LETTER pubs.acs.org/JPCL
Are Roaming and Conventional Saddle Points for H2CO and CH3CHO Dissociation to Molecular Products Isolated from Each Other? Benjamin C. Shepler,† Yongchang Han, and Joel M. Bowman* Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States
bS Supporting Information ABSTRACT: High-level ab initio calculations are performed to examine previously unexplored regions of full-dimensional potential energy surfaces that connect the conventional and recently reported “roaming” saddle points for the H2CO and CH3CHO unimolecular dissociations to form molecular products, H2 þ CO and CH4 þ CO, respectively. The aim of this investigation is to determine whether or not there are large barriers separating these saddle points and their associated intrinsic reaction pathways. The results are of fundamental significance in formulating statistical and reduced dimensionality dynamical approaches to model these reactions, including both pathways. SECTION: Dynamics, Clusters, Excited States
he photodissociation dynamics of formaldehyde118 and acetaldehyde1928 have been the subject of intense experimental and theoretical investigations for several decades. In both systems, the photodissociation dynamics are initiated by absorption of a photoexcitation from the ground electronic state S0 to the first excited singlet state S1. From there, the molecules return to the ground electronic state, nonradiatevely, either via internal conversion or via a multistep intersystem crossing involving the first triplet state, T1. The resulting energized molecules dissociate on the ground electronic state potential energy surface to form molecular and radical products. (Radical products are also formed from dissociation from T1; however, this process is not of further interest here.) It is important to note that all published theoretical dynamical studies of the dissociation to form molecular products focused on the S0, and until fairly recently, these were initiated at the well-known saddle point for these products. Below, we will refer to this as the conventional saddle point and denote it by SPC. In both molecules, two product channels have been the most widely studied. The first is the radical channel for each reaction
T
H2 CO f H þ HCO
ðR1Þ
CH3 CHO f CH3 þ HCO
ðR2Þ
radical channel. Also in both molecules, the molecular channel is nearly isoenergetic with the global minimum of the parent molecules, but there is a large barrier separating these two regions of the potential energy surfaces (PES). Another similarity between these two systems is that the barrierless radical channel threshold energy is only a few kcal/mol higher in energy than the SPc energy, approximately 7 and 3 kcal/mol for H2CO and CH3CHO, respectively. In 1993, van Zee et al.5 reported an unusual feature in the experimental CO rotational distribution following photolysis of H2CO at energies close to the opening of the radical channel R1. This feature is characterized by a shoulder at low jCO, and it is not reproduced by dynamics calculations initiated at SPC.913,15 The origin of this feature was described in 2004 by a joint experimental and theoretical study14 in which the CO and H2 product state distributions were measured/calculated simultaneously. That study reported a second pathway for the formation of molecular products that was named the “roaming” pathway and which bypassed the conventional transition state at large CH distances in the region of HHCO. In this pathway, which has a threshold roughly 100 cm1 below the energetic threshold for formation of the radicals, one hydrogen atom starts to dissociate in the direction of the radical products but does not have enough energy to completely escape. The hydrogen atom and HCO moiety counter-rotate about each other (for the case of zero total angular momentum), and eventually the distant H-atom abstracts the other H-atom to form H2 and CO. This pathway results in highly vibrationally excited H2 and rotationally cold CO. H2 þ CO formed via a pathway that passes through the
and the second is the molecular channel H2 CO f H2 þ CO
ðR3Þ
CH3 CHO f CH4 þ CO
ðR4Þ
In each reaction, the ground-state radical products correlate with both S0 and T1 states of the parent molecule, but the ground-state molecular products correlate exclusively with S0. In both molecules, there is no potential barrier to the production of the r 2011 American Chemical Society
