Ab initio Molecular Dynamics Study of H2 Formation inside POSS

Article pubs.acs.org/JPCA

Ab initio Molecular Dynamics Study of H2 Formation inside POSS Compounds. 2. The Effect of an Encapsulated Hydrogen Molecule Published as part of The Journal of Physical Chemistry virtual special issue “Mark S. Gordon Festschrift”. Takako Kudo,*,† Tetsuya Taketsugu,‡ and Mark S. Gordon§ †

Division of Pure and Applied Science, Graduate School of Science and Technology, Gunma University, Kiryu 376-8515, Japan Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan § Department of Chemistry, Iowa State University, Ames, Iowa 50011-2030, United States ‡

S Supporting Information *

ABSTRACT: The mechanism and dynamics for the formation of a hydrogen molecule in the cavity of POSS (polyhedral oligomeric silsesquioxane) compounds have been investigated by ab initio molecular orbital and ab initio molecular dynamics (AIMD) methods. The host molecules are two POSS compounds, T8 ([HSiO1.5]8) (Oh) and T12 ([HSiO1.5]12) (D2d). To investigate the H2 formation reaction, two approaches were considered: trajectories were initiated by inserting a second hydrogen atom into (I) a hydrogen molecule encapsulated-POSS (H + H2@Tn → (H + H2)@Tn; n = 8, 12), and (II) the hydrogen atom and hydrogen molecule-encapsulated-POSS (H + (H + H2)@Tn → 2 H2@Tn; n = 8 and 12). A wide variety of reactions were observed depending on the system and the initial conditions, especially in process II. Therefore, for reaction II, an energy decomposition analysis was employed to examine the variation of the distribution of the translational, rotational and vibrational kinetic energies of the guest species along the reaction processes, and to determine the role of each energy component for the H2 formation. This analysis provides important insights into the expected kinetic energy distributions among the products of encapsulation reactions.

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method that utilizes ab initio energy gradients on the fly along a trajectory. The previous AIMD study29 provided insight into the effects of the shape and the size of the cage and the initial conditions on H2 formation reactions. The main focus in this work is on the possibility of a hydrogen exchange reaction, (I) H + H2@Tn → (H + H2)@Tn (n = 8 and 12) and on the impact of an additional H2 molecule on H2 formation in the POSS cavity, (II) H + (H + H2)@Tn → 2H2@Tn (n = 8 and 12). Because the cage is crowded, complex reactions may occur. To shed light on the relation between the reaction mechanism and the energy distribution of hydrogen atom pairs and molecules, and on the energy distribution among the product modes, the results of energy decomposition analyses are presented.

INTRODUCTION Polyhedral oligomeric silsesquioxanes (POSS), [RSiO1.5]n (n = 4, 6, 8, 10, 12, ...), referred to as Tn, belong to an important class of hybrid organic−inorganic materials in the silicone industry. POSS have been the focus of considerable experimental and theoretical interest because of their wide variety of practical uses.1−13 The authors have studied various properties and processes14−20 of POSS and related compounds, such as metalsubstituted POSS (metallasilsesquioxanes),18,21−23 particularly the possibility of utilizing a POSS, or substituted POSS, cavity for the storage of hydrogen molecules19,20 or other atoms, ions, and molecules.23−28 For the encapsulation of atoms and molecules, larger cage compounds, such as fullerenes, may be more convenient. However, the flexible siloxane (−Si−O−Si−) framework in POSS compounds makes the POSS cages resilient, which may be an advantage for molecular reactions in the cavity. Furthermore, various types of interactions between polar guest species and the skeletal Siδ+−Oδ− bonds both outside and inside the cage may be important for a fundamental understanding of the chemistry of these compounds. Recently, the present authors investigated the mechanism and dynamics for H2 formation processes inside POSS compounds (H + H@Tn → H2@Tn) using ab initio molecular dynamics (AIMD) calculations.29−35 AIMD, also called the dynamic reaction path (DRP) method,30−35 is a classical trajectory © XXXX American Chemical Society

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COMPUTATIONAL METHODS Ab initio Molecular Orbital Calculations. Geometry optimizations were performed for all of the molecules at least at the Hartree−Fock (HF) level of theory with the 6-31G(d) basis set.36 All optimized structures were characterized as minima or transition states by normal-mode analyses. Single -point energy calculations on the HF optimized geometries were carried out at Received: August 5, 2016 Revised: October 5, 2016

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DOI: 10.1021/acs.jpca.6b07907 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Figure 1. UHF/6-31G(d) and CASSCF(3,3)/6-31G(d) (in italics) optimized geometries of the transition-state structures and the inclusion complexes for reaction I H + H2@Tn → (H + H2)@Tn (n = 8 and 12) in Å and deg. The numbers below the structures are the HF/6-31G(d), MP2/631+G(d,p)//HF/6-31G(d) (in parentheses), and CASSCF(3,3)/6-31G(d) (in italics) energies relative to those of the reactant, H + H2@Tn (n = 8 and 12) in kcal/mol.

the second-order perturbation (MP2)39 level of theory with the 6-31+G(d,p) basis set37,38 to obtain more reliable energetics. MP2/6-31+G(d,p) geometry optimizations were performed for selected structures. The 6-31G(d) basis set was used for all of the AIMD simulations. For T8 AIMD simulations,30−35 the spin-unrestricted Hartree−Fock (UHF) and complete active space multiconfigurational self-consistent field calculations with the active space of three electrons over three orbitals (CASSCF(3,3)) were employed for reaction I: H + H2@T8 → (H + H2)@T8 to describe the H-exchange reaction in the (H + H2) system. For reaction II: H + (H + H2)@T8 → 2H2@T8, both CASSCF(2,2) and CASSCF(4,4) calculations were carried out to treat complicated reactions in the small cage. In the CASSCF(2,2) calculations, the two electrons and two orbitals of the colliding hydrogen atoms comprise the active space, while the four electrons and four orbitals in the two H2 molecules make up the CASSCF(4,4) active space. For the reactions of T12, UHF calculations were carried out for reaction I, whereas CASSCF(2,2) calculations with the same active space as that for T8 were employed for reaction II to conserve computer resources. To confirm if the UHF and CASSCF(2,2) methods for the reactions of interest are reliable, the H-exchange reaction in the (H + H2) system was investigated using the CASSCF(3,3)/6-31G(d) method; the double H-exchange reaction in the (H + H2 + H) system was examined with the CASSCF(4,4)/6-31G(d) level of theory. The UHF and CASSCF(3,3) calculations give essentially the same energetic results for the exchange reaction. Likewise, the CASSCF(2,2)

and CASSCF(4,4) calculations give virtually the same relative energies for the double exchange reaction.40 In the AIMD simulations two different sets of initial conditions are considered for both processes:29 (A) inserting the last hydrogen atom with a small (0.5 kcal/mol) amount of kinetic energy initiated at the transition state or at the center of the H insertion face and (B) inserting the last hydrogen atom with a kinetic energy that is larger than the HF energy barrier for the H insertion reaction (60 kcal/mol for T841 and 23 kcal/mol for T12). The latter trajectory (B) is initiated from 2.5 Å above a POSS face. These two initial conditions will be referred to as processes I(A) and I(B) or II(A) and II(B) in subsequent discussions. Preliminary Considerations. The transition-state structures and the inclusion complexes for the insertion of a hydrogen atom into the two types of POSS (T8 and T12) in processes I and II are shown in Figures 1 and 2, respectively. For process 1 one transition-state (TS) structure and the resultant inclusion complex were located for the insertion of a hydrogen atom into H2@T8. In both structures the hydrogen atom and hydrogen molecule are collinear (L-type transition state). A T-type transition state, in which the reacting H atom forms a T with the H2, was not located for T8 at the UHF or CASSCF(3,3) levels of theory. The effect of electron correlation via MP2 is to stabilize the transition structure and the inclusion complex relative to those of the reactant, H + H2@T8. For the T8 TS and inclusion complex the CASSCF (3,3) configurational mixing is small. The UHF spin contamination is also very small. B


