Structural Dynamics of an ELM-11 Framework Transformation Accompanied with Double-Step CO 2 Gate Sorption: An NMR Spin Relaxation Study

: [Cu(4,4 (cid:48) -bipyridine) 2 (BF 4 ) 2 ] (ELM-11), an elastic layer-structured MOF (metal-organic framework), is expected to be a sophisticated CO 2 reservoir candidate because of its high capacity and recovery e ﬃ ciency for CO 2 sorption. While ELM-11 shows a unique double-step gate sorption for CO 2 gas, the dynamics of the structural transition have not yet been clariﬁed. In this study, the dynamics of the 4,4 (cid:48) -bipyridine linkers and the BF 4 − anions were studied by determining 1 H spin-lattice relaxation times ( T 1 ). The ELM-11 structural transition accompanying CO 2 sorption was also examined through the CO 2 uptake dependence of the 1 H spin–spin relaxation time ( T 2 ), in addition to T 1 . In its closed form, the temperature dependence of the 1 H T 1 of ELM-11 was analyzed by considering the contributions of both paramagnetic and dipolar relaxations, which revealed the isotropic reorientation of BF 4 − and the torsional ﬂipping of the 4,4 (cid:48) -bipyridine moieties. The resultant activation energy of 32 kJ mol − 1 for the isotropic BF 4 − reorientation is suggestive of strong (B-F...Cu 2 + ) interactions between Cu(II) and the F atoms in BF 4 − . Furthermore, the CO 2 uptake dependence of T 1 was found to be dominated by competition between the increase in the longitudinal relaxation time of the electron spins and the decrease in the spin density in the unit cell.


Introduction
Porous metal-organic frameworks (PMOFs) and porous coordination polymers (PCP), which exhibit dynamic structural transitions attributed to soft interactions in their crystal structures, are expected to have sorption properties that are different to those of traditional porous materials [1][2][3]. One of the most interesting phenomena in a flexible MOF is its guest-induced structural transition, which typically occurs at a threshold gas pressure and leads to an abrupt increase in the sorption isotherm, a phenomenon referred to as "breathing" and "gate sorption" [4][5][6][7][8][9][10]. The breathing of Recently, the nuclear spin-lattice relaxation rates in paramagnetic substances have attracted much attention due to interest in distance-geometry [32], MRI-relaxation-agents [33], and quantum-computation [34] applications. The molecular motions, phase transitions, and inter-spin interactions in paramagnetic materials have been discussed through 1 H spin-lattice relaxation times (T 1 ) [35][36][37][38]. Therefore, 1 H nuclear magnetic relaxation in ELM-11 is expected to provide useful information about the structural changes and spin-spin interactions that accompany CO 2 gate sorption.
In this study, we investigated the dynamic behavior of the 4,4 -bipyridine linkers and the BF 4 − anions in the closed form of ELM-11 by determining its temperature-dependent 1 H spin-lattice relaxation times (T 1 ), after which the structural transition of ELM-11 accompanying CO 2 sorption was examined by the CO 2 uptake dependence of the 1 H spin-spin relaxation time (T 2 ) as well as T 1 . Finally, the structural change due to CO 2 sorption was examined in terms of magnetic dipolar interactions between nuclear spins and between paramagnetic spins.

Experimental
ELM-11 was prepared according to the reported method [27]. After pretreatment under vacuum (<0.1 Pa) at 373 K for 10 h, CO 2 sorption isotherms were obtained volumetrically at 273 and 195 K using BELSORP Mini II (MicrotracBEL Corp., Osaka, Japan) instruments. The CO 2 gas was 99.9999% pure.
The NMR sample was prepared as follows: a 300 mg sample of ELM-11 powder was introduced into a glass NMR tube (φ 10 mm) and maintained under vacuum at 373 K for 10 h. CO 2 gas was loaded into the tube at 273 or 195 K and adjusted to the appropriate pressure. The tube was sealed with a valve and then inserted into the NMR spectrometer, with the temperature controlled at 273 or 195 K.
A JNM-MU25 pulse NMR spectrometer (JEOL, Akishima, Tokyo, Japan) with a 1 H resonance frequency of 25 MHz (0.5872 T, permanent magnet) was used to measure 1 H relaxation times. T 1 values were measured with the inversion recovery method using a radio-frequency pulse width of 2 µs, a repetition time of 2 ms, and 50 datapoints with a sampling interval of 30 µs. T 2 values were measured with the solid-echo method using a radio-frequency pulse width of 2 µs, a repetition time of 2 ms, and 500 datapoints with a sampling interval of 0.2 µs.