Received: February 17, 2011 Accepted: March 16, 2011 Published: March 21, 2011 834
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conventional TS, on the other hand, produces vibrationally cold H2 and rotationally hot CO, and this is consistent with results from trajectories initiated at the conventional TS. In 2006, Houston and Kable21 presented experimental results on photodissociation of acetaldehyde that suggested that it was a second example of a roaming mechanism to form molecular products. This was subsequently confirmed by additional experiments and calculations.2327 However, unlike formaldehyde, where roaming is a minor channel relative to the conventional TS pathway, in acetaldehyde, roaming was shown to be the major pathway to the molecular products.23,26,27 Shortly after the initial calculations of roaming in acetaldehyde,23 Harding et al.24 reported new first-order saddle points using second-order multireference perturbation theory29,30 (CASPT2) on the PESs of H2CO and CH3CHO. Harding et al. termed these roaming saddle points (SPRs) because they occur in the region of incipient radical formation, that is at large H 3 3 3 HCO separation in formaldehyde and large CH3 3 3 3 HCO separation in acetaldehyde, in agreement with the results of the earlier trajectory calculations.14,23 Harding also calculated associated intrinsic reaction coordinates (IRCs), and these showed that the steepest-descent path from the SPRs did indeed connect the molecule (formaldehyde or acetaldehyde) to the molecular products (private communication). Further, as noted by Harding, the IRCs for the SPRs do bear a resemblance to roaming trajectories animated from quasiclassical trajectory calculations for both formaldehyde14 and acetaldehyde.23 More recently, roaming IRCs have been reported by Harding and Klippenstein for alkane decomposition.31 Thus, based on this interesting work, it might appear that the SPR and associated IRC present a way to model the roaming dynamics, using statistical theory or other path-oriented theory. However, one issue of concern, noted by Harding et al.,24 for H2CO is several very low frequency (harmonic) modes at the SPR. Using these in a standard statistical estimate of the flux following the roaming IRC versus the conventional one gave an unrealistically large result for the former.25 In view of this issue, a hybrid statistical/trajectory approach was subsequently proposed and applied to CH3CHO.32 In this approach it was assumed, as noted,32 that the roaming and conventional TSs are isolated from each other, presumably by substantial potential barriers. This assumption then justified a reduced dimensionality treatment of the dynamics in which essentially only the radical fragments are treated dynamically up to some dividing surface describing the roaming pathway. Here, we examine whether or not there are large barriers separating the conventional and roaming saddle points. This will be made more precise below. This is of importance not only in examining the assumption of the specific model mentioned above but in hopefully guiding a proper formulation of transition-state theory or other approximate theories to describe both conventional and roaming dynamics. In order to do this, we recalculated the SPRs using multireference configuration interaction33,34 (MRCI) calculations with the correlation-consistent aug-cc-pVTZ (aVTZ) basis set35 for formaldehyde and using CASPT2 with the aVTZ basis for acetaldehyde. All calculations in the current work were carried out with the MOLPRO suite of ab initio programs.36 In the applications here, single-reference methods, for example, CCSD(T), are suitable for calculations of properties of the SPC; however, multireference methods are needed for the SPR.
Table 1. Geometries (bohr and degrees) and Harmonic Frequencies (cm1) of the Conventional (C) and “Roaming” (R) Saddle Points for H2CO, Depicted in Figure 1a ωC
ωR
2.235
1830i
137i
2.101
786
17
869
75
SPC
SPR
r (CO)
2.216
r (CH)
2.071
θ (HCO) r (CH0 ) θ (H0 CO) φ (dihedral)
163.2 3.171 111.1 0.0
125.0 6.906 94.0 70.2
1282
1132
1837 3140
1878 2793
a
Results for SPC are from CCSD(T)/aVTZ calculations, and those for SPR are from present MRCI/aVTZ ones.
Figure 1. Depiction of the H2CO and CH3CHO structures at the conventional tight and roaming saddle points, denoted SRC’s and SPR’s, respectively.
