Letter pubs.acs.org/JPCL
CH+5 : Symmetry and the Entangled Rovibrational Quantum States of a Fluxional Molecule Robert Wodraszka* and Uwe Manthe* Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstraβe 25, D-33615 Bielefeld, Germany S Supporting Information *
ABSTRACT: Protonated methane, CH+5 , is the prototypical example of a fluxional molecular system. The almost unconstrained angular motion of its five hydrogen atoms results in dynamical phenomena not found in rigid or semirigid molecules. Here it is shown that standard concepts to describe rotational quantum states of molecules can not be applied to CH+5 or any other fluxional system of the type ABn or Bn with n > 4 due to fundamental symmetry reasons. Instead, the ro-vibrational states of CH+5 display a unique level scheme, which results from a complex entanglement of rotational and tunneling motions. A detailed analysis of the ro-vibrational quantum states of CH+5 based on full-dimensional quantum dynamics simulations is presented, and the effects of the Pauli principle are considered. The consequences for the interpretation of recent experimental results are highlighted. KEYWORDS: fluxional molecules, quantum dynamics, Pauli principle
S
The infrared (IR) spectroscopy of protonated methane constitutes “one of the Holy Grails of rotation-vibrational spectroscopy”4 and has been intensively studied experimentally and theoretically (see, e.g., refs 5−10 and references therein). While in the past decade low resolution spectra of CH+5 and isotopomers could be successfully studied in the frequency range above 1000 cm−1,7,8 the detailed understanding of the rovibrational states of CH+5 on a quantum state resolved level remained a challenging task for theory and experiment. Theoretically, Wang and Carrington11 succeeded to accurately calculate the lowest 13 vibrational levels of CH+5 for vanishing total angular momentum, J = 0, using a full-dimensional ab initio PES developed by Bowman and co-workers.12,13 Despite interesting studies,14−16 comparable theoretical results for the ro-vibrational (J > 0) states could not yet be obtained. In a pioneering experimental study,10 Schlemmer and co-workers very recently measured the energy differences between the lowest ro-vibrational states of CH+5 . Thus, detailed experimental information on the low-lying ro-vibrational states of protonated methane is now available, but the observed energy differences still have to be assigned. The present work focuses on the challenge presented by the understanding of the ro-vibrational quantum states of CH+5 and, in a more general context, the special properties of fluxional molecular systems of the type ABn or Bn with n > 4. Unlike in typical molecules, here symmetry enforces an entanglement of rotational and vibration motion. In other words, the rovibrational eigenstates, i.e., the eigenstates showing a nonvanishing total angular momentum, can not be approximated as products of vibrational and rotational wave functions. The reasons will be explained in the following.
eparation of rotational and vibrational motion is a key concept in the analysis of the dynamics of isolated molecules. Approximate separability does not only hold for rigid or semirigid molecules, but often works quite well even in the presence of large amplitude motion. While the precise analysis of spectroscopic experiments often requires one to account for rotation−vibration couplings, a separable ansatz typically provides a valid zero order description on which further refinements can be based. Here it will be shown that a fundamentally different situation is encountered if fluxional molecules like ABn or Bn with n > 4 are considered. Then fundamental symmetry arguments require the complete breakdown of the separability of rotational and internal motion. Rotational and internal motion must always be entangled, i.e., ro-vibrational wave functions must always be (systematic) superpositions of product wave functions. In the present work, the resulting phenomena will be explored studying a prototypical example, the CH+5 molecular cation. Protonated methane, CH+5 , is well known as a fluxional molecule where “the concept of a molecular equilibrium geometry is not valid”.1 CH+5 is also a prototypical example of nonclassical binding in an organic molecule. The fluxionality of CH+5 is a consequence of the coexistence of different types of binding, covalent C−H bonds and a two-electron-three-center C−H2 bond, and the symmetry resulting from the presence of five identical atoms in this six atom system. The global minima of the potential energy surface (PES) show local Cs symmetry and a structure that can be described as an H2 moiety attached to a CH3 tripod. Due to the interchangeability of the five hydrogens, there are 5! = 120 such minima. The barriers separating these minima are very low: the barrier for rotation of the H2 moiety relative to the CH3 tripod is 34 cm−1 and the barrier for a flipping motion interchanging hydrogens between the H2 and CH3 moieties is 287 cm−1.2 Consequently, the ground state wave function of CH+5 is completely delocalized.3 © XXXX American Chemical Society