Results and Discussion
3.1. CO 2 Sorption Isotherms CO 2 sorption isotherms for ELM-11 at 273 and 195 K are shown in Figure 1. The CO 2 sorption isotherm of ELM-11 to P/P 0~0 .03 at 273 K reveals a vertical uptake at P/P 0~0 .01 (Figure 1a), which corresponds to gate opening, as previously reported [4,24,26,27]. Another steep increase in sorption is seen at 195 K at P/P 0~0 .3, as shown in Figure 1b. Similar double-step sorption isotherms have previously been reported [13,14,28,29]. Detailed structural analyses showed that ELM-11 absorbs two CO 2 molecules per Cu atom to form 2, with a 28% expansion in the interlayer distance at the first step at 273 K, and absorbs four more CO 2 molecules per Cu atom to form 3, with a 56% expanded layer structure compared to the initial structure at 195 K [28,29].
Crystals 2020, 10,328 3 of 20 Therefore, 1 H nuclear magnetic relaxation in ELM-11 is expected to provide useful information about the structural changes and spin-spin interactions that accompany CO2 gate sorption.
In this study, we investigated the dynamic behavior of the 4,4′-bipyridine linkers and the BF4 − anions in the closed form of ELM-11 by determining its temperature-dependent 1 H spin-lattice relaxation times (T1), after which the structural transition of ELM-11 accompanying CO2 sorption was examined by the CO2 uptake dependence of the 1 H spin-spin relaxation time (T2) as well as T1. Finally, the structural change due to CO2 sorption was examined in terms of magnetic dipolar interactions between nuclear spins and between paramagnetic spins.
The NMR sample was prepared as follows: a 300 mg sample of ELM-11 powder was introduced into a glass NMR tube (ϕ 10 mm) and maintained under vacuum at 373 K for 10 h. CO2 gas was loaded into the tube at 273 or 195 K and adjusted to the appropriate pressure. The tube was sealed with a valve and then inserted into the NMR spectrometer, with the temperature controlled at 273 or 195 K.
A JNM-MU25 pulse NMR spectrometer (JEOL, Akishima, Tokyo, Japan) with a 1 H resonance frequency of 25 MHz (0.5872 T, permanent magnet) was used to measure 1 H relaxation times. T1 values were measured with the inversion recovery method using a radio-frequency pulse width of 2 μs, a repetition time of 2 ms, and 50 datapoints with a sampling interval of 30 μs. T2 values were measured with the solid-echo method using a radio-frequency pulse width of 2 μs, a repetition time of 2 ms, and 500 datapoints with a sampling interval of 0.2 μs.

CO2 Sorption Isotherms
CO2 sorption isotherms for ELM-11 at 273 and 195 K are shown in Figure 1. The CO2 sorption isotherm of ELM-11 to P/P0 ~0.03 at 273 K reveals a vertical uptake at P/P0 ~0.01 (Figure 1a), which corresponds to gate opening, as previously reported [4,24,26,27]. Another steep increase in sorption is seen at 195 K at P/P0 ~0.3, as shown in Figure 1b. Similar double-step sorption isotherms have previously been reported [13,14,28,29]. Detailed structural analyses showed that ELM-11 absorbs two CO2 molecules per Cu atom to form 2, with a 28% expansion in the interlayer distance at the first step at 273 K, and absorbs four more CO2 molecules per Cu atom to form 3, with a 56% expanded layer structure compared to the initial structure at 195 K [28,29].

Calculating the Second Moment Plateau Values
The van Vleck formula can be used to calculate NMR second moments in rigid lattices of solid-state materials with well-known molecular and crystal structures [39,40]. A theoretical description of the NMR second moment is given in Appendix A. Using the above-mentioned formula, we calculated the 1 H and 19 F second moments of the rigid lattices of the three crystal structures of ELM-11. Second-moment reductions were also calculated by taking into account the anisotropy parameter [40,41] associated with the isotropic reorientation of BF 4 − and the torsional flipping of the 4,4 -bipyridine linkers. The second moments in the rigid lattices determined for the 1 H and 19 F nuclei are summarized in Tables A1 and A2  in Appendix A, while Tables 1 and 2 show the evaluated reductions in the 1 H and 19 F second moments. to a large reduction in the second moment, which suggests that spin-lattice relaxation is expected to be effective through a mechanism involving fluctuations in magnetic dipolar interactions that act on 19 F nuclei and control the 1 H spin-lattice relaxation rate through cross-relaxation between the 1 H and 19 F spin systems. In this case, the nuclear spin systems relax through two mechanisms: paramagnetic and dipolar relaxation. In general, relaxation times through paramagnetic ions are one or two orders of magnitude shorter than the relaxation times of diamagnetic substances. According to the multi-paramagnetic-center model, which is preferred for paramagnetic materials with dense paramagnetic-centers, the paramagnetic relaxation rate (R 1p ) is given by [47,48].
