- Email: [email protected]
A direct isomerization path for the H6+ cluster.
21 August 1998
Chemical Physics Letters 293 Ž1998. 59–64
A direct isomerization path for the Hq6 cluster. An ab initio molecular orbital study Yuzuru Kurosaki ) , Toshiyuki Takayanagi AdÕanced Science Research Center, Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki 319-11, Japan Received 27 January 1998; in final form 11 June 1998
Abstract q We carried out ab initio molecular orbital calculations for the Hq 6 cluster and found two isomers of H 6 and the transition state ŽTS. for the direct isomerization. Analysis of the intrinsic reaction coordinate confirmed that the TS is located at the saddle point of the isomerization path. The existence of the direct isomerization path for Hq 6 may be one of the reasons for the observation of Kirchner and Bowers that the yield of Hq is by far the largest among the even-membered Hq 6 n clusters produced. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The structure and stability of hydrogen cluster cations Hq n have long been investigated using both experimental and theoretical techniques. Hq n clusters are produced by electron impact in H 2 gas w1–3x or by the photoionization of neutral ŽH 2 . n clusters w4x and only odd-membered clusters were observed. It has been accepted that the Hq 3 molecule is one of the most stable cation species produced in hydrogen discharge or plasma. Since H 2 molecules are thought to be the most abundant in the usual experimental conditions, some odd-membered clusters such as Hq 5, q Hq and H should easily be produced via colli7 9 sional stabilizations between Hq 3 and H 2 molecules. Wright and Borkman w5x theoretically predicted, however, that even Ž n s 4,6,8. and odd Ž n s 5,7,9.membered Hq n clusters have comparable binding
energies, implying that even-membered clusters should also exist. This prediction encouraged Kirchner and Bowers w6x to carry out experiments to find even-membered Hq n clusters using high-resolution mass spectrometry and they finally succeeded in detecting even-membered Hq clusters as minor n products of the Hq cluster formation. n It is of interest to note that Kirchner and Bowers w6x found the relative yields of the even-membered q q q q Hq n clusters to be H 6 4 H 8 ) H 10) H 4 , i.e. the q yield of H 6 was observed to be by far the largest. As for the mechanism of even-membered cluster formation, they proposed the reaction scheme given as follows: y ) Hq n qe ™Hn q H )n q Hq m ™ Ž H nqm .
)
)
Ž Hqnq m . ™ Hqnqm q hn , ) Corresponding author. E-mail: [email protected]
where n and m are odd. In this scheme an electronically excited odd-membered cluster H )n is first pro-
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 7 2 1 - 0
60
Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
duced via electron capture by ground-state Hq n . Then an electronically excited collisional intermediate ŽHq . ) is produced when H )n collides with annq m other odd-membered cluster Hq m. Finally the intermediate is stabilized to the ground-state Hq n q m via light emission. This scheme well explains the high yield of Hq 6 in even-membered cluster formations since, as stated above, Hq 3 is considered to be one of the most stable species in hydrogen cluster cations, i.e. it is likely that n s m s 3 in this scheme. Although the scheme of Kirchner and Bowers seems reasonable, it is not an easy task to give theoretical grounds for their speculation, since quantitative calculations for electronically excited states must be done. Instead, in this Letter we will focus on the characteristic of the Hq 6 cluster that may qualitatively explain the high yield of Hq 6 from a different point of view and carry out ab initio molecular orbital ŽMO. calculations. We will find two isomers of the Hq 6 cluster and an isomerization path that directly connects them, whereas there seem no direct isomerization paths like Hq 6 for other even-membered clusters such as Hq and Hq 8 10. One can therefore presume that the existence of the direct isomerization path for Hq 6 is one of the reasons for the high yield of Hq because it increases the density of states 6 q q on the Hq 6 surface relative to that for H 8 or H 10.
