Article pubs.acs.org/JPCB
Diffusion Monte Carlo Calculations of Zero-Point Structures of Partially Deuterated Isotopologues of H+7 Chen Qu and Joel M. Bowman* Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States ABSTRACT: Diffusion Monte Carlo calculations for all the deuterated H+7 isotopologues and isotopomers were performed to determine their zero-point energies, and thus the stability of them. Based on these calculations, we conclude that the deuterium atom prefers the unbonded position in the central H+3 , and then the bonded position in H+3 . When two deuterium atoms are in the outer H2 units, forming a D2 is more stable than one deuterium in each H2 unit. We also discovered that some unstable isotopomers can rearrange to a more stable isotopomer through two types of isomerization: one is that a new H+3 core is formed with more deuterium atoms in it; the other is that the deuterium in the central H+3 goes from the interior (the bonded position) to the exterior (the unbonded position) while the number of deuterium atoms in the H+3 does not change. Three transition states related to the isomerization were identified, two of which have not been reported previously. The corresponding reaction paths were also determined.
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However, for H+7 , little theoretical work has been done. A fulldimensional, permutationally invariant PES, which is based on roughly 170 000 MP2/cc-pVQZ ab initio energies, as well as a DFT-based PES, were recently reported.15,16 Path-integral Monte Carlo and Diffusion Monte Carlo (DMC) calculations36 were performed to characterize the ground state properties of H+7 and D+7 , using the MP2 PES. According to this work, the ground state of H+7 /D+7 can be best described by a relatively localized central H+3 /D+3 solvated by two almost freely rotating H2/D2 units. Unlike H+5 , in which the exchange of proton between the two H2 groups is significant, the exchange in H+7 is rare based on these calculations. Following the DMC work, a new PES based on 42 720 CCSD(T)-F12b/cc-pVQZ-F12 energies was reported and used in five-mode calculations of the IR spectra of H+7 and D 7+ . 37 Those calculations were inspired by the recent experimental results of the predissociation action spectroscopy of these ions,38 which is a revisit to the early experiment,39 and by the theoretical work of the vibrational analysis of the Hn+ clusters.40 The calculated spectra were in very good agreement with the experimental ones. Here we revisit the vibrational ground-state properties and check carefully for other possible large-amplitude motions of H+7 , using the latest PES. In addition, another interesting question for these clusters is which position does deuterium prefer in partial deuterated isotopologues. This question is related to the
INTRODUCTION Since the detection of H+3 in the interstellar medium,1,2 the + protonated hydrogen clusters H2n+1 have been studied extensively, due to the important role they play in interstellar chemistry.3−5 Greatly aided by theory, the structure of larger clusters (with n > 1) was characterized as a central H+3 solvated by H2 molecules,6−10 and the first solvation shell can contain at most three H2 molecules, forming H+5 , H+7 , and H+9 , respectively. Characterization of the potential energy surfaces (PESs) of these clusters11−16 illustrates that the potential is very flat in the region around the minimum with several low-lying saddle points, indicating the fluxional nature of these ions. It is always challenging to study the dynamics and spectroscopy of these fluxional (“floppy”) molecules. Therefore, investigation of the vibrational ground states of these floppy molecules, which could provide insights for understanding the spectroscopy and dynamics, is of great importance. For example, the vibrational ground state and large-amplitude motion of CH+5 has been extensively characterized.17−21 As a member