Classical Trajectory Study of Collision Energy Transfer between Ne

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Classical Trajectory Study of Collision Energy Transfer between Ne and C2H2 on a Full Dimensional Accurate Potential Energy Surface Yang Liu, Yin Huang, Jianyi Ma, and Jun Li J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b11483 • Publication Date (Web): 23 Jan 2018 Downloaded from http://pubs.acs.org on January 24, 2018

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The Journal of Physical Chemistry

Submitted to J. Phys. Chem. A, 11/23/2017, revised, 01/13/2018

Classical Trajectory Study of Collision Energy Transfer between Ne and C2H2 on a Full Dimensional Accurate Potential Energy Surface

Yang Liu,1,# Yin Huang,1,# Jianyi Ma,2,* Jun Li1,* 1

School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China

2

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, Sichuan 610065, China

_______ #

: These authors contributed equally in this work.

*: Corresponding authors, email: [email protected](JM), and [email protected](JL).

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Abstract Collision energy transfer plays an important role in gas phase reaction kinetics and relaxation of excited molecules. However, empirical treatments are generally adopted for the collisional energy transfer in the master equation based approach. In this work, classical trajectory approach is employed to investigate the collision energy transfer dynamics in the C2H2-Ne system. The entire potential energy surface is described as the sum of the C2H2 potential, and interaction potential between C2H2 and Ne. It is highlighted that both parts of the entire potential are highly accurate. In particular, the interaction potential is fit to ~41,300 configurations determined at the level of CCSD(T)-F12a/cc-pCVTZ-F12 with the counterpoise correction. Collision energy transfer dynamics are then carried out on this benchmark potential and the widely used Lennard-Jones and Buckingham interaction potentials. Energy transfers and related probability densities at different collisional energies are reported and discussed.


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INTRODUCTION Many reactions are dependent on both temperature and pressure, for instance, unimolecular and chemically activated reactions. In those cases, both the intermolecular collisional energy transfer process and intramolecular reaction should be considered, especially when the time scales of the two processes are of the same order. Master equation approach accomplishes that purpose. The intramolecular reaction kinetics can now be routinely calculated, and well described by the statistical rate theory within the transition state framework with wellvalidated, predictive strategies.1 In contrast, highly empirical treatments are generally adopted to describe the collisional energy transfer in the master equation calculations,2-4 such as neglecting the dependence of the initial internal energy in the target molecule, and assuming the probability distribution of the averaged energy transfer to be exponential, etc. From a theoretical point of view, the collisional energy transfer dynamical outcome relies on the accuracy of the potential energy surface (PES) on which the collision takes place. Consequently, an accurate PES is highly desirable for these dynamical studies. Unfortunately, it is computationally expensive for developing a full-dimensional accurate PES for complicated polyatomic systems. Traditionally, the entire PES for collision dynamics is written as a sum of the intra-molecular PES and its inter-molecular counterpart describing the interaction between the two colliders, with the latter approximated by a sum of pairwise functions dependent on the interatomic distances.4 These intermolecular pairwise functional forms include Lennard-Jones (LJ) and its modifications,5 Buckingham,6 Varandas,7,8 and Tang-Toennies.9 However, it has been argued by many authors that these simplified inter-molecular PESs can introduce errors, in particular, for systems with significant anisotropic inter-molecular interactions.4,10,11 An alternative to obtaining PES is the “on-the-fly” direct dynamics,12 namely, the energies and



forces (namely, the minus gradient of the energy with respect to its atomic coordinates) are computed at each step along the evolution. However, this approach may be very expensive if high-level ab initio electronic calculations or long time propagation are required. For small- and moderate-sized systems, the intra-molecular PESs can now be accurately fit to ab initio data points using many methods, such as permutation invariant polynomial (PIP),13,14 neural network (NN),15 and PIP-NN methods.16-18 The intra-molecular PES might be not sensitive to the average energy transfer, but may affect rare events such as highly efficient collisions, HEC.11 The PIP-NN approach is highly accurate and computationally efficient, rigorous and general in enforcing the permutation invariance in the PES, and simple to implement. In our previous work, we extended the PIP-NN approach to fit the interaction PES, and applied it to developing the PES of Ne-C2H2.19 The intramolecular PES for C2H2 was adopted from the work by Han et al.20 Two different approaches, i.e., the PIP and the modified PIP-NN, were employed to fit the inter-molecular part based on ~42,000 points calculated using the explicitly correlated coupled cluster with singles, doubles, and perturbative triples (CCSD(T)F12a) method21,22 with the correlation-consistent triple-ζ basis set optimized for describing corevalence correlation effects with the explicitly correlated method (cc-pCVTZ-F12).23 It has been shown that the PIP-NN approach can also be employed to construct inter-molecular PESs efficiently and accurately. In this work, the basis set superposition error (BSSE) was corrected by the counterpoise correction for the interaction energies. BSSE can result in an artificially deeper interaction well and a shorter equilibrium distance when describing weakly bound complexes, like here Ne…C2H2. The correction was determined by recalculating the isolated energy for each of the


