A Better Understanding of the Unimolecular Dissociation Dynamics of

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A: Kinetics, Dynamics, Photochemistry, and Excited States

A Better Understanding of the Unimolecular Dissociation Dynamics of Weakly Bound Aromatic Compounds at High Temperature: A Study on CH-CF and Comparison with CH Dimer 6

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Himashree Mahanta, Daradi Baishya, Sk. Samir Ahamed, and Amit Kumar Paul J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b12188 • Publication Date (Web): 08 Mar 2019 Downloaded from http://pubs.acs.org on March 9, 2019

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The Journal of Physical Chemistry

A Better Understanding of the Unimolecular Dissociation Dynamics of Weakly Bound Aromatic Compounds at High Temperature: A Study on C6H6-C6F6 and Comparison with C6H6 Dimer

Himashree Mahanta, Daradi Baishya, Sk. Samir Ahamed, Amit K. Paul*

Department of Chemistry National Institute of Technology Meghalaya Shillong 793003 Meghalaya, India

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT

Chemical dynamics simulations are performed to study the unimolecular dissociation of Benzene (Bz) - Hexafluorobenzene (HFB) complex at five different temperatures ranging from 1000 K to 2000 K and the results are compared with that of Bz dimer at common simulation temperatures. Bz-HFB, in comparison with Bz dimer, possesses a much attractive intermolecular interaction, a very different equilibrium geometry, and a lower average quantum vibrational excitation energy at a given temperature. Six low frequency modes of Bz-HFB are formed by Bz + HFB association which are weakly coupled with the vibrational modes of Bz and HFB. However, this coupling is found much stronger in Bz-HFB compare to the same in Bz dimer. The simulations are done with very good potential energy parameters taken from literature. Considering canonical (TST) model, the unimolecular dissociation rate constant at each temperature is calculated and fitted to the Arrhenius equation. An activation energy of 5.0 kcal/mol and a pre-exponential factor of 2.39 × 1012 s-1 are obtained which are of expected magnitudes. The responsible vibrational mode for dissociation is identified by performing normal mode analysis. Simulation with random excitations of high frequency Bz and HFB modes and low frequency inter-Bz-HFB vibrational modes, respectively of Bz-HFB complex are also performed. The intramolecular vibrational energy redistribution (IVR) time and the unimolecular dissociation rate constants are calculated from these simulations. The latter shows good agreement with the same obtained from simulation with random excitation of all vibrational modes.


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I.

INTRODUCTION The detailed understanding on non-bonded interactions between aromatic molecules is important in determining the crystal structures,1-3 supramolecular chemistry,4-6 and combustion chemistry.7-11 In particular, the study of π-interactions between benzene (Bz) and hexafluorobenzene (HFB) has been a topic of considerable amount of research1,5,6,8-11 since 1960 when Patrick and Prosser reported12 a strong interaction between them. The melting point of Bz-HFB crystal was obtained significantly higher than the same of each of the monomers.12 Most of the studies on Bz-HFB were focused on the intermolecular interaction between these two molecules. One of the particular interests is that the interactions of the Bz-HFB complex is attractive in parallel conformation,17,18 whereas such interaction is unfavourable in Bz dimer.20 Furthermore, the binding energy is almost twice the time lower than the same for Bz dimer. In order to have a better understanding of the interactions between aromatic compounds, the study of Bz-HFB is essential. The temperature dependence on the unimolecular decomposition of this complex can provide important insights on the topic. As mentioned before, the binding of aromatic molecules is also crucial in combustion chemistry, where the nucleation, growth, and coagulation of polycyclic aromatic hydrocarbons (PAHs) are considered to be the important steps for soot formation.7 Hence, the understanding of the interactions between such molecules is essential. In spite of extensive studies on the understanding of the precursor of soot formation,7-11,21-24 the issue remains uncertain and demands further investigations. In order to have a better knowledge on the unimolecular dissociation of PAH dimer, Bz dimer was studied24 at high temperatures and discussed later in this section. With different and significant intermolecular interactions, the unimolecular dissociation dynamics of Bz-HFB, may reveal further information and a generalised dissociation mechanism for weakly bound aromatic compounds may be presumed.