of the protonated hydrogen cluster family and a fluxional ion, H+5 has also attracted many theoretical studies. H+5 has several low-lying stationary points on the PES: the D2v saddle point for proton exchange lies at only 52 cm−1 above the C2v minimum, and the saddle point for the torsion of the outer H2 is roughly 100 cm−1 above the minimum. Vibrational ground state properties of H+5 and its large-amplitude motion have been investigated,22−26 and theoretical simulations of the experimental action spectrum of H+5 has been performed,25,27−35 using the preliminary knowledge obtained from the characterization of the low-lying stationary points on the PES and from the investigation of the large-amplitude motion. © 2014 American Chemical Society
Special Issue: James L. Skinner Festschrift Received: February 7, 2014 Revised: March 23, 2014 Published: March 24, 2014 8221
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Table 1. Zero-Point Energies (cm−1) of H6D+, H5D+2 , H4D+3 , H3D+4 , H2D+5 , and HD+6 Calculated by DMCa isotopologue
isotopomer
DMC (PES I)
DMC (PES II)
harmonic (PES I)
[H2−HDH−H2]+ [H2−DHH−H2]+ [HD−HHH−H2]+
9847.4 ± 2.6 9866.8 ± 1.1b N/Ac
10021.3 ± 3.6 10051.2 ± 7.2 10137.6 ± 7.7b
10315.8 10330.7 10444.8
[H2−DDH−H2]+ [H2−DHD−H2]+ [HD−HDH−H2]+ [HD−DHH−H2]+ [HD−HHD−H2]+ [D2−HHH−H2]+ [HD−HHH−HD]+
9373.8 ± 1.3 9413.3 ± 1.1b 9502.5 ± 0.5 9524.4 ± 2.0b 9525.9 ± 2.1b N/Ac N/Ac
9549.8 ± 6.4 9578.6 ± 7.7b 9674.7 ± 4.1 9693.6 ± 3.5 9698.0 ± 5.3 N/Ac 9798.5 ± 4.0b
9800.4 9824.7 9941.8 9956.1 9957.2 10019.3 10072.3
[H2−DDD−H2]+ [HD−DDH−H2]+ [HD−HDD−H2]+ [HD−DHD−H2]+ [D2−HDH−H2]+ [D2−DHH−H2]+ [D2−HHD−H2]+ [HD−HDH−HD]+ [HD−HHD−HD]+ [D2−HHH−HD]+
8889.6 ± 2.2 9030.4 ± 3.6 9030.0 ± 2.6 9064.7 ± 1.8b 9106.0 ± 5.9b N/Ac N/Ac 9158.3 ± 3.1 9182.0 ± 5.5b N/Ac
9052.8 ± 4.7 9201.8 ± 3.8 9196.4 ± 6.1 9225.1 ± 6.2b 9274.8 ± 6.5 9289.8 ± 6.5b 9292.2 ± 7.4b 9328.1 ± 2.7 9346.6 ± 2.9b N/Ac
9267.6 9424.1 9425.1 9448.8 9514.6 9528.6 9530.4 9567.3 9582.1 9646.3
[HD−DDD−H2]+ [D2−DDH−H2]+ [D2−HDD−H2]+ [HD−DDH−HD]+ [D2−DHD−H2]+ [HD−DHD−HD]+ [D2−HDH−HD]+ [D2−DHH−HD]+ [D2−HHD−HD]+ [D2−HHH−D2]+
8544.1 ± 2.1 8636.8 ± 0.8b 8629.6 ± 2.4 8686.9 ± 1.3 8665.4 ± 1.1b 8719.4 ± 1.5b N/Ac N/Ac N/Ac N/Ac
8701.7 ± 4.9 8791.8 ± 9.3 8784.9 ± 3.5 8845.3 ± 3.7 8822.1 ± 6.6 8876.3 ± 2.9b 8918.0 ± 5.7b 8946.3 ± 5.5b 8942.2 ± 6.4 8992.6 ± 5.7b
8890.3 8995.2 8996.9 9048.4 9020.2 9072.5 9139.7 9154.1 9154.8 9220.0
[D2−DDD−H2]+ [HD−DDD−HD]+ [D2−DDH−HD]+ [D2−HDD−HD]+ [D2−DHD−HD]+ [D2−HDH−D2]+ [D2−DHH−D2]+
8143.2 ± 2.1 8198.0 ± 1.8 8288.8 ± 1.1b 8283.9 ± 2.4 8320.3 ± 2.2b N/Ac 8389.9 ± 3.8b
8286.8 ± 5.4 8348.8 ± 3.3 8445.0 ± 5.0 8435.7 ± 4.2 8469.3 ± 4.2 8517.0 ± 6.0 8534.1 ± 6.4b
8460.4 8512.4 8619.0 8619.7 8643.5 8711.7 8726.5
[D2−DDD−HD]+ [D2−DDH−D2]+ [D2−DHD−D2]+
7797.2 ± 1.4 7889.8 ± 1.5b 7922.0 ± 1.6b
7935.7 ± 3.7 8035.6 ± 5.4 8063.9 ± 6.1
8082.1 8190.0 8214.1
H6D+
H5D+2
H4D+3
H3D+4
H2D+5
HD+6
a
Two PESs are used: “PES I” denotes the CCSD(T)-F12 one and “PES II” is the MP2 one. bIsomerization occurred in the calculations for these species, but we collected adequate data and were able to obtain the ZPE. cIsomerization occurred quickly so that we were not able to obtain the ZPE value.
deuterium fractionation in the interstellar medium.41−48 Complete characterizations for all the possible deuterated counterparts of H+5 has been undertaken using DMC, predicting that hydrogen atoms prefer to be in the middle and deuterium in the outer H2 position.24,26 Here we undertake a similar characterization for the deuterated isotopologues of H+7 . Therefore, in this article we focus on the following two problems: (1) revisiting the vibrational ground state properties of H+7 using the new PES and (2) a complete characterization of the vibrational ground states of H7+ and all its deuterated
isotopologues. To answer these questions, we perform DMC calculations for H+7 and all its possible deuterated isotopologues using the latest CCSD(T)-F12b PES.
