Theoretical Investigation of the Structures and Dynamics of Crystalline

Article pubs.acs.org/JPCC

Theoretical Investigation of the Structures and Dynamics of Crystalline Molecular Gyroscopes Anant Babu Marahatta,† Manabu Kanno,† Kunihito Hoki,‡ Wataru Setaka,§ Stephan Irle,∥ and Hirohiko Kono*,† †

Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Center for Frontier Science and Engineering, University of Electro-Communications, Tokyo 182-8585, Japan § Faculty of Pharmaceutical Sciences, Tokushima Bunri University, Sanuki 769-2193, Japan ∥ Department of Chemistry, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan ‡

ABSTRACT: Recently, molecular rotor systems have been emerging as a promising candidate of functional nanoscale devices. A macroscopic gyroscope like molecule in a crystalline solid is particularly unique owing to its variable physicochemical properties. Setaka et al. have achieved the synthesis of a novel crystalline molecular gyroscope characterized by a closed topology with a phenylene rotator encased in three long siloxaalkane spokes [Setaka, W.; et al. Chem. Lett. 2007, 36, 1076]. We theoretically investigated the underlying mechanism of its rotational dynamics by utilizing the self-consistent-charge density-functional-based tight-binding (DFTB) method for crystal structures. We first found that the DFTB semiquantitatively reproduced the unit cell molecular geometries of all three stable X-ray structures under the periodic boundary condition. From the potential energy surface calculations, the activation barrier for phenylene rotation was estimated to be about 1.2 kcal/mol, which is much lower than those of other, previously synthesized gyroscopic compounds. In comparison to 1,4bis(trimethylsilyl)benzene of a similar crystal structure but of an open topology, the siloxaalkane frame in the crystalline molecular gyroscope under consideration effectively blocks strong intermolecular steric interactions experienced by the phenylene rotator. The molecular dynamics simulations based on the DFTB exemplified facile phenylene flipping between the stable structures, especially at high temperature. The present results demonstrate the remarkable ability of the DFTB method to predict the crystal structures and rotational dynamics of this type of crystalline molecular gyroscopes.

1. INTRODUCTION One of the current goals of nanotechnology is to fabricate various molecular machines that perform desired functions.1,2 Solid-state engineering, which aims at synthesizing molecules whose crystal structures incorporate moving units and at functionalizing them as a crystalline solid device, has become a fundamental tool of molecular machinery.3 Several studies on the rotation of single molecules and molecular arrays bound to surfaces4−7 have already highlighted possible implementations of molecular machines in reality. It is still more challenging to realize the crystalline solid devices with well-defined and controllable molecular motions. The direct application of them in technology is far away because the fundamental mechanisms of functionalizing such systems are not well understood yet.8−10 Currently, the synthesized crystalline macrocyclic compounds with bridged phenylene groups have become an intriguing representative as a potential candidate of molecular machines. As a solid-state molecular gyroscope, Garcia-Garibay et al. first proposed a crystalline macrocyclic compound 1 (Chart 1),10 though its total synthesis has not yet been achieved. This type of macrocyclic compound possesses a © 2012 American Chemical Society

central rotating unit (i.e., rotator) linked by a rotary axle to a shielding frame (i.e., stator) as in a macroscopic gyroscope; the phenylene (or substituted phenylene) ring acts as the rotator. In such molecules, the orientation of the bridged phenylene ring could be controlled by thermal or electromagnetic (photonic) stimuli. To demonstrate gyroscopic behaviors, a molecular analogue must possess a “free volume” unit around the rotating segment. The free volume is defined as the volume where the rotator enclosed by the surrounding static frame moves more or less freely. It is closely related to the packing coefficient Ck of the crystal, which is given by the volume of all the molecules in the unit cell (Vmol) relative to the total cell volume (Vcell): Ck = Vmol/Vcell. A tightly packed crystal lattice usually experiences strong intermolecular steric interactions and hinders gyroscopic rotations. Instead of compound 1, Garcia-Garibay et al. have synthesized related molecular gyroscopes 2a−2d and invesReceived: September 10, 2012 Revised: October 29, 2012 Published: October 31, 2012 24845

dx.doi.org/10.1021/jp308974j | J. Phys. Chem. C 2012, 116, 24845−24854

The Journal of Physical Chemistry C

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Chart 1.