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Figure 2. HF/6-31G(d) and CASSCF(m,m)/6-31G(d) (m = 2 and 4) (in italics) optimized geometries of the transition-state structures and the inclusion complexes for reaction II H + (H + H2)@Tn → 2H2@Tn (n = 8 and 12) in Å and deg. The numbers below the structures are the HF/6-31G(d), MP2/6-31+G(d,p)//HF/6-31G(d) (in parentheses), and CASSCF(4,4)/6-31G(d) for T8 (in italic) and CASSCF(2,2)/6-31G(d) for T12 (in italics) energies relative to those of the reactant, H + (H + H2)@Tn (n = 8 and 12) in kcal/mol.

Figure 3. TS in the middle is a HF transition-state (TS) structure for the exchange of two H atoms (3 and 4) in T8. End points correspond to the ends of the IRC.

For the reaction of the fourth hydrogen atom with the hydrogen atom of the (H + H2)@T8 complex in reaction II (Figure 2), the energy decreases monotonically throughout the insertion process, and no transition-state structure was located at

Two types of transition-state structures (F(ront)-type and U(pper)-type) and one inclusion complex were found for reaction I of T12. The MP2/6-31+G(d,p)//HF/6-31G(d) F-type structure is slightly lower in energy than the U-type structure. C


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Scheme 1. Various Approaches for the Insertion of a Hydrogen Atom in Each Initial Condition for the T8 and T12 Processes I and IIa

a

Detailed explanations are given in Table 1 and in the text. D


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The Journal of Physical Chemistry A the HF, CASSCF(2,2), or CASSCF(4,4) levels of theory. For reaction II, MP2 stabilizes the transition structures and the inclusion complex relative to the reactant as in the case of reaction I. For both CASSCF(2,2) and CASSCF(4,4) the configurational mixing is small, so these levels of theory are essentially the same as HF. Figure 3 shows another type of transition-state structure of T8, in which two H atoms and one H2 are involved in the cage. The HF (MP2/6-31+G(d,p)//HF/6-31G(d)) energy barrier for this insertion is calculated to be 43.6 (57.9) kcal/mol. The intrinsic reaction coordinate (IRC) calculation starting from the transition structure suggests that this is the transition-state structure for the exchange of H atom 3 for H atom 4. There is also a change in the alignment of the guests. This TS structure may suggest the possibility that a T-type insertion process exists for the second hydrogen. This type of structure has not been found at either the CASSCF(2,2) or the CASSCF(4,4) level of theory; however, it was located with MP2/6-31+G(d,p), yielding a 43.7 kcal/mol energy. Therefore, an AIMD study (discussed below) for the T-type insertion reaction was performed to explore this possibility. The T8 inclusion complex in reaction II, in which the two hydrogen molecules are collinear in the cage, is also shown in Figure 2. Despite the remarkable congestion in the cage, this inclusion complex is found to be much more stable than the reactant at all levels of theory used here. As for reaction I, two types of transition-state structures (F-type and U-type) and one inclusion complex were also found for the insertion of a hydrogen atom into the (H + H2)@T12 complex (Figure 2). The two transition-state structures are essentially isoenergetic; the U-type TS is slightly lower in energy than the F-type TS at both the MP2/6-31+G(d,p)//HF/ 6-31G(d) and HF/6-31G(d) levels of theory. In the inclusion complex, two hydrogen molecules are arranged in a twisted structure independent of the direction of the inserting hydrogen atom. Ab initio Molecular Dynamics Calculations. Scheme 1 and Table 1 summarize the AIMD initial conditions for simulations of processes I and II. For reaction I, only one direction (the linear attack) is considered for the T8 processes I(A) and I(B). For T12, two initial directions, F-type and U-type (Figure 1), starting from the transition-state structures, are used for processes I(A)-F and I(A)-U. For the T8 process II, no transition-state structures for insertion were located, but AIMD initial conditions similar to I(A) and I(B) are used: the starting point for II(A) is the center of a D4 (Si4O4) face and the starting point for II(B) is a point 2.5 Å above the center of the D4 face. Another initial condition (II(B+)-T), in which the kinetic energy is larger (120 kcal/mol) and the starting point is the same as that in condition II(B), is also applied for the T-type insertion. For the T12 process II, two initial directions, F-type and U-type (Figure 2), starting from the transition-state structures, are applied for II(A)-F and II(A)-U. The H insertion starts from a point 2.5 Å above the D4 ring surface in each TS in the two directions II(B)-F and II(B)-U. The MD time step was generally taken to be 0.3 fs, although a smaller time step was employed when energy conservation was not sufficient. To discuss the dynamics aspects of the generation and dissociation of hydrogen molecules in reaction II, energy decomposition analyses are performed for various pairs of hydrogen atoms (Hi−Hj; i ≠ j) along the trajectory in AIMD simulations. The details of the energy decomposition approach are described

Table 1. Initial Conditions of AIMD Simulations for the Reactions I H + H2@Tn → (H + H2)@Tn, n = 8 and 12, and II H + (H + H2)@Tn → 2H2@Tn, n = 8 and 12 T8 T12

T8b

T12

I(A) I(B) I(A)-F I(B)-F I(A)-U I(B)-U II(A)-L II(B)-L II(A)-R II(B)-R II(A)-T II(B)-T II(B+)-T II(A)-F II(B)-F II(A)-U II(B)-U

Reaction I TS with 0.5 kcal/mol TS + 2.5 Å with 60 kcal/mola F-type TS with 0.5 kcal/mol F-type TS + 2.5 Å with 23 kcal/mol U-type TS with 0.5 kcal/mol U-type TS + 2.5 Å with 23 kcal/mol Reaction II left center of D4c with 0.5 kcal/mol left center of D4 + 2.5 Å with 60 kcal/mol right center of D4 with 0.5 kcal/mol right center of D4 + 2.5 Å with 60 kcal/mol T-type center of D4 with 0.5 kcal/mol T-type center of D4 + 2.5 Å with 60 kcal/mol T-type center of D4 + 2.5 Å with 120 kcal/mol F-type TS with 0.5 kcal/mol F-type TS + 2.5 Å with 23 kcal/mol U-type TS with 0.5 kcal/mol U-type TS + 2.5 Å with 23 kcal/mol

