cu Kα (λ = 1.54184 Å) radiation monochromated by multilayer mirror optics. Bragg spots were integrated using the CrysAlisPro program package (Rigaku Oxford 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 program64 and refined by a full-matrix least-squares minimization on F2 executed by the SHELXL program65 using the Olex2 software package (OlexSys Ltd.)66 and the ShelXle graphical user interface67. Thermal displacement parameters were refined anisotropically for all non-hydrogen atoms. The data were corrected for scattered electron density in large solvent voids using the PLATON SQUEEZE method68. The crystal structures are shown in Figs. 1d, 7d and Supplementary Fig. 35. The crystallographic data are summarized in Supplementary Table 2.DFT calculationsThe optimized structures of 2-(4-pyridyl)-1,3,5-triazine, 4-phenylpyridine, 4-(2,6-dimethylphenyl)pyridine, [PdPy2]2+,[Pd’Py2]2+, [Pd12]2+, [Pd22]2+ ZnPy2 were obtained using the B3LYP functional69,70 implemented in the Gaussian16 program package71. The SDD basis set72 was used for the Pd and Zn atoms, and the D95** basis set73 was used for the calculations of the H, C, N, and Cl atoms. The 6-31 G** basis set74,75 was employed for the calculations of 2-(4-pyridyl)-1,3,5-triazine, 4-phenylpyridine and 4-(2,6-dimethylphenyl)pyridine. To determine the relative energies depending on θ in the ligands and φ in [PdPy2]2+, [Pd’Py2]2+, and ZnPy2 (Figs. 3a and 6c), geometrical optimizations were performed by fixing the corresponding dihedral angle by inserting opt = modredundant keyword into the input file. Optimized structures and their energies were obtained using B3LYP/def2-SV(P) level of theory in [Pd624](BF4)12, [Pd926](BF4)18, [Pd1228](BF4)24, and [Pd’624](BF4)12. After optimization, single-point calculations were carried out with various functionals of LC-OLYP, M06, and ωB97X, taking into account an implicit solvent effect as a polarizable continuum model (PCM) of nitromethane (ε = 36.562). The cartesian coordinates of the optimized structures are listed in Supplementary Data 1.Geometric analysisFor the calculation of RSU, it is necessary to determine the distance between the ends of the MnLn chain. Given that one end is situated at the origin of the global coordinate system, it suffices to ascertain the coordinates of the other end in the global coordinate system. When there exist coordinate systems A and B with bases \(^{{\rm{A}}}{{\bf{e}}}_{x},^{{\rm{A}}}{{\bf{e}}}_{y},^{{\rm{A}}}{{\bf{e}}}_{z}\) and \(^{{\rm{B}}}{{\bf{e}}}_{x}, ^{{\rm{B}}}{{\bf{e}}}_{y},^{{\rm{B}}}{{\bf{e}}}_{z}\) respectively, and the rotation matrix between them is denoted as \({R}_{{\rm{AB}}}\) i.e.,$$\left({{\scriptstyle{\rm{ B}}\atop}{\!{\bf{e}}}_{x},\, {}^{{\rm{B}}}{{\bf{e}}}_{y},\, {}^{{\rm{B}}}{{\bf{e}}}_{z}}\right)=\left({{\scriptstyle{\rm{ A}}\atop}{\!{\bf{e}}}_{x},\, {}^{{\rm{A}}}{{\bf{e}}}_{y},\, {}^{{\rm{A}}}{{\bf{e}}}_{z}}\right){R}_{{\rm{AB}}}$$
(1)
