A new (multi-reference configuration interaction) potential energy surface for H₂CO and preliminary studies of roaming

Abstract

We report a new global potential energy surface (PES) for H₂CO, based on precise fitting of roughly 67 000 MRCI/cc-pVTZ energies. This PES describes the global minimum, the cis- and trans-HCOH isomers, and barriers relevant to isomerization, formation of the molecular (H₂+CO) and radical (H+HCO) products, and the loose so-called roaming transition-state saddle point. The key features of the PES are reviewed and compared with a previous PES, denoted by PES04, based on five local fits that are ‘stitched’ together by switching functions (Zhang et al. 2004 J. Phys. Chem. A 108, 8980–8986 (doi:10.1021/jp048339l)). Preliminary quasi-classical trajectory calculations are performed at the total energy of 36 233 cm⁻¹ (103 kcal mol⁻¹), relative to the H₂CO global minimum, using the new PES, with a particular focus on roaming dynamics. When compared with the results from PES04, the new PES findings show similar rotational distributions, somewhat more roaming and substantially higher H₂ vibrational excitation.

This article is part of the themed issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.

1. Introduction

Transformations between substances are at the heart of chemistry, and the rates of these processes are of enormous practical consequence. In unimolecular processes, the rate constant as a function of energy is usually predicted accurately by the RRKM (Rice–Ramsperger–Kassel–Marcus) expression, which involves the ratio between, in the numerator, the number of states that can lead to products at the energy of the transition-state barrier and, in the denominator, the product of Planck’s constant and the density of states of the reactant at the excitation energy. The transition-state barrier is usually defined as the highest point on the minimum-energy path between reactants and products. Reactions for which the highest point on the minimum-energy path is the energy of the products can be predicted by choosing the transition state as the point along the path where the flux is a minimum.

As successful as this approach to unimolecular rate constants has been, there are now numerous reaction systems which have been found both experimentally and theoretically to stray very far from the minimum-energy path. In some cases, these ‘roaming’ trajectories comprise a significant part of the reactive flux. For a recent review, see [1]. It is not yet clear either how best to determine the correct transition-state dividing surface for such reactions or whether existing theories will still be predictively effective.

One of the earliest systems exhibiting such roaming behaviour is formaldehyde (H₂CO) [2]. Dating from the 1980s, the photodissociation dynamics of this molecule have been intensively investigated, with a focus on the molecular products, H₂+CO, and the internal state distributions [3–5] that result from unimolecular dissociation from the ground electronic state, S₀. The energy of the transition-state saddle point is roughly 85 kcal mol⁻¹ above the global minimum. This value was determined with ever-increasing accuracy over the years from direct single-point electronic structure calculations [6–8]. This prototypical tight transition state arises from an avoided crossing between the ground and first excited singlet states.

Over the years, the unimolecular dissociation of H₂CO has become the focus of many experimental and theoretical studies, and served as a textbook example of RRKM theory. Indeed, the barrier height was accurately determined from experiment [4], based on RRKM calculations, and this value was used to assess the accuracy of electronic structure calculations of the barrier. The now-widely-used tunnelling correction to RRKM theory, due to Miller, was developed specifically for H₂CO dissociation [9].

Classical dynamics calculations of the unimolecular dynamics were performed over the course of 20 years using semi-global potential energy surfaces (PESs) for H₂CO that included the reaction channel H₂+CO [10–13]. In addition, several classical ‘direct dynamics’ have been reported for this reaction [14,15]. Guided by transition-state theory, these, as well as the calculations using a limited PES [12], were initiated at the conventional tight transition state. Agreement with experiment was generally good for the CO rotational distribution.

There have also been a handful of experimental studies of the radical products H+HCO of the dissociation of H₂CO at high photolysis energies [16–20]. It was noted that these doublet products can be formed from both S₀ and T₁. While the dissociation on S₀ is barrierless (in the potential), there is an exit channel barrier on T₁ [21], and the translational energy distribution of these products then is a signature of the contributions of these two electronic states [22]. Unfortunately, experiments that probe both the molecular and radical channels have not yet been done.