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COMPUTATIONAL DETAILS AND RESULTS
DMC Calculations. DMC is a powerful method to calculate the exact zero-point energy (ZPE) and wave function; it has been applied for floppy molecules such as H+5 and CH+5 .17−21,24−27,49 In this article, we applied DMC to study the isotopologues and 8222
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possible large-amplitude motion of H+7 , and we employed the simplest unbiased algorithm.50−52 Specifically, DMC calculations were performed for all the possible deuterated H 7+ isotopologues and all possible isotopomers. For each isotopomer, 20 000 walkers were propagated for a total of 65 000 steps with step size of 5 au for all calculations. The initial configurations of the walkers were the global minimum structure, but for each distinct isotopomer, hydrogen atoms at different positions were replaced appropriately by deuterium. These walkers were first allowed to equilibrate for 5000 steps, and then the energies of the remaining 60 000 steps were collected to calculate the average, which approximates the ZPE. These calculations were based on the new CCSD(T)-F12b PES. In addition, similar calculations were also performed using the MP2 PES; however, instead of 65 000 steps, the walkers were only propagated for 10 000 steps, and three simulations were carried out for each isotopomer. To conveniently show our results, the notation [H2−HHH−H2]+ is used to represent the H+7 . The first and last “H” in “HHH” represent the two atoms in the central H+3 that are bonded to the diatom H2 units, and the “H” in the middle is the unbonded atom. The ZPEs of all the isotopomers using the CCSD(T)-F12b PES as well as the MP2 one are listed in Table 1. For some isotopomers, we were not able to obtain an accurate ZPE because they isomerized to more stable counterparts very quickly. Though the ZPE values from the two potential energy surfaces differ, relative stability of each isotopomer obtained from two PESs agrees with each other. Based on the ZPEs calculated by DMC, we can summarize the following rules for the position that deuterium prefers. (1) Deuterium prefer to stay in the central H+3 positions rather than the outer H2 position. Furthermore, in H+3 , deuterium prefers the unbonded position. (2) If two deuteriums are in the outer H2, forming D2 is more stable than two HD’s. In addition, if we compare the order of the ZPEs of different isotopomers calculated by DMC with that calculated by harmonic approximation, we find the harmonic approximation gives the correct order. However, as seen, the harmonic ZPEs are much higher than the correct DMC ones. Though H+7 is a molecule with high anharmonicity, the harmonic approximation is adequate to describe the relative stability of the isotopomers, qualitatively. Exchange Motion. When analyzing the DMC results, we found some isotopomers can rearrange to more stable ones on the imaginary-time-scale of the propagations done. All these isomerizations could be classified into two types: the first type is that the unstable isotopomer rearranges to the one with more deuterium atoms in the H+3 core; the other type is that the number of deuterium atoms in the central H+3 does not change, but the deuterium moves from the bonded to the unbounded location in the complex. For example, [HD−HHH−H2]+ → [H2−HDH−H2]+ belongs to the first type, while [H2−DHH− H2]+ → [H2−HDH−H2]+ belongs to the second type. In Figure 1 we show the imaginary time evolution of the reference energy for two DMC simulations. The upper panel shows the isomerization [H2−DHH−H2]+ → [H2−HDH−H2]+; however, since the ZPEs of the two isotopomers differ by only 20 cm−1, one cannot see a significant change in the reference energy. In the lower panel, the initial configuration was [D2−HHH−HD]+, and it first became [D2−DHH−H2]+, and then became [H2−DDD− H2]+. The second type of isomerization can be accomplished through a reaction path that connects two equivalent minima and
Figure 1. Imaginary time evolution of the reference energy for two DMC simulations: (a) [H2−DHH−H2]+ → [H2−HDH−H2]+; (b) [D2−HHH−HD]+ → [D2−DHH−H2]+ → [H2−DDD−H2]+.
the saddle point 4-C2v, which has been reported in the previous MP2 PES16 (see Figure 2 for the structure). The reaction path is shown in Figure 3, with the computational details to determine the path given below. However, this type of isomerization can also be accomplished by simply the torsion of the central H+3 , for which motion the saddle point was not reported in the MP2 PES. Furthermore, none of the 10 stationary points reported in the MP2 PES could be the transition state of the exchange in the first type of isomerization. This motivated us to try to locate the two saddle points for the torsional motion of the central H+3 and for the exchange, respectively. The analytical PES allows us to perform a thorough search for stationary points. Specifically, we used the geometries of the walkers in the DMC calculations as the initial guesses, and then applied Newton’s method to optimize the structure on the analytical PES. If the geometry of the walker was close to a stationary point, Newton’s method converged to that stationary structure quickly; on the other hand, if the walker was far from a stationary point, Newton’s method could not converge and we simply discarded that walker. Most of the walkers were discarded, but since we had thousands of random walkers, we were able to locate several stationary points on the PES. We identified two saddle points that may relate to the isomerization, and their structures were further optimized and then harmonic frequency analysis was carried out using MOLPRO,53 employing the CCSD(T)-F12b/VQZ-F12 level of theory. For the two new saddle points as well as the 4-C2v, “quenched” molecular dynamics calculations were carried out on the CCSD(T)-F12b PES: the molecule was initially at the saddle point and was given small initial internal energy. After each step we reduced the velocity of each atom to 95% so that they would basically follow 8223
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Table 2. The Energy Relative to the Minimum (in cm−1) of the Two New Stationary Points Computed at the Indicated Levels of Theory As Well As the Value from the Analytical MP2 and CCSD(T)-F12 PES config.
MP2/ VQZ
MP2 PESa
CCSD(T)-F12b/VQZF12
CCSD(T)-F12 PESb
4-C2v 11-C2v 12-C2v
831.7 705.4 1462.4
828.3 707.1 1198.2
839.4 661.4 1457.6
847.3 659.4 1457.6
a
Reference 16. bReference 37.