“TS + 2.5 Å with 70 kcal/mol” was applied in the CASSCF(3,3)/ 6-31G(d) simulations.41 bThe TS for the insertion was not located so the center of the D4 face is used as the starting point. cD4 refers to a D4 (Si4O4) face of the T8 cage. a

in the Appendix. The total energy for H2 can be written as E=K+V

(1)

where E, K, and V denote the total energy, the kinetic energy, and the potential energy, respectively. Here V corresponds to the Born−Oppenheimer energy that includes electronic kinetic energy, electron−nuclear attraction, electronic repulsion, and nuclear repulsion terms, and V is set to be zero at the equilibrium structure. The kinetic energy has contributions from the translational, rotational, and vibrational motions: K = K trans + K vib + K rot (2) When the H−H distance becomes the H2 equilibrium bond length along the trajectory, the kinetic energy can be regarded as the total energy of the H2 molecule, because the potential energy is zero at the equilibrium structure. Thus, by plotting each component of the kinetic energy defined in eq 1 along the trajectory, one can discuss the energy transfer among the internal degrees of freedom of H2, as well as the dynamical aspects of the generation and dissociation of the H2 molecule. All calculations were performed with the GAMESS electronic structure code.42,43

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RESULTS AND DISCUSSION 1. Insertion of a Hydrogen Atom into T8 and T12. Table 2 shows the energies of the transition states and products relative to those of the reactants for the insertion of a hydrogen atom into the various POSS systems. (I) H + H2@Tn → (H + H2)@Tn and (II) H + (H + H2)@Tn → 2H2@Tn. For comparison, the reactions studied previously (denoted P), (PI) H + Tn → H@Tn and (PII): H + H@Tn → H2@Tn, are also shown in Table 2. All reactions except for the (PI) reaction are initiated from the Tn cage with guest species inside. Therefore, the relative stability of the product is expected to decrease as the congestion inside the cages increases. However, reactions PII and II are exothermic as they are radical−radical recombination reactions. The MP2 E


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The Journal of Physical Chemistry A Table 2. HF/6-31G(d), MP2/6-31+G(d,p)//HF/6-31G(d),a and CASSCF(m,m)/6-31G(d) (m = 2, 3, and 4)b Energies (kcal/mol) for Transition State (TS) and Products Relative to Those of the Reactants for the Insertion Processes of a Hydrogen Atom into the POSS Tn (n = 8 and 12) T8 (Oh) (PI) H + Tn TS H@Tn (PII) H + H@Tn TS H2@Tn (I) H + H2@Tn TS (H + H2)@Tn (II) H + (H + H2)@Tn TS 2H2@Tn

T12 (D2d)

50.5 (39.9) 14.6 (10.1)

18.7 (14.6) 3.4 (3.7)

55.6 (44.2) −71.3 (−85.4)

19.9 (15.0) −80.9 (−93.1)

54.9 (40.5) 65.6 51.4 (36.5) 53.2

−46.4 (−70.3) −59.0

H + H2 → H···H2 process is repulsive in the gas phase.44 A comparison among the series of H insertion reactions (PI ∼ II) may be useful to better understand the reaction. Therefore, reaction I was also examined. It is not surprising that the energy barrier for the insertion is higher, whereas the relative stability of the inclusion complex is lower, in T8 than in T12 because of the smaller face and cavity of the former. In both POSS cages, the energy barrier for the insertion reaction is approximately insensitive to the guests in the cavity. This observation agrees with a previously reported prediction that the energy barrier for the insertion just depends on the size of the face through which the insertion reaction takes place.19 As shown in Figure 1, the cage of the reaction I T8 inclusion complex is considerably deformed, and consequently, the destabilization energy relative to the reactants is large (ca. 50 kcal/mol). Attempts to find another inclusion complex with the T-shape conformation of H and H2 were unsuccessful. The two hydrogen molecules in the reaction II inclusion complex, 2H2@T8, are also collinear (Figure 2). In contrast, for the T12 inclusion complex in process I ((H + H2)@T12), the H and H2 are T-shaped, rather than linear as in T8 (Figure 1). Attempts to find a collinear structure were unsuccessful. An inclusion complex with a twisted conformation of two hydrogen molecules was located for 2H2@T12 (Figure 2). A twisted conformation of two hydrogen molecules was also observed in the two hydrogen atoms encapsulated in the larger POSS compounds T14 and T16 in a previous study.20 According to MP2/cc-pVTZ, two free hydrogen molecules are found to

19.6 (13.2) F 19.9 (13.6) U 15.1 (10.7) 21.3 (15.6) F 21.0 (15.5) U −75.7 (−99.5) −88.0

a

The values are in parentheses. bThe values are in italics; m = 3(T8) for reaction I and m = 2 (T12) and 4 (T8) for reaction II.

stabilization energy of the product (ΔE = E(2H2@T12) − E(H + (H + H2)@T12)) in the T12 reaction II is larger than that in reaction PII. In contrast, the inclusion complex in reaction I is the least stable among all reactions both for T8 and T12. The release of the hydrogen atom from both cages may be easy even at room temperature. This is in agreement with the observation that the

Figure 4. Changes in the kinetic energies: KE(H+H2) and KE(T8) and potential energy, and H−H distances along the CASSCF(3,3)/6-31G(d) trajectory for the T8 processes I(A) ((a) and (b)) and I(B) ((c) and (d)). F


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Figure 5. Changes in the kinetic energies: KE(Hi-Hj) and KE(T8), and H−H distances along the CASSCF(4,4)/6-31G(d) trajectories for the T8 processes II(A)-L ((a) and (b)) and II(B)-L ((c) and (d)).

Figure 6. Changes in the kinetic energies: KE(Hi-Hj) and KE(T8), and H−H distances along the CASSCF(4,4)/6-31G(d) trajectory for the T8 processes II(A)-R ((a) and (b)) and II(B)-R ((c) and (d)).

form an almost planar structure instead of the twisted structure. However, the energy difference between the planar and the

twisted conformation in 2H2@T12 is only 4.6 kcal/mol, suggesting that the potential energy surface of the H2 dimer is very G


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Figure 7. Changes in the kinetic energies: KE(Hi-Hj) and KE(T8), and H−H distances along the CASSCF(4,4)/6-31G(d) trajectory for the T8 processes II(A)-T ((a) and (b)) and II(B)-T ((c) and (d)).

