Bend Excitation Is Predicted to Greatly Accelerate Isomerization of

Dec 16, 2014 - Using a new potential energy surface, based on fitting around 33 000 CCSD(T)-F12/aug-cc-pVTZ energies, a robust set of predictions is m...
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Bend Excitation Is Predicted to Greatly Accelerate Isomerization of trans-Hydroxymethylene to Formaldehyde in the Deep Tunneling Region Yimin Wang and Joel M. Bowman* Cherry L. Emerson Center for Scientific Computation and Department of Chemistry, Emory University, Atlanta, Georgia 30322, United States ABSTRACT: Using a new potential energy surface, based on fitting around 33 000 CCSD(T)-F12/aug-cc-pVTZ energies, a robust set of predictions is made for mode-specific isomerization of trans-hydroxymethylene to formaldehyde in the deep tunneling region. The calculations make use of a recent projection model for mode-specific tunneling based on the rectilinear Qim path [Wang, Y.; Bowman, J. M. J. Chem. Phys. 2013, 139, 154303]. The most interesting prediction is a large decrease in the half-life from roughly 5 h for the ground vibrational state to roughly 1.5 min and 1 s by excitation of the fundamental and first overtone of the asymmetric bending normal mode, respectively. The properties of the new PES are described along with variational calculations of low-lying vibrational states of trans- and cis-hydroxymethylene.

I

An intriguing question that we address here is the effect of mode-specific vibrational excitation of trans-hydroxymethylene on the isomerization rate to formaldehyde. We do so by making use of our Qim projection approach to describe mode-specific tunneling quantitatively in polyatomic molecules.4 The approach is an extension of the one-dimensional tunneling method that accounts for the coupling of molecular vibrational degrees on the tunneling coordinate. The method was tested on malonaldehyde and shown to produce results (using an accurate potential energy surface) in good agreement with experiment for excitation of modes that produce substantial increases in tunneling. The objective here is to predict which mode excitations lead to a substantial increase in the tunneling rate of trans-hydroxymethylene and to challenge experiment to measure these rates. Formaldehyde is perhaps the most studied tetraatomic molecule from the perspective of both dynamics and spectroscopy. Yet, only one global ab initio potential energy surface exists.5 This PES contains the global minimum and the two high-energy hydroxymethylene isomers and dissociates to the molecular and radical products. This PES has been used in dynamics studies of unimolecular dissociation and also for the H + HCO reaction to form H2 + CO, where the “roaming” pathway to those products was discovered. However, this PES is not of “spectroscopic” accuracy and has not been used in that context. More recently, as noted already, limited ab initio PESs have been reported for the global minimum6 and separately for the trans- and cis-HCOH isomers,2 which are at or near

somerization is a central process in Chemistry, and yet, the experimental determination of isomers and rates of isomerization have proven to be a major challenge. A notable exception to this was the recent detection and determination of the rate of isomerization of hydroxymethylene at near 0 K to formaldehyde in a rare-gas matrix.1 The measured half-life of roughly 2 h is certainly due to a large barrier separating transhydroxymethylene from the much more stable formaldehyde.1 This long half-life was accounted for quantitatively theoretically, based on a standard 1d quantum tunneling calculation making use of the intrinsic reaction path from the saddle point barrier separating the two minima and an effective potential consisting of the potential along the path plus the local harmonic zeropoint energy (ZPE). The remarkable (and almost certainly somewhat fortuitous) near-perfect agreement between theory and experiment certainly demonstrates the power of even a 1d approach, provided that the potential describing the tunneling process is accurate, which was certainly the case in that study. The infrared spectrum of trans-hydroxymethylene was also reported in ref 1, and comparison with high-level theory was a vital precursor to the subsequent rate determination, as this permitted an unambiguous assignment of the species as transhydroxymethylene. Subsequently, the rigorous theoretical determination of the IR spectra of trans-hydroxymethylene and the slightly higher energy cis-hydroxymethylene isomer were reported.2 Even more recently, the IR spectrum of lowlying states of trans-hydroxymethylene was measured in a He nanodroplet.3 The band origins are generally in good agreement with the earlier Ar matrix experiments. All of these theoretical approaches made use of either local potentials or local quartic force fields and thus cannot describe isomerization. © XXXX American Chemical Society