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two species in a specified basis that includes also the functions of the other monomer but neglecting its electrons and nuclei.24 The effect of BSSE can be significant. For each point, the interaction energies with BSSE was calculated at the level of CCSD(T)-F12a/cc-pCVTZ-F12. A new PES, which includes the BSSE correction, was fit with the modified PIP-NN approach, as reported in our previous work.19 Collision energy transfer dynamics were then performed using the classical trajectory method on the new PES. For comparison, the widely used LJ-form interaction potential was also used to study the collision energy transfer dynamics.

FULL-DIMENSIONAL ACCURATE PES As mentioned above, ~41300 ab initio points used in the fit were all adopted from our previous work.19 The interaction energy was determined by the difference between the entire potential and the intramolecular potential, namely, it is not computed by direct ab initio calculations.19 In this work, the interaction energies of all these points are computed at the level of CCSD(T)-F12a/cc-pCVTZ-F12 with BSSE corrected according to the Boys-Bernardi strategy.24 The program MOLPRO25 was used in all ab initio calculations. The entire PES of the Ne…C2H2 system is expressed as

VNe-C2H2 = VC2H2 +Vinter

(1)

where VC2H2 is the isolated six-dimensional PIP-NN PES for C2H2, which has been reported in ref. 20, and Vinter is the interaction PES between Ne and C2H2. As has been discussed in our earlier study,19 the entire PES can be exactly separated into two parts, VC2H2 and Vinter without approximation, as shown in Eq. (2).



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 N l  VNe-C2H2 = V  Sˆ ∏ pijij   i< j  N N   l  l = VC2H2  Sˆ ∏ pijij  + Vinter  Sˆ ∏ pijij  i < j ,∑ li 5 ≠ 0  i < j ,li 5 =0  i 

(2)

   

In Eq. (2), Sˆ , the symmetrization operator, contains all possible relevant nuclear permutation operations in the system, pij=exp(-αrij) are the Morse-like variables with α=1.0 Å-1 and rij the N(N-1)/2 (here N=5) internuclear distances. lij is the degree of pij and M =

∑

N

l the total

i < j ij

degree in each monomial. For convenience, the atoms in Ne-C2H2 are reordered as: H(1)H(2)C(3)C(4)Ne(5). Apparently, the 6D C2H2 PES ( VC2H2 ) is independent of any distances involving Ne, l15=l25=l35=l45=0, i.e., li5=0 (i=1~4). The remaining terms, those with

∑l

i i5

≠ 0,

must be considered in the 9D interaction PES (Vinter). Actually, these PIPs with the condition,

∑l

i i5

≠ 0 , corresponds to the “purified basis” denoted by Bowman and coworkers.26 The PIPs

can be not only employed as a basis to fit the ab initio points, as in the original PIP method,13,14 but also used as symmetry functions to enforce permutation symmetry in NN fitting, as in the PIP-NN method.18 Therefore, the input layer of NN was replaced by low-order PIPs under the constraint

∑l

i i5

≠ 0.

Since details of the PIP-NN fitting approach have been extensively discussed in our recent review,18 only a brief description is presented here. Explicitly, the nine-dimensional (9D) interaction PES Vinter was fit using the following functional form, represented by a feed-forward NN with two hidden layers: K  J   Vinter = b1(3) + ∑  ω1,( 3k) ⋅ f 2  bk( 2) + ∑  ωk( 2, )j ⋅  k =1  j =1  

I   f1  b(j1) + ∑ ω (j1,i) ⋅ Gi     i =1     


,

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(3) J and K are the size number of the neurons of the two hidden layers, respectively; fi is the transfer function taken as hyperbolic tangent function; b(j ) are biases of the jth neurons of the lth l

layer; ω (j ,i) are weights that connect the ith neuron of (l-1)th layer and the jth neuron of the lth l

layer. I denotes the size of the input layer permutation invariant polynomials Gi =Sˆ ∏ i < j pijl under N

the constraint

∑l

i i5

ij

≠ 0 . In this work, the total degree M=3 was used.

