J. Phys. Chem. C 2008, 112, 17721–17725
17721
Can Ti2-C2H4 Complex Adsorb H2 Molecules? Yasuharu Okamoto† Nano Electronics Research Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba, Ibaraki, 305-8501, Japan ReceiVed: April 29, 2008; ReVised Manuscript ReceiVed: August 21, 2008
Ab initio molecular orbital calculations were used to examine the adsorption of H2 molecules on a Ti2-C2H4 complex. Dependence of the adsorption energy on the computational method was elucidated, and the importance of Gibbs free energy correction on the adsorption energy was clarified. However, van der Waals interaction did not play much role in the adsorption. Molecular dynamics simulations were performed to confirm whether the complex adsorbs H2 molecules at a finite temperature. 1. Introduction H2 is the simplest diatomic molecule; however, theoretical calculations of the system containing it often encounter sensitive problems and require careful handling, particularly when the calculation is performed using the methods based on densityfunctional theory (DFT). The H2 molecule challenges the approximations that are usually used in DFT, such as local density approximation (LDA)1 and generalized gradient approximation (GGA).2-5 It has sometimes been observed that LDA based on a uniform electron gas model is unsuitable for application to systems containing H2, which has just two electrons. Even GGA calculations frequently fail to correct the error in LDA calculations, and poor results are obtained for such systems. One simple and impressive example that discloses a failure of the application of DFT in a system containing H2 is Johnson et al.’s systematic study of H abstraction from H2 (H + H2 f H2 + H) using various ab initio methods.6 The authors showed that the barrier for the abstraction obtained by most DFT methods is very low compared to the experimental value. Even the GGA calculations explained only 30% of the barrier height. Moreover, some DFT calculations provided a qualitatively wrong prediction that the transition state is stable relative to the H + H2 asymptote. This failure of the DFT calculation was attributed to the lack of self-interaction correction (SIC)7 in DFT. Another example of unsatisfactory prediction of DFT is H2 in crystalline Si or GaAs.8,9 Experiments showed that the vibrational frequency of H2 in Si10 and GaAs11 shifts downward by 542 and 227 cm-1 compared to that of the isolated molecule in gas, respectively. Although it was possible to predict a downward shift of 960 (Si) and 536 (GaAs) cm-1 using LDA calculations,8 the value is much too large. GGA calculations also provide a fairly large downward shift of 714 (Si) and 373 (GaAs) cm-1.8 Since a hybrid-DFT method such as B3LYP,12 which includes a 20% Hartree-Fock (HF) exchange interaction, predicts a relatively small magnitude of the downward shift,9 SIC might also be important in this system. It is also worthy of notice that GGA and hybrid-DFT cannot properly treat the van der Waals (vdW) interaction between H2 and the material for hydrogen storage. The potential curve between H2 and the graphene flake becomes repulsive in GGA and hybrid-DFT calculations.13 †
E-mail:
[email protected].