Structures of these roaming and conventional saddle points for both molecules, depicted in Figure 1, are in good agreement with the original calculations of Harding et al.24 The Cartesian coordinates of all of these saddle points are given in the Supporting Information (SI). Table 1 gives the geometries of the two saddle points and the corresponding normal-mode harmonic frequencies for H2CO. Inspection of these geometries clearly indicates some significant differences between the two saddle points. Also, as seen, and as a noted previously by Harding et al.,24 SPR has several very low frequency modes. We calculated these IRCs using the same level of theory as that for the optimizations and show the IRC for H2CO associated with the conventional and roaming saddle points. To be clear, both IRCs connect the H2CO global minimum to the H2 þ CO products at their respective minima but via saddle points that are widely different from each other in configuration space, as can be seen in Figure 1. The IRCs are shown in Figure 2, where the flat nature of the PES along the roaming IRC is immediately clear from the shape of this IRC compared to the IRC for the SPC. (Note s = 0 is the saddle point configuration.) This figure also shows nicely the issue that we wish to address, namely, whether the two saddle points and associated IRCs are isolated from each other. By this, we mean whether there are substantial barriers between these paths and, most significantly, between the saddle points. Also, we need to be careful about how we address this question because there is a trivial path connecting the two saddle points with no potential value higher than either SPR. This is a 835
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Table 2. Harmonic Frequencies (cm1) of the Conventional (T) and Roaming (R) Saddle Point for CH3CHOa
Figure 2. Roaming and conventional intrinsic reaction coordinates for H2CO f H2 þ CO.
ωC
ωR
1715i
293i
138 278
34 48
524
81
538
85
743
125
907
571
1063
1112
1437
1415
1439 1802
1432 1857
3023
2804
3144
3125
3161
3300
3168
3346
a
Results for the former are obtained from CCSD(T)/aVTZ calculations, and CASPT2/aVTZ is used for the latter. Structures of the SPC’s and SPR’s are depicted in Figure 1. Cartesian coordinates of these saddle points are given in the SI.
Figure 3. Interpolated path connecting the SPC’s and SPR’s of formaldehyde, with structures indicated along the path.
path that starts at, say, SPR and follows its IRC to the minimum and then joins the other IRC to SPC. Instead, we wish to ask if there exist separate flux-dividing surfaces that contain these saddle points or whether it is more reasonable to consider a single dividing surface that contains both saddle points, and therefore, we focus on paths that are in some sense the continuation of one or the other of these surfaces. Without making specific assumptions about dividing surfaces, for example, the usual ones based on the harmonic analyses, we first investigate the potential along a simple “least-motion” path connecting these two SPs. This path is a simple linear interpolation connecting the SPs and is in terms of the internal coordinates of the SPs given in Table 1. The potential along this path, computed at the MRCI/aVTZ level of theory/basis, is shown in Figure 3. The zero of energy is at SPR; this choice was made because the issue we raise only becomes significant at energies above the higher-energy SP. As seen, there is a small barrier of ∼2.5 kcal/mol along this very simple path connecting roaming and conventional saddle points. (The Cartesian coordinates of the maximum on this path are given in SI.)
Figure 4. Interpolated path connecting the SPC's and SPR's of acetaldehyde, with structures indicated along the path.
To further explore the region of the PES connecting these two saddle points, we constructed a limited analytical PES that focuses solely on this region. This PES is a fit to approximately 5300 MRCI/aVTZ energies distributed at configurations between the two saddle points and without sampling the deep H2CO minimum or the local minima of the HCOH isomers. The PES is a least-squares fit to a basis of permutationally invariant polynomials of total order six in terms of Morse variables of all internuclear distances using procedures developed in our group and described in detail in a recent review by Braams and Bowman.37 A relaxed contour plot obtained from this limited PES is shown in Figure S1 of the SI. The dihedral angle describing the out-of-plane nature of the roaming H-atom is the variable on the y-axis, and the CH bond length of this roaming H-atom is the 836
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be highly interesting, and we plan to investigate this in the future. As a reminder of the radical channel, we include that pathway in the Table of Contents and Abstract graphics.) In any case, it is clear from the above results that there are at most moderate or no potential barriers separating conventional and roaming saddle points. Of course, one must ask, “small compared to what?”. We will attempt to address this next with a schematic model based on Figures 3 and 4. This model is described by representing these figures by an asymmetric double-well potential, shown in Figure 5. The minima of this potential correspond to the conventional and roaming saddle points, and the schematic potential is extended beyond these points to indicate the expected qualitative behavior of the potential away from the saddle points. As with conventional transition-state theory, this potential represents schematically the potential along a dividing surface. If the two saddle points are isolated from each other, then, locally, the dividing surface in the vicinity of each saddle point would be approximated in terms of the corresponding normal modes, and the 1d energy eigenvalues below the barrier, indicated by solid lines, would be well approximated by the usual separable harmonic oscillator model. However, if the two saddle points are not isolated or become interacting at higher energy, then the eigenvalues indicated by dashed lines cannot be represented by local oscillator modes. Of course, it should be noted that unlike a true double-well potential, each minimum indicated in Figure 5 is a first-order saddle point, and therefore, the eigenvalues and eigenfunctions would not be true bound states but quasibound states associated with the “quantized transition states”, which have been discussed extensively in the literature.38,39 These states “gate” the reactive flux through the dividing surface, and the discussion above suggests that this flux might be localized at the two saddle points or delocalized over both. Clearly, developing this idea is the subject of future research, which would also need to acknowledge the possible competition with the radical channel at energies where that channel is open. It is perhaps appropriate to conclude this Letter with a quote from the seminal paper of van Zee et al.5 They gave two possible explanations for the shoulder: “First, anharmonic motion at the transition state may lead the molecule to dissociate from configurations with smaller impact parameters and thus produce broadened rotational distributions. More likely, a second fragmentation path, related to the exit channel of the H þ HCO f H2 þ CO abstraction reaction and accessed through the radical channel, may open.” The present investigation indicates that these two proposals may in fact be closely related to each other. Clearly, further investigations are warranted.