In typical unimolecular dissociation reactions, Rice-Ramsperger-Kassel-Marcus (RRKM) theory,25-30 which is also a microcanonical transition state theory is used to determine the rate of unimolecular dissociation. For E = skBT, (s − 1) ≈ s, and E ≫ E0, with E0 and s are the dissociation energy and total number of vibrational modes, respectively, the microcanonical rate constant k(E) becomes identical to the canonical one k(T).31 In past, the unimolecular dissociation dynamics were studied for van der Waals molecules24,32-35 and ion-molecule27,36,37 complexes and are found different in nature with respect to the prediction of RRKM theory that assumes rapid intramolecular vibrational energy redistribution (IVR) among all the vibrational modes of unimolecular reactant.25,28,37 In case of PAH dimer, a weak coupling between the vibrational modes of PAH monomers and the vibrational modes with much lower frequency formed by the association of two PAH monomers is observed.24 Such ineffective coupling leads to a slower energy transfer from the high frequency modes of PAH monomers to the low frequency modes formed by the association of PAH + PAH resulting a long lived vibrational states with excited high frequency mode. Therefore, with inefficient IVR for PAH2, neither a canonical nor a microcanonical ensemble is maintained during the dynamical process resulting a non-exponential time dependent behavior of the dissociation and the process is highly non-RRKM. Recently, Ma, et al. have performed the unimolecular dissociation of Benzene dimer (Bz2) at high temperatures.24 In this study, they have shown a restricted IVR due to the weak coupling between 60 intra-Bz-Bz vibrational modes (30 from each monomer) and 6 inter-Bz-Bz vibrational modes (formed by the association of Bz + Bz). As a result, the dissociation was biexponential in nature and the dynamics was non-RRKM. Considering canonical (TST) rate constants for the unimolecular dissociation, the Arrhenius parameters were calculated and the values were obtained as expected for Bz2 molecule. The study was further extended to explore the IVR bottleneck, which resulted a non-RRKM nature of the dissociation. Furthermore, the


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intermolecular interactions of Bz-HFB was studied at high level ab initio methods17,18 and the binding energies of local and global minima were reported with the corresponding geometries, the lifetime of 1:1 complex of Bz-HFB complex was studied at a lower temperature range of 125 – 165 K using infrared spectroscopy in solution,15 where a considerable lifetime has been observed in this temperature range. In this article, a detailed study of the unimolecular dissociation of Bz-HFB complex is reported. The simulations are performed with good intramolecular and intermolecular potential energy parameters which represent the geometry and the binding energy of the complex very well and comparable with high level ab initio calculations.17,18 Since the molecular mass of BzHFB is larger than Bz dimer, some of the intra-Bz-HFB vibrational modes (vibrational modes of Bz and HFB monomers) of the former are of with low frequency. This may enhance the coupling between the inter-Bz-HFB (low frequency modes formed by Bz + HFB association) and intra-Bz-HFB modes and the IVR rate. Therefore, different behaviour of dissociation of Bz-HFB with respect to Bz dimer can be expected to observe. In the next Section II the details of the computational procedure are discussed, which includes potential energy functions, simulation methodology, and analysis techniques. In Section III, the results as obtained are discussed in detail in terms of dissociation rate constants, Arrhenius parameters, normal mode analysis, and non random vibrational excitation of Bz-HFB complex. The results for Bz-HFB dissociation are also compared directly with the same obtained for Bz2 dissociation in this section. Finally, the article is summarised in Section IV.

II.

COMPUTATION DETAILS A. Potential Energy The potential energy function for Bz-HFB is written analytically as



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𝑉 = 𝑉𝐵𝑧 + 𝑉𝐻𝐹𝐵 + 𝑉𝐵𝑧 ― 𝐻𝐹𝐵

(1)

where VBZ, and VHFB are the intramoleculer potential energy of benzene, and hexafluorobenzene, respectively. VBZ-HFB is the benzene-hexafluorobenzene intermolecular potential energy. The parameters for the intramolecular potential energy function are obtained from literature38 and are also tested in previous works.39,40 The equilibrium bond lengths re, bond angles θe, force constants for stretching fs, bending fθ, wagging fα, and the torsional barriers V0 for benzene and HFB are tabulated in the following:

Table 1. Intramolecular Potential Energy Parameters of Bz and HFB. Equilibrium Force Field