the coordinates of a vector \({{\bf{r}}}\) expressed as \(^{{\rm{B}}}{{\bf{x}}}=\left({\scriptstyle{\rm{B}}\atop}\!{x},\, ^{{\rm{B}}}y,\, ^{{\rm{B}}}z\right)^{{\rm{T}}}\) in coordinate system B can be calculated in coordinate system A as \(^{{\rm{A}}}{{\bf{x}}}={R}_{{\rm{AB}}} {\scriptstyle{\rm{B}}\atop}{{\bf{x}}}\).$$\because {{\bf{r}}} =\left({{\scriptstyle{\rm{A}}\atop}{\!{\bf{e}}}}_{x},\, ^{{\rm{A}}}{{\bf{e}}}_{y},\, ^{{\rm{A}}}{{\bf{e}}}_{z}\right)\left(\begin{array}{c}{\scriptstyle{\rm{A}}\atop}\!x\\ ^{{\rm{A}}}y\\ ^{{\rm{A}}}z\end{array}\right)\\ =\left({{\scriptstyle{\rm{B}}\atop}{\!{\bf{e}}}}_{x},\, ^{{\rm{B}}}{{\bf{e}}}_{y},\, ^{{\rm{B}}}{{\bf{e}}}_{z}\right)\left(\begin{array}{c}{\scriptstyle{\rm{B}}\atop}\!x\\ {}^{{\rm{B}}}y\\ ^{{\rm{B}}}z\end{array}\right)\\ =\left({{\scriptstyle{\rm{A}}\atop}{\!{\bf{e}}}}_{x},\, ^{{\rm{A}}}{{\bf{e}}}_{y},\, ^{{\rm{A}}}{{\bf{e}}}_{z}\right){R}_{{\rm{AB}}}\left(\begin{array}{c}{\scriptstyle{\rm{B}}\atop}\!x\\ ^{{\rm{B}}}y\\ ^{{\rm{B}}}z\end{array}\right)$$
(2)
Note that vectors are quantities independent of coordinate systems. The vectors represented by \({{\bf{r}}}\) here and later, such as \(\overrightarrow{{\rm{AB}}}\), fall into this category. On the other hand, coordinates are representations of vectors that depend on the choice of coordinate systems and are expressed by arranging the components of the basis. Coordinates will be denoted by \({{\bf{x}}}\).Each unit is assigned points A, B, and C, and coordinate systems A, B1, B2, and C, as illustrated in Fig. 9a–d. Because the positions of the coordinate systems do not affect the subsequent calculations, the coordinate systems can be located anywhere. Only the orientation of coordinate systems is crucial. Two dihedral angles between the rings in the ligand, denoted as θ1 and θ2, are defined as shown in Fig. 9a–d (0 ≤ θ1, θ2 ≤ 90°). To specify the rotation direction of the dihedral angles (R or L), integers j and k are introduced as follows.$$j=\left\{\begin{array}{c}+1\left({{\rm{rotation}}} \, {{\rm{direction}}} \, {{\rm{of}}}\,{\theta }_{1}\,{{\rm{is}}}\,R\right)\\ -1\left({{\rm{rotation}}} \,{{\rm{direction}}} \, {{\rm{of}}}\,{\theta }_{1}\, {{\rm{is}}}\,L\right)\end{array}\right.$$
(3)
$$k=\left\{\begin{array}{c}+1\left({{\rm{rotation}}} \, {{\rm{direction}}} \, {{\rm{of}}}\,{\theta }_{2}\, {{\rm{is}}}\,R\right)\\ -1\left({{\rm{rotation}}} \, {{\rm{direction}}} \, {{\rm{of}}}\,{\theta }_{2}\, {{\rm{is}}}\,L\right)\end{array}\right.$$
(4)
Fig. 9: Definition of inner points and coordinates for ditopic ligands with cis-protected metal ions.The ditopic ligand is depicted as a model of 1,3-di-4-pyridylbenzene. Two cis-protected metal ions with a bent ditopic ligand are shown (a–d). a RR, b RL, c LR, and d LL configurations. Points A and C are located at the center of the metal ions, whereas point B is located at the center of the middle hexagon in the ditopic ligand. The four coordinates are shown as the x, y, and z axes. The cis-protecting groups are omitted. Definitions of L and