Received: August 26, 2015 Accepted: October 8, 2015
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DOI: 10.1021/acs.jpclett.5b01869 J. Phys. Chem. Lett. 2015, 6, 4229−4232
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The Journal of Physical Chemistry Letters In a typical molecule, the ro-vibrational eigenstates Ψ can in good approximation be given as a product of a vibrational wave function ψvib, which is an eigenfunction of the vibrational Hamiltonian obtained for vanishing total angular momentum (J = 0), and a rotational wave function ψrot, which describes the orientation of the chosen body fixed frame in space. The rovibrational states have to respect the molecular symmetry group, which includes the feasible permutations of identical nuclei and the inversion, and the isotropy of space. Thus, the ro-vibrational states have to transform according to an irreducible representation Γ of the molecular symmetry group and, simultaneously, according to an irreducible representation J of the rotation group SO(3). In a rigid or semirigid molecule, a reference frame can be chosen disregarding the permutations of identical atoms, and the molecular symmetry group can be replaced by an appropriate point group. Taking ψvib’s transforming according to the irreducible representation of the point group and ψrot’s transforming according to the irreducible representation of the rotation group, the separable ro-vibrational wave functions Ψ = ψvib·ψrot display proper symmetry behavior. In a fluxional molecule of the type ABn or Bn, the B atoms must take part in the definition of the body fixed frame. Permutations of the B atoms will thus affect the position of the body fixed frame in the space fixed coordinate system. To allow for a separable ro-vibrational wave function, the rotational wave function ψrot must therefore simultaneously transform according to an irreducible representation J of the rotation group SO(3) and an irreducible representation Γ of the molecular symmetry group. One must now note that the permutation groups Sn are not isomorphic to subgroups of the rotation group SO(3) if n > 4. Hence, for n > 4 a homomorphic mapping from the Sn group to a group of so-called “equivalent rotations” (see, e.g., refs 17 and 18) can not be established. Consequently, a separable ro-vibrational wave function (with J > 0) displaying proper symmetry behavior can not exist. (Note that this result is irrelevant for (semi)rigid molecules: if rovibrational states corresponding to different irreducible representations Γ are degenerate due to negligible tunneling coupling, separable wave functions can be obtained by unitary transformations which mix these degenerate states.) To investigate the structure of the ro-vibrational quantum states of CH+5 in detail, full-dimensional quantum-mechanical calculations were performed. First, a large basis of vibrational wave functions was generated using the (multilayer) multiconfigurational time-dependent Hartree (MCTDH) approach19−22 and an iterative diagonalization approach specifically designed for multiwell systems.23 The computations employed the potential energy surface of Bowman and coworkers,12,13 a coordinate system based on a (5 + 1)-Radau construction and a stereographic parametrization24 of the Radau vectors, and a kinetic energy operator analogous to the one developed in ref 25. These calculations yielded a basis of several thousand vibrational wave functions for J = 0, which formed the basis for our subsequent ro-vibrational calculations. The ro-vibrational eigenstates of CH+5 were calculated for total angular momenta J up to 2 using the approach developed in ref 26. The correct symmetry of the computed eigenstates according to the G240 group was explicitly enforced in all calculations. A more detailed description of the calculations including a discussion of the convergence errors due to the limited single-particle function bases employed can be found in the Supporting Information. Here it should only be noted that
the present calculations provide a correct description of the energy level scheme and are sufficiently accurate to allow for a reliable analysis of the ro-vibrational states but are less precise than the results for the 13 lowest vibrational (J = 0) levels obtained by Wang and Carrington.11 The present computational scheme allows one to analyze each ro-vibrational state in terms of contributing vibrational wave functions. Any ro-vibrational wave function ψ(J,M,Γ) is n,γ given as a sum of products of vibrational wave functions ψ n ′(,Jγ ′= 0 , Γ ′ ) and purely rotational wave functions J J D̅ MK = (2J + 1)/4π ·DMK * (DJMK denotes a Wigner rotation matrix): J
ψn(,Jγ, M , Γ) =
∑ ∑
J cn , Γ, γ , J , K , n ′ , Γ′ , γ ′·ψn(′J,=γ ′0, Γ′)·D̅ MK
(1)
Γ′ , γ ′ , n ′ K =−J