Condition & Force

Bz

HFB

re (Å)

1.084

1.327

fs (mdyn Å-1)

5.076

6.640

re (Å)

1.397

1.394

fs (mdyn Å-1)

6.640

5.800

θe (°)

120.0

120.0

0.521

0.800

θe (°)

120.0

120.0

fθ (mdyn Å rad-2)

1.065

1.550

CCH/CCF-wag

fα (mdyn Å rad-2)

0.297

0.297

Torsion Barrier

V0 (kcal mol-1)

24.00

24.00

Const.a CH/CF-stretch CC-stretch CCH/CCF-bend CCC-bend

fθ (mdyn Å

rad-2)

a. The values are obtained from ref. 38.

(

) 2

A Morse potential energy of the form 𝐷𝑒(1 ― 𝑒 ―𝛽 𝑟 ― 𝑟𝑒 ) is considered for CC (HFB), CC (Bz), CH, and CF stretches. β is obtained from

𝑓𝑠 2𝐷 , with fs given in Table 1 and the 𝑒

dissociation energy De values of 124.0, 125.8, and 116.0 kcal/mol for C-C (HFB), C-C (Bz), C-F/C-H, respectively. 6 ACS Paragon Plus Environment

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The Bz-HFB intermolecular potential is represented by OPLS-AA model41 for which the potential energy is written as a sum of two body terms, i.e.,

𝑉𝑖𝑗 =

∑∑ 𝑖

𝑗

𝑞𝑖𝑞𝑗𝑒2 𝑟𝑖𝑗

+

𝐴𝑖𝑗 𝑟𝑖𝑗

― 12

𝐶𝑖𝑗 𝑟𝑖𝑗6

(2)

Where, Aii = 4εiσi12, Cii = 4εiσi6, ε and σ are the Lennard-Jones parameters. The combination rules for the inter-particle parameters are used as Aij = (AiiAjj)1/2 and Cij =(CiiCjj)1/2. In case of HFB the values for εc = 0.07 kcal/mol, σc = 3.55 Å, qc = +0.1300, εF = 0.061 kcal/mol, σF = 2.85 Å, qF = -0.1300 and for Bz the values are εc = 0.07 kcal/mol, σc = 3.55 Å, qc = -0.115 , εH = 0.03 kcal/mol, σH = 2.42 Å, qH = +0.115. The equilibrium geometry of Bz-HFB is parallel stacked and is shown in Figure 1. The center-of-mass distance between Bz and HFB and the binding energy of the complex are obtained as 3.54 Å, and -5.93 kcal/mol, respectively for the potential energy parameters considered here. In comparison, the lowest energy geometry that was obtained from CCSD(T) and with complete basis set limit is the same parallel geometry with a center of mass distance of 3.3 Å and the binding energy of -6.2 kcal/mol.18 In some of the calculations, a slipped parallel geometry with similar centre-of-mass distance and binding energy is also observed.17,19 Therefore, it is evident from the observations that the OPLS-AA intermolecular potential energy parameters represent the Bz-HFB system very well and are considered for the simulations.



Figure 1. The equilibrium geometry of Bz-HFB complex

The average intermolecular potential energy versus center-of-mass distance of Bz and HFB is shown in Figure 2. This potential energy is calculated by averaging over random orientations of Bz and HFB keeping center-of-mass separation fixed. The calculation is carried out from a center-of-mass separation of 6 Å to 16 Å. The curve shows that the potential energy is lowest with a value of -0.5 kcal/mol at a center-of-mass separation of ~7.1 Å. This is to note here that the same average potential energy has a minimum value of -0.54 kcal/mol for Bz224 in spite of a much higher global minima value of -2.32 kcal/mol compare to -5.93 kcal/mol for Bz-HFB complex. The reason may be due to the higher energy differences among different relative orientations of Bz and HFB, which is also evident from the ab initio calculations, where the T-shaped complexes are having energies between -0.88 - -1.74 kcal/mol.17 Such energy difference may not be that prominent in case of Bz dimer. For the center-of-mass distance of 7.1 Å, among many random orientations, three orientations are also shown in Figure 2, where the calculated potential energies values are (a) highest, (b) lowest, and (c) intermediate. The 8 ACS Paragon Plus Environment