R are shown in the main text (Fig. 3d). A metal center with two coordinating rings is shown (e–h). e type FF, f type FB, g type BF, and h type BB. The point Ci is shown in the center of the Pd(II) ion, and the two coordinates are shown in the center of the coordinating rings. The coordinate is defined such that the front face of the coordinating ring is placed on the positive side of the z-axis. The cis-protecting ligand was omitted. Definitions of F and B are shown in the main text (Fig. 3d).Definitions of L and R are shown in the main text (Fig. 3d).What is ultimately required is only the transformation matrix between coordinate systems A and C, and the displacement vector between points A and C. Other coordinate systems, such as B1 and B2, are essentially not crucial. To determine the transformation matrix from A to C, the auxiliary coordinate systems B1 and B2 are introduced. Note that the z-axis of coordinate systems A and C always protrudes towards the front face.Let the rotation matrix from A to B1 be denoted as \({R}_{{\rm{A}}{{\rm{B}}}_{1}}\), from B1 to B2 as \({R}_{{{\rm{B}}}_{1}{{\rm{B}}}_{2}}\), and from B2 to C as \({R}_{{{\rm{B}}}_{2}{\rm{C}}}\). Each matrix can be expressed as follows:$${R}_{{\rm{A}}{{\rm{B}}}_{1}}\left(j,\,{\theta }_{1}\right)=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos {\theta }_{1} & -j\sin {\theta }_{1}\\ 0 & j\sin {\theta }_{1} & \cos {\theta }_{1}\end{array}\right)$$
(5)
$${R}_{{{\rm{B}}}_{1}{{\rm{B}}}_{2}}\left(j\right)=\left(\begin{array}{ccc}-\cos 120^{\circ} & -j\sin 120^{\circ} & 0\\ j\sin 120^{\circ} & -\cos 120^{\circ} & 0\\ 0 & 0 & 1\end{array}\right)$$
(6)
$${R}_{{{\rm{B}}}_{2}{\rm{C}}}\left(j,\, k,\, {\theta }_{2}\right)=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & -{jk}\cos {\theta }_{2} & j\sin {\theta }_{2}\\ 0 & -j\sin {\theta }_{2} & -{jk}\cos {\theta }_{2}\end{array}\right)$$
(7)
Since consecutive rotations can be expressed as the product of rotation matrices, the rotation matrix from A to C, denoted as \({R}_{{\rm{AC}}}\), can be expressed as follows:$${R}_{{\rm{AC}}}\left(j,k,{\theta }_{1},{\theta }_{2}\right)={R}_{{\rm{A}}{{\rm{B}}}_{1}}\left(j,{\theta }_{1}\right){R}_{{{\rm{B}}}_{1}{{\rm{B}}}_{2}}\left(j\right){R}_{{{\rm{B}}}_{2}{\rm{C}}}\left(j,k,{\theta }_{2}\right)$$
(8)
Let us now determine the components of the vector \(\overrightarrow{{\rm{AC}}}\) as seen in the local coordinate system A. As \(\overrightarrow{{\rm{AC}}}=\overrightarrow{{\rm{AB}}}+\overrightarrow{{\rm{BC}}}\) holds, it is necessary to express the right-hand side components \(\overrightarrow{{\rm{AB}}}\) and \(\overrightarrow{{\rm{BC}}}\) as coordinates seen from coordinate system A.Since \(\left|\overrightarrow{{\rm{AB}}}\right|=\left|\overrightarrow{{\rm{BC}}}\right|=1\) by definition, \(\overrightarrow{{\rm{AB}}}\) is expressed in coordinate system A as \(^{{\rm{A}}}{{\bf{x}}}_{\overrightarrow{{\rm{AB}}}}={\left(1,0,0\right)}^{{\rm{T}}}\), and \(\overrightarrow{{\rm{BC}}}\) is expressed in the B2 coordinate system as \(^{{{\rm{B}}}_{2}}{{\bf{x}}}_{\overrightarrow{{\rm{BC}}}}={\left(1,0,0\right)}^{{\rm{T}}}\). When converted to coordinates in the A system, \(\overrightarrow{{\rm{BC}}}\) is expressed as \(^{{\rm{A}}}{{\bf{x}}}_{\overrightarrow{{\rm{BC}}}}={R}_{{\rm{AB}}_{2}}\, {\scriptstyle{{\rm{B}}}_{2}\atop}{\!