where C and D is the efficiency of direct relaxation and the diffusion coefficient for spin diffusion, respectively, and N p is the number of paramagnetic centers per unit volume of the sample. In the powder sample, C is represented by where γ S and γ I are the gyromagnetic ratios of the electron spin and resonant nuclei, respectively, S is the spin of the paramagnetic ion, τ e is the correlation time for the z-component of the paramagnetic spin (longitudinal relaxation time for the electron spin), and ω I is the resonance frequency of a resonant nucleus. According to Bloembergen [49], D = a 2 /50T 2 , where a is the average 1 H-1 H distance (0.551 nm for 1) and T 2 is the 1 H spin-spin relaxation time (average of experimental values;~22 µs). As a result, D = 2.87 × 10 −16 m 2 s −1 for 1. This is reasonable because it is of the same order of magnitude as the D value (6.25 × 10 −16 m 2 s −1 ) for the high spin state of [Fe(ptz) 6 ](BF 4 ) 2 (ptz = 1-n-propyl-1H-tetrazole) [35]. Furthermore, we evaluated N p as 1.91 × 10 27 m −3 for the body-centered lattice formed by the Cu 2+ ions in 1. Thus, R 1p depends strongly on τ e . On the other hand, the dipolar relaxation rate (R 1d ) is mainly controlled by fluctuations in the magnetic dipolar interactions among the 1 H (I = 1/2), 19 F (S = 1/2), 10 B (S = 3), and 11 B (S = 3/2) spins. In such a multi-spin system, cross relaxation between the 1 H, 19 F, 10 B, and 11 B nuclei are taken into account [40]. Here, assuming that both the 1 H and 19 F nuclei dominantly contribute to cross relaxation because of their large gyromagnetic ratios, the actual relaxation rates are given by the eigenvalues of the relaxation matrix R [43,44,[50][51][52]: Crystals 2020, 10, 328 7 of 21 In general, these relaxation rates lead to the non-exponential recovery of magnetization: however, the 1 H magnetization recovers exponentially in ELM-11. In this context, as mentioned in Appendix B, we can regard R HH , R FF ≈ R FH , R HF ; hence one of the two eigenvalues is almost zero. The observed relaxation rate then takes the following form where R HH and R FF are diagonal elements of the relaxation matrix R. In this case, R HH and R FF are given by [39,40]: The analytical formulas for g 1 (ω i , τ i ) and g 2 ω i , ω j , τ i are given by [40,50]: Assuming that a thermal activation process is responsible for the fluctuation in the internuclear vector, the temperature dependence of τ i (i = H, F) is given by the Arrhenius equation, as follows where E a,i (i = H, F) is the activation energy for BF 4 − and 4,4 -bipyridine. Consequently, we analyzed the temperature dependence of 1 H T 1 using the sum of the contributions from both paramagnetic relaxation (R 1p ) and dipolar relaxation (R 1d ): The experimental data were fitted to Equation (8), the results of which are shown in Figure 2a,b. The R 1p component was optimized at τ e = 1.22 × 10 −11 s, resulting in a T 1p value more than one order of magnitude smaller than T 1d . The evaluated τ e value is reasonable because typical τ e values for paramagnetic metal ions range between 10 −8 s and 10 −12 s [53]; it is also sufficiently fast to average out the width of the 1 H resonance line due to 1 H-electron dipolar interactions. As described below, the average value of 1 H T 2 is about 22 µs, which corresponds to a full width at half maximum (FWHM) of 15 kHz, where FWHM = 1/πT 2 . This value is much narrower than the linewidth (~500 kHz) caused by the average local magnetic field between interlayer Cu-H pairs. Table 3  parameters. Therefore, in order to improve the reliability of the optimization results and to guarantee that the parameters have physical meaning, we assumed an E a value for the torsional flipping of the 4,4 -bipyridine. In fact, Moreau et al. reported that the torsional barrier for phenylene rings within linkers in a series of isoreticular octacarboxylate MOFs depended on the steric hindrance around the linkers, as well as the electronic structure of the framework [54]. Furthermore, Inukai et a kind of flexible PCP referred to as "CID-5/6", the energy barrier for the rotation of the pyridyl ring depended on the steric hindrance around the linkers: the E a values for the 4-site and 2-site flip rotations are 20 and 25 kJ mol −1 for CID-5/6 (x = 0.55), and 32 and 27 kJ mol −1 for CID-5/6 (x = 0.37) [55]. In the latter case, the intermolecular distances between 4,4 -bipyridine linkers in CID-5 and 6 are 4.11 Å and 3.91 Å, whereas it is 6.21 Å in the closed form of ELM-11, which suggests that there is less steric hindrance between the linkers in ELM-11. Therefore, we referred to the E a value as reported in the gas phase (4.0 kcal mol −1 ) [45] for simplicity, and then fixed the E a value to be close to this value during our T 1 analysis. As a result, the E a value (32 kJ mol −1 ) obtained for the isotropic reorientation of BF 4 − is slightly larger than those (10-26 kJ mol −1 ) reported in various systems [40,[42][43][44]. The relatively short Cu-F interatomic distance of 2.404 Å facilitates the formation of a strong hydrogen-bond-like interaction (C-F...M + [56]) between Cu(II) and a F atom in BF 4 − (B-F...Cu 2+ ). As a result, the BF 4 − isotropic reorientation in ELM-11 has a large E a value. The gate phenomenon is closely associated with lattice vibration as well as the diffusivity of gas molecules. The rotational flipping of the 4,4 -bipyridine moiety is a type of phonon acoustic lattice-vibration mode of ELM-11. Gas molecules, such as CO 2 , perturb the rotational motion of the 4,4 -bipyridine moiety through molecular collisions. In particular, the inelastic collisions between gas molecules and the ELM-11 framework is considered to effectively perturb the thermally activated rotational motion of the 4,4 -bipyridine moiety, which then triggers the structural transition for gate opening. Thus, energy-transfer efficiency between the gas molecules and the ELM-11 framework determines the gate-opening pressure.