Figure 1. X-ray crystallography of the experimentally synthesized siloxaalkane molecular gyroscope 3a with three stable positions of phenylene taken from the Supporting Information of ref 19. The phenylene dihedral angle is formed by the spin axis (Si−C bond), the C atom next to the Si atom in a siloxaalkane spoke (yellow spheroid), and the ortho C atom in phenylene (other colored spheroids). The blue, green, and purple spheroids represent the ortho C atoms for the three stable positions A, B, and C, respectively. Hydrogen atoms are omitted for clarity. The rotator, stator, and spin axis are encircled. 24846



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tigated their dynamical behaviors in the solid state.10−15 The phenylene group in compound 2a is almost fixed at the equilibrium structure owing to the comparatively high Ck value of the crystal:14 the phenylene rotation is strongly hindered. In contrast, compound 2b obtained by substituting methyl and propyl groups into the stator of 2a, i.e., the 9-triptycyl group, achieved a significantly low Ck value, which is favorable for phenylene rotation.14 This is due to a higher efficiency of the bulky stator of 2b to prevent intermolecular interdigitation in the crystal. A bulky stator thus blocks the intermolecular steric interaction experienced by the rotator effectively and hence lowers the energy barrier Ea for phenylene rotation:16 crystalline 2b- and 2c-based molecular gyroscopes have much lower Ea values (4.414 and 5.0 kcal/mol,15 respectively) than that of compound 2d (12.8 kcal/mol10,13,15). Garcia-Garibay et al. have also revealed the direct effects of stators on the flipping rates k of rotators experimentally. The flipping rate of compound 2b is higher (4.6 MHz at 173 K14) than that of 2d (1.3 MHz at 348 K10), in agreement with the fact that 2b has bulkier stators than 2d. Thus, there exists a reasonable correlation between k and the shielding efficiency of the stators, while the rotational dynamics in these crystals also depends on other factors such as the flexibility of the rotator and stator segments and the strength of the coupling between them.7 Recently, Garcia-Garibay et al. have succeeded in synthesizing a few types of bridged molecular gyroscopes,17,18 though their Ea values have not yet been reported. Molecular crystals with very low Ck values tend to be unstable and collapse into more densely packed forms; in some cases they become amorphous. The strategy to design molecular gyroscopes should therefore not only be concentrated on reducing Ck values but also on maintaining the robustness of the crystal. Furthermore, for functional crystalline free rotators, Ea must be as small as thermal energy at ambient temperature (kBT = 0.60 kcal/mol at 300 K, where kB and T are the Boltzmann constant and absolute temperature, respectively). The key to all these prerequisites is developing highly efficient encapsulating frames around the rotators of properly designed molecular gyroscopes. Setaka et al. have recently proposed a gyroscopic molecular model 3 of a closed topology that has a large free volume to facilitate flipping of rotators.19 Following the concept of the model, they synthesized its molecular analogue, namely, a siloxaalkane molecular gyroscope 3a, and determined its structure by X-ray crystallography. To the best of our knowledge, it is the first synthesized silicon-based molecular gyroscope possessing a phenylene rotator encased inside three long siloxaalkane spokes. The X-ray structure of this molecule with three stable phenylene positions is shown in Figure 1, where the rotator, stator, and spin axis are indicated. The large free volume allows the phenylene rotator to flip frequently at T ≥ 223 K. It is experimentally observed that the crystal structure changes from a monoclinic to triclinic one, while the temperature is decreased from 223 to 173 K. In the triclinic crystal, the siloxaalkane spokes are largely deformed with slight reduction in the unit cell volume. As a consequence, the phenylene ring is confined in the areas around three equilibrium structures at T ≤ 173 K. A related molecular gyroscope 3b even shows abnormal crystal volume variation due to temperature change;20 unusually large thermal expansion of the crystal volume was observed, indicating a new dynamic state of the crystalline molecular gyroscope. The temperature-dependent behavior of the rotator also gives rise to

the thermal modulation of birefringence in silicon-based molecular gyroscopes.21 The siloxaalkane spokes are robust and transparent to UV/ visible light. Setaka et al. have stated that optical control of phenylene rotation may be feasible by introducing polar substituents in the rotating unit.22 The manipulated rotation or orientation of the rotator is expected to vary the optical properties of crystalline molecular gyroscopes such as dichroism and birefringence. The design and synthesis of molecular gyroscopes that respond instantly to an external electric field will lead to potential applications in nanotechnology, e.g., polarization-controllable optical materials such as line switchers for optical communication operated by polarization modulation or birefringence change. Even though such novel properties are expected for crystalline molecular gyroscopes, the detailed underlying mechanism of their rotational dynamics has not yet been revealed. Complementary theoretical studies aimed to elucidate the structures and rotational dynamics of the molecular gyroscopes are highly necessary. In this paper, we present the results of our theoretical investigation of the crystalline siloxaalkane molecular gyroscope 3a. Its structures and dynamics were examined by a theoretical method called the density-functional-based tight-binding (DFTB)23−27 approximation for electronic structure calculations. The DFTB method is known to achieve decent accuracy at a low computational cost. The structure of this paper is as follows. The theoretical approach and computational method we employed are outlined in section 2. The results and discussions are presented in section 3. A summary and conclusions are given in section 4.