{\bf{x}}}_{\overrightarrow{{\rm{BC}}}}={R}_{{\rm{A}}{{\rm{B}}}_{1}}{R}_{{{\rm{B}}}_{1}{{\rm{B}}}_{2}}\,{\scriptstyle{{\rm{B}}}_{2}\atop}{\!{\bf{x}}}_{\overrightarrow{{\rm{BC}}}}={R}_{{\rm{A}}{{\rm{B}}}_{1}}{R}_{{{\rm{B}}}_{1}{{\rm{B}}}_{2}}{\left(1,0,0\right)}^{{\rm{T}}}\). Therefore, the components of \(\overrightarrow{{\rm{AC}}}\) as seen in the local coordinate system A are indicated as follows:$${\scriptstyle{\rm{A}}\atop}{\!{\bf{x}}}_{\overrightarrow{{\rm{AC}}}}={\scriptstyle{\rm{A}}\atop}{\!{\bf{x}}}_{\overrightarrow{{\rm{AB}}}}+{\scriptstyle{\rm{A}}\atop}{\!{\bf{x}}}_{\overrightarrow{{\rm{BC}}}}={\left(1,0,0\right)}^{{\rm{T}}}+{R}_{{\rm{A}}{{\rm{B}}}_{1}}{R}_{{{\rm{B}}}_{1}{{\rm{B}}}_{2}}{\left(1,\, 0,\, 0\right)}^{{\rm{T}}}$$
(9)
Consider two units, labeled as unit \(i\) and unit \(i+1\). The two units are connected, and the unit numbers are indicated with superscripts in the alphabet (e.g., point A in unit \(i\) is denoted as point \({{\rm{A}}}^{i}\), and the coordinate system of unit \(i\) is represented as coordinate system \({{\rm{A}}}^{i}\)). Figure 9e–h shows an enlarged view of the junction part. φ was set to 90° throughout the calculation, which was demonstrated by DFT calculations (Fig. 3a) as the most stable for pyridyl groups coordinating to cis-protected Pd(II) center with TMEDA. While points \({{\rm{C}}}^{i}\) and \({{\rm{A}}}^{i+1}\) represent the same point, the coordinate systems \({{\rm{C}}}^{i}\) and \({{\rm{A}}}^{i+1}\) have different orientations.Let the bite angle be denoted as δ (0° < δ < 180°). Additionally, to specify the relative orientations (F or B) of the connecting units, we defined integers \(l\) and \(m\) as follows:$$l=\left\{\begin{array}{c}+1\left({{\rm{orientation}}} \,{{\rm{of}}} {{\rm{unit}}}\,i\,{{\rm{is}}}\, F \right) \\ -1\left({{\rm{orientation}}} \,{{\rm{of}}} \,{{\rm{unit}}} \,i\,{{\rm{is}}}\, B \right)\end{array}\right.$$
(10)
$$m=\left\{\begin{array}{c}+1\left({{\rm{orientation}}}\, {{\rm{of}}}\, {{\rm{unit}}}\,i+1 \,{{\rm{is}}}\,F\right)\\ -1\left({{\rm{orientation}}} \,{{\rm{of}}}\, {{\rm{unit}}}\,i+1 \,{{\rm{is}}}\, B\right)\end{array}\right.$$
(11)
Definitions of F and B are shown in the main text (Fig. 3d).The rotation matrix \({R}_{{{\rm{C}}}^{i}{{\rm{A}}}^{i+1}}\) can be expressed as follows:$${R}_{{{\rm{C}}}^{i}{{\rm{A}}}^{i+1}}\left(l,m,\delta \right)=\left(\begin{array}{ccc}-\cos \delta & 0 & -m\sin \delta \\ 0 & {lm} & 0\\ l\sin \delta & 0 & -{lm}\cos \delta \end{array}\right)$$
(12)