Furthermore, the torsional flipping and/or rotational motion of the 4,4 -bipyridine moiety also affects the orientational selectivity of the CO 2 molecules toward molecular diffusion and arrangement in 1 at the first gate opening. Torsional flipping gives rise to an excluded volume for the pyridyl ring that is larger than the rigid one. This reduces the effective free volume along the b-axis because twisted 4,4 -bipyridine moieties lie along the b-axis. As a result, the accessible space for the CO 2 molecules elongates along the b-axis as a prolate spheroid, which not only affects the molecular orientation when CO 2 molecules penetrate into the ELM-11 crystal lattice, but also facilitates the alignment of CO 2 molecules along the b-axis. In fact, the CO 2 molecules are accommodated in the interlayer void spaces formed between the neighboring layered square grids in 2, which results in the alignment of the molecular axes with the b-axis.  [55]. In the latter case, the intermolecular distances between 4,4′-bipyridine linkers in CID-5 and 6 are 4.11 Å and 3.91 Å , whereas it is 6.21 Å in the closed form of ELM-11, which suggests that there is less steric hindrance between the linkers in ELM-11. Therefore, we referred to the Ea value as reported in the gas phase (4.0 kcal mol −1 ) [45] for simplicity, and then fixed the Ea value to be close to this value during our T1 analysis.
The gate phenomenon is closely associated with lattice vibration as well as the diffusivity of gas molecules. The rotational flipping of the 4,4'-bipyridine moiety is a type of phonon acoustic latticevibration mode of ELM-11. Gas molecules, such as CO2, perturb the rotational motion of the 4,4'bipyridine moiety through molecular collisions. In particular, the inelastic collisions between gas molecules and the ELM-11 framework is considered to effectively perturb the thermally activated rotational motion of the 4,4'-bipyridine moiety, which then triggers the structural transition for gate opening. Thus, energy-transfer efficiency between the gas molecules and the ELM-11 framework determines the gate-opening pressure.
Furthermore, the torsional flipping and/or rotational motion of the 4,4'-bipyridine moiety also affects the orientational selectivity of the CO2 molecules toward molecular diffusion and arrangement in 1 at the first gate opening. Torsional flipping gives rise to an excluded volume for the pyridyl ring that is larger than the rigid one. This reduces the effective free volume along the b-axis because twisted 4,4'-bipyridine moieties lie along the b-axis. As a result, the accessible space for the CO2 molecules elongates along the b-axis as a prolate spheroid, which not only affects the molecular orientatio    Figure 3a,b shows the dependence of 1 H T 1 on the amount of CO 2 sorbed into ELM-11 at 273 and 195 K, respectively. The T 1 value was observed to decrease in a stepwise manner at 273 K, from 500 to 455 µs at P/P 0 = 0.01. On the other hand, the T 1 value decreased in a stepwise manner at 195 K, from 532 to 490 µs at P/P 0 = 0.01, and then increased again to 529 µs in the 0.2-0.4 P/P 0 range. These observed changes are in good agreement with the stepwise increases in the uptake of CO 2 shown in the sorption isotherms ( Figure 1). The crystal structure of ELM-11 changes through the stepwise sorption of CO 2 , resulting in an increase in the interlayer distance. Therefore, this feature suggests that variations in T 1 due to CO 2 sorption are closely related to the structural changes undergone by ELM-11. Table 4 lists the T 1 values for each ELM-11 structure at 273 and 195 K. The T 1 changes observed between 529 and 455 µs are due to structural changes, and the change in T 1 during a one-step structural change is in the 39-45 µs range.     T 1 appears to depend on CO 2 uptake, which is ascribable to: (1) an increase in the interlayer distance, and (2) an increase in the chemical pressure due to the impact of CO 2 on the molecular motions of BF 4 − and 4,4 -bipyridine. The change in T 1 in 1 in moving from 250 to 323 K is about 32 µs, which is smaller than those observed for the CO 2 -uptake dependence. Since ∆M F11B 2 dominates ∆M 2 , a further increase in ∆M F11B 2 is required in order to explain the relationship between T 1 and CO 2 uptake. However, the structure of BF 4 − is not significantly affected by changes in the crystal structure of ELM-11; consequently, isotropic BF 4 − reorientation cannot be used to reasonably explain the observed change in T 1 due to CO 2 sorption. On the other hand, the increase in the interlayer distance between the stacked two-dimensional [Cu(bpy) 2 2+ ] n sheets increases the unit cell volume and the interlayer Cu-Cu distance; these affect N p and τ e , which dominate R 1p . Since R 1p depends on N p 2 and N p 4/3 [47], an increase in the cell volume decreases N p (see Table 4), resulting in a decrease in R 