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corresponding values are 4.15, -1.10, and -0.27 kcal/mol, respectively. In (a) one of the hydrogen of Bz and fluorine in HFB come very close (1.95 Å) and hence, the potential energy is repulsive. Such repulsive interaction is absent in (b) and an electrostatic interaction between closer H and F plays the role to result an attractive potential energy. A slipped parallel geometry is shown in (c). This is a favorable interaction for Bz-HFB at smaller center-of-mass distance (~3.5 Å), but due to large center-of-mass distance (7.1 Å), the potential energy is not such attractive. The dissociation threshold for the simulation is considered when the center-of-mass distance between Bz and HFB achieves a value of 14 Å.

Figure 2. Intermolecular potential energy versus center-of-mass distance between Bz and HFB. Panel (a), (b), and (c) are three orientations at the center-of-mass distance (RCOM) of 7.1 Å for which the potential energy values are 4.15, -1.10, and 0.27 kcal/mol, respectively.

The frequencies for all the Bz-HFB vibrational modes are listed in Table 2. The frequencies of the six inter-Bz-HFB vibrational modes are very low, e.g., 1.7, 18.8, 18.8, 56.7, 56.7, and 67.7 cm-1, respectively and the corresponding potential energy curves must be very flat and highly anharmonic. It can be seen from the same table that there are some of the intraBz-HFB vibrational modes which are of very low frequency, namely, υ5 – υ7 compare to Bz2. 9 ACS Paragon Plus Environment


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These low frequency intra-Bz-HFB modes can be expected to be coupled more strongly with the inter-Bz-HFB modes and may play vital role in the IVR process, which considers energy redistribution among all vibrational modes including inter-Bz-HFB and intra-Bz-HFB modes during dissociation dynamics. Table 2: The Frequencies of the Vibrational Modes of Bz-HFB Complex. Modea

Frequency Mode Frequency Mode Frequency (cm-1) (cm-1) (cm-1) υ1 1.7 υ12 (2) 403.3 υ23 926.5 υ2 (2) 18.8 υ13 (2) 444.2 υ24 (2) 929.0 υ3 (2) 56.7 υ14 478.0 υ25 (2) 991.0 υ4 67.7 υ15 591.7 υ26 (2) 1015.0 υ5 (2) 126.4 υ16 (2) 611.0 υ27 1025.3 υ6 171.8 υ17 (2) 616.6 υ28 1070.8 υ7 204.3 υ18 658.7 υ29 (2) 1132.8 υ8 (2) 270.7 υ19 681.1 υ30 1172.2 υ9 277.5 υ20 755.4 υ31 (2) 1181.7 υ10 (2) 285.1 υ21 773.8 υ32 1374.5 υ11 (2) 383.4 υ22 (2) 833.8 υ33 1455.4 a.Degeneracy of the modes is given in the parenthesis.

Mode υ34 (2) υ35 (2) υ36 υ37 υ38 (2) υ39 υ40 (2) υ41 (2) υ42 υ43 (2) υ44

Frequency (cm-1) 1529.9 1596.0 1606.7 1697.7 1738.9 1749.5 1819.7 3054.5 3054.6 3056.9 3058.0

B. Simulation Methodology Chemical dynamics simulations are performed for unimolecular dissociation of Bz-HFB complex using VENUS, a general chemical dynamics code.42,43 The dissociation reactions are studied for a set of temperatures of 1000, 1200, 1500, 1800, and 2000 K. The initial conditions for the simulations are chosen from quasiclassical microcanonical (QM) normal mode sampling,44 The average quantum vibrational energies which explicitly consider zero point energy (ZPE) are determined from the corresponding simulation temperatures by considering the harmonic oscillator model. These energies, corresponding to the temperatures of 1000 , 1200, 1500, 1800, and 2000 K, are obtained from the following equation,45

𝑠

𝐸𝑣𝑖𝑏 = 𝑁

∑

(

𝑖=1

ℎ𝜈𝑖 2

+

ℎ𝜈𝑖 ℎ𝜈𝑖

)

𝑒𝑘𝐵𝑇 ― 1 10

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(3)