The specific form of the coordinates \({\scriptstyle{{\rm{A}}}^{i+1}\atop}{{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{i+1}{{\rm{C}}}^{i+1}}}\) in the coordinate system \({{\rm{A}}}^{i+1}\) for \(\overrightarrow{{{\rm{A}}}^{i+1}{{\rm{C}}}^{i+1}}\) is already known. The representation of \(\overrightarrow{{{\rm{A}}}^{i+1}{{\rm{C}}}^{i+1}}\) in the coordinate system \({{\rm{A}}}^{i}\) is obtained by multiplying \({\scriptstyle{{{\rm{A}}}^{i+1}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{i+1}{{\rm{C}}}^{i+1}}}\) from the left with the rotation matrix \({R}_{{{\rm{A}}}^{i}{{\rm{A}}}^{i+1}}\). Therefore, the calculation is given by \({\scriptstyle{{{\rm{A}}}^{i}\atop}{{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{i+1}{{\rm{C}}}^{i+1}}}={R}_{{{\rm{A}}}^{i}{{\rm{A}}}^{i+1}}{{{\cdot }}}\,{\scriptstyle{{{\rm{A}}}^{i+1}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{i+1}{{\rm{C}}}^{i+1}}}={R}_{{{\rm{A}}}^{i}{{\rm{C}}}^{i}}{R}_{{{\rm{C}}}^{i}{{\rm{A}}}^{i+1}}{{{\cdot }}}{\scriptstyle{{{\rm{A}}}^{i+1}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{i+1}{{\rm{C}}}^{i+1}}}\).Let us consider a chain consisting of n units (Unit 1, Unit 2,…, Unit n). The point A of Unit 1 (i.e., point \({{\rm{A}}}^{1}\)) is located at the origin of the global coordinate system, and the orientation of the local coordinate system of Unit 1 (i.e., coordinate system \({{\rm{A}}}^{1}\)) coincides with the global coordinate system.Using the previously calculated \({\scriptstyle{{{\rm{A}}}^{i}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{i}{{\rm{C}}}^{i}}},{R}_{{{\rm{A}}}^{i}{{\rm{C}}}^{i}}\), and \({R}_{{{\rm{C}}}^{i}{{\rm{A}}}^{i+1}}\), the position vector of the chain’s endpoint (i.e., point \({{\rm{C}}}^{n}\)) in the global coordinate system can be expressed in terms of components as follows.$${\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{n}}}= {\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}+{\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}}+\ldots+{\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{n}{{\rm{C}}}^{n}}}\\= {\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}+{R}_{{{\rm{A}}}^{1}{{\rm{A}}}^{2}}{\scriptstyle{{\rm{A}}}^{2}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}}+\ldots+{R}_{{{\rm{A}}}^{1}{{\rm{A}}}^{2}}{R}_{{{\rm{A}}}^{2}{{\rm{A}}}^{3}}\ldots {R}_{{{\rm{A}}}^{n-1}{{\rm{A}}}^{n}}{\scriptstyle{{\rm{A}}}^{n}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{n}{{\rm{C}}}^{n}}}\\= {\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}+{R}_{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}{R}_{{{\rm{C}}}^{1}{{\rm{A}}}^{2}}{\scriptstyle{{\rm{A}}}^{2}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}}+\ldots \\ +{R}_{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}{R}_{{{\rm{C}}}^{1}{{\rm{A}}}^{2}}{R}_{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}{R}_{{{\rm{C}}}^{2}{{\rm{A}}}^{3}}\ldots {R}_{{{\rm{A}}}^{n-1}{{\rm{C}}}^{n-1}}{R}_{{{\rm{C}}}^{n-1}{{\rm{A}}}^{n}}{\scriptstyle{{\rm{A}}}^{n}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{n}{{\rm{C}}}^{n}}}$$
(13)