1p . In contrast, τ e is affected by interactions between electron spins (dipolar interactions and/or exchange interactions) and, as a first approximation, 1/τ e is proportional to the magnetic dipolar and/or exchange interaction [53]. The average Cu-Cu distance in a [Cu(bpy) 2 2+ ] n layer is 1.11 nm, whereas the average Cu-Cu distance between layers is 0.9105 nm in 1, 0.9959 nm in 2, and 1.0692 nm in 3. This feature strongly suggests that interlayer spin-spin interactions dominate more than intralayer ones. Since the magnetic dipolar and exchange interactions decay with increasing inter-spin distance, τ e increases with inter-spin distance. Consequently, the change in T 1 due to CO 2 sorption can be examined using τ e as a variable. Table 4 summarizes the experimental and calculated values of T 1 for each crystal structure. T 1p, calc was calculated using τ e as a variable so as to reproduce T 1p, exp . At 273 K, the experimental value for 1 is somewhat smaller than the calculated one; this difference stems from the contribution of R 1d . Compared to τ e at 195 K, a longer interlayer Cu-Cu distance leads to a longer τ e . Thus, expansion of the unit cell due to CO 2 sorption decreases the spin density, whereas elongation of the interlayer distance increases τ e . These two effects act on T 1p in opposite directions, and in ELM-11 they are balanced and determine the total T 1p of the system. The T 1p of 2 is shorter than that of 1 because the contribution of τ e is rather large. On the other hand, both effects are comparable in 3 and, as a result, its T 1p is almost the same as that of 1.

Spin-Spin Relaxation Time (T 2 ) in ELM-11
Figure 4a,b shows the dependence of T 2 on the amount of CO 2 sorbed at 273 K and 195 K. At 273 K, ELM-11 shows a stepwise increase in T 2 at P/P 0~0 .01, despite a decrease in T 1 . On the other hand, ELM-11 shows two stepwise increases in T 2 at 195 K, at P/P 0~0 .01 and~0.3. These changes in T 2 also correspond to the gate sorption of CO 2 , as was observed for T 1 , which accompanies a structural change in the crystal structure, in particular, an increase in the interlayer distance. The spin system satisfies a condition that ω H τ >> 1 in these temperature regions, because T 1 T 2 and T 2 << T 1 ; hence T 2 is governed by the local magnetic field at the 1 H nuclei (1/T 2 ∝ B 2 loc ). The local magnetic field caused by a spin with magnetic moment µ at a position far from the spin, is given by (µ 0 /4π)(µ/r 3 )(3cos 2 θ -1) [39]. Here, θ is the angle between the inter-spin vector and the external magnetic field and µ 0 is the magnetic permeability of a vacuum. The 1 H, 19 F, and electron spins contribute to the local magnetic field in ELM-11.
The magnitude of the local magnetic field is inversely proportional to the cube of the inter-spin distance. The contribution of Cu 2+ can be evaluated from the average Cu-H distance between the stacked two-dimensional [Cu(bpy) 2 2+ ] n sheets, which is 0.7801 nm in 1, 0.8668 nm in 2, and 0.9702 nm in 3. The square of the local magnetic field, B 2 loc , is evaluated using these distances to be 383 × 10 −8 T 2 , 203 × 10 −8 T 2 , and 103 × 10 −8 T 2 , respectively. Consequently, extending the interlayer distance results in a decrease in B 2 loc to 53% in 2, and 27% in 3, of that of 1. Actually, the magnetic moment of Cu 2+ is partially averaged out by the fast flip-flopping of the electron spin; hence, the net magnetic moment of Cu 2+ reduces B 2 loc to B 2 loc . The contributions from the 1 H and 19 F magnetic moments can also be evaluated through the second moments in the rigid lattices (see Tables A1 and A2 2,intra , and M F11B 2,intra are almost identical in the rigid lattices of the three substances, which suggests that the increase in the interlayer distance affects the intermolecular 1 H-19 F dipolar interactions little. Therefore, 1 H-1 H dipolar interactions are also considered to be among the factors that affect T 2 through the local magnetic field. Crystals 2020, 10, 328 11 of 20 magnetic permeability of a vacuum. The 1 H, 19 F, and electron spins contribute to the local magnetic field in ELM-11. The magnitude of the local magnetic field is inversely proportional to the cube of the inter-spin distance. The contribution of Cu 2+ can be evaluated from the average Cu-H distance between the stacked two-dimensional [Cu(bpy)2 2+ ]n sheets, which is 0.7801 nm in 1, 0.8668 nm in 2, and 0.9702 nm in 3. The square of the local magnetic field, loc 2 , is evaluated using these distances to be 383 × 10 −8 T 2 , 203 × 10 −8 T 2 , and 103 × 10 −8 T 2 , respectively. Consequently, extending the interlayer distance results in a decrease in loc 2 to 53% in 2, and 27% in 3, of that of 1. Actually, the magnetic moment of Cu 2+ is partially averaged out by the fast flip-flopping of the electron spin; hence, the net magnetic moment of Cu 2+ reduces loc 2 to 〈 loc 2 〉.