After several testing, the final interaction PES was selected to be of two hidden layers with 10 and 90 neurons (J=10, K=90), respectively, resulting in 1781 parameters. The final overall root mean square error (RMSE) is 1.04 meV with the maximum deviation (MAD) of 27.1 meV. The fitting errors are presented in Figure 1. It is evident the small errors are evenly distributed in the energy range of 0~6 eV (only the range of 0~1.5 eV is shown). The distributions of the unsigned fitting errors are also shown in the lower panel of Figure 1: ~24,000 points have fitting errors less than 0.2 meV, ~7,800 points within 0.2-0.4 meV, and ~3,400 points within 0.4-0.6 meV. The level of fitting accuracy is comparable to that of the PIPNN PES of isolated C2H2 (1.18 meV).20 Figure 2 shows several one-dimensional (1D) cuts for the interaction between Ne and acetylene (HCCH fixed at its equilibrium geometry) at different configurations from various levels: “PIP-NN PES with BSSE” denotes the current PES which includes the BSSE correction, “PIP-NN PES without BSSE” for the PES which does not include the BSSE correction, “LJ 12-6 PES” for the interaction PES based on tabulated LJ parameters from ref. 27, “Exp-6 PES” for the interaction based on the modified Buckingham (Exp-6) potential,4 “Ab initio with BSSE” for the results calculated at the level of CCSD(T)-F12a/cc-pCVTZ-F12 with BSSE correction. One



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can see that, the fitting PIP-NN PESs, with or without BSSE correction, are all in excellent agreement with the ab initio ones for all regions including the repulsive walls, the interaction region, and the asymptotic limit, manifesting the ultra-flexibility of NN. A closer comparison for the attractive regions is shown in the right column of the same figure. Evidently, PIP-NN PES with BSSE correction agrees quite well with the BSSE corrected ab initio ones. As discussed above, if the BSSE was not corrected, the interaction well is artificially deeper with a shorter equilibrium distance. The magnitude of BSSE is small, 20~30 cm-1, and mostly in the attractive regions. One the other hand, apparently, the LJ 12-6 PES cannot describe the shape of the repulsive wall well, as found earlier by Jasper and Miller.4 Indeed, LJ 12-6 PES seems to be too stiff in the repulsive region, which has been pointed out half a century ago.28,29 The LJ 12-6 PES does not describe the attractive region well either: the well can be overestimated or underestimated at different configurations. This can be attributed to the simple function form of the LJ 12-6 PES, namely,  σ V2,LJ ( ri ,Ne ) =4ε   ri ,Ne 

12

  σ   −     ri ,Ne 

6

   

(4)

The entire potential energy is then expressed as the sum of intramolecular potential of the target molecule C2H2, and the intermolecular potential between Ne and C2H2, i.e., Ventire = VC2 H 2 + Vinter

(5)

The latter, the intermolecular potential, is further approximated as a sum of pair-wise atom-atom interactions, Vinter ≈

∑ V (r ) 2,LJ

i ,Ne

i = C, H


(6)

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In addition, a function form based on the Buckingham potential (denoted as “Exp-6 PES” hereafter) is also employed to fit the intermolecular interaction.30 Explicitly, Vinter ≈

∑V

2,Exp-6

i = C, H

(r ) i ,Ne

V2,Exp-6 ( ri ,Ne ) =Ai ,Ne exp ( − Bi ,Ne ri ,Ne ) + Ci ,Ne ri 6,Ne

(7)

(8)

In order to fit the Exp-6 PES, a total of 107 points were sampled with Ne approaching HCCH (44 points) and H2CC (63 points) from random spatial orientations. Geometries of the HCCH or the H2CC molecule were kept at their equilibrium. For those points, the interaction energies were then determined at the level of CCSD(T)-F12a/cc-pCVTZ-F12 with BSSE correction. A nonlinear least-squares approach was then employed to determine the parameters AC,Ne, BC,Ne, CC,Ne, and AH,Ne, BH,Ne, CH,Ne, which minimize the differences between the Exp-6 PES and the ab initio data. The total fitting RMSE is 10.5 meV. The fitting RMSEs for HCCH and H2CC are 3.9

meV, and 13.3 meV, respectively. They are much larger than those for PIP-NN PES. The existing of the two isomers (HCCH and H2CC) for the target molecule makes it hard to fit with only six parameters. Therefore, these selected points are of interaction energy up to only 5 kcal/mol. Thanks to the physically meaningful parameters in the Buckingham form, the resulting Exp-6 PES is reasonably consistent with the PIP-NN PES, as shown in Figure 2, even in the high energy regions. In other words, it can be reasonably extrapolated in regions without points sampled for the Exp-6 PES and the LJ 12-6 PES, which are very simple with only several adjusting parameters. The Exp-6 PES is apparently much better than the LJ 12-6 PES. Indeed, the Exp-6 PES seems very accurate, in both low and high energy regions, as shown in Figure 2, although obvious deviations do exist. Similarly, several 1D cuts for the interaction between Ne and vinylidene (H2CC fixed at



its equilibrium geometry) at different configurations are presented in Figure 3, 4 and 5. They all show that the PIP-NN PES can reproduce the ab initio values for all regions very well, while the LJ 12-6 PES cannot describe the repulsive and attractive regions well. On the other hand, Exp-6 PES is again much better than the LJ 12-6 PES, and comparable to the PIP-NN PES. It should be borne in mind that in LJ 12-6 PES and Exp-6 PES, the intermolecular interaction energies are assumed to be only dependent on the internuclear distances between the atom Ne and one atom in C2H2. While the PIP-NN interaction PES is full-dimensional and accurate, and expected to perform well for complicated interactions, for instance, the orientation dependences between CH4 and H2, N2, CO, or H2O, which cannot be described well by the Exp-6 separable pairwise model.4 Two polar contour plots of the interaction PESs along the variables R and θ are shown in