ExampleThe method described above is employed to determine the distance between the endpoints of the M3L2 chain, illustrated through a specific example using the RR(FF)RL chain whose dihedral angles are all 30°. In this chain, the ditopic ligands with conformations RR and RL are connected to a cis-protected Pd(II) center with an FF relative orientation. Initially, the first unit (RR with both dihedral angles at 30°) is placed at the origin of the global coordinate system (i.e., \({\theta }_{1}=30^{\circ},\,{\theta }_{2}=30^{\circ},\, j=+ 1,\,k=+1\)).First, compute \({R}_{{{\rm{A}}}^{1}{{\rm{B}}}_{1}^{1}}\), \({R}_{{{\rm{B}}}_{1}^{1}{{\rm{B}}}_{2}^{1}}\) and \({R}_{{{\rm{B}}}_{2}^{1}{{\rm{C}}}^{1}}\) as follows.$${R}_{{{\rm{A}}}^{1}{{\rm{B}}}_{1}^{1}}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos {\theta }_{1} & -j\sin {\theta }_{1}\\ 0 & j\sin {\theta }_{1} & \cos {\theta }_{1}\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \sqrt{3}/2 & -1/2\\ 0 & 1/2 & \sqrt{3}/2\end{array}\right)$$
(14)
$${R}_{{{\rm{B}}}_{1}^{1}{{\rm{B}}}_{2}^{1}}=\left(\begin{array}{ccc}-\cos 120^{\circ} & -j\sin 120^{\circ} & 0\\ j\sin 120^{\circ} & -\cos 120^{\circ} & 0\\ 0 & 0 & 1\end{array}\right)=\left(\begin{array}{ccc}1/2 & -\sqrt{3}/2 & 0\\ \sqrt{3}/2 & 1/2 & 0\\ 0 & 0 & 1\end{array}\right)$$
(15)
$${R}_{{{\rm{B}}}_{2}^{1}{{\rm{C}}}^{1}}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & -{jk}\cos {\theta }_{2} & j\sin {\theta }_{2}\\ 0 & -j\sin {\theta }_{2} & -{jk}\cos {\theta }_{2}\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & -\sqrt{3}/2 & 1/2\\ 0 & -1/2 & -\sqrt{3}/2\end{array}\right)$$
(16)
Next, the components of the vector \(\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}\) as viewed from the local coordinate system \({{\rm{A}}}^{1}\), denoted as \({{\scriptstyle{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}\), is calculated as follows:$${{\scriptstyle{\rm{A}}}^{1}\atop}{\!{\bf{x}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}} ={\left(1,0,0\right)}^{{\rm{T}}}+{R}_{{{\rm{A}}}^{1}{{\rm{B}}}_{1}^{1}}{R}_{{{\rm{B}}}_{1}^{1}{{\rm{B}}}_{2}^{1}}{\left(1,0,0\right)}^{{\rm{T}}}\\ =\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)+\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \sqrt{3}/2 & -1/2\\ 0 & 1/2 & \sqrt{3}/2\end{array}\right)\left(\begin{array}{ccc}1/2 & -\sqrt{3}/2 & 0\\ \sqrt{3}/2 & -1/2 & 0\\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)\\ =\left(\begin{array}{c}3/2\\ 3/4\\ \sqrt{3}/4\end{array}\right) \sim \left(\begin{array}{c}1.5\\ 0.75\\ 0.43\end{array}\right)$$
(17)
Finally, the components of the position vector of point \({{\rm{C}}}^{1}\) as seen from the global coordinate system, denoted as \({{\scriptstyle{\rm{global}}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}\), is computed. Since the local coordinate system \({{\rm{A}}}^{1}\) of Unit 1 is equivalent to the global coordinate system,$${{\scriptstyle{\rm{global}}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}={{\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}=\left(\begin{array}{c}3/2\\ 3/4\\ \sqrt{3}/4\end{array}\right) \sim \left(\begin{array}{c}1.5\\ 0.75\\ 0.43\end{array}\right)$$