2. Methods of calculation In the present calculations the wavefunctions for closed and open-shell molecules were calculated based on spin-restricted and unrestricted Hartree– Fock ŽHF. wavefunctions, respectively, although prefixed letters R and U indicating spin-restricted and unrestricted wavefunctions are not explicitly denoted if not necessary. All ab initio MO calculations were carried out using the GAUSSIAN 94 program w7x. Geometry optimizations were done by the Berny optimization algorithm w8x at the second-order Møller–Plesset perturbation ŽMP2. w9–11x level with the cc-pVTZ basis set w12x. Harmonic vibrational frequencies were computed analytically at the MP2rcc-pVTZ level w13x in order to characterize the optimized geometries as potential minima or saddle points. Single-point calculations for the MP2rcc-pVTZ geometries were
also carried out by the following post HF methods with the cc-pVTZ basis set: MP3 w14,15x; MP4 including single, double, triple and quadruple substitutions ŽMP4ŽSDTQ.. w16,17x; quadratic configuration interaction including single and double substitutions and perturbative triple substitutions ŽQCISDŽT.. w18x. Since spin-unrestricted MPn ŽUMPn. wavefunctions for open-shell molecules are in general not eigenfunctions of the spin operator S 2 , spin-projected UMPn ŽPMPn. wavefunctions w19x were calculated and the spin contamination was completely removed. The intrinsic reaction coordinate ŽIRC. w20–22x was also calculated at the MP2rcc-pVTZ level with a step-size of 0.1 amu1r2 bohr. Potential energy values along the IRC were corrected at several points by adding zero-point vibrational energies ŽZPE. of the orthogonal harmonic vibrations obtained by diagonalizing the projected force constant matrix w23x.
3. Results and discussion 3.1. Geometries and energetics The Hq 6 cluster was predicted to have two isomers with C s and D 2d symmetries, which is consistent with the computational result of Montgomery and Michels w24x. Especially interesting in the present study is that we found the transition state ŽTS. for the direct Hq 6 isomerization. Optimized geometries for the three stationary points are depicted in Fig. 1. For comparison, the optimized geometries for q the Hq 4 and H 5 clusters obtained at the same level of theory are also given. Harmonic vibrational frequencies are shown in Table 1, which characterize the optimized stationary points as a minimum or saddle point of the potential surface. As shown in Ž . Fig. 1a, Hq 6 C s was optimized to be a complex where H 2 and H are weakly bound to the apices of the Hq 3 core. It is worth noting that the lengths of Ž . weak bonds in Hq 6 C s indicated by ‘ PPP ’ in the figure were predicted to be somewhat longer than the q corresponding bonds in the Hq 4 and H 5 clusters and qŽ . that the bond length of H 2 in H 6 C s was predicted to be slightly shorter than that in Hq 5 . This result Ž . means that the interaction in Hq 6 C s between the Hq 3 core and weakly bound species is weaker than
Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
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˚ Ž . Ž . qŽ . Ž . Ž . q Ž . q Fig. 1. Optimized geometries at the MP2rcc-pVTZ level: Ža. Hq 6 C s , b H 6 D 2d , c TS, d H 4 , e H 5 . Bond lengths in A and angles in degree.
q qŽ . that in Hq 4 or H 5 . As shown in Fig. 1b, H 6 D 2d was optimized to be a complex where it is seen that two H 2 molecules are weakly bound to the Hq 2 core. The bond lengths of the isolated H 2 and Hq 2 were ˚ respectively, calculated to be 0.737 and 1.057 A, qŽ . which are close to those of H 2 and Hq 2 in H 6 D 2d . It should be emphasized, however, that the bond Ž . length of H 2 in Hq 6 D 2d was calculated to be 0.785
˚ which is slightly longer than that of H 2 in A, q Ž . Hq 6 C s and in H 5 and that the distance between H 2 q ˚ Ž . and the H 2 core in Hq 6 D 2d to be 1.154 A, which is significantly shorter than that between H 2 and the q qŽ . Hq 3 core in H 6 C s and in H 5 . This result clearly shows that the interaction between the core and Ž . weakly bound species in Hq 6 D 2d is stronger than Ž . that in Hq C . 6 s
Table 1 Harmonic vibrational frequencies estimated at the MP2rcc-pVTZ level Frequencies Žcmy1 . H2 Hq 2 Hq 4 Hq 5 Ž . Hq 6 Cs Ž . Hq 6 D 2d Ž . Hq 6 TS
4525 Ž sg . 2334 Ž sg . 579 Ža 1 . 200 Ža 2 . 2187 Žb 2 . Y 123 Ža . Y 765 Ža . 68 Žb1 . 1004 Žb 2 . X 1027i Ža . X 891 Ža .