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where s is the vibrational degree of freedom, νi is the frequency of ith vibrational mode, kB is the Boltzmann constant, T is the temperature, and N is the Avogadro number. The corresponding energies are 164.6, 186.4, 220.8, 256.5, and 280.9 kcal/mol, respectively, including the zero point energy (ZPE) of 95.8 kcal/mol for Bz-HFB complex. This is to note here that considering classical average vibrational energy (sRT) expression, the above energies represent higher temperatures than what are considered here. In this sampling method, the normal mode vibrational energy levels are sampled randomly. The normal mode coordinates, Q’s and momenta, P’s are obtained from the normal mode energies using standard algorithm.46 Using the normal mode eigenvectors, these P’s and Q’s are then converted to Cartesian p’s and q’s which are then used in integration of the Hamilton’s equation of motion. Since the simulations are done at very high energies and, as shown in Table 2, the inter-Bz-HFB vibrational frequencies are very low for Bz-HFB complex, high potential energies cannot be sampled correctly for these low frequency modes because the converted Cartesian displacements become substantially large, inaccurate, and problematic for numerical calculations.47 To add high energy to those modes, the energies are added in the kinetic energy part to maintain a correct energy distribution for the microcanonical ensemble. The angular momentum and the rotational energies are also sampled from angular momentum distribution at simulation temperatures. The vibrational and rotational energies are then transform to momentum (p’s) and positions (q’s) using standard algorithms. The time step size of 1.0 fs is used for the simulations and the trajectories are integrated up to 30 ps using 6th order symplectic integrator.48 Two simulations are also performed for T = 1500 K, in which, the energies are not randomly sampled to the all vibrational modes of Bz-HFB complex but to the inter-Bz-HFB and intra-Bz-HFB modes, respectively. For the former simulation, an amount of 113.42 kcal/mol energy is added only to the inter-Bz-HFB modes, whereas, 203.19 kcal/mol is added



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to the intra-Bz-HFB modes in the latter simulation. These energies are obtained from same eq 3 by using the inter-Bz-HFB, and intra-Bz-HFB vibrational frequencies, respectively. For all the simulations reported here, a total of 1000 trajectories are integrated at each temperature. C. Analysis of the Results Since the sampling technique used here does not sample the Bz-HFB complex at larger distance, there is a time-lag (t0) since when the complex of the first trajectory is dissociated. The relative numbers of dissociated trajectories, i.e., [N(t0) – N(t-t0)]/N(t0), where N(t-t0) is the number of trajectories remain undissociated at time t and N(t0) is one less than the total number of trajectories, are calculated versus simulation time for all temperatures and fitted with biexponential function of the form

𝑁(𝑡0) ― 𝑁(𝑡 ― 𝑡0) 𝑁(𝑡0)

= 1 ― 𝑓1exp ( ― 𝑘1(𝑡 ― 𝑡0)) + 𝑓2exp ( ― 𝑘2(𝑡 ― 𝑡0))

(4)

with f1 + f2 = 1. In the previous Bz dimer dissociation this bi-exponential nature of relative dissociation versus time was also observed.24 The non-exponential behaviour of Bz-HFB dissociation implies that a microcanonical ensemble is not maintained throughout the dynamics. However, it can be assumed that the initial microcanonical ensemble is retained at short time t = t0 and the dynamics is RRKM in nature. At high Bz-HFB energies, used in the simulations, and large number of vibrational modes (s) of the Bz-HFB complex, the classical RRKM (microcanonical) rate constants at t = t0, i.e., k(t0,E) become similar to the transition state theory (TST) (canonical) rate constants k(t0,T).31 The vibrational energies considered in the microcanonical simulations are calculated from eq 3 for the respective simulation temperature. The k(t0,T) is obtained by taking the derivative of eq 4 with respect to t and then setting the limit t→t0 considering a microcanonical ensemble is maintained at time t0. The resulted k(t0,T) is obtained as f1k1 + f2k2. ln k(t0) is then fit to the Arrhenius equation, ln k(t0) = 12 ACS Paragon Plus Environment

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ln υ – Ea/kBT (see eq 8 in Section III) to obtain the Arrhenius parameters, namely, the activation energy Ea and the frequency factor υ.

III.