Nevertheless, a reliable calculation of the system containing H2 is important because it is a fundamental molecule in chemistry and physics. Moreover, the calculation may have a potential impact on an envisaged hydrogen society. Based on DFT calculations, Durgun et al. indicated the possibility of a Ti-ethylene (Ti2-C2H4) complex as a new hydrogen storage medium.14 Their calculations showed that the complex can store five H2 molecules per Ti atom, which leads to 14 wt % storage capacity. The value is extraordinarily large compared to metal alloys for hydrogen storage. It remains at 7.6 wt % even in MgH2. In terms of practical convenience and efficiency, it is desirable that adsorption and desorption of H2 molecules occur somewhat above room temperature in material for hydrogen storage. Since 0.1 eV corresponds to 1160 K, accurate evaluation of H2 adsorption energy is indispensable for judging the capacity of the materials. However, ab initio calculation with accuracy of less than 0.1 eV may be a difficult task. A comparative study of various computational methods such as LDA, GGA, hybridDFT, HF, and Mo¨ller-Plesset perturbation theory (MP2) would be helpful to avoid coming to misleading conclusions in this precarious situation. In this paper, we explain the ab initio molecular orbital calculations that were done to elucidate how the description of H2 adsorption on a Ti2-C2H4 complex depends on the employed computational methods. In addition to a comparative study of various computational methods, we emphasized the importance of including Gibbs free energy for accurate evaluation of H2 adsorption because the sum of translational and rotational contributions of H2 in gas is as much as 0.4 eV in the standard state, which cannot be ignored. 2. Computational Method All ab initio molecular orbital (MO) calculations reported in this paper were carried out with the Gaussian 03 program.15 The valence double-ζ and polarization functions (6-31G(d,p)) were used as the basis set of Ti, C, and H atoms. Molecular dynamics (MD) simulations were done using atom-centered density matrix propagation (ADMP).16 The time step (∆t) of ADMP-MD was set at 0.20 fs, and the temperature was maintained using the velocity scaling method during the MD simulations. Reliability of the localized basis set in the MO calculations was assessed by comparing the adsorption of H2 molecules on
10.1021/jp803766z CCC: $40.75 2008 American Chemical Society Published on Web 10/17/2008
17722 J. Phys. Chem. C, Vol. 112, No. 45, 2008
Figure 1. Optimized geometry of Ti2-C2H4-10H2complex in PBE (left) and in MP2 (right). Red arrows designate apical H2 molecules. Symbols a-j correspond to trajectories in Figure 6.
the Ti2-C2H4 complex with that obtained from pseudopotential plane wave (PSPW) calculations carried out using GGA)PBE2,3 method. The STATE program package was used to perform PSPW calculations.17 The ultrasoft pseudopotentials for C, H, and Ti atoms were generated according to the process described by Vanderbilt.18 The partial core correction scheme described by Louie et al.19 was used for C and Ti atoms. Plane-wave basis sets with cutoff energies of 25 and 225 Ry were respectively used for the expansion of wave functions and charge density. The size of a computational cell was (15.88 × 15.88 × 15.88 Å) under a three-dimensional periodic boundary condition, and Γ point sampling was used for Brillouin zone integration. In the PSPWMD calculation, we set ∆t at 0.61 fs. 3. Results and Discussion First, we optimized the geometry of a Ti2-C2H4 complex on which ten H2 molecules were adsorbed at 0 K in PBE and MP2 calculations. We found that the triplet spin state was more stable than the singlet state and that the geometries belonging to a high symmetry point group such as D2h had as many as ten imaginary frequencies. This means that these geometries were not even the local minimum of the potential energy surface. By starting the optimization process from a deformed geometry belonging to a C1 point group, we obtained the geometry of a Ti2-C2H4-10H2 complex having no imaginary frequency, as shown in Figure 1. The argument also applies to a Ti2-C2H4 complex where the triplet spin state was more stable than the singlet state, and an optimization process that started from a D2h geometry resulted in a geometry having two imaginary frequencies. The distance between the center of mass of an H2 molecule and a Ti atom (RH2) is longer in the two apical H2 molecules designated by red arrows in Figure 1 (1.94 Å in PBE and 2.01 Å in MP2) than those in the other eight equatorial H2 molecules (1.80-1.84 Å in PBE and 1.81-1.87 Å in MP2). The H-H bond length (dH2) of the apical H2 molecules is 0.81 Å in PBE and 0.77 Å in MP2, and that of the equatorial H2 molecules is 0.84-0.86 Å in PBE and 0.80-0.83 Å in MP2. The H-H vibrational frequency (νH2) of the apical H2 molecules is about 3300 cm-1 in PBE and 4000 cm-1 in MP2 and that of the equatorial H2 molecules is 2500-3000 cm-1 in PBE and 3080-3550 cm-1 in MP2. It is evident that the interaction between equatorial H2 molecules and a Ti atom is stronger than that between apical H2 molecules and a Ti atom from the calculated results of RH2, dH2, and νH2. It is noted that the downward shift of the vibrational frequency predicted by PBE is significantly larger than that by MP2, which is similar to a trend observed in H2 in Si.9 Then, the averaged adsorption energy of ten H2 molecules without Gibbs free energy correction (∆Eads) and with Gibbs
Okamoto
Figure 2. Molecular orbital calculation concerning temperature dependence of averaged H2 adsorption energy on Ti2-C2H4 complex with including Gibbs free energy correction (∆Gads) by PBE and MP2 methods.