The contributions from the 1 H and 19 F magnetic moments can also be evaluated through the second moments in the rigid lattices (see Tables A1 and A2). 1 and 2 contain two kinds of 4,4'bipyridine linkers with different conformations, whereas 3 has four kinds of 4,4'-bipyridine linker.
values for the two conformers of 1 are 7.348 × 10 −8 T 2 and 2.086 × 10 −8 T 2 , while in 2 they are 5.683 × 10 −8 T 2 and 1.958 × 10 −8 T 2 , and they are 8.454 × 10 −8 T 2 , 6.202 × 10 −8 T 2 , 6.093 × 10 −8 T 2 , and 2.357 × 10 −8 T 2 , for the four conformers of 3. In each case, the conformer with the somewhat smaller torsion angle, in which 1 H-1 H distances are relatively short, gives a larger 2, value than that with the larger torsion angle. Furthermore, the values of 2, of the planar and twisted conformers are similar in each compound, but 2, decreases in the order: 1 > 2 > 3, which indicates that the intermolecular 1 H-1 H dipolar interaction is affected little by the conformation of the 4,4'-bipyridine moiety, but decreases due to the increase in the interlayer distance. On the other hand,  In terms of the structural changes that occur in going from 1 to 2 and then from 2 to 3, increases in the interlayer distance and the conformational changes undergone by the 4,4'-bipyridine linkers decrease both the 1 H-electron and 1 H-1 H dipolar interactions, i.e., the local magnetic field around the In terms of the structural changes that occur in going from 1 to 2 and then from 2 to 3, increases in the interlayer distance and the conformational changes undergone by the 4,4 -bipyridine linkers decrease both the 1 H-electron and 1 H-1 H dipolar interactions, i.e., the local magnetic field around the protons, resulting in an increase in T 2 . In addition, at 195 K, T 2 is somewhat lower for CO 2 sorption between the first and the second steps. Since no lattice shrinkage was observed by powder XRD to accompany the decrease in interlayer distance during this process, we infer that the decrease in T 2 is not related to a change in interlayer distance (i.e., the 1 H-electron distance). In fact, the closest 1 H-1 H distance in 4,4 -bipyridine, pairs of which contribute the most to the local magnetic field, changes periodically with torsion angle. The local field is smallest at a twist angle of 90 • , in which two pyridine rings are perpendicular to each other, and is largest for the planar structure, with a twist angle of 0 • or 180 • . Hence, we speculate that the conformational change undergone by the 4,4 -bipyridine linkers is one of the origins of the observed decrease in T 2 between the first and the second CO 2-sorption steps. The CO 2 uptake during the first gate sorption is estimated to be 160 mg g -1 , which corresponds to the sorption of two CO 2 molecules per [Cu(bpy) 2 ](BF 4 ) 2 formula unit, after which the CO 2 uptake increases gradually with P/P 0 , to a value of 230 mg g −1 just prior to the second gate sorption. This uptake corresponds to the sorption of 2.9 CO 2 molecules per ELM-11 formula unit. Furthermore, uptake was observed to increase to 500 mg g −1 following the second gate sorption, which corresponds to the sorption of 6.2 CO 2 molecules per ELM-11 formula unit. Hiraide et al. reported the crystal structures of 2 and 3, and revealed that the torsion angle around the C-C axis becomes small as the structure transforms from 2 into 3 [11,29]. This feature is considered to avoid repulsion between CO 2 and 4,4 -bipyridine, which increases the amount of sorbed CO 2 because the planar 4,4 -bipyridine structure has less free volume around its linkers than the other conformers. In fact, the conformation of the 4,4 -bipyridine linkers reportedly approaches that of the planar conformer by reducing the torsional angles from 0.74 • and 70.64 • in 2 to 0.14 • and 68.74 • in ELM-11⊃3CO 2 [11,29]. As the molecular structure of 4,4 -bipyridine approaches planarity, the intramolecular 1 H-1 H distances (particularly, at the 2,6 and 2',6' positions) become shorter, which increases the 1 H-1 H magnetic dipolar interactions. This conclusion is also supported by the M HH 2,intra values of the 4,4 -bipyridine moieties, which are significantly different for the planar (5.683 × 10 −8 T 2 ) and twisted (1.958 × 10 −8 T 2 ) orientations. These M HH 2,intra values correspond to T 2 contributions of 13 and 22 µs. Therefore, the increase in the intramolecular 1 H-1 H dipolar interaction is regarded as a possible explanation for the decrease in T 2 observed between the first and second sorption steps.