RESULTS AND DISCUSSION A. Relative Dissociated Trajectories Versus Time The analysis of individual trajectory shows that there are few trajectories where the complex has a significant lifetime before those trajectories report the dissociation. As a result, the exponential function could not fit the [N(t0) – N(t-t0)]/N(t0) exactly. However, low frequency intra-Bz-HFB modes of the complex may enhance the dissociation rate due to a relatively faster IVR compare to Bz dimer. This can be observed by comparing unimolecular dissociation rate constants between Bz-HFB and Bz2 which is explored in Section III.C. Due to inefficient coupling between intra-Bz-HFB and inter-Bz-HFB vibrational modes, the dynamics is intrinsically non-RRKM. As obtained in Bz dimer dissociation, for Bz-HFB complex also, most of the trajectories are dissociated directly. At 1500 K temperature, 88% of the trajectories dissociate within 2 ps, and 95% of the trajectories report dissociation at 4 ps. The probability of s vibrational modes initially obtain energy excess of a threshold energy (E0) is given by:

𝑃(𝐸) =

∫

∞

𝐸𝑠 ― 1𝑒

𝐸0(𝑠

―

𝐸 𝑘𝐵𝑇

𝑑𝐸 ― 1)!(𝑘𝐵𝑇)𝑠

(5)

where kB is the Boltzmann constant and T is the temperature. Using eq 5, and considering the threshold energy as the dissociation energy, 5.93 kcal/mol minus ZPE of six inter-Bz-HFB modes, the probability is obtained as 0.98 at 1500 K. In other words, the probability that these six modes gets an energy in excess of the dissociation energy is 0.98, which justifies the direct 13 ACS Paragon Plus Environment


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dissociation trajectories observed in the dynamics. The N(t-t0)/N(t0) for each temperature are calculated and good fits are obtained with bi-exponential expression given in eq 4. In Figure 3, [N(t0) - N(t-t0)]/N(t0) is plotted for all five temperatures along with the fit to eq 4.

Figure 3. [N(t0)-N(t-t0)]/N(t0) versus simulation time for temperature 1000, 1200, 1500, 1800, and 2000 K and the fit to eq 4

The fit parameters are given in Table 3. The larger one (k1) between the two rate constants varies from 1.5 ps-1 to 3.1 ps-1 for the temperature range of 1000 - 2000 K, respectively and the probability (f1) of k1 is varied from 0.64 at 1000 K to 0.94 at 2000 K, respectively. The ratio of k1 and k2 for all temperatures is almost similar and varies between 7 and 8.

Table 3. Fit Parameters for Simulation [N(t0) – N(t-t0)]/N(t0) to eq 4 T (K)

f1

f2

k1 (ps-1)

k2 (ps-1)


k(t0) (ps-1)

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1000

0.64

0.36

1.5

0.18

1.02

1200

0.74

0.26

1.8

0.24

1.43

1500

0.85

0.15

2.4

0.35

2.09

1800

0.89

0.11

2.7

0.38

2.44

2000

0.94

0.06

3.1

0.46

2.94

t0 values are the time when the first trajectory is dissociated at each temperature. The t0 value is 0.87 at 1000 K and decreases consistently up to 0.60 at 2000 K. As expected, at higher temperature, the t0 is lower as it takes lesser time to reach the dissociating distance, i.e., 14 Å between the centre-of-mass of Bz and HFB. B. Arrhenius Parameters The microcanonical rate constants, k(t0,E) for the unimolecular dissociation of Bz-HFB are obtained from the bi-exponential expression given in eq 4. As discussed earlier in Section II.C. that at high energy for the Bz-HFB complex and large number of vibrational degrees of freedom (s), i.e., s - 1  s, the harmonic microcanonical RRKM rate constant

𝐸 ― 𝐸0

(

𝑘(𝐸) = 𝜈

𝐸

𝑠―1

)

(6)

becomes identical with canonical harmonic TST rate constants31 as per Arrhenius equation