Figure 6. R is the distance between Ne and the center of C-C bond, and θ is the angle between the direction of R and the C-C axis. The target molecules HCCH and H2CC are fixed at their respective equilibrium geometries. Note that the BSSE is considered in this work, resulting in 20~30 cm-1 shallower for the well depth than on the PES without BSSE correction.19

DYNAMICAL CALCULATIONS The energy transfer dynamics were simulated by the standard classical trajectory calculations with the PIP-NN PES with BSSE correction (denoted as PIP-NN PES hereafter) and the LJ 12-6 PES interfaced to the VENUS chemical dynamics program.12 The initial vibrational energy and rotational energy of HCCH was set to 20 kcal/mol (including the zero point energy), and 0 kcal/mol, respectively. The initial vibrational energy of each normal mode is determined randomly using the harmonic oscillator approximation at constant total vibrational energy of


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HCCH. Then the coordinates and momenta of each normal mode are chosen by the microcanonical sampling technique.31 The initial orientation between reactants were chosen randomly by means of rotation through Euler angles.32 The initial and final separations between the target molecule HCCH and the neon atom were set to 8 Å, which is sufficient long to avoid significant inter-molecular interaction. The combined fourth-order Runge-Kutta and sixth-order Adams-Moulton algorithms were used for the integration of the trajectories33 and the propagation time step was selected to be 0.1 fs. The gradients of the PES with respect to atomic coordinates were calculated numerically. Almost all trajectories conserved energy within a chosen criterion (10-4 kcal/mol). Simulations were performed at two different initial collision energies: 0.5 and 5 kcal/mol. For a collision process without complex or reaction, the maximum of the impact parameter, bmax, cannot be uniquely defined, since energy transfer approaches 0 only asymptotically. Therefore, there existed several different suggestions for the choice for bmax.34-36 In this work, the methods by Conte et al.,11 and by Lendvay and Schatz35 were adopted. In short, three criteria were monitored. 1) First, the average total energy transfer should approach 0 when b is large enough. The b-dependent average total energy transfer is defined as Eq. (4) with ∆Ei ( b ) = Ei′ ( b ) − Ei ( b )

being the total energy transfer for ith trajectory.

∆E ( b ) =

1 Nb ∑ ∆Ei ( b ) Nb i =1

(4)

2) Second, if b is large enough, the cumulative total energy transfer should become almost constant, indicating that further contributions for larger values of b can be neglected as they are smaller and smaller. The cumulative total energy transfer, is defined as the following



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integrand,35 i.e.,

∆E

( b′) =

1 π b′2

∫

b′

0

2π b E ( b ) db

(5)

The cumulative total energy transfer, Eq. (5), is a double average, and is determined as an integrand averaged limited to the bꞌ value considered. When bꞌ is set as bmax, Eq. (5) becomes flat, and is just the average total energy transfer for the specified collision energy. In this work, a different criterion was used, i.e., cumulative numerator,

∆E

b′

c

= ∫ 2π b E ( b ) db 0

(6)

This is because both the numerator, namely, Eq. (6), and the denominator, bꞌ2, will be increased along with bꞌ. However, the increase in the denominator is faster than the numerator, leading to decline of Eq. (5) as bꞌ is increased. The integration in Eq. (6) was determined numerically by an extended four-point Newton-Cotes scheme. 3) Third, if b is larger and larger, the interaction between Ne and C2H2 would be weaker and weaker. When the desired bmax is reached, the interaction becomes almost negligible and the total simulation time eventually converged to the simulation time of non-interacting particles. Therefore, the mean total simulation time, as defined in Eq. (7), is expected to drop steeply when some desired value of bmax is reached.

1 Nb T (b) = ∑ ti ( b ) Nb i =1

(7)

As shown in Figure 7 and Figure 8, the impact parameter b was scanned from 0 to 8 Å with a step size of 0.25 Å at Ec=0.5 and 5 kcal/mol, respectively. At each b, 5000 trajectories were calculated to keep good statistical accuracy. As it can be seen, (b) decreases with b increased. (b) decreases rapidly along with b for b4 Å. The mean total simulation time is oscillated at small b, and then drops steeply for b>5 Å.