TABLE 1: Averaged H2 Adsorption Energy on Ti2-C2H4 Complex without Gibbs Free Energy Correction ∆Eads and with Gibbs Free Energy Correction in the Standard State ∆Gads(298.15 K) (in eV) ∆Eads ∆Gads(298.15 K)
PBE
MP2
0.580 (0.517)a 0.118
0.195 -0.294b
a Value in parentheses means the energy obtained by PSPW calculation. b Minus sign means the adsorption is endothermic.
free energy correction in the standard state (∆Gads) was calculated. ∆Eads was defined as follows:
∆Eads ) {E(Ti2-C2H4) + 10E(H2) E(Ti2-C2H4-10H2)}/10 Here, E(X) stands for the total energy of X. ∆G ads was defined similarly.
∆Gads ) {G(Ti2-C2H4) + 10G(H2) G(Ti2-C2H4-10H2)}/10 Here, G(X) stands for the Gibbs free energy of X. G(X) and Gibbs free energy correction Gcorr(X) are defined as follows:
G(X) ) E(X) + Gcorr(X) Gcorr(X) ) Ezpe(X) + Ethm(X) + kBT - TS(X) Here, Ezpe(X) is zero-point energy, Ethm(X) is a sum of thermal energy correction for vibrational, rotational, and translational motion of X, kB is Boltzmann constant, and S(X) is a sum of entropy for vibrational, rotational, and translational motion of X. The results are given in Table 1. The Gibbs free energy correction, which mainly stems from translational and rotational entropy of reactant H2 molecules in gas, significantly reduced the averaged adsorption energy. It is noted that MP2 calculation obviously provides smaller adsorption energy than PBE calculation does. Even if a thermal fluctuation of the energy at T ) 298.15 K (kBT ) 0.026 eV) is considered, the ∆Gads (298.15 K) of PBE calculation indicates that H2 molecules are bound by the Ti2-C2H4 complex. However, ∆Gads (298.15 K) of the MP2 calculation indicates that H2 adsorption is not energetically favorable. It is also noted that the vdW interaction is included in MP2, and it is not expected to be included in the PBE calculation. Table 1 indicates that the vdW interaction does not play much of a role in the adsorption of H2 on the Ti2-C2H4 complex, which is quite different from H2 on nanocarbon fragments.13
Ti2-C2H4 Complex Adsorption of H2 Molecules
J. Phys. Chem. C, Vol. 112, No. 45, 2008 17723
Figure 3. Mulliken atomic charge versus averaged H2adsorption energy on Ti2-C2H4 complex.