Here J and M denote the quantum numbers corresponding to the total angular momentum and its space-fixed z-component, respectively, Γ and Γ′ label irreducible representations of the molecular symmetry group G240, γ and γ′ number the components of the irreducible representations, and n and n′ account for different states corresponding to the same irreducible representation. Consequently, J
Nn , Γ, J , n ′ , Γ′ =
∑ ∑ γ , γ ′ K =−J
|cn , Γ, γ , J , K , n ′ , Γ′ , γ ′|2 (2)
measures the contribution of the vibrational state (n′,Γ′) to the ro-vibrational level (n,Γ,J). The calculated vibrational and ro-vibrational energy levels for total angular momenta J = 0 and J = 1 are depicted in Figure 1
Figure 1. Vibrational (J = 0) and ro-vibrational (J = 1) energy levels of CH+5 labeled according to the molecular symmetry group G240. All computed levels are given, i.e., the Pauli principle and the nuclear spin statistics are not considered. The energy of the lowest level (1A1+(J = 0)) defines the zero of energy in the figure. The decompositions of the three lowest ro-vibrational eigenstates in terms of vibrational wave functions is indicated by dashed lines, and numbers beside the lines denote multiple contributions (see text for details). 4230
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The physically existing states obtained by the present calculations are depicted in Figure 2. Since the interconversion
(at this point the nuclear spin states and the Pauli principle are not yet considered). While the vibrational ground state is nondegenerate and totally symmetric, the lowest J = 1 state is (for each value of M) a 4-fold degenerate state transforming according to the G−1 irreducible representation. It should be noted that similar energy level schemes were previously seen in CH+5 -inspired models investigated by Deskevich et al.14 The energy difference between the lowest J = 0 and J = 1 states, 36 cm−1, exceeds any reasonable estimate of a rotational energy difference. Using a simple rigid rotator estimate and ignoring the small differences between the three principle moments of inertia, one would expect rotational energies of B·J(J + 1) with B ≈ 4 cm−112 and thus an energy difference between the lowest J = 0 and J = 1 states of about 8 cm−1. These findings clearly demonstrate the absence of any approximate separability of the rotational and vibrational motion. Based on these results, one might be tempted to simply abandon any picture which distinguishes between rotational and internal motion in CH+5 . However, a detailed analysis of the ro-vibrational wave functions reveals intriguing structures that indicate a specific entanglement of rotational motion and tunneling in CH+5 . Inspecting the Nn,Γ,J,n′,Γ′ (see eq 2), one can approximately assign the ro-vibrational levels to linear combinations of products of vibrational wave functions and purely rotational wave functions. The lowest ro-vibrational level, 1G−1 (J = 1), for example, is 4-fold degenerate (for a fixed value of M). Its eigenfunctions can approximately be described as a linear combination of four products of vibrational wave functions and purely rotational wave functions. Two of these products include the 1A+1 (J = 0) wave function, one includes the 1G−2 (J = 0) wave function, and another one includes the 1H+1 (J = 0) wave function. In Figure 1, these correlations are depicted as lines connecting the G−1 (J = 1) level with the different J = 0 levels. Multiple contributions, i.e., Nn,Γ,J,n′,Γ′’s larger than 1, are indicated by additional numbers beside these lines. As further examples, analogous decompositions of the 1I+(J = 1) and 1G+2 (J = 1) levels are displayed in Figure 1. Thus, the present results clearly demonstrate that the coupling of rotational and internal motion in CH+5 is not an unstructured state mixing, but a specific entanglement related to the system’s symmetry. Up to now, our discussion has focused on the entanglement of rotational and tunneling motion in CH+5 and did not consider the nuclear spin statistics and the Pauli principle. Due to the Fermionic nature of the five hydrogen nuclei, the total wave function, which is a product of the nuclear spin wave function and the ro-vibrational wave function, must be antisymmetric with respect to any odd permutation of the five hydrogen atoms. Nuclear spin wave functions describing the coupled spin state of the five protons transform according to the A1, G1, or H1 irreducible representation of the S5 group.27 Since Pauliallowed total wave functions transform according to the totally antisymmetric representation A2, the total wave function combines a nuclear spin wave function transforming according to A1, G1, or H1 with a ro-vibrational wave function transforming according to A2, G2, or H2, respectively. Thus, ro-vibrational states transforming according to the A1± , G1± , H1± , or I± representations (which have been included in Figure 1) do not exist in nature. However, it should be noted that Pauli-forbidden vibrational (J = 0) wave functions contribute to Pauli-allowed ro-vibrational J > 0 wave functions (see, e.g., the composition of the 1G+2 (J = 1) level given in Figure 1).