The first (and until now only) global PES that contains the two dissociation channels, as well as the isomers cis- and trans-HCOH, was reported in 2004 [23]. The PES was a fit to around 80 000 CCSD(T)/aug-cc-pVTZ energies for the bound region and the H₂+CO channel. For the radical dissociation to H+HCO, the electronic energy could not be characterized using the single-reference method, and roughly 53 000 MRCI/aug-cc-pVTZ energies were sampled in this region. The final global PES is a combination of five local fits, joined by switching functions. The switching functions smoothly connected the local fits; however, connection between fits introduced small ‘bumps’ in the potential. Another problem of this PES is that more than 10% of the classical trajectories failed due to the unphysical regions of the PES, which makes the surface unreliable for quantum calculations, which typically sample a much broader region of configuration space than classical trajectories at the same total energy. Nevertheless, this PES has been used in many trajectory studies of the dissociation dynamics, including those that uncovered the roaming pathway to the molecular products [2].

Given the limitations of this previous PES, denoted PES04, and the difficulty in improving it, we were motivated to develop a new global PES that includes all stationary points of the H₂CO system and dissociation channels. In order to avoid the switching between the single-reference and multi-reference calculations, we used the MRCI (multi-reference configuration interaction) method for the whole surface, except for the radical channel, H+HCO, where we do switch to a previous high-level PES for HCO [24]. In §2, we describe the details of the electronic energy calculations and fitting approach, followed by an evaluation of the properties of the new PES. Preliminary quasi-classical trajectory calculations of H₂CO dissociation to molecular products, with a focus on roaming, are performed using the new surface, and the results are provided in §3, along with a comparison of previous results using PES04. Finally, we summarize this study and comment on future work using the new PES.

2. Potential energy surface

As noted above, PES04 used local fits of single-reference (CCSD(T)) and multi-reference (MRCI) electronic structure energies, depending on the region of the potential. For example, the electronic energy calculation of the radical dissociation of H₂CO to H+HCO required the multi-reference method. It should also be noted that a loose saddle point, termed the ‘roaming transition state’, was reported in 2007 [25]. This saddle point was not incorporated in PES04 and so it does not contain this saddle point.

To avoid two sets of electronic structure calculations, we chose to use an MRCI approach with the cc-pVTZ basis for the entire surface. The active space consists of eight electrons in eight active orbitals. The effect of quadruple excitations are approximately described by the well-known Davidson correction. The MOLPRO 2010 package was used for all the ab initio calculations.

Although the MRCI method can describe both single- and multi-reference regions of the PES, there is some expected sacrifice in accuracy in the single-reference regions when compared with the CCSD(T) method. This is investigated in table 1, which shows a comparison of stationary-point energies (relative to the global minimum) using the two methods. The configurations of stationary points were optimized using the MRCI method, and the comparison is with previous CCSD(T) calculations [23,26].

Table 1.

Comparison of the stationary-point energies (upper row in cm⁻¹ and lower row in kcal mol⁻¹) using present MRCI/pVTZ and previous CCSD(T)/aVTZ [23] and CCSD(T)-F12/aVTZ [26] methods. The configurations are directly optimized from the ab initio calculation.

	min	trans	cis	H+HCO	H2+CO	TS1	TS2	TS3
MRCI	0	18 190	19 803	32 849	1208	28 657	30 124	30 168
	0	52.1	56.7	94.1	3.5	82.1	86.3	86.4
CCSD(T)a	0	17 940	19 600	33 239b	1931b	28 247	29 939	30 261
	0	51.4	56.2	95.2b	5.5b	80.9	85.8	86.7
CCSD(T)-F12c	0	18 078	19 753	—	—	28 374	30 049	30 258
	0	51.8	56.6	—	—	81.3	86.1	86.7

^a[23]. ^bOur calculation. ^c[26].