Figure 2. The structures of the global minimum and 4-C2v, 11-C2v and 12-C2v saddle points.
Figure 4. Reaction paths to the minimum that start from (a) 11-C2v and (b) 12-C2v. As mentioned in the text, the paths are obtained from MD simulations, and the geometries shown along the path correspond to the red circles on the path.
Therefore, for partially deuterated isotopologues, isomerization between the isotopomers is possible at vibrational ground state. Now the question arises whether the two types of isomerization could also occur at the vibrational ground state of H+7 . To answer this question, we analyzed the 98 714 walkers collected from 5 DMC trajectories of H+7 using the CCSD(T)F12b PES. Among these walkers, we found about 2000 of them has hydrogen atom at the interior that moved to the exterior, and about 100 walkers with the exchange occurred. These results indicate that H+7 is indeed fluxional in the ground vibrational state. However, based on the rarity of the events leading to these isomerizations, we cautiously assert that H+7 is not as fluxional as CH+5 , which isomerizes readily, owing to much smaller barriers separating the isomers.
Figure 3. Reaction path obtained from “quenched” MD simulation that starts from 4-C2v. The geometries shown along the path correspond to the red circles on the path.
the potential gradient, and the trajectory is a good approximation to the reaction path. Figure 2 shows the structures of the global minimum and the 4-C2v, 11-C2v, as well as 12-C2v saddle points. As one can see, 11C2v can be viewed as one H2 molecule bonded to the H+5 at its D2h configuration; thus this exchange is very similar to that in H+5 . The exchange saddle is 661.4 cm−1 above the minimum, and the barrier height for the H+3 torsion is about 1450 cm−1. Both 11-C2v and 12-C2v have one imaginary frequency, corresponding to the proton exchange mode and the torsional motion of the central H+3 , respectively. These two saddle points are now also included in our new CCSD(T)-F12b/VQZ-F12 PES, and their energies as well as the 4-C2v are listed in Table 2. The reaction paths from 11C2v and 12-C2v to the minimum are shown in Figure 4.
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SUMMARY AND CONCLUSIONS The stability and zero-point structures of different isotopomers of partially deuterated H+7 were investigated using the ZPE values obtained from the DMC calculations. Our results from the DMC 8224
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calculations predict that the most stable isotopomer first maximizes the number of deuterium atoms in the central H+3 position and then maximizes the deuterium atoms in one H2. Furthermore, for the two different positions in the central H+3 , the unbonded position is preferred over the two equivalent bonded positions. The harmonic ZPE can correctly predict the relative stability of isotopomers, though not quantitatively. DMC calculations confirm that H+7 is also a fluxional molecule, and two types of isomerization exist: one is an exchange of H between the central H+3 and the H2 unit, and the other is either the torsion of the central H+3 or the migration of one H2 to another corner of the H+3 . Even at the vibrational ground state, the two isomerizations are possible.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank NASA for financial support through Grant No. 370NNX12AF42G from the NASA Astrophysics Research and Analysis program.
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REFERENCES
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(36) Barragán, P.; Pérez de Tudela, R.; Qu, C.; Prosmiti, R.; Bowman, J. M. Full-Dimensional Quantum Calculations of the Dissociation Energy, Zero-Point, and 10 K Properties of H7+/D7+ Clusters Using an Ab Initio Potential Energy Surface. J. Chem. Phys. 2013, 139, 024308/1−8. (37) Qu, C.; Prosmiti, R.; Bowman, J. M. MULTIMODE Calculations of the Infrared Spectra of H7+ and D7+ Using Ab Initio Potential Energy and Dipole Moment Surfaces. Theor. Chem. Acc. 2013, 132, 1413/1−7. (38) Young, J. W.; Cheng, T. C.; Bandyopadhyay, B.; Duncan, M. A. IR Photodissociation Spectroscopy of H7+, H9+, and Their Deuterated Analogues. J. Phys. Chem. A 2013, 117, 6984−6990. (39) Okumura, M.; Yeh, L. I.; Lee, Y. T. Infrared spectroscopy of the cluster ions H3+(H2)n. J. Chem. Phys. 1988, 88, 79−91. (40) Barbatti, M.; Nascimento, M. A. C. Vibrational analysis of small Hn+ hydrogen clusters. J. Chem. Phys. 2003, 119, 5444−5448. (41) Giles, K.; Adams, N. G.; Smith, D. A Study of the