Received: October 29, 2014 Accepted: December 16, 2014

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spectroscopic accuracy. However, these local PESs cannot be used to study the unimolecular dissociation of H2CO or even the more limited isomerization dynamics, which is the subject of this Letter. As just noted, the existing global PES, while reasonably accurate for the energies and barrier related to isomerization, is not quantitatively accurate (see below). Thus, we have developed a new full-dimensional PES for H2CO that spans the global minimum and trans- and cis-hydroxymethylene isomers and accurately describes the barriers between them. This PES is used here in calculations of mode-specific isomerization of trans-hydroxymethylene using the simple Qim projection method.4 The new PES can of course be used in fulldimensional or reduced-dimensional quantum calculations and/or a variety of quasi-classical and semiclassical calculations of the isomerization, and that is certainly a future goal for use by us and other groups. However, it is extremely difficult to converge quantum calculations in the deep tunneling region of interest here, and therefore, the Qim projection approach is used. A brief description of the new PES and its properties is given next, followed by the relevant details of the projection theory, followed by results and discussion. For completeness, fulldimensional vibrational calculations using a localized approach for the trans- and cis-hydroxymethylene isomers are also reported and compared to experiment for trans-hydroxymethylene. The new PES is a permutationally invariant fit to 33 047 electronic energies obtained using CCSD(T)-F12a/aug-ccpVTZ theory7−9 implemented in the MOLPRO suite of quantum chemistry programs.10 The level of theory is certainly sufficient for the present objectives, and it is efficient enough for this number of energies to be performed in order to obtain the new PES. The data set of electronic energies was obtained as follows. First, classical molecular dynamics were performed on the two previous PESs, the global one5 and the recent local ones for the two hydroxymethylene isomers,2 to sample the local minimum regions, that is, the formaldehyde global minimum and the trans- and cis-hydroxymethylene isomers. Gaussian random sampling in normal coordinates of each stationary point was employed to weight those important regions of the PES. Lastly, with the present and future quantum application in mind, we also did random sampling in various types of Jacobi coordinates to ensure a good coverage of transitional areas between local regions characterized by stationary points as well as regions of higher potential energy. The final data set is roughly equally distributed among formaldehyde and the trans- and cis-hydroxymethylene isomers, with roughly 13 000 configurations in the formaldehyde well and 20 000 configurations for the hydroxymethylene. The rootmean-square fitting error of the PES is 0.1 kcal/mol for energies up to 63 kcal/mol (above the global minimum), 0.6 kcal/mol for energies between 63 and 126 kcal/mol, and 2.6 kcal/mol for energies between 126 and 314 kcal/mol. Under 2% of the entire data set have energies above 314 kcal/mol. A schematic of the PES is given in Figure 1, and more details of the properties of the PES, for example, energies of minima, barriers, and harmonic frequencies, are given in Tables 1 and 2. Consider the results in Table 1 first, which includes results from previous calculations. The energies in the column labeled “CCFPA” are considered the benchmark values as they come from high-level coupled cluster focal point analysis.1 PES04 refers to the global PES mentioned above.5 As seen, the new PES

Figure 1. Schematic diagram of the PES. The potential (kcal/mol) is relative to the H2CO global minimum. Note, TS3 is the saddle point to the H2+CO products.

Table 1. Comparison of Energies (kcal/mol) of the Present PES with the Present CCSD(T)-F12/aug-cc-pVTZ Calculations and Previous CCSD(T)/aug-cc-pVTZ and the PES of Reference 5 (denoted PES04) and Benchmark HighLevel Coupled Cluster Focal Point Analysis (denoted CCFPA) (eef 1) Values, Relative to the Global Minimum for the Indicated Stationary Points