591 Žb 2 . 497 Ža 1 . 3730 Ža 1 . X 183 Ža . Y 971 Ža . 365 Že. 1219 Že. Y 115 Ža . Y 1193 Ža .
771 Žb1 . 824 Žb 2 . 4220 Ža 1 . X 405 Ža . X 2256 Ža . 365 Že. 1219 Že. X 400 Ža . X 1388 Ža .
2361 Ža 1 . 880Žb1 .
2414 Žb 2 . 1196 Žb1 .
3492 Ža 1 . 1876 Žb1 .
Y 524 Ža . X 2479 Ža . 783 Že. 2089 Ža 1 . Y 629 Ža . X 2156 Ža .
X 541 Ža . X 3366 Ža . 783 Že. 3865 Žb 2 . X 794 Ža . X 2250 Ža .
X 736 Ža . X 4313 Ža . 904 Ža 1 . 3941 Ža 1 . Y 806 Ža . X 4057 Ža .
Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
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Table 2 Total energies and zero-point vibrational energies ŽZPE. for the MP2rcc-pVTZ geometries Žhartree. Method
H2
Hq 2
Ž . Hq 6 Cs
Ž . Hq 6 D 2d
Ž . Hq 6 TS
MP2 ²S 2 :a ZPE b PMP2 MP3 PMP3 MP4ŽSDTQ. PMP4ŽSDTQ. QCISDŽT.
y1.16465 0.0 0.01031 y1.16465 y1.17026 y1.17026 y1.17172 y1.17172 y1.17232
y0.60224 0.75 0.00532 y0.60224 y0.60224 y0.60224 y0.60224 y0.60224 y0.60224
y3.01507 0.75042 0.03796 y3.01513 y3.02702 y3.02705 y3.03022 y3.03025 y3.03141
y3.01812 0.76122 0.03783 y3.01985 y3.03177 y3.03265 y3.03584 y3.03671 y3.03764
y3.00503 0.78465 0.03344 y3.00927 y3.01875 y3.02133 y3.02327 y3.02584 y3.02620
a b
The expectation value of spin operator S 2 for the MP2 wavefunctions. The ZPE was estimated at the MP2 level of theory.
Total energies of the optimized geometries for q H 2 , Hq 2 and the three stationary points of H 6 are listed in Table 2. The potential energy diagram for Ž . the Hq 6 isomerization calculated at the QCISD T q ZPE level is depicted in Fig. 2, with the zero-point of the diagram being the Hq 2 q 2H 2 energy. The Ž . barrier height from the Hq 6 C s side was calculated Ž . to be 0.4 kcal moly1 and that from the Hq 6 D 2d side y1 qŽ to be 4.4 kcal mol . It was predicted that H 6 D 2d . Ž . is lower in energy by 4.0 kcal moly1 than Hq 6 Cs . This prediction accords qualitatively with the computational result obtained by Montgomery, and Michels w24x using a lower level of theory than the present one.
Fig. 2. Potential energy diagram for the Hq 6 isomerization at the QCISDŽT.rcc-pVTZqZPE level.
3.2. IRC analysis for the isomerization of H6q It was confirmed that the IRC trajectories initiated Ž . and Hq Ž . from the TS towards the Hq 6 Cs 6 D 2d
Fig. 3. Potential energy profile along the IRC for the Hq 6 isomerization. The solid curve is the bare potential and the dashed curve the corrected potential including ZPEs of the orthogonal harmonic vibrations.
Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
directions finally reached the optimized geometries qŽ Ž . . of Hq 6 C s and H 6 D 2d , respectively. This result indicates that the optimized TS is located at the saddle point on the isomerization path that directly qŽ Ž . . connects Hq 6 C s and H 6 D 2d . The potential energy profile along the IRC is shown in Fig. 3, where s denotes the IRC. The TS is set to be at s s 0.0 qŽ Ž . . amu1r2 bohr and Hq 6 C s and H 6 D 2d are in the s - 0.0 and s ) 0.0 regions, respectively. In this figure the solid curve is the bare potential obtained at the MP2rcc-pVTZ level and the dashed curve the corrected potential, including ZPEs of the orthogonal vibrational modes. The ZPE correction reduces the barrier height by about 2 kcal moly1 . A small plateau is seen in the corrected potential curve around s s 0.5 amu1r2 bohr, suggesting that a critical change in geometry occurs around this point. Fig. 4 shows geometrical changes along the IRC: Ža. bond length and Žb. bond angle. It is seen that during the isomerqŽ Ž . . ization from Hq 6 C s to H 6 D 2d both the H 2 H 3
63
and H 2 H 4 bonds in the Hq 3 core break and the weak H 1 H 2 bond becomes a strong chemical bond, while the H 3 H 4 and H 4 X bonds retain their characters. It Ž . is now clear that the H 3 H 4 bond in Hq 6 C s correqŽ . lates with the Hq core in H D and the weak 2 6 2d Ž . H 1 H 2 bond and the H 5 H 6 molecule in Hq 6 Cs correlate with the two H 2 molecules weakly bound qŽ . to both sides of the Hq 2 core in H 6 D 2d . One can consider the region around s s 0.5 amu1r2 bohr as a geometrical critical point, since it is seen from the figure that the H 2 H 4 bond completely breaks and at the same time the H 1 H 2 bond completely forms at this point. It is thought that this critical change in geometry significantly affects the vibrational frequencies of the orthogonal modes and the small plateau emerges in the corrected potential curve as shown above. One cannot think of direct isomerization paths as q in Hq 6 for other even-membered clusters such as H 8 q and H 10. Therefore the prediction that only the Hq 6 cluster has a direct isomerization path suggests that the lifetime of Hq 6 is long compared to the other even-membered clusters, because the density of states in the Hq 6 cluster may be large. This suggestion is thought to be one of the reasons for the observation of Kirchner and Bowers w6x that the yield of Hq 6 is by far the largest among even-membered clusters.
4. Concluding remarks
Fig. 4. Geometrical changes along the IRC for the Hq 6 isomerization: Ža. bond length and Žb. bond angle.
In the present study, ab initio MO calculations were carried out for the Hq 6 isomerization. It was predicted that Hq 6 has two isomers with C s and D 2d symmetries; the C s isomer has the Hq 3 core and the D 2d one has the Hq 2 core, and the D 2d isomer is a few kcal moly1 lower in energy than the C s isomer. This result is consistent with the theoretical prediction of Montgomery, Jr. and Michels w24x. Interestingly, we found the TS for the direct Hq 6 isomerization and the IRC analysis confirmed that the TS is located at the saddle point on the isomerization path. There seem no direct isomerization paths as in Hq 6 for other even-membered Hq n clusters. Therefore the lifetime of Hq 6 may be long as compared to other even-membered clusters, which may be one of the reasons for the high yield of Hq 6 observed by Kirchner and Bowers w6x.
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Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
References w1x M. Saporoschenko, Phys. Rev. A 139 Ž1965. 349. w2x K. Hiraoka, Kebarle, J. Chem. Phys. 62 Ž1974. 2267. w3x R. Johnson, C. Huang, M. Biondi, J. Chem. Phys. 65 Ž1976. 1539. w4x S. Anderson, T. Hirooka, P. Tiedemann, B. Mahan, Y. Lee, J. Chem. Phys. 73 Ž1980. 4779. w5x L.R. Wright, R.F. Borkman, J. Chem. Phys. 77 Ž1982. 1938. w6x N.J. Kirchner, M.T. Bowers, J. Chem. Phys. 86 Ž1987. 1301. w7x M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. DeFrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, GAUSSIAN 94, Revision D.3, Gaussian, Pittsburgh, PA, 1995. w8x H.B. Schlegel, J. Comp. Chem. 3 Ž1982. 214. w9x M. Head-Gordon, J.A. Pople, M.J. Frisch, Chem. Phys. Lett. 153 Ž1988. 503.