2. COMPUTATIONAL DETAILS 2.1. DFTB Method. This method is widely used to study crystals with very large unit cells.24 It is computationally extremely fast and semiquantitatively reproduces the results of the density functional theory (DFT).28,29 In the DFTB method, the electronic energy of a system is obtained by solving the tight-binding eigenvalue equation, and the parameters involved are determined by DFT calculations. The DFTB method is based on the zeroth- or second-order expansion of the Kohn− Sham total energy with respect to electron density fluctuation. The former is denoted as “non-self-consistent-charge DFTB” (NCC-DFTB),28 whereas the latter is denoted as “selfconsistent-charge DFTB” (SCC-DFTB).29,30 The NCCDFTB method is the traditional tight-binding method in which an eigenvalue of the Hamiltonian operator is a noniterative solution. It is very successful in nonpolar systems and homonuclear systems, such as carbon fullerene clusters.31 On the other hand, in the SCC-DFTB method, the charge distribution in a molecule, represented by point charges, is obtained in an iterative self-consistent manner by taking into account charge interactions between atoms. It thus enables one to cope with systems where the charge balance between atoms is crucial, as for instance in biomolecules and other heteroatomic molecular systems.32 In this study, we have applied both schemes to perform geometry optimizations and potential energy surface (PES) calculations. We have only used the NCC-DFTB scheme for the molecular dynamics (MD) simulation (hereafter denoted by NCC-DFTB/MD). The computational cost of the NCC-DFTB method is less than 1/10th of that of SCC-DFTB (the difference is roughly the number of iterations required to obtain self-consistent charges). 24847



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We used the DFTB+ program package27 with the parameter sets mio-1−129 and pbc-0−333,34 to carry out all DFTB calculations. We employed the Slater−Kirkwood model26 for the evaluation of dispersion energy corrections in the DFTB: the van der Waals interactions between the phenylene rotator and the siloxaalkane spokes (and surrounding molecules) were taken into account. This model requires parameters: the atomic polarizabilities α, Slater−Kirkwood effective numbers Neff of electrons, and cutoff interaction ranges rc for all the atoms.26 In this work, we constructed a set of the parameters for Si atoms and tested the validity for molecular gyroscopes. We set α as previously reported for the sp3 hybridized Si atom of SiH4:35 α = 2.0 Å3. The value of Neff was calculated from the relation Neff = 1.17 + 0.33nν with nν being the number of valence electrons proposed by Halgren:26,36 Neff = 2.5 (nν = 4). For rc the standard value for the sp3 hybridized C atom was adopted: rc = 4.0 Å. 2.2. Geometry Optimizations. As the starting (trial) structures for geometry optimization of 3a, the initial coordinates of atoms were taken directly from the X-ray crystallographic data at 223 K (monoclinic crystal). Since this molecular gyroscope has three equilibrium structures with different orientations of the phenylene ring, we fully optimized each of them under the periodic boundary condition (PBC) by using the NCC- and SCC-DFTB methods while fixing the experimental unit cell parameters. 2.3. Potential Energy Surface (PES) Calculations. We investigated the rotational dynamics of the phenylene rotator in a crystalline molecular gyroscope. To that end, we calculated the rotational energy barrier experienced by the rotator, i.e., the electronic energy as a function of a particular dihedral angle between the phenylene ring and the reference plane of a siloxaalkane spoke. Prior to this calculation, we at first computed DFTB and DFT (B3LYP/6-31G**) energies of an isolated siloxaalkane molecule at specified dihedral angles of the phenylene ring and found good agreement between the two methods. The dihedral angle that designates the orientation of the phenylene ring was chosen as follows. The axis of rotation around which the dihedral angle is defined is the Si−C bond that links the phenylene ring to the siloxaalkane frame. The remaining two atoms to form the dihedral angle are the C atoms adjacent to the Si−C bond, i.e., the relatively stationary C atom next to the Si atom in one of the three siloxaalkane spokes, and the one at the ortho position of the phenylene ring. These C atoms are represented by colored spheroids in Figure 1. To estimate the rotational barrier accurately, the electronic energy must be calculated as a function of the dihedral angle. While the dihedral angle is fixed at a value, the other degrees of freedom are then fully relaxed to minimize the energy. This type of optimization is not routinely carried out in the DFTB+ program. We therefore utilized the molecular geometry optimizer implemented in GAUSSIAN 03.37 GAUSSIAN has a routine to optimize the positions of all the atoms along with minimizing the energy at each dihedral angle. Via the keyword “external”, the code can use a standardized interface (a script) to run an external program providing the energy and gradient at each geometry of the optimization. In this study, the external program is the DFTB+ by which all the prerequisite parameters used in GAUSSIAN, such as energies and gradients under the PBC, were computed. We refer to this methodology as “Gaussian-external”.