Figure 2. Pauli-allowed (ro-)vibrational energy levels of CH+5 . (Ro)vibrational states transforming according to the A2, G2, and H2 representations of the S5 group (which must be combined with nuclear spin states transforming according A1, G1, and H1, respectively) are depicted in separate groups. Solid and dashed lines are used for (ro)vibrational states with even or odd parity, respectively. States with total angular momenta J = 0, 1, or 2 are depicted by black, red, or blue lines, respectively. The same definition of the zero of energy as in Figure 1 is employed.
of states showing different nuclear spin symmetry tends to be very slow, molecules in different nuclear spin states can be considered different species under many typical experimental conditions. Therefore, (ro-)vibrational states transforming according to the A2, G2, and H2 representations (which must be combined with A1, G1, and H1 symmetric nuclear spin wave functions, respectively) are displayed in separate groups. Inspecting Figure 2, one finds that the combined effect of the entanglement of rotational and tunneling motions and the Pauli principle results in a completely unintuitive energy level pattern. The lowest (ro-)vibrational state of CH+5 showing A2 symmetry, for example, is a ro-vibrational state with J = 1 and an energy more than 100 cm−1 above the (G−2 -symmetric) ground state. The present results highlight the challenge presented by the interpretation of the combination differences determined by Schlemmer and co-workers.10 Selection rules for optical transitions in CH+5 require that the initial and final states show opposite parity and identical nuclear spin symmetry. Schlemmer and co-workers obtained energy differences between states that can undergo optical transitions into the same excited state. Consequently, only energy differences between (ro-)vibrational states transforming according to the same irreducible representation of the permutation-inversion group G240 are detected in the experiment. In Figure 2, these combination differences thus correspond to energy differences between levels depicted by the same type of line (full or dashed) and displayed in the same group (A2, G2, or H2). It 4231
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Breaking Explains Infrared Spectra of CH+5 . Nat. Chem. 2010, 2, 298− 302. (9) Huang, X.; McCoy, A. B.; Bowman, J. M.; Johnson, L. M.; Savage, C.; Dong, F.; Nesbitt, D. J. Quantum Deconstruction of the Infrared Spectrum of CH5+. Science 2006, 311, 60−63. (10) Asvany, O.; Yamada, K. M. T.; Brünken, S.; Potapov, A.; Schlemmer, S. Experimental Ground-state Combination Differences of CH+5 . Science 2015, 347, 1346−1349. (11) Wang, X.-G.; Carrington, T. Vibrational Energy Levels of CH+5 . J. Chem. Phys. 2008, 129, 234102. (12) Brown, A.; McCoy, A. B.; Braams, B. J.; Jin, Z.; Bowman, J. M. Quantum and Classical Studies of Vibrational Motion of CH5+ on a Global Potential Energy Surface obtained from a Novel Ab Initio Direct Dynamics Approach. J. Chem. Phys. 2004, 121, 4105−4116. (13) Jin, Z.; Braams, B. J.; Bowman, J. M. An Ab Initio Based Global Potential Energy Surface Describing CH+5 → CH+3 + H2. J. Phys. Chem. A 2006, 110, 1569−1574. (14) Deskevich, M. P.; McCoy, A. B.; Hutson, J. M.; Nesbitt, D. J. Large-amplitude Quantum Mechanics in Polyatomic Hydrides. II. A Particle-on-a-Sphere Model for XHn (n = 4,5). J. Chem. Phys. 2008, 128, 094306. (15) Hinkle, C. E.; McCoy, A. B. Characterizing Excited States of CH+5 with Diffusion Monte Carlo. J. Phys. Chem. A 2008, 112, 2058− 2064. (16) Hinkle, C. E.; Petit, A. S.; McCoy, A. B. Diffusion Monte Carlo Studies of Low Energy Ro-vibrational States of and its Deuterated Isotopologues. J. Mol. Spectrosc. 