w10x M.J. Frisch, M. Head-Gordon, J.A. Pople, Chem. Phys. Lett. 166 Ž1990. 275. w11x M.J. Frisch, M. Head-Gordon, J.A. Pople, Chem. Phys. Lett. 166 Ž1990. 281. w12x T.H. Dunning, J. Chem. Phys. 90 Ž1989. 1007. w13x M. Head-Gordon, T. Head-Gordon, Chem. Phys. Lett. 220 Ž1994. 122. w14x J.A. Pople, R. Seeger, R. Krishnan, Int. J. Quant. Chem. Symp. 11 Ž1977. 149. w15x J.A. Pople, J.S. Binkley, R. Seeger, Int. J. Quant. Chem. Symp. 10 Ž1976. 1. w16x R. Krishnan, J.A. Pople, Int. J. Quant. Chem. 14 Ž1978. 91. w17x R. Krishnan, M.J. Frisch, J.A. Pople, J. Chem. Phys. 72 Ž1980. 4244. w18x J.A. Pople, M. Head-Gordon, K. Raghavachari, J. Chem. Phys. 87 Ž1987. 5968. w19x H.B. Schlegel, J. Chem. Phys. 84 Ž1986. 4530. w20x K. Fukui, J. Phys. Chem. 74 Ž1970. 4161. w21x C. Gonzalez, H.B. Schlegel, J. Chem. Phys. 90 Ž1989. 2154. w22x C. Gonzalez, H.B. Schlegel, J. Phys. Chem. 94 Ž1990. 5523. w23x W.H. Miller, N.C. Handy, J.E. Adams, J. Chem. Phys. 72 Ž1980. 99. w24x J.A. Montgomery Jr., H.H. Michels, J. Chem. Phys. 87 Ž1987. 771.
Chemical Physics Letters 293 Ž1998. 59–64
A direct isomerization path for the Hq6 cluster. An ab initio molecular orbital study Yuzuru Kurosaki ) , Toshiyuki Takayanagi AdÕanced Science Research Center, Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki 319-11, Japan Received 27 January 1998; in final form 11 June 1998
Abstract q We carried out ab initio molecular orbital calculations for the Hq 6 cluster and found two isomers of H 6 and the transition state ŽTS. for the direct isomerization. Analysis of the intrinsic reaction coordinate confirmed that the TS is located at the saddle point of the isomerization path. The existence of the direct isomerization path for Hq 6 may be one of the reasons for the observation of Kirchner and Bowers that the yield of Hq is by far the largest among the even-membered Hq 6 n clusters produced. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The structure and stability of hydrogen cluster cations Hq n have long been investigated using both experimental and theoretical techniques. Hq n clusters are produced by electron impact in H 2 gas w1–3x or by the photoionization of neutral ŽH 2 . n clusters w4x and only odd-membered clusters were observed. It has been accepted that the Hq 3 molecule is one of the most stable cation species produced in hydrogen discharge or plasma. Since H 2 molecules are thought to be the most abundant in the usual experimental conditions, some odd-membered clusters such as Hq 5, q Hq and H should easily be produced via colli7 9 sional stabilizations between Hq 3 and H 2 molecules. Wright and Borkman w5x theoretically predicted, however, that even Ž n s 4,6,8. and odd Ž n s 5,7,9.membered Hq n clusters have comparable binding
energies, implying that even-membered clusters should also exist. This prediction encouraged Kirchner and Bowers w6x to carry out experiments to find even-membered Hq n clusters using high-resolution mass spectrometry and they finally succeeded in detecting even-membered Hq clusters as minor n products of the Hq cluster formation. n It is of interest to note that Kirchner and Bowers w6x found the relative yields of the even-membered q q q q Hq n clusters to be H 6 4 H 8 ) H 10) H 4 , i.e. the q yield of H 6 was observed to be by far the largest. As for the mechanism of even-membered cluster formation, they proposed the reaction scheme given as follows: y ) Hq n qe ™Hn q H )n q Hq m ™ Ž H nqm .