Reactions of H3+, H2D+, HD2+, and D3+ with H2, HD, and D2 Using a VariableTemperature Selected Ion Flow Tube. J. Phys. Chem. 1992, 96, 7645− 7650. (42) Paul, W.; Schlemmer, S.; Lücke, B.; Gerlich, D. Deuteration of Positive Hydrogen Cluster Ions H5+ to H17+ at 10 K. Chem. Phys. 1996, 209, 265−274. (43) Gerlich, D.; Schlemmer, S. Deuterium Fractionation in Gas-phase Reactions Measured in the Laboratory. Planet. Space Sci. 2002, 50, 1287−1297. (44) Gerlich, D.; Herbst, E.; Roueff, E. H3+ + HD − H2D+ + H2: Lowtemperature Laboratory Measurements and Interstellar Implications. Planet. Space Sci. 2002, 50, 1275−1285. (45) Millar, T. J. Modelling Deuterium Fractionation in Interstellar Clouds. Planet. Space Sci. 2002, 50, 1189−1195. (46) Roberts, H.; Herbst, E.; Millar, T. J. Enhanced Deuterium Fractionation in Dense Interstellar Cores Resulting From Multiply Deuterated H3+. Astrophys. J. 2003, 591, L41−L44. (47) Roberts, H.; Herbst, E.; Millar, T. J. The Chemistry of Multiply Deuterated Species in Cold, Dense Interstellar Cores. Astron. Astrophys. 2004, 424, 905−917. (48) Hugo, E.; Asvany, O.; Schlemmer, S. H3+ + H2 Isotopic System at Low Temperatures: Microcanonical Model and Experimental Study. J. Chem. Phys. 2009, 130, 164302/1−21. (49) McCoy, A. B. Diffusion Monte Carlo Approaches for Investigating the Structure and Vibrational Spectra of Fluxional Systems. Int. Rev. Phys. Chem. 2006, 25, 77−107. (50) Anderson, J. B. A Random-Walk Simulation of the Schrödinger Equation: H3+. J. Chem. Phys. 1975, 63, 1499−1503. (51) Anderson, J. B. Quantum Chemistry by Random Walk. H 2P, H3+ D3h 1A1′, H2 3Σu+, H4 1Σg+, Be 1S. J. Chem. Phys. 1976, 65, 4121−4127. (52) Kosztin, I.; Faber, B.; Schulten, K. Introduction to the Diffusion Monte Carlo Method. Am. J. Phys. 1996, 64, 633−644. (53) Werner, H.-J. et al. MOLPRO, version 2010.1, a package of ab initio programs; see http://www.molpro.net.
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Diffusion Monte Carlo Calculations of Zero-Point Structures of Partially Deuterated Isotopologues of H+7 Chen Qu and Joel M. Bowman* Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States ABSTRACT: Diffusion Monte Carlo calculations for all the deuterated H+7 isotopologues and isotopomers were performed to determine their zero-point energies, and thus the stability of them. Based on these calculations, we conclude that the deuterium atom prefers the unbonded position in the central H+3 , and then the bonded position in H+3 . When two deuterium atoms are in the outer H2 units, forming a D2 is more stable than one deuterium in each H2 unit. We also discovered that some unstable isotopomers can rearrange to a more stable isotopomer through two types of isomerization: one is that a new H+3 core is formed with more deuterium atoms in it; the other is that the deuterium in the central H+3 goes from the interior (the bonded position) to the exterior (the unbonded position) while the number of deuterium atoms in the H+3 does not change. Three transition states related to the isomerization were identified, two of which have not been reported previously. The corresponding reaction paths were also determined.
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However, for H+7 , little theoretical work has been done. A fulldimensional, permutationally invariant PES, which is based on roughly 170 000 MP2/cc-pVQZ ab initio energies, as well as a DFT-based PES, were recently reported.15,16 Path-integral Monte Carlo and Diffusion Monte Carlo (DMC) calculations36 were performed to characterize the ground state properties of H+7 and D+7 , using the MP2 PES. According to this work, the ground state of H+7 /D+7 can be best described by a relatively localized central H+3 /D+3 solvated by two almost freely rotating H2/D2 units. Unlike H+5 , in which the exchange of proton between the two H2 groups is significant, the exchange in H+7 is rare based on these calculations. Following the DMC work, a new PES based on 42 720 CCSD(T)-F12b/cc-pVQZ-F12 energies was reported and used in five-mode calculations of the IR spectra of H+7 and D 7+ . 37 Those calculations were inspired by the recent experimental results of the predissociation action spectroscopy of these ions,38 which is a revisit to the early experiment,39 and by the theoretical work of the vibrational analysis of the Hn+ clusters.40 The calculated spectra were in very good agreement with the experimental ones. Here we revisit the vibrational ground-state properties and check carefully for other possible large-amplitude motions of H+7 , using the latest PES. In addition, another interesting question for these clusters is which position does deuterium prefer in partial deuterated isotopologues. This question is related to the