TABLE 2: Averaged H2 Adsorption Energy on Ti2-C2H4 Complex without Gibbs Free Energy Correction ∆Eads Calculated Using Seven Methods (in eV) SVWN PBE BLYP PBE1 B3LYP BHandHLYP MP2 ∆Eads
1.020
0.580 0.407 0.462
0.352
0.200
0.195
We confirmed that PSPW-PBE calculation gives a similar geometry of the complex and adsorption energy to the MOPBE calculation. The RH2 in two apical H2 molecules were 1.97 and 1.94 Å, respectively, in the PSPW and MO calculations. The RH2 in eight equatorial H2 molecules were 1.83-1.84 Å and 1.80-1.84 Å in PSPW and MO calculations, respectively. The dH2 of the apical H2 molecules were 0.81 Å in PSPW and 0.81 Å in MO calculations and that of equatorial H2 molecules were 0.84-0.87 Å in PSPW and 0.84-0.86 Å in MO calculations. Moreover, the averaged adsorption energy from the PSPW calculation agreed well with the energy from the MO calculation (Table 1). These results show that the difference between localized and plane wave basis sets was smaller than that between PBE and MP2 calculations. Next, we considered the effect of Gibbs free energy correction on the adsorption of H2 molecules on the complex at a finite temperature. Figure 2 shows how ∆Gads depends on temperature. It was found that H2 adsorption is energetically unfavorable in MP2 calculation, unless the temperature is very low ( 4.48 Å). This evidently shows that the inclusion of electron correlation is indispensable for evaluating H2 adsorption. Now we are in a position to confirm whether the Ti2-C2H4 complex can store H2 molecules at a finite temperature by using ADMP-MD simulations. We first examined the time evolution of Ti2-C2H4-10H2 by using the PBE and MP2 methods. The simulation was started from the optimized geometries obtained from the respective methods. The time evolution of RH2 in PBE calculation at T ) 300 K is shown in Figure 4. We found that the complex binds the ten H2 molecules within 1 ps. The RH2 of the ten H2 molecules oscillates around 2 Å. We tried other ADMP simulations starting from a different initial geometry and observed that at least eight H2 molecules were retained. We also checked that the eight H2 molecules remained adsorbed on the complex using a PSPW-MD calculation with the PBE method. This also indicated that the difference between a localized and a plane wave basis set did not affect the calculated results very much. By contrast, according to a snapshot at 0.3 ps in the ADMPMD calculation using the MP2 method shown in Figure 5a, nine H2 molecules desorbed from the Ti2-C2H4 complex and one H2 molecule resulted in dissociative adsorption on a Ti atom. The time evolution of RH2 in MP2 calculation showed successive desorption of H2 from the complex. Figure 5b shows that the first molecules to desorb were four H2 molecules (a-d) that interact with the upper Ti atom. Then, four H2 molecules that interact with the lower Ti atom desorbed. From Figure 2, it is presumed that MP2 calculation predicts that H2 molecules cannot be stored on the complex, even at relatively low temperature. To examine this, we carried out MP2-ADMP-MD simulation at 150 K. Figure 6 shows that ten H2 molecules were kept for up to 0.3 ps, then at least five molecules desorbed from the Ti atoms, and two molecules
17724 J. Phys. Chem. C, Vol. 112, No. 45, 2008
Okamoto and four H2 molecules remained in a BHandHLTP-MD simulation. This obviously shows that the prediction of H2 adsorption on the Ti2-C2H4 complex by using DFT-based methods depends on which exchange correlation functional is employed in the calculation. 4. Summary
Figure 5. (a) Snapshot of Ti2-C2H4-10H2 complex at 0.3 ps calculated using ADMP-MD with the MP2 method. Red arrows designate dissociative adsorption of an H2 molecule. (b) Trajectories of time evolution of RH2 in Ti2-C2H4-10H2 complex at T ) 300 K calculated using ADMP-MD with the MP2 method. Symbols a-j correspond to H2molecules in the right image of Figure 1.