)
)
Ž Hqnq m . ™ Hqnqm q hn , ) Corresponding author. E-mail: [email protected]
where n and m are odd. In this scheme an electronically excited odd-membered cluster H )n is first pro-
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 7 2 1 - 0
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Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
duced via electron capture by ground-state Hq n . Then an electronically excited collisional intermediate ŽHq . ) is produced when H )n collides with annq m other odd-membered cluster Hq m. Finally the intermediate is stabilized to the ground-state Hq n q m via light emission. This scheme well explains the high yield of Hq 6 in even-membered cluster formations since, as stated above, Hq 3 is considered to be one of the most stable species in hydrogen cluster cations, i.e. it is likely that n s m s 3 in this scheme. Although the scheme of Kirchner and Bowers seems reasonable, it is not an easy task to give theoretical grounds for their speculation, since quantitative calculations for electronically excited states must be done. Instead, in this Letter we will focus on the characteristic of the Hq 6 cluster that may qualitatively explain the high yield of Hq 6 from a different point of view and carry out ab initio molecular orbital ŽMO. calculations. We will find two isomers of the Hq 6 cluster and an isomerization path that directly connects them, whereas there seem no direct isomerization paths like Hq 6 for other even-membered clusters such as Hq and Hq 8 10. One can therefore presume that the existence of the direct isomerization path for Hq 6 is one of the reasons for the high yield of Hq because it increases the density of states 6 q q on the Hq 6 surface relative to that for H 8 or H 10.
2. Methods of calculation In the present calculations the wavefunctions for closed and open-shell molecules were calculated based on spin-restricted and unrestricted Hartree– Fock ŽHF. wavefunctions, respectively, although prefixed letters R and U indicating spin-restricted and unrestricted wavefunctions are not explicitly denoted if not necessary. All ab initio MO calculations were carried out using the GAUSSIAN 94 program w7x. Geometry optimizations were done by the Berny optimization algorithm w8x at the second-order Møller–Plesset perturbation ŽMP2. w9–11x level with the cc-pVTZ basis set w12x. Harmonic vibrational frequencies were computed analytically at the MP2rcc-pVTZ level w13x in order to characterize the optimized geometries as potential minima or saddle points. Single-point calculations for the MP2rcc-pVTZ geometries were
also carried out by the following post HF methods with the cc-pVTZ basis set: MP3 w14,15x; MP4 including single, double, triple and quadruple substitutions ŽMP4ŽSDTQ.. w16,17x; quadratic configuration interaction including single and double substitutions and perturbative triple substitutions ŽQCISDŽT.. w18x. Since spin-unrestricted MPn ŽUMPn. wavefunctions for open-shell molecules are in general not eigenfunctions of the spin operator S 2 , spin-projected UMPn ŽPMPn. wavefunctions w19x were calculated and the spin contamination was completely removed. The intrinsic reaction coordinate ŽIRC. w20–22x was also calculated at the MP2rcc-pVTZ level with a step-size of 0.1 amu1r2 bohr. Potential energy values along the IRC were corrected at several points by adding zero-point vibrational energies ŽZPE. of the orthogonal harmonic vibrations obtained by diagonalizing the projected force constant matrix w23x.
3. Results and discussion 3.1. Geometries and energetics The Hq 6 cluster was predicted to have two isomers with C s and D 2d symmetries, which is consistent with the computational result of Montgomery and Michels w24x. Especially interesting in the present study is that we found the transition state ŽTS. for the direct Hq 6 isomerization. Optimized geometries for the three stationary points are depicted in Fig. 1. For comparison, the optimized geometries for q the Hq 4 and H 5 clusters obtained at the same level of theory are also given. Harmonic vibrational frequencies are shown in Table 1, which characterize the optimized stationary points as a minimum or saddle point of the potential surface. As shown in Ž . Fig. 1a, Hq 6 C s was optimized to be a complex where H 2 and H are weakly bound to the apices of the Hq 3 core. It is worth noting that the lengths of Ž . weak bonds in Hq 6 C s indicated by ‘ PPP ’ in the figure were predicted to be somewhat longer than the q corresponding bonds in the Hq 4 and H 5 clusters and qŽ . that the bond length of H 2 in H 6 C s was predicted to be slightly shorter than that in Hq 5 . This result Ž . means that the interaction in Hq 6 C s between the Hq 3 core and weakly bound species is weaker than
Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
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˚ Ž . Ž . qŽ . Ž . Ž . q Ž . q Fig. 1. Optimized geometries at the MP2rcc-pVTZ level: Ža. Hq 6 C s , b H 6 D 2d , c TS, d H 4 , e H 5 . Bond lengths in A and angles in degree.