2011, 268, 189−198. (17) Bunker, P. R.; Jensen, P. Molecular Symmetry and Spectroscopy; NRC Research: Ottawa, Canada, 1998. (18) Hougen, J. T. Permutation-Inversion Groups. J. Mol. Spectrosc. 2009, 256, 170. (19) Meyer, H.-D.; Manthe, U.; Cederbaum, L. S. The MultiConfigurational Time-Dependent Hartree Approach. Chem. Phys. Lett. 1990, 165, 73−78. (20) Manthe, U.; Meyer, H.-D.; Cederbaum, L. S. Wave-packet Dynamics within the Multiconfiguration Hartree Framework: General aspects and Application to NOCl. J. Chem. Phys. 1992, 97, 3199−3213. (21) Wang, H.; Thoss, M. Multilayer Formulation of the Multiconfiguration Time-Dependent Hartree Theory. J. Chem. Phys. 2003, 119, 1289−1299. (22) Manthe, U. A. Multilayer Multiconfigurational Time-Dependent Hartree Approach for Quantum Dynamics on General Potential Energy Surfaces. J. Chem. Phys. 2008, 128, 164116. (23) Wodraszka, R.; Manthe, U. A Multi-Configurational TimeDependent Hartree Approach to the Eigenstates of Multi-Well Systems. J. Chem. Phys. 2012, 136, 124119. (24) Schiffel, G.; Manthe, U. Quantum Dynamics of the H+CH4 → H2+CH3 Reaction in Curvilinear Coordinates: Full-Dimensional and Reduced Dimensional Calculations of Reaction Rates. J. Chem. Phys. 2010, 132, 084103. (25) Wodraszka, R.; Palma, J.; Manthe, U. Vibrational Dynamics of the CH4 · F− Complex. J. Phys. Chem. A 2012, 116, 11249−11259. (26) Wodraszka, R.; Manthe, U. Iterative Diagonalization in the Multi-Configurational Time-Dependent Hartree Approach: Ro-Vibrational Eigenstates. J. Phys. Chem. A 2013, 117, 7246. (27) Bunker, P. A. Preliminary Study of the Proton Rearrangement Energy Levels and Spectrum of CH+5 . J. Mol. Spectrosc. 1996, 176, 297−304.
seems obvious from the above discussion that any simple scheme to assign these combination differences using, e.g., typical spectroscopic model Hamiltonians must fail. Furthermore, also the assignment based on rigorous numerical calculations could only rely on the comparison of “meaningless numbers”. The numerical accuracy of the present calculations is not sufficient to facilitate such an assignment since the numerical inaccuracies of the computed energies are in the 10 wavenumber range. However, the present results clearly contradict simple interpretations based on pure rotational excitations discussed in ref 10. In conclusion, formal symmetry arguments and numerical calculations demonstrated a specific entanglement of the rotational and internal (tunneling) motion in CH+5 . This entanglement is a generic, symmetry-induced phenomenon related to the fact that permutation groups with more than four elements are not isomorphic to any subgroup of the rotation group SO(3). Thus, a similar entanglement should also be present in other fluxional molecules or clusters of the type ABn or Bn with n > 4. Analogous phenomena might therefore be observed, e.g., in helium clusters doped with a single atom or ion.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b01869. Numerical details of the calculation and convergence tests (PDF)
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors want to thank Tucker Carrington and Stefan Schlemmer for helpful discussions. The Paderborn Center for Parallel Computing is gratefully acknowledged for providing computational resources.
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REFERENCES
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