The cumulative numerator is increased rapidly for b4 Å. The results on the LJ 12-6 PES are similar to those on the accurate PIP-NN PES, however, the deviations are evident. Figure 7(a) shows that averaged total energy transfer on the PIP-NN PES is remarkably larger that on the LJ 12-6 PES for small b. Their trends along with b on the two PESs are also different. is increasingly attenuated along with b for small b on the PIP-NN PES. While on the LJ 12-6 PES, is nearly constant for small b, and then

drops suddenly at about b=4 Å. At large b, on both PESs are very small, as expected. The corresponding results on the Exp-6 PES agree well with those on the PIP-NN PES. As shown in Figures 7 and 8, they are essentially same to each other, with exceptions for (b) at small b. Considering all the criteria discussed above, bmax was set large enough to include all significant energy transfer contributions. Once bmax is determined, 100 000 trajectories were carried out at Ec=0.5 and 5 kcal/mol with b of each trajectory sampled according to the formula,

b=bmax rnd , where rnd denotes random numbers distributed uniformly in the interval [0, 1]. Consequently, the probability distribution for some specified b in [0, bmax] should be

PD=

1 2 π bmax

∫

b

0

2π bdb

(8)

In this way, the averaged energy transfer can be obtained straightforward by

∆E =

1 N ∑ ∆Ei N i =1

(9)

Figures 9, 10 and 11 present histograms for the vibrational, rotational and total energy transfers, respectively, on the PIP-NN PES, Exp-6 PES, and LJ 12-6 PES. The binning was done equally for each trajectory. Note that if b is sampled uniformly, the binning should be done 13 ACS Paragon Plus Environment


assigning every energy transfer a weight equal to the impact parameter of the corresponding trajectory, as used by Conte et al.11 The binning width was set as 10 cm-1 for energy transfers at Ec=0.5 kcal/mol. For Ec=5 kcal/mol, different binning width was used: 20, 50, and 50 cm-1 were used for the vibrational, rotational, and translational energy transfer, respectively. These parameters have been tested for convergence. One can see that most collision events yield small energy transfers at both collision energies. The histograms show exponential-like decays, so events involving large energy transfer are rare. In addition, an increment in initial collision energy results in more positive energy transfer. Besides, as shown in Figures 9, 10 and 11, the results on the PIP-NN PES, LJ 12-6 PES, and Exp-6 PES are quite similar in shape, however, some differences can be clearly seen, especially in magnitude. For instance, as shown in Figure 9, at Ec=0.5 kcal/mol, the decay of the vibrational energy transfer on the PIP-NN PES is significantly faster than that on the LJ 12-6 PES. This is mainly caused by the artificially deeper well depth on the LJ 12-6 PES, as discussed above. The results on the Exp-6 PES, on the other hand, agree well with those on the PIP-NN PES, in both shape and magnitude. At Ec=5.0 kcal/mol, the difference of the results on the three PESs is smaller, but not negligible, especially for the LJ 12-6 PES. Similar trends can be found for rotational and total energy transfers, as shown in Figures 10 and 11, respectively. On the other hand, the increase in the initial translational energy, namely from Ec=0.5 to Ec= 5.0 kcal/mol, results in increase in all vibrational, rotational and total energy transfers, as expected. Note that the target molecule, acetylene, cannot experience dramatic geometric changes, like isomerization to vinylidene, under the current condition, since the combination of the initial collisional energy Ec=0.5 or 5 kcal/mol, and initial vibrational energy 20 kcal/mol, is lower than the isomerization barrier, ~47 kcal/mol.20


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A summary of relevant dynamic outcome on the three PESs at two collision energies was given in Table 1. bmax is decreased with increasing collision energy. This is expected, since at lower energies, motion can be perturbed by lower interaction energies. The various average collison energy transfer values were obtained from Eq. (5) at b=bmax. These average values, calculated from trajectory simulations, are referenced to the hard sphere collision rate (ZHS) evaluated at bmax,37 2 ZHS = π bmax

( 8π kBTbath ) πµ

(10)

where µ is the reduced mass of Ne-C2H2. The collision energies correspond to the mean translational energies related to the temperature of the bath by Ec=2kBTbath38 and can be used to obtain the average energy transfer rate,

r∆E = ZHS ∆E

(11)

This is the observable that is not dependent on the bmax value (given that it is large enough to include all sizable energy transfers) or on the collision rate. The initial rotational energy was set to zero, thus, the average rotational energy transfer can only be positive. As shown in Table 1, the increase in Ec results in larger average rotational energy transfer. The average vibrational energy transfer is negative. The average rotational and vibrational energy transfers on the PIP-NN PES and Exp-6 PES are significantly smaller (in magnitude), respectively, than those on the LJ 12-6 PES for the two collision energies. Again, the results on the Exp-6 PES are in good agreement with those on the PIP-NN PES. The average total energy transfer on all three PESs are quite similar. With the collision energy increased from 0.5 to 5 kcal/mol, the average total energy transfer is increased by about 7~8 times on all three PESs. For ZHS, its values increase with the collision energy or equivalently, the temperature of