Conclusions
We calculated the 1 H and 19 F second moments in the rigid lattices of the three crystal structures of ELM-11, and the reductions in the second moments due to both isotropic BF 4 − reorientation and the torsional flipping of the 4,4 -bipyridine linkers. 1 H second-moment reductions of (0.4-2) × 10 −8 T 2 were determined, indicative of a low contribution to the total magnetic dipolar relaxation rate. On the other hand, reductions of (21-24) × 10 −8 T 2 were determined for the 19 F second moment. These large reductions suggested that 1 H spin-lattice relaxation effectively takes place through fluctuations in the magnetic dipolar interactions that act on 19 F nuclei through cross-relaxation between the 1 H and 19 F spin systems. The temperature dependence of 1 H T 1 in the closed form of ELM-11 was analyzed using the sum of the contributions from both paramagnetic relaxation (R 1p ) and dipolar relaxation (R 1d ). We found that R 1p makes a dominant contribution to the total 1 H spin-lattice relaxation rate, but the T 1 minimum observed at 323 K is mainly due to the averaging of 19 F-19 F and 19 F-11 B magnetic dipolar interactions through isotropic BF 4 − reorientation. The large E a value (32 kJ mol −1 ) obtained for the isotropic BF 4 − reorientation supports the formation of a strong hydrogen-bond-like interaction (B-F...Cu 2+ ) between Cu(II) and a F atom in BF 4 -. We also discussed the role that torsional flipping of the 4,4 -bipyridine moiety plays in relation to the gate-opening phenomenon, as well as the orientational selectivity of the CO 2 molecules in relation to their diffusion and arrangement in the lattice.
The dependence of T 1 on CO 2 uptake is the result of a corresponding increase in the interlayer distance. The increase in the unit cell volume due to CO 2 sorption led to a decrease in spin density, whereas an increase in the interlayer distance resulted in an increase in the longitudinal relaxation time of the electron spins (τ e ). These two effects, which act on T 1p in opposite directions, balance each other and control the T 1 value.
The local magnetic field at the 1 H nuclei governs the T 2 value, and a decrease in the local magnetic field increases the T 2 value. The local magnetic field associated with the net magnetic moments of Cu 2+ and the intermolecular 1 H dipolar interaction decreases with increasing interlayer distance in ELM-11, leading to an increase in T 2 . Furthermore, the conformational change in the 4,4 -bipyridine unit, from the twisted form to the planar form, enables the intramolecular 1 H dipolar interaction to increase, which shortens T 2 .

Acknowledgments:
The authors sincerely thank the late Mamoru Imanari (Center for Analytical Instrumentation, Chiba University) for his assistance with the NMR experiments, and Hideki Tanaka (Shinshu University) for providing detailed structural data for ELM-11.

Conflicts of Interest:
The authors declare no conflicts of interest.

Appendix A. Calculating Second Moments for ELM-11
Appendix A.1. Theoretical Description of the NMR Second Moment NMR second moments in a rigid lattice of a solid-state material with a well-known molecular and crystal structure can be calculated using the van Vleck formula. In a powdered sample, the van Vleck formula can be represented for like spins and unlike spins as follows [39,40]: where I and S are the spins of NMR-active nuclei, γ I and γ S are the gyromagnetic ratios of nuclear spins I and S, respectively, r j,k and r j, m are internuclear distances, and N I and N S are the number of I and S spins, respectively. In this study, 1 H (I = 1/2) is the observed resonant nucleus. The NMR second moments for the observed nucleus are given as a sum of the respective contributions from the like spins and the unlike spins: Here, we take into account three kinds of nucleus as unlike spins, namely 19 F (S = 1/2), 10 B (S = 3), and 11 B (S = 3/2). When molecules that include the observed nuclei move, the NMR second moments are lowered in a manner that depends on their motional modes. If the internuclear vector undergoes isotropic rotation, such as a molecule in an isotropic liquid, the second moment in the rigid lattice is completely averaged out to zero (M I 2,ave = 0). This situation corresponds to the isotropic reorientation of BF 4 − in the ELM-11 crystal. When the molecular motion is anisotropic (for example, by rotation about one axis), the second moment in the rigid lattice is partially averaged out. In this case, the anisotropy parameter, q 2 , which represents the degree of the motion anisotropy, is defined as follows: where M I 2,ave is the second moment after motional averaging. Using the reduction in the second moment, ∆M I 2 = M I 2,rigid − M I 2,ave and ∆M I 2 = q 2 M I 2,rigid , q 2 is determined by the nature of the molecular motion. When the internuclear vector jumps at a flip angle φ while maintaining angle θ with respect to a fixed axis, q 2 is given by [40,41]: Using Equations (A1)−(A4), we calculated the 1 H and 19 F second moments in the rigid lattices for the three crystal structures of ELM-11, as well as the reductions in the second moments when BF 4 − isotropically reorients and when the 4,4 -bipyridine linkers flip. The second moments in the rigid lattices determined for the 1 H and 19 F nuclei are summarized in Tables A1 and A2 in Appendix A.