q qŽ . that in Hq 4 or H 5 . As shown in Fig. 1b, H 6 D 2d was optimized to be a complex where it is seen that two H 2 molecules are weakly bound to the Hq 2 core. The bond lengths of the isolated H 2 and Hq 2 were ˚ respectively, calculated to be 0.737 and 1.057 A, qŽ . which are close to those of H 2 and Hq 2 in H 6 D 2d . It should be emphasized, however, that the bond Ž . length of H 2 in Hq 6 D 2d was calculated to be 0.785
˚ which is slightly longer than that of H 2 in A, q Ž . Hq 6 C s and in H 5 and that the distance between H 2 q ˚ Ž . and the H 2 core in Hq 6 D 2d to be 1.154 A, which is significantly shorter than that between H 2 and the q qŽ . Hq 3 core in H 6 C s and in H 5 . This result clearly shows that the interaction between the core and Ž . weakly bound species in Hq 6 D 2d is stronger than Ž . that in Hq C . 6 s
Table 1 Harmonic vibrational frequencies estimated at the MP2rcc-pVTZ level Frequencies Žcmy1 . H2 Hq 2 Hq 4 Hq 5 Ž . Hq 6 Cs Ž . Hq 6 D 2d Ž . Hq 6 TS
4525 Ž sg . 2334 Ž sg . 579 Ža 1 . 200 Ža 2 . 2187 Žb 2 . Y 123 Ža . Y 765 Ža . 68 Žb1 . 1004 Žb 2 . X 1027i Ža . X 891 Ža .
591 Žb 2 . 497 Ža 1 . 3730 Ža 1 . X 183 Ža . Y 971 Ža . 365 Že. 1219 Že. Y 115 Ža . Y 1193 Ža .
771 Žb1 . 824 Žb 2 . 4220 Ža 1 . X 405 Ža . X 2256 Ža . 365 Že. 1219 Že. X 400 Ža . X 1388 Ža .
2361 Ža 1 . 880Žb1 .
2414 Žb 2 . 1196 Žb1 .
3492 Ža 1 . 1876 Žb1 .
Y 524 Ža . X 2479 Ža . 783 Že. 2089 Ža 1 . Y 629 Ža . X 2156 Ža .
X 541 Ža . X 3366 Ža . 783 Že. 3865 Žb 2 . X 794 Ža . X 2250 Ža .
X 736 Ža . X 4313 Ža . 904 Ža 1 . 3941 Ža 1 . Y 806 Ža . X 4057 Ža .
Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
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Table 2 Total energies and zero-point vibrational energies ŽZPE. for the MP2rcc-pVTZ geometries Žhartree. Method
H2
Hq 2
Ž . Hq 6 Cs
Ž . Hq 6 D 2d
Ž . Hq 6 TS
MP2 ²S 2 :a ZPE b PMP2 MP3 PMP3 MP4ŽSDTQ. PMP4ŽSDTQ. QCISDŽT.
y1.16465 0.0 0.01031 y1.16465 y1.17026 y1.17026 y1.17172 y1.17172 y1.17232
y0.60224 0.75 0.00532 y0.60224 y0.60224 y0.60224 y0.60224 y0.60224 y0.60224
y3.01507 0.75042 0.03796 y3.01513 y3.02702 y3.02705 y3.03022 y3.03025 y3.03141
y3.01812 0.76122 0.03783 y3.01985 y3.03177 y3.03265 y3.03584 y3.03671 y3.03764
y3.00503 0.78465 0.03344 y3.00927 y3.01875 y3.02133 y3.02327 y3.02584 y3.02620
a b
The expectation value of spin operator S 2 for the MP2 wavefunctions. The ZPE was estimated at the MP2 level of theory.
Total energies of the optimized geometries for q H 2 , Hq 2 and the three stationary points of H 6 are listed in Table 2. The potential energy diagram for Ž . the Hq 6 isomerization calculated at the QCISD T q ZPE level is depicted in Fig. 2, with the zero-point of the diagram being the Hq 2 q 2H 2 energy. The Ž . barrier height from the Hq 6 C s side was calculated Ž . to be 0.4 kcal moly1 and that from the Hq 6 D 2d side y1 qŽ to be 4.4 kcal mol . It was predicted that H 6 D 2d . Ž . is lower in energy by 4.0 kcal moly1 than Hq 6 Cs . This prediction accords qualitatively with the computational result obtained by Montgomery, and Michels w24x using a lower level of theory than the present one.