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the bath. The average total energy transfer rates, also listed in Table 1, are affected by the collision energy significantly, on all three PESs. Energy transfer probability densities P(E, Eꞌ) were generally assumed to be independent on the initial energy, but only dependent on the energy transferred, namely, P(E, Eꞌ)=P(∆E). Here, the P(∆E) in the histograms were fitted by the following analytic bi-exponential function form,

P(∆E ) = c exp ( − ∆E a ) + d exp ( − ∆E b )

(12)

The “up” and “down” wings of the energy transfer populations were fitted separately with the central bin (i.e., the averaged energy transfer is zero) excluded. Table 2 reports the fitting parameters for the average rotational, vibrational, and total energy transfers at the two collision energies on the two PESs. Parameters for the “up” wing are denoted with prime (aꞌ, bꞌ, cꞌ, dꞌ). c and d denotes the weight parameters, and a and b are the exponential parameters which characterize (following Troe’s terminology39), the “strong” and “weak” components of the energy transfer, respectively. As shown in Table 2, “strong” exponential parameters, a, are much (typically 4-12 times) larger than the corresponding “weak” ones, b. Another interesting phenomena is the highly efficient collision (HEC), in which the energy transferred is at least 5 times the average energy down, following Clary’s criterion.40 The contributions of HEC on the PIP-NN PES are 0.12% (nHEC/ntotal) and 0.28% at Ec = 0.5 and 5 kcal/mol, respectively. The HEC thresholds are -16.4 and -21.7 cm-1 at Ec = 0.5 and 5 kcal/mol, respectively, on the PIP-NN PES. The corresponding numbers on the LJ 12-6 PES are 0.55% and 0.96%, respectively. The HEC thresholds are -52.0 and -151.8 cm-1 at Ec = 0.5 and 5 kcal/mol, respectively, on the LJ 12-6 PES. On the Exp-6 PES, HEC fractions are 0.12% at Ec=0.5 kcal/mol, and 0.40% at Ec=5.0 kcal/mol, respectively, with the thresholds being -12.6 and -21.2 16 ACS Paragon Plus Environment

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cm-1. Again, Exp-6 PES performs very well for HEC. This is clearly seen in Figure 12 that HEC is sensitive to the accuracy of the underlying PES, as HEC or collisions with large energy transfers are rare events. As shown in Figure 12, the populations of HEC at the two collisional energies are significantly affected by the underlying PES: the populations of HEC on the Exp-6 PES are in good agreement with those on the PIP-NN PES, with apparent deviations around the HEC threshold limit; while results on the LJ 12-6 PES are significantly different from those on the Exp-6 and the PIP-NN PES, due to the more stiff repulsive interaction potential of the LJ 126 PES.

CONCLUSIONS Classical trajectory approach is employed to study the energy transfer dynamics in the C2H2-Ne system based on full-dimensional accurate BSSE corrected PES. This new PES, denoted as PIP-NN PES, is fit to ~41,300 configurations computed at the level of CCSD(T)F12a/cc-pCVTZ-F12 with the BSSE correction. The fitting RMSE for the interaction potential is only 1.04 meV, due to the ultra-flexibility of the PIP-NN non-linear fitting approach. Then the energy transfer dynamics are studied on this benchmark PES, the LJ-form and Exp-6 interaction potentials. It has been found that, the LJ-form interaction potential cannot describe the attractive and the repulsive regions well, and cannot produce reliable dynamical results, especially for rare events, such as highly efficiently collisions. The Exp-6 PES works much better than the LJ 12-6 PES.

Nonetheless, it should be borne in mind that in LJ 12-6 PES and Exp-6 PES, the

intermolecular interaction energies are assumed to be only dependent on the internuclear distances between the atom Ne and one atom in C2H2. While the PIP-NN interaction PES is fulldimensional and accurate, and is free of such drawback: it is also accurate for HCCH or H2CC



different from their equilibrium. The benchmark PES guarantees the reliability of the current results and allows more systematic work. Therefore, the PIP-NN PES can be used as a benchmark when approximated interaction model was used. In the future, we plan to investigate the energy collision dynamics extensively, including effects of the initial internal energies, the colliders (He, Ar, Kr, etc.), and the isomers (HCCH vs. H2CC). In summary, we have represented collision energy transfer dynamical simulations based on an accurate full dimensional PES for the system C2H2-Ne, which are expected to stimulate relevant experimental and theoretical studies.

Acknowledgments: This work was supported by the National Natural Science Foundation (21573027) of China to JL. JM thanks the National Natural Science Foundation of China (Contract, No. 91441107) for support.