H2CO transHCOH cis-HCOH TS1 TS2 TS3

CCFPA

CCSD(T)-F12/ aVTZ

PES

CCSD(T)/ aVTZ

PES04

0.0 52.4

0.0 51.8

0.0 51.8

0.0 51.3

0.0 51.4

57.3 81.9 86.1

56.6 81.3 86.1 86.7

56.6 81.3 86.1 86.7

56.0 80.8 85.6 86.5

55.8 78.2 86.2 87.4

energies are quite close to the benchmark ones and are a significant improvement over those from PES04. Next, consider the comparison of harmonic frequencies, given in Table 2. Note that the mode numbering is listed in two ways; one is just in increasing order of frequency, and the other is spectroscopic. As expected, the new PES results are much closer to the ab initio ones than are the PES04 ones. However, because the current ab initio F12 method is presumably more accurate than the previous method used to fit PES04 (and this is borne out by the results in Table 1), the frequencies from the PES are almost certainly more accurate than those from PES04. Now, we give a brief description of the Qim projection approach, which has been described in detail previously,4,11,12 and give specific details for the present application. In this approach, the relaxed potential along the path given by the mass-scaled, rectilinear imaginary frequency normal mode of the relevant saddle point separating two minima is determined. In the present case, the saddle point is the one separating transhydroxymethylene and formaldehyde; this is TS2 depicted in Figure 1. The relaxed potential, denoted V(Qim), is given in Figure 2. This potential is the primary quantity in the projection approach. It is used to determine the tunneling rate for both the ground vibrational state of trans-hydroxymethylene and for vibrationally excited states. For the ground 125

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Table 2. Comparison of Harmonic Frequencies (cm−1) of the Present PES at the Global Minimum, trans-HCOH, cis-HCOH, and Saddle Points with the Present CCSD(T)-F12/aug-cc-pVTZ (denoted ab initio) Calculations and Results of Previous PES from Reference 5 (denoted PES04) ab initio mode mode mode mode mode mode

1 2 3 4 5 6

mode mode mode mode mode mode

1 2 3 4 5 6

(ν6) (ν5) (ν4) (ν3) (ν2) (ν1)

1187 1269 1534 1779 2933 3005 1494i 732 1191 1394 2734 3873

PES H2CO 1187 1269 1535 1780 2935 3007 TS1 1373i 740 1184 1407 2735 3864

PES04

ab initio

1164 1337 1623 1806 2949 2810

1090 1217 1327 1511 2874 3756

1074i 968 1208 1465 2541 3531

2171i 735 1308 1405 2607 2911

PES trans-HCOH 1095 1214 1323 1499 2872 3753 TS2 2163i 735 1303 1405 2617 2907

PES04

ab initio

1108 1281 1369 1714 2927 4098

1009 1233 1327 1477 2782 3648

2068i 725 1259 1454 2598 2992

1834i 788 869 1288 1854 3141

PES cis-HCOH 990 1219 1328 1468 2770 3670 TS3 1820i 788 903 1325 1855 3147

PES04 1221 1342 1440 1608 2741 3839 1824i 589 652 1351 1861 3039

calculation due to the asymmetry of the potential. In full dimensionality, there are thousands of formaldehyde states below the first localized trans-hydroxymethylene state. As noted above, the ground vibrational state isomerization rate (actually half-life) had been reported previously using a potential along the much more widely used intrinsic reaction coordinate, with the addition of the local ZPE, and amazingly good agreement with experiment was found.1 We tested this approach as well as the Qim one for the ground-state splittings in full-dimensional malonaldehyde and d1-malonaldehyde against accurate full-dimensional ones for the ground vibrational state. The results for malonaldehyde showed definitively that the intrinsic reaction coordinate (IRC) approach (with no curvature correction) is very inaccurate whereas the Qim one gave results in near-quantitative agreement with benchmarks ones.4 In the case of malonaldehyde, there are substantial differences in the barrier width along the IRC and the Qim paths, and that substantively accounted for the inaccuracy of the IRC path. For the present case of trans-hydroxymethylene isomerization, the two paths are very similar, and therefore, calculations of the ground-state tunneling rates using the two paths are quite similar. Note, we do not include the local ZPE in the Qim theory. However, we do note that we had done that in the previous tests of the tunneling in malonaldehyde and found the results to be slightly less accurate for both malonaldehyde and d1-malonaldehyde than simply using V(Qim). In the mode-specific projection theory, the objective is to determine the change in classical turning points upon mode excitation of the “product” molecule, in this case, transhydroxymethylene. As we have shown, this is easily done for the rectilinear Qim path.4 For excitations of a normal mode i, this change is given by4