For evaluating the shielding efficiency of the static frame in the siloxaalkane molecular gyroscope 3a, we took 1,4bis(trimethylsilyl)benzene38 in a monoclinic crystal structure as a reference compound 4 (Figure 2). It has an open topology

Figure 2. Molecular structure formula of 1,4-bis(trimethylsilyl)benzene (a reference compound 4).

in the sense that the phenylene ring is relatively naked in comparison to 3a. We also used the Gaussian-external methodology to estimate the rotational energy barrier of 4. 2.4. Molecular Dynamics (MD) Simulations. The dynamics of crystalline molecular gyroscopes are investigated under the PBC. SCC-DFTB/MD simulations are computationally very expensive and do not improve rotational barriers much. We therefore performed only NCC-DFTB/MD simulations for 3a, neglecting relatively small effects of charge polarization and van der Waals interaction on the rotational (flipping) dynamics of the phenylene ring. The simulations were carried out at constant energy (an NVE ensemble); the effective temperature T was defined as the mean kinetic temperature. The optimized geometry of the most stable structure was used as the initial positions of nuclei; the initial velocities of nuclei were given at random, satisfying the mean kinetic temperature requirement. The velocities and positions of nuclei during the MD were computed by the velocity Verlet algorithm.39 The values of the time increment and duration used for MD runs were 0.2 fs and 250 ps, respectively. The dihedral angle of the phenylene rotator against a siloxaalkane spoke was calculated at each MD step. Running many DFTB/ MD trajectories in such a large system to take an ensemble average is difficult with current computational resources. For this reason, we only ran several trajectories with different initial velocities at a given kinetic temperature to qualitatively analyze the temperature dependence of phenylene rotation. Typical examples of the trajectories will be presented in section 3.4.

3. RESULTS AND DISCUSSION 3.1. Comparison between DFTB and DFT. In general, theoretical calculations based on the DFT methods provide trustworthy benchmarks for all other semiempirical methods. To confirm a qualitative agreement between DFTB and DFT 24848



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Figure 3. (a) DFT (B3LYP/6-31G**) and (b) SCC-DFTB optimized geometries of an isolated siloxaalkane molecule viewed along the spin axis. The blue, cyan, red, and gray spheroids represent hydrogen, carbon, oxygen, and silicon atoms, respectively. The phenylene rings are encircled.

characterize the experimentally synthesized siloxaalkane molecular gyroscope 3a. 3.2. Optimized Geometries and Energies. To begin with, geometry optimization under the PBC was carried out without dispersion energy correction. We have confirmed that the same converged structure and energy are obtained for the optimizers of DFTB+ and GAUSSIAN 03 (i.e., the Gaussianexternal methodology). We could theoretically locate three structures that correspond to the three equilibrium structures determined by X-ray analysis. The phenylene dihedral angles for the three X-ray equilibrium structures are 0.78π, 0.55π, and 0.19π (modulo π), of which structures or dihedral angles are denoted by A, B, and C, respectively (Table 1). The ortho C atoms for the dihedral angles A, B, and C are represented by blue, green, and purple spheroids, respectively, in Figure 1. In the NCC-DFTB method, these angles are converged to 0.77π, 0.35π, and 0.05π, respectively. The dihedral angles of the three X-ray structures are also well reproduced by the SCC-DFTB method. As shown in Table 1, the experimentally determined Si−O− Si bond angles of three siloxaalkane spokes are ∼0.9π (162°). It is known that the vibrational frequency of the Si−O−Si bending decreases as the Si−O−Si angle increases.40 The large bond angle hence indicates the soft deformation of the Si−O− Si angle, i.e., the flexibility of the siloxaalkane spoke chains, which is attributed to partial delocalization of lone electron pairs on the O atom into vacant π* or d orbitals of Si.41 The Si−O−Si angle of each siloxaalkane spoke is found to be slightly smaller in the optimized geometries than in the X-ray geometries. Considering the flexibility of the Si−O−Si angle, the agreement between experiment and theory is fairly good. To estimate the “free volume” unit available around the phenylene rotator, i.e., the intervening space that exists between the O atom of each siloxaalkane arm and the nearest C atom of the phenylene ring, we compared the C−O distance of each optimized structure with that of the corresponding X-ray equilibrium structure. The C−O distances of the equilibrium structures, denoted by dCO, are listed in Table 1. The values dCO of the optimized structures are in close agreement with the