The reduction in the second moment due to intramolecular 1 H-1 H magnetic dipolar interactions, ∆M HH 2, intra , was evaluated using Equation (A4) using the ideal 4,4 -bipyridine molecular structure when the pyridine ring flips around the C-C axis. Figure  The reduction in the second moment due to intramolecular 1 H-1 H magnetic dipolar interactions, ∆ 2, , was evaluated using Equation (A4) using the ideal 4,4'-bipyridine molecular structure when the pyridine ring flips around the C-C axis. Figure A1 shows the ∆ 2, value as a function of the flip angle. The ∆ 2, value increases with increasing flip angle to a maximum at 22°, and then decreases gradually. The torsion angle of the 4,4'-bipyridine unit is 54.6° in 1 and 70.64° in 2. When the ring flips with these torsion angles, the ∆ 2, value is expected to be (0.2-0.4) × 10 −8 T 2 . In contrast, the torsion angles are 14.98° and 17.52° in 3, and ring flipping with these torsion angles is expected to give ∆ 2, values of (1-1.3) × 10 −8 T 2 .

Appendix B. Theoretical Background for NMR Spin-Lattice Relaxation of Multi-Spins
There are four kinds of NMR active nucleus in ELM-11, namely 1 H (I = 1/2), 19 F (S = 1/2), 10 B (S = 3), and 11 B (S = 3/2). In such a multi-spin system, fluctuations in the magnetic dipolar interactions between like and unlike spins causes magnetic relaxation between spin systems and the lattice. In particular, the 1 H and 19 F nuclei, which have relatively large gyromagnetic ratios, have large magnetic dipolar interactions with other spins. In such a case, the effect of cross relaxation, which involves relaxation through other spins, in addition to the direct relaxation from each spin system to the lattice, cannot be ignored. The effect of cross relaxation imparts non-exponential behavior on the recovery of both 1 H and 19 F magnetizations. In ELM-11, the 1 H and 19 F nuclei are regarded to contribute to cross relaxation because 1 H-10 B and 1 H-11 B magnetic dipolar interactions are much smaller than 1 H-19 F magnetic dipolar interactions. However, since the 19 F-10 B and 19 F-11 B magnetic dipolar interactions are somewhat larger than the 1 H-1 H and 1 H-19 F dipolar interactions, the interactions between 19 F and 10,11 B are treated as contributing to the 19 F relaxation rate. Now, in such a system, the recovery rate of the magnetization of different spins obeys the following differential equation [40]: If the magnitudes of the off-diagonal elements means that they cannot be ignored, then cross relaxation needs to be taken into account. In that case, the 1 H and 19 F magnetizations are expected to recover non-exponentially. The diagonal and off-diagonal elements in matrix R represent the spin-lattice relaxation rates caused by fluctuations in the magnetic dipolar interactions between the like-and unlike-spins as follow [40]: Here, we ignore the contribution of the cooperative motion between BF 4 and 4,4 -bipyridine (1/τ c = p/τ H + (1 − p)/τ F ; 0 < p < 1). The analytical formulas for g 1 (ω i , τ i ), g 2 ω i , ω j , τ i , and g 3 ω i , ω j , τ i are given by [11,14,15,21,23]: Assuming a thermal activation process for the fluctuation of the internuclear vector, the temperature dependence of τ i (i = H, F) is given by the Arrhenius equation: where E a,i (i = H, F) is the activation energy for BF 4 − and 4,4 -bipyridine.
The reductions in the second moment, ∆M ii 2 and ∆M ij 2 , can be calculated from the crystal structure by assuming the appropriate motional mode. In this study, we evaluated ∆M ii 2 and ∆M ij 2 for the isotropic reorientation of BF 4 − and the torsional flipping of 4,4 -bipyridine around the C-C axis.
However, we treat ∆M ii 2 and ∆M ij 2 as variables during actual data analysis and then optimize the above equations to fit the experimental T 1 data. As a result, the validity of the motional mode is discussed by comparing the ∆M ii 2 and ∆M ij 2 values obtained with the calculated ones. Moreover, in a multinuclear spin system containing both 1 H and 19 F nuclei, the nuclei relax with relaxation rates R and R", which are the eigenvalues of the relaxation matrix R. As a result, both magnetizations recover non-exponentially. However, in the case of the closed form of ELM-11, the 1 H magnetization recovered exponentially, as shown in Figure A2. In multinuclear spin systems containing both 1 H and 19 F nuclei, the magnetization recovery curves exhibit single exponential