(18)
For calculations for subsequent units, the rotation matrix \({R}_{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}\) from the global coordinate system to the \({{\rm{C}}}^{1}\) coordinate system is calculated as follows.$${R}_{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}= {R}_{{{\rm{A}}}^{1}{{\rm{B}}}_{1}^{1}}{R}_{{{\rm{B}}}_{1}^{1}{{\rm{B}}}_{2}^{1}}{R}_{{{\rm{B}}}_{2}^{1}{{\rm{C}}}^{1}}\\= \left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \sqrt{3}/2 & -1/2\\ 0 & 1/2 & \sqrt{3}/2\end{array}\right)\left(\begin{array}{ccc}1/2 & -\sqrt{3}/2 & 0\\ \sqrt{3}/2 & 1/2 & 0\\ 0 & 0 & 1\end{array}\right)\\ \left(\begin{array}{ccc}1 & 0 & 0\\ 0 & -\sqrt{3}/2 & 1/2\\ 0 & -1/2 & -\sqrt{3}/2\end{array}\right)\\= \left(\begin{array}{ccc}1/2 & 3/4 & -\sqrt{3}/4\\ 3/4 & -1/8 & 3\sqrt{3}/8\\ \sqrt{3}/4 & -3\sqrt{3}/8 & -5/8\end{array}\right) \sim \left(\begin{array}{ccc}0.5 & 0.75 & -0.43\\ 0.75 & -0.13 & 0.65\\ 0.43 & -0.65 & -0.63\end{array}\right)$$
(19)
Next, consider connecting a new Unit 2 (RL with both dihedral angles at 30°) to the tip (point C) of Unit 1 (i.e., \({\theta }_{1}=30^{\circ},\, {\theta }_{2}=30^{\circ},\,\, j=1,\, \, k=-1\)). The relative orientation between the units is FF, and the bite angle is set at 90° (i.e., \(l=1,\, m=1,\, \delta=90^{ \circ}\)).First, calculate \({R}_{{{\rm{A}}}^{2}{{\rm{B}}}_{1}^{2}}\), \({R}_{{{\rm{B}}}_{1}^{2}{{\rm{B}}}_{2}^{2}}\), \({R}_{{{\rm{B}}}_{2}^{2}{{\rm{C}}}^{2}}\), and \({R}_{{{\rm{C}}}^{1}{{\rm{A}}}^{2}}\) as follows.$${R}_{{{\rm{A}}}^{2}{{\rm{B}}}_{1}^{2}}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos {\theta }_{1} & -j\sin {\theta }_{1}\\ 0 & j\sin {\theta }_{1} & \cos {\theta }_{1}\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \sqrt{3}/2 & -1/2\\ 0 & 1/2 & \sqrt{3}/2\end{array}\right)$$
(20)
$${R}_{{{\rm{B}}}_{1}^{2}{{\rm{B}}}_{2}^{2}}=\left(\begin{array}{ccc}-\cos 120^{\circ} & -j\sin 120^{\circ} & 0\\ j\sin 120^{\circ} & -\cos 120^{\circ} & 0\\ 0 & 0 & 1\end{array}\right)=\left(\begin{array}{ccc}1/2 & -\sqrt{3}/2 & 0\\ \sqrt{3}/2 & 1/2 & 0\\ 0 & 0 & 1\end{array}\right)$$
(21)
$${R}_{{{\rm{B}}}_{2}^{2}{{\rm{C}}}^{2}}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & -{jk}\cos {\theta }_{2} & j\sin {\theta }_{2}\\ 0 & -j\sin {\theta }_{2} & -{jk}\cos {\theta }_{2}\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \sqrt{3}/2 & 1/2\\ 0 & -1/2 & \sqrt{3}/2\end{array}\right)$$
(22)
$${R}_{{{\rm{C}}}^{1}{{\rm{A}}}^{2}}=\left(\begin{array}{ccc}-\cos \delta & 0 & -m\sin \delta \\ 0 & {lm} & 0\\ l\sin \delta & 0 & -{lm}\cos \delta \end{array}\right)=\left(\begin{array}{ccc}0 & 0 & -1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{array}\right)$$
(23)