Figure 8. Changes in the kinetic energies: KE(H+H2) and KE(T12) and the potential energy, and the H−H distances along the UHF/6-31G(d) trajectory for the T12 processes I(A)-F ((a) and (b)) and I(B)-F ((c) and (d)). H


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Figure 9. Changes in the kinetic energies: KE(H+H2) and KE(T12) and the potential energy, and the H−H distances along the UHF/6-31G(d) trajectory for the T12 processes I(A)-U ((a) and (b)) and I(B)-U ((c) and (d)).

flat and conformational changes seem to take place easily depending on the environment.20 2. T8 Reaction I: T8: H + H2@T8 → (H + H2)@T8. Now consider the UHF and CASSCF(3,3) AIMD simulations for T8 reaction I with initial conditions (A). The results obtained using the two methods are basically the same, so only the CASSCF (3,3) results are discussed here. The UHF results are displayed in Figure S1 in the Supporting Information. Figure 4a shows the variation in the kinetic energies of three hydrogen atoms (the sum of H and H2) and T8 as a function of time, whereas Figure 4b shows the variation in the distances between the three hydrogen atoms for t = 0−400 fs. The plotted kinetic energy is a sum of the kinetic energies of the respective atoms included in the hydrogen pairs. The hydrogen atom labels are displayed in Scheme 1. As seen from Figure 4b, the distances between the hydrogen atom and the hydrogen molecule (r(H2−H3) and r(H1−H3)) undergo small changes, while the H2 interatomic distance remains very close to the equilibrium distance (req(HH) = 0.753 Å). This result suggests that collisions between the hydrogen atom and the hydrogen molecule do not take place in this simulation. Therefore, energy transfer from the guests to the host is not observed during the time period (Figure 4a). The same reaction with initial condition (B) results in collisions between H and H2, and the exchange of a hydrogen atom takes place. As Figure 4c shows, the potential and kinetic energy curves of the three hydrogen atoms are found to vibrate considerably after the insertion of the third hydrogen atom into the cage. The times of ca. 20 and 45 fs (Figure 4c) correspond to the insertion of a hydrogen atom into the cage and the release of a dissociated hydrogen atom from the cage, respectively (the corresponding UHF trajectories are shown in Figure S1 in the Supporting Information). The energy barrier for the H atom

insertion is 65.6 kcal/mol at the CASSCF(4,4)/6-31G(d) level as shown in Table 2, suggesting that the energy barrier for the hydrogen exchange (∼10 kcal/mol in gas phase) is quite small compared with the energy of hydrogen atoms inside the POSS. The three hydrogen atoms attain equal distances at ca. 19 fs (Figure 4d). Starting at that point, the inclusion complex between the newly formed H2 and the dissociated H atom is completed at ca. 30 fs (Figure 4c). The inclusion complex is kinetically unstable, and the distance between the inserted hydrogen atom (H3) and encapsulated hydrogen pair (H1 and H2) oscillates several times even after the formation of the inclusion complex, as shown in Figure 4d. This is not observed in the UHF simulations (Figure S1(d) in the Supporting Information). The spin contamination along the UHF trajectory does not change much according to the S2 expectation value (0.764−0.766). Although a transition-state structure for the insertion of a hydrogen atom through the side face (the direction perpendicular to the molecular axis of the encapsulated H2) was not located, simulations were performed with several kinds of initial conditions. However, even starting with 80 kcal/mol of kinetic energy at a point 2.5 Å from the side D4 face, the second hydrogen atom was not able to insert into the cage but instead rebounded from the face. Therefore, a T-type insertion in this case is presumed to be rare. 3. T8 Reaction II: H + (H + H2)@T8 → 2H2@T8. As noted above, no transition-state structure has been located for the insertion of a hydrogen atom into (H + H2)@T8, using insertion points at the centers of the three different D4 faces. Two kinds of initial conditions (A) and (B) (II(A) and II(B)) were employed for each point (Scheme 1). The results of the CASSCF(4,4) AIMD simulations are shown in Figures 5−7. The -L in the figures means that the inserting I


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Figure 10. Changes in the kinetic energies: KE(Hi-Hj) and KE(T12), and the H−H distances along the CASSCF(2,2)/6-31G(d) trajectories for the T12 processes II(A)-F ((a) and (b)) and II(B)-F ((c) and (d)).

0.724 Å, r(H1−H3) = 1.167 Å and r(H3−H4) = 0.800 Å, respectively, for the process II(A)-R. The distance between the two hydrogen atoms (H3 and H4) is very close to the equilibrium distance of the hydrogen molecule. Therefore, as Figure 6b shows, the formation of a new hydrogen molecule completes quickly. As a result, the original hydrogen molecule (H1−H2) and a new hydrogen molecule exist together within the cage, with little residual vibrational excitation. For the reaction II(B)-R (Figures 6c,d), the new hydrogen pair (H3−H4) undergoes frequent collisions. As a result, the original hydrogen molecule (H1−H2) escapes at ca. 260 fs after several collisions with H3−H4, moving rapidly (Figure 6c). The short distance between the hydrogen molecule (H1−H2) and the new hydrogen pair (H3−H4) in the small cavity makes the interaction between the two species stronger, suggested by the bonding interaction between H1 and H3 in the CASSCF(4,4) active space. The CASSCF(2,2) active space is too small to address such an interaction (Supporting Information). Hydrogen exchange is not observed in the reactions II(A)-R and II(B)-R. The T8 “T-type” insertion was examined because an unexpected TS structure was located with HF/6-31G(d), MP2/ 6-31G(d) and MP2/6-31+G(d,p). The CASSCF (4,4) dynamics were investigated for the associated process. Panels a−d of Figure 7 show the variations of energies and distances for the processes II(A)-T and II(B)-T. The energy transfer to the cage is remarkably large compared to the linear insertions. This type of insertion seems to activate various motions of the guest species, so the frequency of collision of the guests with the host cage is high. It is especially interesting that the kinetic energy of the cage

H atom is on the left side of the line H2−H1···H3. H4. The variations of the energies and distances with initial condition II(A)-L are displayed in Figure 5a,b, respectively. The H−H-distances at the starting point are r(H1−H2) = 0.724 Å, r(H1−H3) = 1.167 Å and r(H2−H4) = 1.086 Å, respectively. In the early stage of the simulation, the four hydrogen atoms move left and right alternately on a line, and two new hydrogen molecules (H1−H3 and H2−H4) are gradually formed. At 230 fs (Figure 5b), one of the new H2 molecules (H2−H4) is finally pushed out of the cage. Concomitantly, the original H2 molecule (H1−H2) dissociates, in spite of the small initial kinetic energy (0.5 kcal/mol). As a result, the exchange of hydrogen atoms was observed. The movement of the H atoms in the cage is not large, but some of the energy of the H atom is transferred to the cage, as shown in Figure 5a, probably because of the congestion. The “left” insertion under the initial condition II(B)-L with r(H2−H4) = 3.586 Å is somewhat different. Two new hydrogen pairs (H2−H4 and H1−H3) are formed by the H-exchange reaction. However, in this case, the H2−H4 pair is not pushed out immediately but rather stays with the H1−H3 pair for ca. 470 fs after the third H (H4) insertion as shown in Figure 5d. Furthermore, the H1−H3 hydrogen (not the H2−H4) pair escapes from the front D4 face. During this time period, the H−H distance of the original hydrogen molecule (H1−H2) seems to oscillate though the molecule has already dissociated. The energy transfer to the cage is much lager (∼50 kcal/mol) than in the case of II(A)-L. Next, consider the insertion from the “right”, the H atom side. The H−H-distances at the starting point are r(H1−H2) = J


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Figure 11. Changes in the kinetic energies: KE(Hi-Hj) and KE(T12), and the H−H distances along the CASSCF(2,2)/6-31G(d) trajectories for the T12 processes II(A)-U ((a) and (b)) and II(B)-U ((c) and (d)).