Figure 5. Schematic of the potential along a hypothetical dividing surface containing conventional SPCs and SPRs indicated as minima.
variable on the x-axis. All other internal coordinates vary as a linear function of the roaming CH bond length as one moves from the SPR at rCH = 6.91 bohr to the SPC at rCH = 3.17 bohr. The zero of energy in the plot is the energy of the SPC. The SPR is at the bottom left-hand corner of this contour plot, and the SPC is in the upper right-hand corner. The interpolated path shown in Figure 3 is the diagonal line connecting these two corners. As is clear from the contour plot, the potential around the SPR is extremely flat. In fact, one can rotate the roaming H-atom about the HCO fragment by scanning the dihedral angle with almost no energy penalty; this is illustrated in Figures S2 and S3 (SI). Therefore, on the basis of these results, we are strongly leaning to the conclusion that, at least in configuration space, a single dividing surface containing these two SPs is more sensible than two separate, noninteracting ones. We return to this suggestion after we present similar results for CH3CHO next. Table 2 contains harmonic frequencies of the conventional and roaming saddle points for CH3CHO, (R4). As seen, there are a number of low frequencies at SPR, and the imaginary frequency is much smaller than the one at the conventional SPC. Figure 4 shows the CASPT2/aVTZ potential energy along an interpolated path between the two saddle points, analogous to the one for H2CO. As seen, the potential at SPR is slightly below the one at SPC, and the barrier on this path, roughly 6 kcal/mol, is greater than the one seen in the analogous path for H2CO. In this plot, the energies are relative to the global minimum; this is done to indicate how high in energy the SPs are and to see that the barrier is roughly 7% of the total potential energy. We have not explored more of the configuration space between the two saddle points as we did for H2CO, owing to the much greater dimensionality of the space for CH3CHO. (The Cartesian coordinates of the maximum on this path are given in SI.) To sum up thus far, we have found barriers of 2 and 6 kcal/mol along interpolated paths connecting the conventional and roaming saddle points for H2CO and CH3CHO, respectively. (We have not investigated whether these are high-order saddle points.) The potentials along these paths were obtained using MRCI/aVTZ and less-accurate CASPT2/aVTZ calculations. We have not searched thoroughly for other reasonable paths between these saddle points, and therefore, we can only conclude that the barriers on these paths are upper limits for the barriers separating these saddle points. In the SI, we do show another path that is somewhat contrived but with no barrier between the H2CO saddle points. (We also note that the SPRs discussed here are located in the region of incipient radical formation, and therefore, it is important to keep in mind the possible connectivity to that channel. Indeed, the fate of trajectories initiated at the SPR would
’ ASSOCIATED CONTENT
bS
Supporting Information. Cartesian coordinates of conventional and roaming first-order saddle points and the barrier along the paths depicted in the Letter for H2CO and CH3CHO dissociation to molecular products. Also, the relaxed contour plot of the region of the potential energy surface relevant to these two saddle points for H2CO and a depiction of a barrierless path connecting these saddle points is given. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 837
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Present Addresses †
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Georgia Gwinnett College.
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