As can be seen from table 1, the MRCI barrier heights differ from the CCSD(T) ones by roughly 1% or less. The dissociation energies (D_e) of H₂CO using MRCI are underestimated compared with the CCSD(T)/aVTZ results by 723 cm⁻¹ (2.1 kcal mol⁻¹) for the H₂+CO and 390 cm⁻¹ (1.1 kcal mol⁻¹) for the H+HCO channels, respectively. For the H₂+CO molecular channel, the products have a large amount of available energy in experiments, owing to the high barrier (TS₃) between H₂CO and these products. Therefore, the 723 cm⁻¹ difference of D_e will have a nearly negligible effect on the dynamics. By contrast, the energy of radical products, H+HCO, is very high, and the available energy of these products is small at the excitation energies we are interested in. In this case, the 390 cm⁻¹ difference in D_e can significantly affect the branching to these products. In order to obtain the correct D_e of H+HCO dissociation and also to make use of a previous accurate PES for HCO [24], we use one switching function to connect to this PES and with the correct D_e. The switching between two PESs is in the range from 8.0 to 10.0 bohr of R_HH. The function switches smoothly from 1 to 0 and we use the function used in PES04 [23].

To develop the new PES, many configurations are needed in order to cover all the stationary points and reaction channels for the global PES. Most geometries were sampled efficiently by running classical trajectories starting from various minima and saddle points using the previous H₂CO surface [23]. More geometries were generated randomly by sampling near the configurations of both the stationary points and the roaming saddle point reported in the literature [6–8]. The electronic energies of the sampled points were calculated using the MRCI/pVTZ method and then used for the PES fitting. Configurations with energies up to roughly 52 000 cm⁻¹ (149 kcal mol⁻¹) above the global minimum were included in the dataset for the fit. A total of 67 193 energies were used to construct the PES that is permutationally invariant with respect to interchange of the hydrogen atoms. Details of the fitting procedure have been described in the literature [27,28]. In brief, Morse-type variables, , are used in the basis functions for PES fitting, where r_i is an internuclear distance between two nuclei and α equals 2.0 bohr. In this case, where the symmetric group is of order two, a simple monomial symmetrization representation was used, i.e.

2.1

where the indices 1,2,…,6 refer to the HH′, CO, HC, HO, H′C and H′O pairs, respectively.

We use a maximum polynomial order, N, of 8 for the fitting, which results in 1561 coefficients. The total root mean square error (RMSE) of the PES is about 256 cm⁻¹ (0.73 kcal mol⁻¹), but the error depends on the energy range of the fit, as shown in figure 1.

Figure 1.

Root mean square error (RMSE) and the number of energy points as a function of the energy range of the points with respect to the energy of the H₂CO global minimum. The number of points listed on the right-hand axis is the accumulated count for all energies below the energy range on the abscissa. (Online version in colour.)

A schematic showing the relevant stationary points and the energies from the PES and those from using the MRCI/pVTZ calculations is shown in figure 2. As seen, the PES values are within roughly 100 cm⁻¹ (roughly 0.3 kcal mol⁻¹) of the MRCI ones. There have been more recent and more accurate calculations of the TS₁ and TS₂ barriers [26,29] and the benchmark values are 28 269 and 29 281 cm⁻¹ (81.0 and 83.0 kcal mol⁻¹), respectively.

Figure 2.

Energy schematic diagram of the stationary points for the H₂CO system. The values in parentheses are the results from MRCI/pVTZ calculations; and the values without parentheses are the optimization results using the fitted potential energy surface. Note that the values given are in 1000 cm⁻¹. (Online version in colour.)