calculations, we optimized the geometry of an isolated siloxaalkane molecule using the SCC-DFTB and B3LYP/631G** methods. The optimized structures are comparable to each other, except for the Si−O−Si bond angles of three siloxaalkane spokes as shown in Figure 3. The flexibility of the Si−O−Si angles will be discussed in section 3.2. We also performed single-point calculations of an isolated siloxaalkane molecule based on the DFTB (NCC- and SCC-) and B3LYP/ 6-31G** levels of theory at different dihedral angles of the phenylene ring. The calculated energies of the molecule are plotted in Figure 4 as a function of the dihedral angle up to 1π

Figure 4. Potential energy as a function of the phenylene dihedral angle of an isolated siloxaalkane molecule computed by single-point calculations. The thin solid, dotted, and thick solid lines denote the PESs at the NCC-DFTB, SCC-DFTB, and B3LYP/6-31G** levels of theory, respectively.

(the other degrees of freedom were fixed to those of a single molecule extracted from the X-ray structure). The main features of the PES obtained by the DFT method are qualitatively reproduced by either of the DFTB methods, though the rotational barriers obtained by the DFTB methods are slightly lower than that of the DFT calculation. These results demonstrate that the DFTB methods can be utilized to 24849



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Table 1. Optimized Parameters of Three Different Equilibrium Structuresa structure A

structure B

structure C

ϕ

θ

dCO

Erel

ϕ

θ

dCO

Erel

ϕ

θ

dCO

Erel

X-ray

0.78

0.55

0

0.05

(b) SCC

0.77

0.57

0.35

0

0.04

Gaussian-external: (a) NCC-DFTB

0.76

0.41

0.35

0

0.05

(b) SCC-DFTB

0.76

0.57

0.35

0

0.01

0.90 0.96 0.94 0.77 0.81 0.81 0.78 0.82 0.82 0.77 0.81 0.81 0.79 0.82 0.82

3.9 4.3 5.1 4.4 4.9 5.0 4.4 5.0 5.1 4.4 5.1 5.1 4.6 5.1 5.1

―

0.35

3.9 4.3 5.1 4.1 4.1 5.2 4.1 4.1 5.3 4.1 4.1 5.2 4.1 4.1 5.3

0.19

0.41

0.90 0.96 0.94 0.79 0.78 0.81 0.80 0.79 0.83 0.79 0.79 0.81 0.79 0.79 0.83

―

0.77

3.9 4.5 4.6 4.1 4.1 4.9 4.1 4.1 4.9 4.1 4.1 4.9 4.1 4.1 4.9

―

DFTB: (a) NCC

0.90 0.96 0.94 0.77 0.77 0.79 0.78 0.78 0.80 0.77 0.77 0.79 0.79 0.79 0.80

methods

0.48

0.60

0.48

0.60

ϕ = phenylene dihedral angle (π); θ = Si−O−Si angle of each arm (π); dCO = distance between the O atom of each siloxaalkane spoke and the nearest C atom of phenylene (Å); Erel = relative energy to that of the structure B (kcal/mol). The three data sets for one method correspond to three siloxaalkane spokes.

a

DFTB). One possible reason for this discrepancy may be the mixing of the triclinic crystal, which is observed predominantly at 173 K, into the monoclinic one. We will further investigate the structures at the stable phenylene positions and their energies in the future work to dissolve the discrepancy in ΔHBC between experiment and theory. 3.3. Potential Energy Surfaces. To determine the rotational energy barrier of 3a under the PBC, we scanned the PES by the DFTB methods with and without dispersion energy correction. The PESs for three cases are plotted in Figure 6 against the dihedral angle up to 2π: NCC-DFTB without dispersion correction (thin solid line); SCC-DFTB without dispersion correction (dotted line); SCC-DFTB with dispersion correction (thick solid line). In the third case, the PES was obtained by single-point calculations at the geometries derived in the second case. All the three cases clearly indicate that three stable structures appear on the PES. The three structures are denoted by α, β, and γ in Figure 6. The dihedral angles of the three minima in the NCC-DFTB case are 0.78π, 0.35π, and 0.01π, respectively, which are close to the corresponding values 0.78π, 0.55π, and 0.19π of the X-ray structures A, B, and C, as already shown in Table 1. The counterpositions reached by 1π flip of the phenylene ring are denoted by α′, β′, and γ′, respectively. In addition to the existence of three equilibrium structures, we found a characteristic feature in these PESs, namely, the appearance of significantly low activation barriers for flipping. The activation energy Ea for the flipping from β to γ is ∼0.8 kcal/mol for the SCC-DFTB, while it increases to ∼1.2 kcal/ mol by including the dispersion correction. The low values of the energy barrier for flipping imply that both the intramolecular steric interaction between the phenylene ring and the siloxaalkane spokes and the intermolecular interaction between the phenylene ring and the neighboring molecules are small enough to allow phenylene flipping at room temperature. The small intra- and intermolecular interactions accord with the crystalline structure shown in Figure 5 where the spokes efficiently block the interaction between the phenylene rotator and the neighboring molecules.