encapsulated hydrogen pair. Therefore, to conserve computation time four UHF/6-31G(d) AIMD simulations were performed on this T12 reaction I system. The variations of energies and distances for the F-type insertions, the processes I(A)-F and I(B)-F, are displayed in Figure 8a−d, whereas those of the processes I(A)-U and I(B)-U are shown in Figure 9a−d, respectively. As Figure 8b shows the distances between the inserted H atom (H3) and each H atom (H1, H2) in the hydrogen molecule (r(H1−H3) and r(H2−H3)) are the same in the F-type insertion for the process I(A). The distance between the H atom and the H2 hydrogen atoms does change slightly, but the H2 interatomic distance is constant. This can be interpreted to mean that the H and H2 do not collide with each other in the cage, because there is apparently no H2 vibrational activation. Additionally, the variations of the energy and energy transfer from the guests to T12 are small during the time period (Figure 8a). Similar behaviors of H and H2 are observed in the process I(B) after the absorption of the third H atom into the cage, as depicted in Figure 8c,d. However, the variations of each kind of energy as well as the energy transfer to the cage are larger than those in I(A), and the H2 rotational modes are activated, as may be seen from the difference between r(H1−H3) and r(H2−H3) in Figure 8d. The collision between a hydrogen atom and a hydrogen molecule does not take place for the U-type insertion. Therefore, the variations of the energies and distances (Figure 9a−d) are similar to those of the F-type process. However, the H2 rotation does not contribute to the energy transfer to the cage, suggesting

increases monotonically irrespective of the initial condition as seen in Figure 7a,c. For the process II(B)-T there appear to be several points at which the oscillations of the kinetic energy and the H−H distance of the new hydrogen pair (H3−H4) stop (Figure 7c,d). This is in contrast to the simulations discussed above, in which continuous oscillations of the kinetic energy were observed and the H−H distance changes monotonically. In agreement with the HF IRC calculations, the original hydrogen molecule (H1−H2) does not dissociate or leave the cage. However, a new hydrogen pair (H3−H4) is formed and does not dissociate, even using the initial condition II(B)-T with a larger kinetic energy provided to the inserting hydrogen atom (H4). The formation of another hydrogen molecule is a new feature that does not appear in the IRC calculation. Of course, the IRC corresponds to motion along an infinitely damped minimum energy path, whereas the AIMD simulations are only constrained by the amount of kinetic energy that is provided. Consequently, the AIMD simulations can reveal important features that do not appear in an IRC calculation. The results of the CASSCF (2,2) simulations are similar to those obtained using CASSCF (4,4). 4. T12 Reaction I: H + H2@T12 → (H + H2)@T12. The H2 encapsulated T12 (D2d) cage, has a larger cavity than T8 and a more complex shape. There are two-types of transition-state structures (F-type and U-type) for reaction I: H + H2@T12 → (H + H2)@T12. The UHF and CASSCF(3,3) results are basically the same as those for T8, except for the behavior of the shortlived complex between an inserted hydrogen atom and the K


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Figure 12. CASSCF(4,4)/6-31G(d) energy decomposition analyses for selected hydrogen pairs along the trajectory for the T8 processes II(A)-L ((a) and (b)) and II(B)-L ((c) and (d)). The analyses were carried out at selected points in the vicinity of the H−H equilibrium distance (0.67−0.75 Å).

new hydrogen pair (H3−H4) are remarkably small. This is discussed in the next section. 6. Energy Decomposition Analysis. As seen in previous sections, the formation of a H2 molecule/pair was observed for all cases of reaction II. To understand the energy distribution during the reactions and to specify which energy components contribute to H2 formation, an energy decomposition analysis for the kinetic energy of hydrogen molecules (H pairs) (KEtot) was performed. From the analysis, the change of the distribution of various contributions to the kinetic energy, are observed along the H2 formation processes. The analysis was applied for the CASSCF(4,4) simulations of T8 and the CASSCF(2,2) calculations of T12. Shown in Figure 12a−d are the results for the two hydrogen pairs in the T8 reactions II(A)-L and II(B)-L. For the II(A)-L reaction, the total energy of both H−H pairs is mainly controlled by the vibrational kinetic energy (KEvib) at least within the cage. After the H2−H4 pair is released from the cage at ca. 230 fs, KEtrans becomes the main component of the total kinetic energy of the pair. For II(B)-L, despite the larger kinetic energy of the inserted hydrogen (H4), the newly formed two H−H pairs stay together within the cage for a relatively long time (until 470 fs: Figure 5d). As frequent collisions take place between the pairs during the 470 fs time period, KErot, as well as KEvib, begins to play an important role for the H2−H4 pair. On the contrary, the escaping H1−H3 pair seems to maintain its identity as a hydrogen molecule, as the total kinetic energy (KEtot) together with KEvib gradually decreases, as shown in Figure 12d.

that the activation of the rotational mode depends on the direction of approach of the H atom. 5. T12 Reaction II: H + (H + H2)@T12 → 2H2@T12. The CASSCF (2,2) trajectory for T12 reaction II, H + (H + H2)@T12 → 2H2@T12, is initiated from the F- and U-type transition-state structures with initial conditions II(A) and II(B), as illustrated in Figures 10 and 11. In both types of reaction, a new hydrogen pair is formed between the two H atoms. For the II(A)-F, the kinetic energy and the H−H distance of the new hydrogen pair (H3− H4) decreases, while the energy of the host (T12) increases gradually with time. This is similar behavior to the reaction (PII): H + H@T12 → [email protected] In contrast, the energy of the preincluded hydrogen molecule (H1−H2) undergoes little change during this time period. The presence of the H1−H2 molecule decreases the available room in the cage and enhances the energy transfer from a hydrogen atom to the cage. For II(B)-F, the variations of the kinetic energy and the H−H distance of the newly formed hydrogen pair (H3−H4) apparently stop at ca. 400 and again at 650 fs, as shown in Figure 10c,d. The energy transfer to the cage is smaller than that in II(A)-F despite the larger initial energy of the fourth H atom. In addition, the energy transfer to the original H2 is very small. The variations of the energies and distances for the U-type insertion with the initial conditions II(A) and II(B) are displayed in Figure 11a−d. They are similar to those of the F-type processes discussed above. The energy transfer to the hydrogen molecule is also small. Again, there are some points at which the oscillations of the kinetic energy and the H−H distance of the L


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Figure 13. CASSCF(4,4)/6-31G(d) energy decomposition analyses for selected hydrogen pairs along the trajectory for the T8 processes II(A)-R ((a) and (b)) and II(B)-R ((c) and (d)). The analyses were carried out at selected points with the H−H distance close to the equilibrium distance (0.67−0.75 Å) for (b) and (d).