INTRODUCTION Since the detection of H+3 in the interstellar medium,1,2 the + protonated hydrogen clusters H2n+1 have been studied extensively, due to the important role they play in interstellar chemistry.3−5 Greatly aided by theory, the structure of larger clusters (with n > 1) was characterized as a central H+3 solvated by H2 molecules,6−10 and the first solvation shell can contain at most three H2 molecules, forming H+5 , H+7 , and H+9 , respectively. Characterization of the potential energy surfaces (PESs) of these clusters11−16 illustrates that the potential is very flat in the region around the minimum with several low-lying saddle points, indicating the fluxional nature of these ions. It is always challenging to study the dynamics and spectroscopy of these fluxional (“floppy”) molecules. Therefore, investigation of the vibrational ground states of these floppy molecules, which could provide insights for understanding the spectroscopy and dynamics, is of great importance. For example, the vibrational ground state and large-amplitude motion of CH+5 has been extensively characterized.17−21 As a member of the protonated hydrogen cluster family and a fluxional ion, H+5 has also attracted many theoretical studies. H+5 has several low-lying stationary points on the PES: the D2v saddle point for proton exchange lies at only 52 cm−1 above the C2v minimum, and the saddle point for the torsion of the outer H2 is roughly 100 cm−1 above the minimum. Vibrational ground state properties of H+5 and its large-amplitude motion have been investigated,22−26 and theoretical simulations of the experimental action spectrum of H+5 has been performed,25,27−35 using the preliminary knowledge obtained from the characterization of the low-lying stationary points on the PES and from the investigation of the large-amplitude motion. © 2014 American Chemical Society
Special Issue: James L. Skinner Festschrift Received: February 7, 2014 Revised: March 23, 2014 Published: March 24, 2014 8221
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Table 1. Zero-Point Energies (cm−1) of H6D+, H5D+2 , H4D+3 , H3D+4 , H2D+5 , and HD+6 Calculated by DMCa isotopologue
isotopomer
DMC (PES I)
DMC (PES II)
harmonic (PES I)
[H2−HDH−H2]+ [H2−DHH−H2]+ [HD−HHH−H2]+
9847.4 ± 2.6 9866.8 ± 1.1b N/Ac
10021.3 ± 3.6 10051.2 ± 7.2 10137.6 ± 7.7b
10315.8 10330.7 10444.8
[H2−DDH−H2]+ [H2−DHD−H2]+ [HD−HDH−H2]+ [HD−DHH−H2]+ [HD−HHD−H2]+ [D2−HHH−H2]+ [HD−HHH−HD]+
9373.8 ± 1.3 9413.3 ± 1.1b 9502.5 ± 0.5 9524.4 ± 2.0b 9525.9 ± 2.1b N/Ac N/Ac
9549.8 ± 6.4 9578.6 ± 7.7b 9674.7 ± 4.1 9693.6 ± 3.5 9698.0 ± 5.3 N/Ac 9798.5 ± 4.0b
9800.4 9824.7 9941.8 9956.1 9957.2 10019.3 10072.3
[H2−DDD−H2]+ [HD−DDH−H2]+ [HD−HDD−H2]+ [HD−DHD−H2]+ [D2−HDH−H2]+ [D2−DHH−H2]+ [D2−HHD−H2]+ [HD−HDH−HD]+ [HD−HHD−HD]+ [D2−HHH−HD]+
8889.6 ± 2.2 9030.4 ± 3.6 9030.0 ± 2.6 9064.7 ± 1.8b 9106.0 ± 5.9b N/Ac N/Ac 9158.3 ± 3.1 9182.0 ± 5.5b N/Ac
9052.8 ± 4.7 9201.8 ± 3.8 9196.4 ± 6.1 9225.1 ± 6.2b 9274.8 ± 6.5 9289.8 ± 6.5b 9292.2 ± 7.4b 9328.1 ± 2.7 9346.6 ± 2.9b N/Ac
9267.6 9424.1 9425.1 9448.8 9514.6 9528.6 9530.4 9567.3 9582.1 9646.3
[HD−DDD−H2]+ [D2−DDH−H2]+ [D2−HDD−H2]+ [HD−DDH−HD]+ [D2−DHD−H2]+ [HD−DHD−HD]+ [D2−HDH−HD]+ [D2−DHH−HD]+ [D2−HHD−HD]+ [D2−HHH−D2]+
8544.1 ± 2.1 8636.8 ± 0.8b 8629.6 ± 2.4 8686.9 ± 1.3 8665.4 ± 1.1b 8719.4 ± 1.5b N/Ac N/Ac N/Ac N/Ac
8701.7 ± 4.9 8791.8 ± 9.3 8784.9 ± 3.5 8845.3 ± 3.7 8822.1 ± 6.6 8876.3 ± 2.9b 8918.0 ± 5.7b 8946.3 ± 5.5b 8942.2 ± 6.4 8992.6 ± 5.7b
8890.3 8995.2 8996.9 9048.4 9020.2 9072.5 9139.7 9154.1 9154.8 9220.0
[D2−DDD−H2]+ [HD−DDD−HD]+ [D2−DDH−HD]+ [D2−HDD−HD]+ [D2−DHD−HD]+ [D2−HDH−D2]+ [D2−DHH−D2]+
8143.2 ± 2.1 8198.0 ± 1.8 8288.8 ± 1.1b 8283.9 ± 2.4 8320.3 ± 2.2b N/Ac 8389.9 ± 3.8b
8286.8 ± 5.4 8348.8 ± 3.3 8445.0 ± 5.0 8435.7 ± 4.2 8469.3 ± 4.2 8517.0 ± 6.0 8534.1 ± 6.4b
8460.4 8512.4 8619.0 8619.7 8643.5 8711.7 8726.5
[D2−DDD−HD]+ [D2−DDH−D2]+ [D2−DHD−D2]+
7797.2 ± 1.4 7889.8 ± 1.5b 7922.0 ± 1.6b
7935.7 ± 3.7 8035.6 ± 5.4 8063.9 ± 6.1
8082.1 8190.0 8214.1
H6D+
H5D+2
H4D+3
H3D+4
H2D+5
HD+6
a
Two PESs are used: “PES I” denotes the CCSD(T)-F12 one and “PES II” is the MP2 one. bIsomerization occurred in the calculations for these species, but we collected adequate data and were able to obtain the ZPE. cIsomerization occurred quickly so that we were not able to obtain the ZPE value.