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Table 1. Collision energy, maximum impact parameter, average rotational, vibrational, and total energy transfers, the hard sphere collision rate ZHS, the averaged total energy transfer rate r∆Etot, and HEC fraction. PES PIP-NN Exp-6 LJ 12-6 a

Ec a 0.5 5.0 0.5 5.0 0.5 5.0

bmax b 4.75 3.8 5 4 5.0 4.2

c 34.86 261.51 34.79 261.48 63.69 312.97

c -4.57 -25.49 -5.41 -33.32 -33.65 -92.58

c 30.29 236.02 29.37 228.16 30.04 220.38

. kcal/mol; b. Å; c. cm-1; d. 10-10 cm3molecule-1s-1; e. cm-1cm3s-1.


ZHSd 2.57 5.20 2.85 5.77 2.85 7.30

r∆Etote 77.84 1227.31 83.71 1316.47 85.32 1608.80

%HEC 0.12 0.28 0.12 0.40 0.55 0.96


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Table 2. Fit parameters for the total, vibrational, and rotational energy transfer histograms. Parameters a, b, aꞌ and bꞌ are in cm-1.

Etot

Evib

Erot

Etot

Evib

Erot

PES PIP-NN Exp-6 LJ 12-6 PIP-NN Exp-6 LJ 12-6 PIP-NN Exp-6 LJ 12-6

a 82.4782 74.3053 57.603 22.1648 27.0624 57.1765 75.6964 71.7968 77.6379

b 6.3844 4.2466 5.9042 5.8371 5.0165 7.3830 6.5770 6.7185 6.7185

PIP-NN Exp-6 LJ 12-6 PIP-NN Exp-6 LJ 12-6 PIP-NN Exp-6 LJ 12-6

504.9806 509.5003 476.8079 91.3355 121.5096 168.3785 517.8509 526.7157 569.1357

46.4075 48.2407 46.4797 16.5665 19.7552 20.207 48.1399 51.5541 48.6875

Ec=0.5 kcal/mol c d 0.0646 0.7115 0.0743 1.3656 0.1182 0.5121 0.0635 0.7580 0.0514 0.8832 0.0554 0.4435 0.0673 0.5576 0.0740 1.0002 0.091 0.3949 Ec=5.0 kcal/mol 0.0562 0.228 0.0536 0.142 0.0611 0.142 0.0533 0.3461 0.0318 0.2358 0.0401 0.1652 0.0543 0.2161 0.0504 0.1924 0.0535 0.1442

aꞌ 12.0899 47.7618 20.9498 11.8289 15.9605 13.2469

bꞌ 2.4766 2.6556 3.4027 4.2924 4.0616 3.3118

cꞌ 0.0016 0.0001 0.0264 0.0278 0.0076 0.0664

dꞌ 1.2791 0.7395 0.2982 1.0291 0.8050 0.805

73.5091 66.5673 73.4473 42.6219 42.0427 62.8769

1.0000 14.0416 1.0000 11.7016 12.6354 12.2916

0.0254 0.0008 0.0254 0.0431 0.0416 0.0411

1.0000 0.0863 1.0000 0.3543 0.2311 0.1375


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Figure Captions Figure 1. Upper panel: fitting errors of the full-dimensional PIP-NN PES for the interaction between Ne and C2H2 as a function of the target energy (in eV, the BSSE corrected ab initio interaction). Lower panel: population of the unsigned fitting errors.

Figure 2. Comparison of several 1-dimensional cuts for the interaction energy between C2H2 (fixed at equilibrium) and Ne. The energies are in cm-1 relative to the asymptote Ne + C2H2. The corresponding x-axis is displayed by the solid green line in the atomic configurations with C2H2 fixed at its equilibrium. The left three panels are for the energy up to 10000 cm-1, while the right three panels show the details of energy less than 120 cm-1.

Figure 3. Similar to Figure 2, but for the interaction energy between H2CC (fixed at equilibrium) and Ne: (in cm-1 relative to the asymptote Ne+H2CC). The corresponding x-axis is displayed by the solid green line in the atomic configurations with H2CC fixed at its equilibrium. The left panels are for larger energy range, the right panels for the details of the low energy range.

Figure 4. Similar to Figure 2, but for the interaction energy between H2CC (fixed at equilibrium) and Ne: (in cm-1 relative to the asymptote Ne + H2CC). The corresponding x-axis is displayed by the solid green line in the atomic configurations with H2CC fixed at its equilibrium. The left panels are for larger energy range, the right panels for the details of the low energy range.

Figure 5. Similar to Figure 2, but for the interaction energy between H2CC (fixed at equilibrium) and Ne: (in cm-1 relative to the asymptote Ne + H2CC). The corresponding x-axis is displayed by the solid green line in the atomic configurations with H2CC fixed at its equilibrium. The left panels are for larger energy range, the right panels for the details of the low energy



range.