This is observed when the kinetic energy of the hydrogen pair is relatively large. The energy decomposition analyses for the T12 F- and U-type insertions of process II are similar to the results of the T8 T-type insertion. KEvib decreases as the new H−H pair is formed. Interestingly, the symmetric change between KEvib and KErot of the new hydrogen pair (H3−H4) is observed not only for the processes II(B)-F and II(B)-U but also (albeit less noticeably) for the processes II(A)-F and II(A)-U. These results suggest that the rotation of the H2 pair is activated more easily in T12 than in T8, as the former cage is larger. The vibration (H---H stretching) is important for H−H bond formation. However, under the initial conditions with larger kinetic energy (i.e., B), the guest species move vigorously within the cage, and this causes activation of other kinds of motions, such as rotation and translation in T12. Consequently, the kinetic energy of the new H−H pair is distributed not only to the vibrational modes but also to other modes that are not effective for H2 formation. This is why a new hydrogen pair is not formed rapidly, even if the inserted hydrogen atom has a large kinetic energy in the processes II(B)-F and II(B)-U in T12. Recall (Figures 10c,d and 11c,d) that there are periods in the trajectories in which the oscillations in KE and the H3−H4 distance stop. It is now clear that these “dead” periods correspond to areas in which KEvib and KErot of the newly formed hydrogen pair (H3−H4) change gently (no oscillation) before the exchange of these two KE components.

For the insertion from the H atom side (“right” insertion), KEtrans is the main component of the total energy of the original hydrogen molecule (H1−H2) in both II(A)-R and II(B)-R (∼260 fs) reactions, as shown in Figure 13a,c. For the II(A)-R reaction, the hydrogen molecule (H1−H2) just moves along the molecular axis, maintaining the H−H distance as the kinetic energy and space are small. KEvib and KErot are also important components of the total kinetic energy of the newly formed hydrogen pair (H3−H4) for both processes (Figure 13b,d). For the II(A)-R reaction, the total kinetic energy and the vibrational energy for the H3−H4 pair decrease monotonically, suggesting a smooth formation of the H3−H4 hydrogen molecule. For the II(B)-R reaction, the main component of the H3−H4 pair changes from KEvib to KErot in a symmetric manner at ca. 110 fs, as may seen from Figure 13d. At around this point in the trajectory, the change in the H3−H4 distance decreases (Figure 6d). Panels a−d of Figure 14 depict the results for the T8 processes II(A)-T and II(B)-T. A new hydrogen pair (H3−H4) is formed between the two separated H atoms while the existing hydrogen molecule (H1−H2) remains intact. For the hydrogen molecule, KEtot oscillates frequently, and KErot is a primary energy component in both initial conditions, which means the rotational mode of the forming H2 is activated in these processes. For the new hydrogen pair (H3−H4), the total kinetic energy decreases with time, particularly in the II(A)-T reaction. In the early part of the simulations, the main component of the total kinetic energy is KEvib for II(A)-T, but the contribution from KErot increases after about 300 fs. In the II(B)-T simulation the contribution of KEvib decreases rapidly and KErot becomes the main component (Figure 14d). This means that the vibrational mode of the hydrogen pair competes with the rotational mode.

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CONCLUDING REMARKS In the present study, reactions between hydrogen atoms and hydrogen molecules in POSS cages, (I) H + H2@Tn→ (H + H2) M


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Figure 14. CASSCF(4,4)/6-31G(d) energy decomposition analyses for selected hydrogen pairs along the trajectory for the T8 processes II(A)-T ((a) and (b)) and II(B)-T ((c) and (d)). The analyses were carried out at selected points with the H−H distance close to the equilibrium distance (0.67−0.75 Å) for (b) and (d).

Table 3. Predictions of the CASSCF/6-31G(d) AIMD Simulations of T8 for Reactions I ((3,3) active space) and II ((4,4) Active Space) reaction of T8 I

II

a

H2 formation

H-exchange

I(A) I(B)

TS with 0.5 kcal/mol TS + 2.5 Å with 70 kcal/mol

no reaction, H2(1−2) kept H2(2−3) formed, H(1) escaped

yes

II(A)-L

left center of D4a with 0.5 kcal/mol

H2(2−4) escaped, H2(1−3) formed

yes

II(B)-L

left center of D4 + 2.5 Å with 60 kcal/mol

H2(2−4) formed, H2(1−3) escaped

yes

II(A)-R

right center of D4 with 0.5 kcal/mol

H2(3−4) formed, H2(1−2) kept

II(B)-R

right center of D4 + 2.5 Å with 60 kcal/mol

H2(3−4) formed, H2(1−2) escaped

II(A)-T

T-type center of D4 with 0.5 kcal/mol


II(B)-T

T-type center of D4 + 2.5 Å with 60 kcal/mol


II(B+)-T

T-type center of D4 + 2.5 Å with 120 kcal/molb

H2(3−4) escaped, H2(1−2) kept

D4 means a D4 (Si4O4) face of the T8 cage. b(2,2) active space

@Tn (n = 8 and 12) and (II) H + (H + H2)@Tn → 2H2@Tn (n = 8 and 12), were investigated. The results of both processes for T8 and T12 are summarized in Tables 3 and 4, respectively. The effect of the encapsulated hydrogen molecule on the H2 formation reaction (in process II) within POSS was determined.

The presence of the hydrogen molecule enhances the energy transfer from the hydrogen atom to the cage and the new H2 formation, even with the small kinetic energy provided by initial condition A. A larger amount of the energy from the inserting hydrogen is transferred to the cage than to the encapsulated N


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The Journal of Physical Chemistry A Table 4. Predictions of the UHF T12 AIMD Simulations for Reactions I and II H2 formation

H-exchangea

no reaction, H2(1−2) kept no reaction, H2(1−2) kept no reaction, H2(1−2) kept no reaction, H2(1−2) kept H2(3−4) formed, H2(1−2) kept H2(3−4) formed, H2(1−2) kept H2(3−4) formed, H2(1−2) kept H2(3−4) formed, H2(1−2) kept

-

reaction of T12 I

II

a

I(A)-F I(B)-F I(A)-U I(B)-U II(A)-F II(B)-F II(A)-U II(B)-U

F-type TS with 0.5 kcal/mol F-type TS + 2.5 Å with 23 kcal/mol U-type TS with 0.5 kcal/mol U-type TS + 2.5 Å with 23 kcal/mol F-type TS with 0.5 kcal/mol F-type TS + 2.5 Å with 23 kcal/mol U-type TS with 0.5 kcal/mol U-type TS + 2.5 Å with 23 kcal/mol

The H-exchange was not observed for the reactions considered here.

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APPENDIX Here the energy decomposition analyses for a kinetic energy of a pair of hydrogen atoms, H1 and H2, along the trajectory in AIMD simulations are explained in detail. Atomic Cartesian coordinates and velocities for Hn atom (n = 1, 2) are denoted as (xn, yn, zn) and (vxn, vyn, vzn), respectively. The kinetic energy for a pair of H1 and H2 is represented as m K = (vx12 + vy12 + vz12 + vx 2 2 + vy2 2 + vz 2 2) (A1) 2