Figure 2. Qim path from trans-HCOH to the H2CO global minimum. The potential is relative to the H2CO global minimum. ZPE denotes the energy level of the first localized trans-HCOH eigenstate.

vibrational state, the well-known semiclassical expression for the tunneling rate is given by ω k(E) = exp( −2θ ) 2π b 1 θ= dQ im 2[V (Q im) − E] (1) ℏ a

∫

where ω is twice the ZPE (in atomic units) of the transhydroxymethylene minimum on V(Qim) and a and b are the classical turning points, such that for the ground vibrational state V(a) = V(b) = ℏω/2. These turning points are indicated in the plot of V(Qim) in Figure 2. Instead of using the harmonic approximation to obtain the ZPE of trans-hydroxymethylene, it is obtained by solving the 1d Schroedinger equation with the relaxed Qim potential shown in Figure 2 using the equally spaced “discrete variable representation” (DVR) of the Cartesian kinetic energy operator due to Colbert and Miller.13 To compare with the harmonic ZPE of trans-HCOH, we define the DVR ZPE to be the energy of the first localized transHCOH DVR eigenstate relative to the trans-HCOH minimum. The difference from the harmonic ZPE is roughly 200 cm−1, which is around 16% of the DVR ZPE. Also, it is perhaps worth noting that there are 12 vibrational states below the ground state of the trans-hydroxymethylene minimum in the formaldehyde well from the present one-dimensional DVR

tp ΔQ im (i) = |q Timq i|(|Q itp(n)| − |Q itp(n = 0)|)

(2)

qTimqi

where is the projection of the imaginary-frequency normal-mode eigenvector qim onto qi, the normal mode eigenvector of the trans-hydroxymethylene minimum, and the second term is just the change in turning point of mode i upon excitation to state n. In the harmonic approximation, which we make here, for any n, turning points of mode i are given by Qtp i = ±[(2n + 1)/ωi]1/2. To justify the use of the harmonic approximation in our model, we note that first the harmonic 126

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approximation of energies of vibrational states ν5, ν4, ν3, and 2ν5 in Table 2 agree well with converged energies (Table 5) obtained using the exact Watson Hamiltonian to within 3%. Second, the leading coefficients of basis functions for vibrational states ν5, ν4, ν3, and 2ν5 are 0.988, 0.992, 0.995, and 0.958, respectively, which is an indication of very small mixing of other zero-order states in those vibrational eigenstates. Thus, the harmonic approximation used in the projection theory is well justified. Clearly, the central quantity in this equation is the projection, qTimqi. If it is zero, or nearly so, there is no predicted change in the classical turning points and thus no enhancement of the tunneling, that is, an increase in the isomerization rate. Before presenting results from this theory, we note that Qim projection theory was tested for the demanding example of malonaldehyde, where it was shown to give mode-specific tunneling splittings in good agreement with near-converged, full-dimensional quantum calculations using the same PES. The results are also in good agreement with experiment.4 In addition, it has been applied to mode-specific, double-well tunneling in porphycene using a V(Qim) path and potential based on high-level electronic energies. The results are again in good agreement with experiment.12 The theory has also been used to study mode-specific tunneling in unimolelecular dissociation of cis-HOCO to H + CO2.11 Note that projections T qim qi alone had been suggested independently and used extensively by Guo and co-workers in the context of modespecfic effects in bimolecular reactions14 and “selection rules” for tunneling splittings.15 The full overlap matrix between trans-hydroxymethylene and the saddle point normal-mode eigenvectors is given in Table 3;

Figure 4. Snapshots of the molecule on the reaction path from transHCOH to the H2CO global minimum.

below. In Table 4, we give the change in turning point for the indicated excitations of these three modes and the half-lives, Table 4. Predicted Rates of Isomerization for the Ground Vibrational State and Mode Excitationa

qTim

qT2

qT3

qT4

qT5

qT6

q1 q2 q3 q4 q5 q6

0.00 −0.83 0.30 0.47 0.04 0.02

0.97 0.00 0.00 0.00 0.00 0.00

0.00 −0.52 −0.61 −0.56 0.15 0.16

0.00 0.11 −0.72 0.66 −0.15 0.08

0.00 0.14 0.14 0.09 0.37 0.89

0.00 −0.09 0.09 −0.15 −0.90 0.36

description

ΔQtp im

t1/2(s)

ZPE ν5 2ν5 ν4 ν3

− asym δ(OH/CH) − ν(C−O) sym δ(OH/CH)

0.00 8.15 13.77 2.79 4.18

18 244 97 1 3708 1547

a

The change in the classical turning point on the Qim path, explained in the text, is also given.