Figure 6. Polar contour plots of the interaction potential with C2H2 (upper panel) and H2CC (lower panel) fixed at their equilibrium structures. The energies are in cm-1 relative to the asymptotic Ne + C2H2 and H2CC, respectively, with an interval of 5 cm-1.

Figure 7. Average total energy transfer (a) and mean total simulation time (b) at fixed impact parameters, and cumulative numerator (c) for different impact parameters at Ec = 0.5 kcal/mol on the PIP-NN PES, LJ 12-6 PES, and Exp-6 PES, respectively.

Figure 8. Average total energy transfer (a) and mean total simulation time (b) at fixed impact parameters, and cumulative numerator (c) for different bmax at Ec = 5 kcal/mol on the PIP-NN PES, LJ 12-6 PES, and Exp-6 PES, respectively.

Figure 9. Comparison of the populations of the vibrational energy transfer on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively.

Figure 10. Comparison of the populations of the rotational energy transfer on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively.

Figure 11. Comparison of the populations of the total energy transfer on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively.

Figure 12. Comparison of the populations of HEC on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively.


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Figure 1.



Figure 2.


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Figure 3.



Figure 4.


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Figure 5.



Figure 6.


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Figure 7.



Figure 8.


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Figure 9.



Figure 10.


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Figure 11.



Figure 12.


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TOC Graphic



Figure 1. Upper panel: fitting errors of the full-dimensional PIP-NN PES for the interaction between Ne and C2H2 as a function of the target energy (in eV, the BSSE corrected ab initio interaction). Lower panel: population of the unsigned fitting errors. 201x141mm (300 x 300 DPI)

ACS Paragon Plus Environment

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Figure 2. Comparison of several 1-dimensional cuts for the interaction energy between C2H2 (fixed at equilibrium) and Ne. The energies are in cm-1 relative to the asymptote Ne + C2H2. The corresponding xaxis is displayed by the solid green line in the atomic configurations with C2H2 fixed at its equilibrium. The left three panels are for the energy up to 10000 cm-1, while the right three panels show the details of energy less than 120 cm-1. 190x132mm (300 x 300 DPI)



Figure 3. Similar to Figure 2, but for the interaction energy between H2CC (fixed at equilibrium) and Ne: (in cm-1 relative to the asymptote Ne+H2CC). The corresponding x-axis is displayed by the solid green line in the atomic configurations with H2CC fixed at its equilibrium. The left panels are for larger energy range, the right panels for the details of the low energy range. 190x132mm (300 x 300 DPI)


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Figure 4. Similar to Figure 2, but for the interaction energy between H2CC (fixed at equilibrium) and Ne: (in cm-1 relative to the asymptote Ne + H2CC). The corresponding x-axis is displayed by the solid green line in the atomic configurations with H2CC fixed at its equilibrium. The left panels are for larger energy range, the right panels for the details of the low energy range. 191x137mm (300 x 300 DPI)



Figure 5. Similar to Figure 2, but for the interaction energy between H2CC (fixed at equilibrium) and Ne: (in cm-1 relative to the asymptote Ne + H2CC). The corresponding x-axis is displayed by the solid green line in the atomic configurations with H2CC fixed at its equilibrium. The left panels are for larger energy range, the right panels for the details of the low energy range. 191x137mm (300 x 300 DPI)


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Figure 6. Polar contour plots of the interaction potential with C2H2 (upper panel) and H2CC (lower panel) fixed at their equilibrium structures. The energies are in cm-1 relative to the asymptotic Ne + C2H2 and H2CC, respectively, with an interval of 5 cm-1. 338x190mm (96 x 96 DPI)



Figure 7. Average total energy transfer (a) and mean total simulation time (b) at fixed impact parameters, and cumulative numerator (c) for different impact parameters at Ec = 0.5 kcal/mol on the PIP-NN PES, LJ 12-6 PES, and Exp-6 PES, respectively. 184x141mm (300 x 300 DPI)


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Figure 8. Average total energy transfer (a) and mean total simulation time (b) at fixed impact parameters, and cumulative numerator (c) for different bmax at Ec = 5 kcal/mol on the PIP-NN PES, LJ 12-6 PES, and Exp-6 PES, respectively. 184x138mm (300 x 300 DPI)



Figure 9. Comparison of the populations of the vibrational energy transfer on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively. 185x139mm (300 x 300 DPI)


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Figure 10. Comparison of the populations of the rotational energy transfer on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively. 184x137mm (300 x 300 DPI)



Figure 11. Comparison of the populations of the total energy transfer on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively. 186x143mm (300 x 300 DPI)


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Figure 12. Comparison of the populations of HEC on the LJ 12-6 PES, Exp-6 PES, and PIP-NN PES, respectively, at Ec = 0.5 and 5 kcal/mol, respectively. 179x128mm (300 x 300 DPI)


Classical Trajectory Study of Collision Energy Transfer between Ne

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