experimental values. The existence of a large free volume that allows the phenylene ring to rotate is hence theoretically confirmed. The X-ray and SCC-DFTB optimized structures of type B in a unit cell are displayed in Figures 5a and 5b, respectively. By comparing the two figures, one can reconfirm the abovementioned features for the optimized structures that the phenylene dihedral angle slightly deviates from that of the X-ray structure and that the Si−O−Si bond angles of three siloxaalkane spokes are smaller than those of the X-ray structure. Besides, we noticed that the orientation and conformation of the methyl groups attached to the Si−O−Si part of the siloxaalkane spokes differ between the X-ray and optimized structures, which is caused by the differences in Si− O−Si angles. On the whole, the X-ray observed and SCCDFTB optimized structures are analogous to each other. The structural resemblance between Figures 5a and 5b signifies that the DFTB methods semiquantitatively reproduce the structures of crystalline molecular gyroscopes. Setaka et al. determined the site occupancy factors of the three stable phenylene positions A, B, and C at 223 and 303 K19 by X-ray analysis and found that the structure B is most populated in the range between 223 and 303 K. We applied the van’t Hoff equation to estimate the enthalpy change ΔH for the phenylene flipping from B to A ln(K 2/K1) = − (ΔH /kB)(1/T2 − 1/T1)

(1)

where K1 and K2 are the experimentally determined equilibrium constants [A]/[B] at absolute temperature T1 and T2, respectively. For T1 = 223 K and T2 = 303 K, we have K1 = 0.51 and K2 = 0.78. The value of ΔHAB obtained from eq 1 is 0.71 kcal/mol, which is as low as the energy at room temperature. This magnitude of the enthalpy change is more or less consistent with the results of DFTB calculations; the energy of the structure A was 0.41 kcal/mol higher than that of B for the NCC-DFTB (0.57 kcal/mol higher for the SCCDFTB). We also determined the experimental enthalpy change from B to C to be ΔHCB = −0.28 kcal/mol, by using eq 1. The energy of the structure C was 0.48 kcal/mol higher than that of B for the NCC-DFTB (0.60 kcal/mol higher for the SCC24850



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Figure 5. (a) X-ray observed and (b) SCC-DFTB optimized unit cell geometries for the most stable structure B of 3a viewed along the spin axis. The blue, cyan, red, and gray spheroids represent hydrogen, carbon, oxygen, and silicon atoms, respectively. The phenylene rings are encircled.

shown by the dotted line, the PES only slightly changes by increasing α to 2.5 Å3. The change was also negligible when α was reduced to 1.5 Å3. The thin solid line for a longer interaction range rc = 5.0 Å overlaps with the thick solid line for the standard set, indicating that the van der Waals energy nearly converges for rc ≥ 4.0 Å. These results illustrate that the van der Waals interaction energy evaluated by the Slater−Kirkwood model is not so sensitive to the choice of parameters in the vicinity of α = 2.0 Å3 and that the interaction energy converges at large cutoffs of rc ≥ 4.0 Å. The PES obtained by the SCCDFTB with dispersion correction, shown in Figure 6, is thus considered accurate in the framework of the Slater−Kirkwood model. Another interesting feature of the PESs in Figure 6 is an asymmetric nature of the slopes in the PESs. For instance, at the NCC- and SCC-DFTB levels, the phenylene rotator experiences a lower barrier while turning toward γ′ than turning

The present analysis of the calculated PESs suggests a prerequisite for functional molecular gyroscopes: introduction of an intervening static frame such as spokes that reduces the steric interaction of the rotator with the surrounding molecules. The structural elements of the frame around the rotator should be skillfully arranged42−44 because the intra- and intermolecular steric repulsive interactions increase steeply as the distance r decreases, according to an inverse power of 1/r. The increase in Ea by the inclusion of dispersion correction indicates the existence of van der Waals interactions between the phenylene rotator and surrounding spokes. To evaluate the validity of the dispersion parameters α, Neff, and rc for Si, we calculated the PES for three different sets of parameters. The three PESs obtained are plotted in Figure 7 for the structures obtained by the SCC-DFTB method used in Figure 6. The thick solid line represents the PES for the standard set of parameters for Si: α = 2.0 Å3, Neff = 2.5, and rc = 4.0 Å. As 24851