Fig. 2. Potential energy diagram for the Hq 6 isomerization at the QCISDŽT.rcc-pVTZqZPE level.
3.2. IRC analysis for the isomerization of H6q It was confirmed that the IRC trajectories initiated Ž . and Hq Ž . from the TS towards the Hq 6 Cs 6 D 2d
Fig. 3. Potential energy profile along the IRC for the Hq 6 isomerization. The solid curve is the bare potential and the dashed curve the corrected potential including ZPEs of the orthogonal harmonic vibrations.
Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
directions finally reached the optimized geometries qŽ Ž . . of Hq 6 C s and H 6 D 2d , respectively. This result indicates that the optimized TS is located at the saddle point on the isomerization path that directly qŽ Ž . . connects Hq 6 C s and H 6 D 2d . The potential energy profile along the IRC is shown in Fig. 3, where s denotes the IRC. The TS is set to be at s s 0.0 qŽ Ž . . amu1r2 bohr and Hq 6 C s and H 6 D 2d are in the s - 0.0 and s ) 0.0 regions, respectively. In this figure the solid curve is the bare potential obtained at the MP2rcc-pVTZ level and the dashed curve the corrected potential, including ZPEs of the orthogonal vibrational modes. The ZPE correction reduces the barrier height by about 2 kcal moly1 . A small plateau is seen in the corrected potential curve around s s 0.5 amu1r2 bohr, suggesting that a critical change in geometry occurs around this point. Fig. 4 shows geometrical changes along the IRC: Ža. bond length and Žb. bond angle. It is seen that during the isomerqŽ Ž . . ization from Hq 6 C s to H 6 D 2d both the H 2 H 3
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and H 2 H 4 bonds in the Hq 3 core break and the weak H 1 H 2 bond becomes a strong chemical bond, while the H 3 H 4 and H 4 X bonds retain their characters. It Ž . is now clear that the H 3 H 4 bond in Hq 6 C s correqŽ . lates with the Hq core in H D and the weak 2 6 2d Ž . H 1 H 2 bond and the H 5 H 6 molecule in Hq 6 Cs correlate with the two H 2 molecules weakly bound qŽ . to both sides of the Hq 2 core in H 6 D 2d . One can consider the region around s s 0.5 amu1r2 bohr as a geometrical critical point, since it is seen from the figure that the H 2 H 4 bond completely breaks and at the same time the H 1 H 2 bond completely forms at this point. It is thought that this critical change in geometry significantly affects the vibrational frequencies of the orthogonal modes and the small plateau emerges in the corrected potential curve as shown above. One cannot think of direct isomerization paths as q in Hq 6 for other even-membered clusters such as H 8 q and H 10. Therefore the prediction that only the Hq 6 cluster has a direct isomerization path suggests that the lifetime of Hq 6 is long compared to the other even-membered clusters, because the density of states in the Hq 6 cluster may be large. This suggestion is thought to be one of the reasons for the observation of Kirchner and Bowers w6x that the yield of Hq 6 is by far the largest among even-membered clusters.
4. Concluding remarks
Fig. 4. Geometrical changes along the IRC for the Hq 6 isomerization: Ža. bond length and Žb. bond angle.
In the present study, ab initio MO calculations were carried out for the Hq 6 isomerization. It was predicted that Hq 6 has two isomers with C s and D 2d symmetries; the C s isomer has the Hq 3 core and the D 2d one has the Hq 2 core, and the D 2d isomer is a few kcal moly1 lower in energy than the C s isomer. This result is consistent with the theoretical prediction of Montgomery, Jr. and Michels w24x. Interestingly, we found the TS for the direct Hq 6 isomerization and the IRC analysis confirmed that the TS is located at the saddle point on the isomerization path. There seem no direct isomerization paths as in Hq 6 for other even-membered Hq n clusters. Therefore the lifetime of Hq 6 may be long as compared to other even-membered clusters, which may be one of the reasons for the high yield of Hq 6 observed by Kirchner and Bowers w6x.
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Y. Kurosaki, T. Takayanagir Chemical Physics Letters 293 (1998) 59–64
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