H2 molecule in all cases, suggesting that the POSS skeleton composed of the flexible siloxane bonds is efficient as an energy acceptor. According to the energy decomposition analysis, the energy of vibration decreases as the formation of a new hydrogen molecule proceeds. On the contrary, under the initial condition B or when the kinetic energy of the hydrogen atom is larger than that in B, the energy is distributed not only to vibrational modes but also to other motions such as translation or rotation, so the H2 formation is not enhanced. This seems to be brought about by collisions with the included H2 molecule. This phenomenon is especially noticeable in the reaction in T12 with larger space than that of T8, which allows various motions of the guest species. This analysis provides important insights into the expected kinetic energy distributions among the products of encapsulation reactions. Another important role of the encapsulated hydrogen molecule is to donate a hydrogen atom to a free hydrogen atom and promote the formation of a new hydrogen molecule. This is the hydrogen-exchange reaction that was observed in the T8 processes I(B), II(A)-L, and II(B)-L (Table 3). Therefore, the hydrogen-exchange reaction was found to take place only when the entering hydrogen atom approaches the H2 side linearly. The molecular shape of T8 is convenient for this purpose. The T-type insertion in T8, such as in the II(A)-T, II(B)-T, and II(B+)-T processes, is efficient for the formation of a hydrogen molecule regardless of the initial conditions. The hydrogenexchange process did not occur even under the initial conditions with significantly large kinetic energy (II(B+)-T). This type of insertion seems to activate various motions of the guest species immediately, so the energy transfer to the host molecule takes place faster than the linear insertion. However, the activation of various motions of the guest species does not always promote the formation of a hydrogen molecule, as shown for T12 (Table 4). T12 has more room than T8, which allows diverse motions of guest species. The vibrational motions of a two-hydrogen pair are crucial for H−H bond formation, but bond formation competes with the rotational motions. As a result, the H2 formation is not as enhanced as expected, even under the initial conditions with larger kinetic energy. As the number of the guest species increases, reactions within the cage are expected to become more diversified and complicated than the processes considered here. However, the basic observations, such as energy transfer from the guest to the host and the change of the energy distributions depending on the initial conditions suggested by the present study, is not expected to change qualitatively. Finally, it is stressed that although the number of trajectories described in this work may not be statistically sufficient, they do provide meaningful chemical and physical insights into the processes of interest.

where m is the mass of a hydrogen atom. In a mass-weighted sixdimensional coordinate space, the conjugate momenta for H1− H2 can be represented as follows: P = ( m vx1 ,

m vy1 ,

m vz1 ,

m vx 2 ,

m vy2 ,

m vz 2) (A2)

Then, the kinetic energy can be expressed as K=

1 p·p 2

(A3)

In this six -dimensional representation, a set of basis unit vectors for x-, y-, and z-translational modes are given as follows: ⎛ 1 ⎞ 1 e Tx = ⎜ , 0, 0, , 0, 0⎟ ⎝ 2 ⎠ 2

(A4)

⎛ ⎞ 1 1 e Ty = ⎜0, , 0, 0, , 0⎟ ⎝ 2 2 ⎠

(A5)

⎛ 1 1 ⎞ ⎟ e Tz = ⎜0, 0, , 0, 0, ⎝ 2 2⎠

(A6)

Then, momenta for x-, y-, and z-translational modes (pTx, pTy, pTz) are expressed as pTx = e Tx ·p (A7) pTy = e Ty ·p

(A8)

pTz = e Tz ·p

(A9)

and the translational energy, Ktrans, is written as K trans =

1 (p 2 + pTy 2 + pTz 2 ) 2 Tx

(A10)

A basis unit vector for H−H vibrational mode is defined as r e vib = vib |rvib| (A11) O


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(6) Choi, J.; Harcup, J.; Yee, A. F.; Zhu, Q.; Lain, R. M. Organic/ Inorganic Hybrid Composites from Cubic Silsesquioxanes. J. Am. Chem. Soc. 2001, 123, 11420−11430. (7) Pescarmona, P. P.; Maschmeyer, T. Review: Oligomeric Silsesquioxanes:Synthesis, Characterization and Selected Applications. Aust. J. Chem. 2001, 54, 583−596. (8) Li, G.; Wang, L.; Ni, H.; Pittman, C. U., Jr. Polyhedral Oligomeric Silsesquioxane (POSS) Polymers and Copolymers: A Review. J. Inorg. Organomet. Polym. Mater. 2001, 11, 123−154. (9) Kim, K. M.; Chujo, Y. Organic-inorganic Hybrid Gels Having Functionalized Silsesquioxanes. J. Mater. Chem. 2003, 13, 1384−1391. (10) Phillips, S. H.; Haddad, T. S.; Tomczak, S. J. Developments in Nanoscience: Polyhedral Oligomeric Silsesquioxane (POSS)-Polymers. Curr. Opin. Solid State Mater. Sci. 2004, 8, 21−29. (11) Fina, A.; Tabuani, D.; Frache, A.; Camino, G. Polypropylenepolyhedral Oligomeric Silsesquioxanes (POSS) Nanocomposites. Polymer 2005, 46, 7855−7866. (12) Yu, X.; Zhong, S.; Li, X.; Tu, Y.; Yang, S.; Van Horn, R. M.; Ni, C.; Pochan, D. J.; Quirk, R. P.; Wesdemiotis, C.; Zhang, W.-B.; Cheng, S. Z. D. A Giant Surfactant of Polystyrene-(Carboxylic Acid-Functionalized Polyhedral Oligomeric Silsesquioxane) Amphiphile with Highly Stretched Polystyrene Tails in Micellar Assemblies. J. Am. Chem. Soc. 2010, 132, 16741−16744. (13) Wang, F.; Lu, X.; He, C. Some Recent Developments of Polyhedral Oligomeric Silsesquioxane (POSS)-based Polymeric Materials. J. Mater. Chem. 2011, 21, 2775−2782. (14) Kudo, T.; Gordon, M. S. Theoretical Studies of the Mechanism for the Synthesis of Silsesquioxanes. 1. Hydrolysis and Initial Condensation. J. Am. Chem. Soc. 1998, 120, 11432−11438. (15) Kudo, T.; Gordon, M. S. Theoretical Studies of the Mechanism for the Synthesis of Silsesquioxanes. 2. Cyclosiloxanes (D3 and D4). J. Phys. Chem. A 2000, 104, 4058−4063. (16) Kudo, T.; Gordon, M. S. Exploring the Mechanism for the Synthesis of Silsesquioxanes. 3. The Effect of Substituents and Water. J. Phys. Chem. A 2002, 106, 11347−11353. (17) Kudo, T.; Machida, K.; Gordon, M. S. Exploring the Mechanism for the Synthesis of Silsesquioanes. 4. The Synthesis of T8. J. Phys. Chem. A 2005, 109, 5424−5429. (18) Kudo, T.; Akasaka, M.; Gordon, M. S. Ab Initio Molecular Orbital Study on the Ge-, Sn-, Zr- and Si/Ge-Mixed Silsesquioxanes. J. Phys. Chem. A 2008, 112, 4836−4843. (19) Kudo, T.; Akasaka, M.; Gordon, M. S. Ab initio Molecular Orbital Study of the Insertion of H2 into POSS Compounds. Theor. Chem. Acc. 2008, 120, 155−166. (20) Kudo, T. Ab Initio Molecular Orbital Study of the Insertion of H2 into POSS Compounds 2: The Substituent Effect and Larger Cages. J. Phys. Chem. A 2009, 113, 12311−12321. (21) Kudo, T.; Gordon, M. S. Structures and Stabilities of Titanium Silsesquioxanes. J. Phys. Chem. A 2001, 105, 11276−11284. (22) Kudo, T.; Gordon, M. S. Ab Initio Study of the Catalytic Reactivity of Titanosilsesquioxanes and Titanosiloxanes. J. Phys. Chem. A 2003, 107, 8756−8762. (23) Komagata, Y.; Iimura, T.; Shima, N.; Kudo, T. A Theoretical Study of the Insertion of Atoms and Ions into Titanosilsesquioxane (TiPOSS) in Comparison with POSS. Int. J. Polym. Sci. 2012, 2012, 391325. (24) Tejerina, B.; Gordon, M. S. Insertion Mechanism of N2 and O2 into Tn(n = 8, 10, 12)-Silsesquioxane Framework. J. Phys. Chem. B 2002, 106, 11764−11770. (25) Park, S. S.; Xiao, C.; Hagelberg, F.; Hossain, D.; Pittman, C. U., Jr.; Saebo, S. Endohedral and Exohedral Complexes of Polyhedral Double Four-Membered-Ring (D4R) Units with Atomic and Ionic Impurities. J. Phys. Chem. A 2004, 108, 11260−11272. (26) Hossain, D.; Pittman, C. U., Jr.; Saebo, S.; Hagelberg, F. Structures, Stabilities, and Electronic Properties of Endo- and Exohedral Complexes of T10-Polyhedral Oligomeric Silsesquioxane Cages. J. Phys. Chem. C 2007, 111, 6199−6206. (27) Hossain, D.; Pittman, C. U., Jr.; Hagelberg, F.; Saebo, S.; Endohedral. and Exohedral Complexes of T8-Polyhedral Oligomeric