Ab initio MO calculations were performed to examine possibilities for the adsorption of H2 molecules on a Ti2-C2H4 complex. It was shown that the averaged H2 adsorption energy is considerably affected by the exchange-correlation functional employed in DFT-based calculations. Moreover, the adsorption energy obtained from a hybrid-DFT method depends on the contribution of HF exchange interaction. The greater the HF exchange contribution the method contains, the lower the adsorption energy becomes. Gibbs free energy correction (∆Eads-∆Gads) is as large as 0.46 eV in the standard state. The correction is comparable or larger than the adsorption energy evaluated by the difference in total energy, which means that the correction is indispensable for accurate description of the adsorption. MD simulations with a PBE functional showed that most of the H2 molecules remain adsorbed on the complex at room temperature. On the other hand, MD simulations with the MP2 method showed that molecular adsorption of H2 on the complex is unlikely to occur unless the temperature is very low. It was also shown that the use of the localized basis set in MO calculations does not affect the conclusion in the present study by comparing the adsorption energy and time evolution of H2 molecules obtained from PSPW calculations. Acknowledgment. PSPW calculations were performed on the Hitachi SR11000 in the supercomputing center at The University of Tokyo. A part of this research was done in “Advanced Large-scale Computational Simulation Services” supported by Open Advanced Facilities Initiative for Innovation (Strategic Use by Industry) of Ministry of Education, Culture, Sports, Science and Technology (MEXT). References and Notes
Figure 6. Trajectories of time evolution of RH2 in Ti2-C2H4-10H2 complex at T ) 150 K calculated using ADMP-MD with the MP2 method.
underwent dissociative adsorption on the Ti atoms from 0.3 to 0.4 ps. The occurrence of desorption at low temperature means that there is no substantial barrier for the desorption other than the adsorption energy. This result indicates that the complex is not suitable for storage for H2 even at low temperature based on MP2 calculation. We also checked that hybrid-DFT calculations such as PBE1 and BHandHLYP could indicate whether ten H2 molecules could be kept in an MD simulation for 0.8 ps at T ) 300 K. The former and latter methods provided averaged adsorption energy of H2 comparable to PBE and MP2 from Table 2, respectively. It was found that the time evolution of RH2 reflects the difference in the adsorption energy. The ten H2 molecules remained in a PBE1-MD simulation, whereas five molecules desorbed from the Ti atoms, one molecule underwent dissociative adsorption,
(1) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (2) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865. (3) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1997, 78, 1396. (4) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (5) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (6) Johnson, B. G.; Gonzales, C. A.; Gill, P. M. W.; Pople, J. A. Chem. Phys. Lett. 1994, 221, 100. (7) Perdew, J. P.; Zunger, A. Phys. ReV. B 1981, 23, 5048. (8) Okamoto, Y.; Saito, M.; Oshiyama, A. Phys. ReV. B 1997, 56, R10016. (9) Okamoto, Y.; Saito, M.; Oshiyama, A. Phys. ReV. B 1998, 58, 7701. (10) Pritchard, R. E.; Ashwin, M. J.; Tucker, J. H.; Newman, R. C.; Lightowlers, E. C.; Binns, M. J.; McQuaid, S. A.; Falster, R. Phys. ReV. B 1997, 56, 13118. (11) Vetterho¨ffer, J.; Wagner, J.; Weber, J. Phys. ReV. Lett. 1996, 77, 5409. (12) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (13) Okamoto, Y.; Miyamoto, Y. J. Phys. Chem. B 2001, 105, 3470. (14) Durgun, E.; Ciraci, S.; Zhou, W.; Yildirim, T. Phys. ReV. Lett. 2006, 97, 226102. (15) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski,
Ti2-C2H4 Complex Adsorption of H2 Molecules V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03; Gaussian, Inc.: Pittsburgh, PA, 2003. (16) Schlegel, H. B.; Iyengar, S. S.; Li, X.; Millam, J. M.; Voth, G. A.; Scuseria, G. E.; Frisch, M. J. J. Chem. Phys. 2002, 117, 8694.
J. Phys. Chem. C, Vol. 112, No. 45, 2008 17725 (17) Morikawa, Y.; Iwata, K.; Terakura, T. Appl. Surf. Sci. 2001, 169170, 11–15. (18) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892. (19) Louie, S. G.; Froyen, S.; Cohen, M. L. Phys. ReV. B 1982, 26, 1738. (20) Frisch, Æ.; Frisch, M. J.; Trucks, G. W. Gaussian 03 User’s Reference; Gaussian Inc.: Carnegie, PA, 2003; p 77.
JP803766Z