Table 3. Overlap Matrix of Normal-Mode Vectors of transHydroxymethylene and the Saddle Point (TS2 in Figure 1) Separating It from the Formaldehyde Minimum trans\TS2

state

t1/2, along with calculated t1/2 for the ground vibrational state. First, note that for the ground state, the t1/2 of 18 244 s (5 h) is in good agreement with the previous calculation of roughly 2 h. The largest enhancement for a fundamental excitation is predicted to be a factor of 187 for ν5. For the overtone of that mode, t1/2 is predicted to be around 1 s, a three-order-ofmagnitude enhancement of the rate. Significant, but less dramatic decreases in the t1/2 are predicted for ν3 and ν4. Note these modes are all infrared-active. The large predicted increase in the isomerization rate for ν5, an HOC and HCO bend, is qualitatively understandable from the depiction of the molecular motion along the Qim path in Figure 4. This clearly starts out as the in-plane asymmetric bend, depicted as ν5 in Figure 3. To complete the current investigation of vibrational dynamics using the new PES, full-dimensional vibrational calculations at trans- and cis-hydroxymethylene isomers are done using the single-reference version of MULTIMODE,16

the results in the first column are the ones of prime interest. As seen, normal modes 2, 3 and 4 have substantial overlap with the imaginary frequency mode eigenvector. These modes are depicted in Figure 3 (using spectroscopic notation), and indeed, they all look somewhat similar to the isomerization reaction motion shown in Figure 4. We discuss the physical content of these modes’ motions on the isomerization rate

Figure 3. Normal-mode vectors of ν5, ν4, and ν3 of trans-HCOH. 127

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ACKNOWLEDGMENTS J.M.B. thanks the Department of Energy (DE-FG0297ER14782), and Y.W. thanks the Army Research Office (W911NF-14-1-0208) for financial support.

denoted MM-SR. The exact Watson Hamiltonian in massscaled, rectilinear normal modes of a reference configuration is solved with an n-mode representation (nMR) of potential and rotational−vibrational coupling in finite basis representation. Here, results of two MM-SR calculations done at the trans and cis reference configurations are reported in Tables 5 and 6.

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ν6 ν5 ν4 ν3 ν2 ν3 + ν4 ν1 a

Ar matrixa

He dropletb

1050.5 1182.6 1289.1 1478.2 2702.2 2774.0 3550.4

1048.5 1183.2 1297.1 1465.5 2703.3 2776.2 3500.6

1046.69

2698.18 2771.72 3541.01

Reference 1. bReference 3.

Table 6. Comparison of Fundamentals and ZPEs (cm−1) at cis-HCOH from MULTIMODE Calculations Using the Present PES and the PES of Reference 2, Denoted PES08 ZPE ν6 ν5 ν4 ν3 ν2 ν1

PES

PES08

5631.1 977.4 1206.7 1290.2 1436.1 2584.7 3380.6

5633.3 977.9 1189.3 1298.8 1442.3 2552.2 3397.5

Convergence was confirmed with respect to nMR in both potential and effective inverse moment of inertia terms in the vibrational angular momentum coupling, as well as the number of quadrature points in each dimension and the size of finite basis representation. The reported eigenvalues were obtained from diagonalizing the two Cs symmetry blocks of the Hamiltonian matrix with sizes of 20 259 and 10 327 using 4MR of the potential and 3MR of the effective moment of inertia. In each dimension of the final 6D direct-product basis, 14 basis functions on 20 Gaussian quadrature points were used. The results in Table 5 are in very good agreement with ones reported previously using MM-SR on a local PESs,1,2 and as seen, they are in good agreement with experiment.1,3 The results for cis-HCOH, shown in Table 6, agree well with previous calculations;2 no experimental results are available yet for comparison with theory. In summary, a robust prediction is made of the orders-ofmagnitude increase in the isomerization rate of trans-hydroxymethylene to formaldehdye upon excitation of the infraredactive bending normal mode, based on a new PES that describes this isomerization in full dimensionality. We hope that this prediction will stimulate new experiments to investigate this acceleration of the rate.