where rvib = (x1 − x 2 , y1 − y2 , z1 − z 2 , x 2 − x1, y2 − y1 , z 2 − z1) (A12)

Then, the momentum for H−H vibration, pvib, and the corresponding kinetic energy, Kvib, are expressed as pvib = e vib·p (A13)

K vib =

pvib 2 2

(A14)

The momentum vector for the rotational mode, prot, can be written as the remaining momentum as prot = p − pTx e Tx − pTy e Ty − pTz e Tz − pvib e vib (A15) and the rotational energy, Krot, is expressed as

K rot =

Prot 2 2

(A16)

Then, the kinetic energy for a pair of H1 and H2 can be written as a sum of translational, vibrational, and rotational energy of the H2 molecule.

■

K = K trans + K vib + K rot

(A17)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b07907. Results of the AIMD simulations for reactions I and II (Tables S1 and S2), changes in the kinetic energies, potential energy, H-H distances, and CASSCF(2,2)/6-31G(d) energy decomposition analyses (Figures S1−S8) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*T. Kudo. Tel: +81-277-30-1935. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by a Grant-in-Aid on PriorityArea-Research: Molecular Theory for Real Systems (461) (T.K. and T.T.), the “Element Innovation” Project from the Ministry of Education, Culture, Sports, Science, and Technology of Japan (T.K.), and by the US Air Force Office of Scientific Research (M.S.G.: AFOSR Award No. FA9550-14-1-0306).

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REFERENCES

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The Journal of Physical Chemistry A Silsesquioxane (POSS) with Transition Metal Atoms and Ions. J. Phys. Chem. C 2008, 112, 16070−16077. (28) Hossain, D.; Gwaltney, S. R.; Pittman, C. U., Jr.; Saebo, S. Insertion of Transition Metal Atoms and Ions into the Nanoscale Dodecahedral Silsesquioxane (T12-POSS) cage: Structures, Stabilities and Electronic Properties. Chem. Phys. Lett. 2009, 467, 348−353. (29) Kudo, T.; Taketsugu, T.; Gordon, M. S. Ab Initio Molecular Dynamics Study of H2 Formation Inside POSS Compounds. J. Phys. Chem. A 2011, 115, 2679−2691. (30) Chen, W.; Hase, W. L.; Schlegel, H. B. Ab Initio Classical Trajectory Study of H2CO -> H2+CO Dissociation. Chem. Phys. Lett. 1994, 228, 436−470. (31) Taketsugu, T.; Gordon, M. S. Dynamic Reaction Coordinate Analysis: An Application to SiH4 + H−.fwdarw.SiH5−. J. Phys. Chem. 1995, 99, 8462−8471. (32) Taketsugu, T.; Gordon, M. S. Dynamic Reaction Path Study of SiH4 + H−.fwdarw.SiH5− and the Berry Pseudorotation Mechanism. J. Phys. Chem. 1995, 99, 14597−14604. (33) Gordon, M. S.; Chaban, G.; Taketsugu, T. Interfacing Electronic Structure Theory with Dynamics. J. Phys. Chem. 1996, 100, 11512− 11525. (34) Taketsugu, T.; Hirano, T. Perturbation Theory and Dynamic Reaction Path Analysis of Intramolecular Vibration Mixing: An Application to the Case of H2O. J. Chem. Phys. 1996, 104, 6081−6088. (35) Taketsugu, T.; Yanai, T.; Hirao, K.; Gordon, M. S. Dynamic Reaction Path Study of SiH 4 +F − -> SiH 4 F − and the Berry Pseudorotation with Valley-ridge Inflection. J. Mol. Struct.: THEOCHEM 1998, 451, 163−177. (36) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; DeFrees, D. J.; Pople, J. A. Self-consistent Molecular Orbital Methods. XXIII. A Polarization-type Basis Set for Second-row Elements. J. Chem. Phys. 1982, 77, 3654−3665. (37) Clark, T.; Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P. v. R. Efficient Diffuse Function-augmented Basis Sets for Anion Calculations. III. The 3-21+G Basis Set for First-row Elements, Li-F. J. Comput. Chem. 1983, 4, 294−301. (38) Frisch, M. J.; Pople, J. A.; Binkley, J. S. Self-consistent Molecular Orbital Methods 25. Supplementary Functions for Gaussian Basis sets. J. Chem. Phys. 1984, 80, 3265−3269. (39) Pople, J. A.; Seeger, R.; Krishnan, R. Variational Configuration Interaction Methods and Comparison with Perturbation Theory. Int. J. Quantum Chem. 1977, 12, 149−163. (40) For these investigations, the structures of some stationary points in the reaction of T8 were also optimized at the CASSCF(3,3) or CASSCF(4,4) levels of theory whereas the product of the reaction II for T12 was optimized at the CASSCF(2,2) level, which was necessary to explain the results of AIMD simulations. (41) The energy barrier for the reaction I of T8 was calculated to be 65.6 kcal/mol at the CASSCF(3,3)/6-31G(d) level. Therefore, 70 kcal/mol of the kinetic energy was applied for the AIMD simulations at the CASSCF(3,3)/6-31G(d) level. (42) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A., Jr. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347−1363. (43) Gordon, M. S.; Schmidt, M. W. Advances in Electronic Structure Theory: GAMESS a Decade Later”, Theory and Applications of Computational Chemistry, Ch. 41, Dykstra, C. E.; Frenking, G.; Kim, K. S.; Scuseria, G. E., Eds., Elsevier, 2005. (44) Amdur, I. Average H-H2 Repulsive Potential from Collision Cross Section Measurements. J. Chem. Phys. 1943, 11, 157−159.

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Ab initio Molecular Dynamics Study of H2 Formation inside POSS

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