Next, calculate the components of the vector \(\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}\) as viewed from the local coordinate system \({{\rm{A}}}^{2}\), denoted as \({{\scriptstyle{{\rm{A}}}^{2}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}}\):$${{\scriptstyle{{\rm{A}}}^{2}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}} ={\left(1,0,0\right)}^{{\rm{T}}}+{R}_{{{\rm{A}}}^{2}{{\rm{B}}}_{1}^{2}}{R}_{{{\rm{B}}}_{1}^{2}{{\rm{B}}}_{2}^{2}}{\left(1,0,0\right)}^{{\rm{T}}}\\ =\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)+\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \sqrt{3}/2 & -1/2\\ 0 & 1/2 & \sqrt{3}/2\end{array}\right)\left(\begin{array}{ccc}1/2 & -\sqrt{3}/2 & 0\\ \sqrt{3}/2 & 1/2 & 0\\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)\\ =\left(\begin{array}{c}3/2\\ 3/4\\ \sqrt{3}/4\end{array}\right) \sim \left(\begin{array}{c}1.5\\ 0.75\\ 0.43\end{array}\right)$$
(24)
Subsequently, the coordinates of the vector \(\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}\) as seen from the global coordinate system (i.e., the local coordinate system \({{\rm{A}}}^{1}\)) can be computed as follows.$${{\scriptstyle{\rm{global}}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}} ={{\scriptstyle{{\rm{A}}}^{1}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}}={R}_{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}{R}_{{{\rm{C}}}^{1}{{\rm{A}}}^{2}}\,{{\scriptstyle{{\rm{A}}}^{2}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}}\\ =\left(\begin{array}{ccc}1/2 & 3/4 & -\sqrt{3}/4\\ 3/4 & -1/8 & 3\sqrt{3}/8\\ \sqrt{3}/4 & -3\sqrt{3}/8 & -5/8\end{array}\right)\left(\begin{array}{ccc}0 & 0 & -1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{array}\right)\left(\begin{array}{c}3/2\\ 3/4\\ \sqrt{3}/4\end{array}\right)\\ =\left(\begin{array}{ccc}1/2 & 3/4 & -\sqrt{3}/4\\ 3/4 & -1/8 & 3\sqrt{3}/8\\ \sqrt{3}/4 & -3\sqrt{3}/8 & -5/8\end{array}\right)\left(\begin{array}{c}-\sqrt{3}/4\\ 3/4\\ 3/2\end{array}\right) \sim \left(\begin{array}{c}-0.30\\ 0.56\\ -1.6\end{array}\right)$$
(25)
Therefore, the components of the position vector of point \({{\rm{C}}}^{2}\) as viewed from the global coordinate system (i.e., the local coordinate system \({{\rm{A}}}^{1}\)) are given by the following:$${{\scriptstyle{\rm{global}}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{2}}}= {{\scriptstyle{\rm{global}}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{1}{{\rm{C}}}^{1}}}+{{\scriptstyle{\rm{global}}\atop}{\!{\bf{x}}}}_{\overrightarrow{{{\rm{A}}}^{2}{{\rm{C}}}^{2}}}\\ \sim \left(\begin{array}{c}1.5\\ 0.75\\ 0.43\end{array}\right)+\left(\begin{array}{c}-0.30\\ 0.56\\ -1.6\end{array}\right)=\left(\begin{array}{c}1.2\\ 1.3\\ -1.2\end{array}\right)$$
(26)
Therefore, the distance between the endpoints of the chain (tilt angle = 30°, bite angle = 90°, conformation id = RR(FF)RL is found to be$$\overrightarrow{\left|{{\rm{A}}}^{1}{{\rm{C}}}^{2}\right|} \sim \sqrt{{1.2}^{2}+{1.3}^{2}+{\left(-1.2\right)}^{2}}=2.1.$$
(27)
Rational design of metal–organic cages to increase the number of components via dihedral angle control