Note also the ‘roaming SP’ in figure 2. This saddle point is described in the new PES and its energy is roughly 7.8 kcal mol⁻¹ higher than that of the conventional tight SP to the molecular products, labelled TS₃. It has several low-frequency modes, i.e. 18 and 135 cm⁻¹, and a small imaginary frequency of magnitude 142 cm⁻¹. These results are in good agreement with previous calculations [25,30]. As noted above, PES04 does not contain this SP and yet the roaming pathway was discovered using this PES. Thus, the implication is that this SP is of minor importance to the dynamics of roaming in this reaction. Nevertheless, it is a step forward to have a new PES that contains this SP.

The major usage of the current PES will be in a variety of dynamics calculations, and we next demonstrate this use in one set of calculations that specifically examine the molecular products and the now well-known signatures of roaming encoded in the final state properties of CO and H₂. These are given in the next section and compared with results using PES04.

3. Trajectory calculations

We performed standard quasi-classical trajectory calculations of the dissociation dynamics of H₂CO, following the methods used in previous papers [2,31,32], which basically follow the procedures described in the Hase ‘handbook’ [33]. In brief, microcanonical sampling of initial momenta was done with coordinates corresponding to the global minimum of H₂CO. The total energy was chosen to be 36 233 cm⁻¹ relative to the bottom of the formaldehyde well, an energy that corresponds approximately to excitation of the 2¹4³ band of formaldehyde used in many experiments that report roaming. About 5000/12 800 trajectories on the old/new surface were calculated, and the CO rotational and H₂ vibrational distributions were determined using Gaussian binning for the H₂ vibrations and standard histogram binning for the CO vibrations and all rotations. Trajectories that follow the roaming path were determined as those that produced H₂ vibrational levels of v=6 or higher.

Figures 3 and 4 show some key differences between the new and old PESs for formaldehyde. Figure 3 compares the CO rotational distributions for the new (solid) and old (dashed) surfaces for both TS trajectories (blue) and roaming trajectories (red). Although the rotational distributions are similar, the branching fraction to roaming is significantly higher for the new surface.

Figure 3.

COrotational distribution for the new (solid) and old (dotted) potential energy surfaces. The blue data are for TS trajectories and the red are for roaming trajectories. (Online version in colour.)

Figure 4.

(a)Total H₂ vibrational distribution for trajectories on the new (solid) and old (dotted) formaldehyde potential energy surfaces. (b) Component distributions for TS trajectories (blue) and roaming trajectories (red) on the new (solid) and old (dashed) formaldehyde potential energy surfaces. (Online version in colour.)

Figure 4 displays H₂ vibrational distributions for the new (solid) and old (dashed) surfaces. Figure 4a presents the total distribution, from which it can be observed that the H₂ vibrational excitation on the new surface is considerably stronger than on the old one, particularly for higher vibrational levels. Figure 4b shows the breakdown of the total distribution into TS results (blue) and roaming results (red). The vibrational excitation on the new surface is particularly stronger for the roaming channel, which primarily populates more excited vibrational levels.

4. Summary and conclusion

A new global PES has been determined for H₂CO. This PES, based on a fit to approximately 67 000 MRCI/cc-pVTZ energies, describes the global minimum; the cis- and trans-HCOH isomers; and all barriers relevant to the isomerization, to the formation of the molecular and radical products, and to the ‘roaming’ saddle point. Transition states and minima for this surface are described in figure 2. The new PES is found to be an improvement on a previous version [23]. Relative to the findings on PES04, preliminary quasi-classical trajectory calculations at a total energy of 36 233 cm⁻¹ with respect to the H₂CO global minimum have found similar CO rotational distributions, somewhat more roaming and substantially more H₂ vibrational excitation.

Data accessibility

The PES is available upon request to the authors.

Authors' contributions

X.W. carried out the calculations, X.W. and P.L.H. did the analysis, J.M.B. conceived the research, and X.W., P.L.H. and J.M.B. co-wrote the paper. All authors gave final approval for publication.

Competing interests

The authors declare that there are no competing interests.

Funding

J.M.B. and X.W. thank the Army Research Office (W911NF-14-1-0208) for financial support.