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Unlike in the siloxaalkane molecular gyroscope 3a, the phenylene rotation in the reference compound 4 has a considerably high activation energy of ∼5.1 kcal/mol. This is due to the fact that there is no proper static frame that can keep the phenylene ring apart from the surrounding molecules. Although 4 has trimethylsilyl groups at the periphery of the phenylene ring, the volume of trimethylsilyl groups is not large enough to block strong intermolecular steric interactions. The phenylene ring of 4 is thus prevented from flipping at ambient temperature, as experimentally confirmed by the 13C CP/MAS technique.19 For 4, the dispersion energy correction does not make any significant differences in the activation energies Ea for flipping. The intra- and intermolecular van der Waals interactions are overwhelmed by strong steric effects and are relatively negligible. 3.4. MD Simulations. We performed MD simulations to obtain several trajectories at different temperatures. The force constants used for the MD simulations were obtained by the NCC-DFTB method. We have observed the flipping motion of a phenylene ring in real time. The notable findings are summarized below. 3.4.1. Temporal Behavior of the Rotational Angle. The dihedral angles of the phenylene ring in typical trajectories at four different kinetic temperatures T are plotted as a function of time in Figure 8. The lifetimes for the phenylene rotator to occupy stable angles in Figure 8 range from subpicoseconds to picoseconds. They correspond to the locations of the minima (i.e., α, β, and γ) of the PES in Figure 6. At high temperatures, the phenylene ring flips frequently in the picosecond regime. An example for T = 1200 K is shown in Figure 8a. The phenylene rotator at first reaches the position γ from its initial position β in ∼500 fs and mostly remains there for about 10 ps before undergoing ∼0.8π clockwise flipping to the local minimum position α. The rotator then stays for more than 20 ps around α and flips by 1π to its degenerate position α′. Similarly, at 800 K, the rotator flips at intervals of several tens of picoseconds, as shown in Figure 8b (less frequently than in the case of T = 1200 K). Inversion of the phenylene ring, i.e., 1π flipping, occurs at these high temperatures. Flipping between stable positions is also observed at lower temperatures; however, whether or not the rotator flips by more than 1π in the picosecond regime depends on the initial velocities of nuclei. In Figures 8c and 8d, we present examples of trajectories that do not undergo an inversion of the phenylene ring within ∼250 ps. At 600 K, the phenylene ring flips between the positions β, γ, and α′ several times. At room temperature (T = 300 K), the phenylene rotator tries to flip toward α from its original position β but fails to cross the potential well that exists between them. 3.4.2. Estimation of Energy Barriers from Flipping Rates. MD simulations of the phenylene flipping allow us to estimate the activation energy Ea by utilizing the Arrhenius equation

Figure 6. Potential energy as a function of the phenylene dihedral angle of 3a under the PBC. The thin solid, dotted, and thick solid lines denote the PESs obtained by the NCC-DFTB, SCC-DFTB, and SCCDFTB with dispersion corrrection, respectively. The symbols α, β, and γ stand for three stable positions of the phenylene ring, and α′, β′, and γ′ represent their respective degenerate positions.

Figure 7. Dependence of the PES obtained by the SCC-DFTB with dispersion correction on three parameters: the polarizability α, Slater− Kirkwood effective number Neff of electrons, and cutoff interaction range rc for Si. Three different cases are compared: α = 2.0 Å3, Neff = 2.5, and rc = 4.0 Å (thick solid line); α = 2.5 Å3, Neff = 2.5, and rc = 4.0 Å (dotted line); α = 2.0 Å3, Neff = 2.5, and rc = 5.0 Å (thin solid line).

toward β from α. The asymmetry in the PES can be ascribed to unequal interactions between the phenylene rotator and three siloxaalkane spokes; the three spokes are unequally spaced around the rotator (at the optimized structure B, the distances between the O atoms of the spokes are 8.4, 8.6, and 8.8 Å). The asymmetry becomes more pronounced when dispersion correction is included, whereas in this case the barrier height for α → γ′ exceeds that for α → β. This suggests that the intraand intermolecular van der Waals interactions affect the dynamic behavior of the phenylene rotator, though the stable structures are insensitive to the presence of the van der Waals interactions. It is a challenge to design or synthesize a new system with an asymmetric torsional potential that may lead to a possibility of unidirectional rotation of the phenylene ring16,45−48 by the application of external electric fields, though more detailed theoretical and experimental investigations are required to justify the idea.