deuterium fractionation in the interstellar medium.41−48 Complete characterizations for all the possible deuterated counterparts of H+5 has been undertaken using DMC, predicting that hydrogen atoms prefer to be in the middle and deuterium in the outer H2 position.24,26 Here we undertake a similar characterization for the deuterated isotopologues of H+7 . Therefore, in this article we focus on the following two problems: (1) revisiting the vibrational ground state properties of H+7 using the new PES and (2) a complete characterization of the vibrational ground states of H7+ and all its deuterated
isotopologues. To answer these questions, we perform DMC calculations for H+7 and all its possible deuterated isotopologues using the latest CCSD(T)-F12b PES.
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COMPUTATIONAL DETAILS AND RESULTS
DMC Calculations. DMC is a powerful method to calculate the exact zero-point energy (ZPE) and wave function; it has been applied for floppy molecules such as H+5 and CH+5 .17−21,24−27,49 In this article, we applied DMC to study the isotopologues and 8222
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possible large-amplitude motion of H+7 , and we employed the simplest unbiased algorithm.50−52 Specifically, DMC calculations were performed for all the possible deuterated H 7+ isotopologues and all possible isotopomers. For each isotopomer, 20 000 walkers were propagated for a total of 65 000 steps with step size of 5 au for all calculations. The initial configurations of the walkers were the global minimum structure, but for each distinct isotopomer, hydrogen atoms at different positions were replaced appropriately by deuterium. These walkers were first allowed to equilibrate for 5000 steps, and then the energies of the remaining 60 000 steps were collected to calculate the average, which approximates the ZPE. These calculations were based on the new CCSD(T)-F12b PES. In addition, similar calculations were also performed using the MP2 PES; however, instead of 65 000 steps, the walkers were only propagated for 10 000 steps, and three simulations were carried out for each isotopomer. To conveniently show our results, the notation [H2−HHH−H2]+ is used to represent the H+7 . The first and last “H” in “HHH” represent the two atoms in the central H+3 that are bonded to the diatom H2 units, and the “H” in the middle is the unbonded atom. The ZPEs of all the isotopomers using the CCSD(T)-F12b PES as well as the MP2 one are listed in Table 1. For some isotopomers, we were not able to obtain an accurate ZPE because they isomerized to more stable counterparts very quickly. Though the ZPE values from the two potential energy surfaces differ, relative stability of each isotopomer obtained from two PESs agrees with each other. Based on the ZPEs calculated by DMC, we can summarize the following rules for the position that deuterium prefers. (1) Deuterium prefer to stay in the central H+3 positions rather than the outer H2 position. Furthermore, in H+3 , deuterium prefers the unbonded position. (2) If two deuteriums are in the outer H2, forming D2 is more stable than two HD’s. In addition, if we compare the order of the ZPEs of different isotopomers calculated by DMC with that calculated by harmonic approximation, we find the harmonic approximation gives the correct order. However, as seen, the harmonic ZPEs are much higher than the correct DMC ones. Though H+7 is a molecule with high anharmonicity, the harmonic approximation is adequate to describe the relative stability of the isotopomers, qualitatively. Exchange Motion. When analyzing the DMC results, we found some isotopomers can rearrange to more stable ones on the imaginary-time-scale of the propagations done. All these isomerizations could be classified into two types: the first type is that the unstable isotopomer rearranges to the one with more deuterium atoms in the H+3 core; the other type is that the number of deuterium atoms in the central H+3 does not change, but the deuterium moves from the bonded to the unbounded location in the complex. For example, [HD−HHH−H2]+ → [H2−HDH−H2]+ belongs to the first type, while [H2−DHH− H2]+ → [H2−HDH−H2]+ belongs to the second type. In Figure 1 we show the imaginary time evolution of the reference energy for two DMC simulations. The upper panel shows the isomerization [H2−DHH−H2]+ → [H2−HDH−H2]+; however, since the ZPEs of the two isotopomers differ by only 20 cm−1, one cannot see a significant change in the reference energy. In the lower panel, the initial configuration was [D2−HHH−HD]+, and it first became [D2−DHH−H2]+, and then became [H2−DDD− H2]+. The second type of isomerization can be accomplished through a reaction path that connects two equivalent minima and
Figure 1. Imaginary time evolution of the reference energy for two DMC simulations: (a) [H2−DHH−H2]+ → [H2−HDH−H2]+; (b) [D2−HHH−HD]+ → [D2−DHH−H2]+ → [H2−DDD−H2]+.