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REFERENCES

(1) Schreiner, P. R.; Reisenauer, H. P.; Pickard, F. C., IV; Simmonett, A. C.; Allen, W. D.; Matyus, E.; Csaszar, A. G. Capture of Hydroxymethylene and Its Fast Disappearance through Tunnelling. Nature 2008, 453, 906−909. (2) Koziol, L.; Wang, Y.; Braams, B. J.; Bowman, J. M.; Krylov, A. I. The Theoretical Prediction of Infrared Spectra of trans- and cisHydroxycarbene Calculated Using Full Dimensional Ab Initio Potential Energy and Dipole Moment Surfaces. J. Chem. Phys. 2008, 128, 204310. (3) Leavitt, C. M.; Moradi, C. P.; Stanton, J. F.; Douberly, G. E. Helium Nanodroplet Isolation and Rovibrational Spectroscopy of Hydroxymethylene. J. Chem. Phys. 2014, 140, 171102. (4) Wang, Y.; Bowman, J. M. Mode-Specific Tunneling Using the Qim Path: Theory and an Application to Full-Dimensional Malonaldehyde. J. Chem. Phys. 2013, 139, 154303. (5) Zhang, X.; Zou, S.; Harding, L. B.; Bowman, J. M. A Global Ab Initio Potential Energy Surface for Formaldehyde. J. Phys. Chem. A 2004, 108, 8980−8986. (6) Yachmenev, A.; Yurchenko, S. N.; Jensen, P.; Thiel, W. A New “Spectroscopic” Potential Energy Surface for Formaldehyde in Its Ground Electronic State. J. Chem. Phys. 2011, 134, 244307. (7) Knizia, G.; Adler, T. B.; Werner, H.-J. Simplified CCSD(T)-F12 Methods: Theory and Benchmarks. J. Chem. Phys. 2009, 130, 054104. (8) Dunning, T. H., Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007−1023. (9) Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron Affinities of the First-Row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96, 6796−6806. (10) Werner, H.-J.; Knowles, P. J.; Manby, F. R.; Schütz, M.; Celani, P.; Knizia, G.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G.; et al. Molpro, version 2010.1, A Package of ab Initio Programs; 2010. (11) Wang, X.; Bowman, J. M. Mode-Specific Tunneling in the Unimolecular Dissociation of cis-HOCO to H + CO2. J. Phys. Chem. A 2014, 118, 684−689. (12) Homayoon, Z.; Bowman, J. M.; Evangelista, F. A. Calculations of Mode-Specific Tunneling of Double-Hydrogen Transfer in Porphycene Agree with and Illuminate Experiment. J. Phys. Chem. Lett. 2014, 5, 2723−2727. (13) Colbert, D. T.; Miller, W. H. A Novel Discrete Variable Representation for Quantum Mechanical Reactive Scattering via the SMatrix Kohn Method. J. Chem. Phys. 1992, 96, 1982−1991. (14) Jiang, B.; Guo, H. Mode Specificity, Bond Selectivity, and Product Energy Disposal in the X + CH4 (X = H, F, O(3P), Cl, and OH) Reactions, Perspective from Sudden Vector Projection Model. J. Chin. Chem. Soc. 2014, 6, 841−859. (15) Siebrand, W.; Smedarchina, Z.; Fernán dez-Ramos, A. Communication: Selection Rules for Tunneling Splitting of Vibrationally Excited Levels. J. Chem. Phys. 2013, 139, 021101. (16) Bowman, J. M.; Carter, S.; Huang, X. C. MULTIMODE: A Code to Calculate Rovibrational Energies of Polyatomic Molecules. Int. Rev. Phys. Chem. 2003, 22, 533−549.

Table 5. Comparison of Fundamentals and Combination Bands (cm−1) at trans-HCOH of the Present MULTIMODE Calculation and Experimental Measurements MM-SR

Letter

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 128

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