ln k = ln A − Ea /kBT

(2)

where k is the flipping rate and A is the pre-exponential factor. It was difficult in the present simulations to observe similar types of phenylene flipping at all four different temperatures. We therefore estimate Ea by using the rates of similar types of phenylene flipping at only two different temperatures 800 and 1200 K. From the results of MD simulations, the average lifetimes for the flipping from β to α at 800 and 1200 K were estimated to be 56 and 48 ps, respectively. The corresponding flipping rates k at 800 and 1200 K are 0.018 and 0.021 ps−1, 24852



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Figure 8. Dihedral angles of phenylene as a function of time at (a) 1200, (b) 800, (c) 600, and (d) 300 K. The symbols α, β, γ, etc. represent stable angular positions of the phenylene rotator.

are not yet established, no significant dependence of the barrier height on the parameters was observed within the Slater− Kirkwood model. A remarkable achievement of a significantly low rotational barrier (1.2 and 0.8 kcal/mol with and without dispersion correction, respectively) indicates a promising architecture of molecular gyroscopes. To the best of our knowledge, these values are the lowest barrier heights obtained so far for phenylene-based molecular gyroscopes; the lowest barrier height achieved by the pioneer research group of GarciaGaribay is still at the limit of about 4.4 kcal/mol.10,12−15 The NCC-DFTB/MD simulations demonstrated that the phenylene ring undergoes several flipping events inside the siloxaalkane cage especially at temperatures as high as 800− 1200 K. On the way of turning around, the rotator finds several stops at different angles which are in excellent agreement with the locations of the minima on the PES. The barrier height evaluated from the Arrhenius equation for the flipping from β to α is almost equal to the corresponding height of the PES (∼0.7 kcal/mol). The most important conclusion of this work is that in the presence of a highly efficient encapsulating frame around the rotating segment the rotational dynamics of crystalline molecular gyroscopes can be dramatically improved with an extremely low activation barrier. Understanding such microscopic mechanisms of rotations with the help of reasonably simple theoretical methods is highly beneficial for the development of nanoscale devices based on assemblies of molecular gyroscopes. The DFTB approach, which has been already applied to the study of geometries and dynamics of large molecules, clusters, and solid-state complexes,24,49−51 provided a superb theoretical description of the experimental observations of crystalline molecular gyroscopes. Ultimately, it would become an indispensable tool to open a molecular machinery world.

respectively. The activation energy can be roughly determined by inserting these values into eq 2: for the flipping from β to α, Ea ∼ 0.73 kcal/mol. This value is in good agreement with the activation barrier height ∼0.7 kcal/mol calculated by the NCCDFTB method (Figure 6). By using the determined A and Ea, we estimated the flipping rate at 300 K to be 0.0084 ps−1. This small value is consistent with the finding from MD simulations at low temperatures; whether flipping occurs or not in a few hundreds of picoseconds at T ∼ 300 K depends on the initial conditions such as the initial velocities.

4. CONCLUSIONS The crystal structures and rotational dynamics of the siloxaalkane molecular gyroscope 3a have been investigated theoretically by employing the computationally efficient DFTB method. We first confirmed good agreement between the DFTB and DFT (B3LYP/6-31G**) for the potential energy of an isolated siloxaalkane molecule. We then applied the DFTB to the crystalline siloxaalkane molecular gyroscope 3a with the periodic boundary condition; the DFTB could locate three stable structures α, β, and γ that are consistent with the three Xray structures A, B, and C reported in ref 19. The analysis of the intervening space between the phenylene ring and siloxaalkane spokes has corroborated the availability of a sufficient clearance around the rotator. We also scanned the potential energy surface (PES) as a function of the dihedral angle between the phenylene ring and a surrounding spoke by the DFTB methods with and without dispersion energy correction. The three counterpositions (degenerate positions) reached by 1π flip are also confirmed. The PES clearly shows an asymmetric nature with respect to the dihedral angle. Inclusion of the dispersion energy for van der Waals interaction increases the barrier heights for flipping by ∼0.4 kcal/mol; the intra- and intermolecular van der Waals interactions influence the dynamic behavior of the phenylene rotator, though the stable structures are insensitive to the van der Waals interactions. Although the parameters for the dispersion energy of a Si atom 24853



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by a Grant-in-Aid for Challenging Exploratory Research (No. 226550003) from the Japan Society for the Promotion of Science.

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Theoretical Investigation of the Structures and Dynamics of Crystalline

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