the saddle point 4-C2v, which has been reported in the previous MP2 PES16 (see Figure 2 for the structure). The reaction path is shown in Figure 3, with the computational details to determine the path given below. However, this type of isomerization can also be accomplished by simply the torsion of the central H+3 , for which motion the saddle point was not reported in the MP2 PES. Furthermore, none of the 10 stationary points reported in the MP2 PES could be the transition state of the exchange in the first type of isomerization. This motivated us to try to locate the two saddle points for the torsional motion of the central H+3 and for the exchange, respectively. The analytical PES allows us to perform a thorough search for stationary points. Specifically, we used the geometries of the walkers in the DMC calculations as the initial guesses, and then applied Newton’s method to optimize the structure on the analytical PES. If the geometry of the walker was close to a stationary point, Newton’s method converged to that stationary structure quickly; on the other hand, if the walker was far from a stationary point, Newton’s method could not converge and we simply discarded that walker. Most of the walkers were discarded, but since we had thousands of random walkers, we were able to locate several stationary points on the PES. We identified two saddle points that may relate to the isomerization, and their structures were further optimized and then harmonic frequency analysis was carried out using MOLPRO,53 employing the CCSD(T)-F12b/VQZ-F12 level of theory. For the two new saddle points as well as the 4-C2v, “quenched” molecular dynamics calculations were carried out on the CCSD(T)-F12b PES: the molecule was initially at the saddle point and was given small initial internal energy. After each step we reduced the velocity of each atom to 95% so that they would basically follow 8223
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Table 2. The Energy Relative to the Minimum (in cm−1) of the Two New Stationary Points Computed at the Indicated Levels of Theory As Well As the Value from the Analytical MP2 and CCSD(T)-F12 PES config.
MP2/ VQZ
MP2 PESa
CCSD(T)-F12b/VQZF12
CCSD(T)-F12 PESb
4-C2v 11-C2v 12-C2v
831.7 705.4 1462.4
828.3 707.1 1198.2
839.4 661.4 1457.6
847.3 659.4 1457.6
a
Reference 16. bReference 37.
Figure 2. The structures of the global minimum and 4-C2v, 11-C2v and 12-C2v saddle points.
Figure 4. Reaction paths to the minimum that start from (a) 11-C2v and (b) 12-C2v. As mentioned in the text, the paths are obtained from MD simulations, and the geometries shown along the path correspond to the red circles on the path.
Therefore, for partially deuterated isotopologues, isomerization between the isotopomers is possible at vibrational ground state. Now the question arises whether the two types of isomerization could also occur at the vibrational ground state of H+7 . To answer this question, we analyzed the 98 714 walkers collected from 5 DMC trajectories of H+7 using the CCSD(T)F12b PES. Among these walkers, we found about 2000 of them has hydrogen atom at the interior that moved to the exterior, and about 100 walkers with the exchange occurred. These results indicate that H+7 is indeed fluxional in the ground vibrational state. However, based on the rarity of the events leading to these isomerizations, we cautiously assert that H+7 is not as fluxional as CH+5 , which isomerizes readily, owing to much smaller barriers separating the isomers.
Figure 3. Reaction path obtained from “quenched” MD simulation that starts from 4-C2v. The geometries shown along the path correspond to the red circles on the path.
the potential gradient, and the trajectory is a good approximation to the reaction path. Figure 2 shows the structures of the global minimum and the 4-C2v, 11-C2v, as well as 12-C2v saddle points. As one can see, 11C2v can be viewed as one H2 molecule bonded to the H+5 at its D2h configuration; thus this exchange is very similar to that in H+5 . The exchange saddle is 661.4 cm−1 above the minimum, and the barrier height for the H+3 torsion is about 1450 cm−1. Both 11-C2v and 12-C2v have one imaginary frequency, corresponding to the proton exchange mode and the torsional motion of the central H+3 , respectively. These two saddle points are now also included in our new CCSD(T)-F12b/VQZ-F12 PES, and their energies as well as the 4-C2v are listed in Table 2. The reaction paths from 11C2v and 12-C2v to the minimum are shown in Figure 4.
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SUMMARY AND CONCLUSIONS The stability and zero-point structures of different isotopomers of partially deuterated H+7 were investigated using the ZPE values obtained from the DMC calculations. Our results from the DMC 8224
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calculations predict that the most stable isotopomer first maximizes the number of deuterium atoms in the central H+3 position and then maximizes the deuterium atoms in one H2. Furthermore, for the two different positions in the central H+3 , the unbonded position is preferred over the two equivalent bonded positions. The harmonic ZPE can correctly predict the relative stability of isotopomers, though not quantitatively. DMC calculations confirm that H+7 is also a fluxional molecule, and two types of isomerization exist: one is an exchange of H between the central H+3 and the H2 unit, and the other is either the torsion of the central H+3 or the migration of one H2 to another corner of the H+3 . Even at the vibrational ground state, the two isomerizations are possible.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank NASA for financial support through Grant No. 370NNX12AF42G from the NASA Astrophysics Research and Analysis program.
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