𝑘(𝑇) = 𝜈𝑒

―

𝐸𝑎 𝑘𝐵𝑇

(7)

or ln 𝑘(𝑇) = ln 𝜈 ―

𝐸𝑎 𝑘𝐵𝑇


(8)


with = skBT, where kB is the Boltzmann constant. With s = 66 for Bz-HFB complex and high vibrational excitation energy, the microcanonical unimolecular rate constants k(t0,E) are considered to be identical with the canonical ones k(t0,T) at t = t0. In Figure 4, ln k(t0,T) is plotted with 1/T and fit with eq 8 to obtained the Arrhenius parameters, ν and Ea. The fit parameter ν and Ea are obtained as 2.39 × 10-12 s-1 and 5.0 kcal/mol, respectively. The activation energy is close to the dissociation energy of 5.9 kcal/mol of the Bz-HFB complex. A temperature dependent anharmonicity which was important for sodium ion - Bz complexs,49 is not significant in Bz-HFB dissociation. The frequency factor (ν) is three - four order of magnitude lower than C-C bond dissociation of alkanes and is as expected for the dissociation of aromatic dimers or complexes.24 The calculated Arrhenius parameters justify the correctness of the simulations done in this work.

Figure 4. ln k(t0,T) versus 1/T for simulation and fit to eq 8 to obtain Arrhenius parameters (see text)

C. Comparison with Benzene Dimer Dissociation 16 ACS Paragon Plus Environment

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In an intrinsic RRKM dynamics, efficient IVR is mandatory so that the microcanonical ensemble is maintained throughout the dynamics and the exponential dissociation rate is obtained. With the bi-exponential dissociation rate the Bz-HFB dissociation dynamics is nonRRKM due to restricted IVR caused by inefficient coupling between intra-Bz-HFB and interBz-HFB vibrational modes. However, this coupling will be shown to be stronger in Bz-HFB in comparison with that of Bz dimer, which is identified from a direct comparison between the dissociation rate constants of Bz2 and Bz-HFB complexes as given below. In Table 4, the microcanonical/canonical unimolecular rate constants for Bz dimer24 and Bz-HFB are compared for different simulation temperatures. Due to lower intra-Bz-HFB frequency values of Bz-HFB complex compared to Bz dimer, the initial vibrational energies which are the average quantum energy at a given temperature calculated from eq 3 are smaller. For example, at 1000, 1200 and 1500 K temperatures, the average vibrational energies of Bz2 and Bz-HFB are 182.4, and 164.6, 202.2, and 186.4, and 234.3 and 220.8, respectively. On the contrary, the unimolecular dissociation rate constants for Bz-HFB are significantly large compare to the same of Bz dimer. At 1000, 1200, and 1500 K, the rate constants for Bz-HFB dissociation are increased by 0.14, 0.41, and 0.85 ps-1, respectively. At higher temperature the difference is quite prominent. It can also be noted here that the absolute value of the binding energy of Bz-HFB is almost two times higher than that of Bz dimer. The binding energy of the former is 5.93 kcal/mol, whereas the same for the latter is 2.32 kcal/mol.24 Despite of the facts that the binding energy is much higher and the vibrational excitation energies are lower in BzHFB complex with respect to Bz dimer, the unimolecular rate constants are increased significantly. This is due to the development of a stronger coupling between the intra-Bz-HFB and inter-Bz-HFB modes in this complex with respect to Bz dimer.

However, the bi-

exponential nature of the relative dissociation trajectories versus time and detailed analysis of



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the trajectories suggest that there is still an IVR bottleneck which could not be overcome completely and discussed in detail in Section III.E. for Bz-HFB complex. Table 4. Comparison of Initial Vibrational Energy and Unimolecular Rate Constants between Bz224 and Bz-HFB Simulations at Different Temperatures. T (K) 700 800 900 1000 1200 1500 1800 2000

Initial Energya Bz2 Bz-HFB 156.1 164.3 173.1 182.4 202.2 234.3 ---

---164.6 186.4 220.8 256.5 280.9

k(t0,E) (ps-1) Bz2 Bz-HFB 0.56 0.69 0.80 0.88 1.02 1.24 ---

---1.02 1.43 2.09 2.44 2.94

a The

energies are in kcal/mol. Bz2 dissociation was not performed at 1800, and 2000 K, and Bz-HFB at 700, 800, and 900 K.

D. Normal Mode Analysis To identify the responsible inter-Bz-HFB vibrational mode for the dissociation, the energy of each inter-Bz-HFB mode is monitored during the dynamics. The normal mode Hamiltonian of the form 𝐻=

1

∑2(𝑃

2 𝑘

+ 𝜆𝑘𝑄𝑘2)

(9)

𝑘

is considered for this analysis where Pk, and Qk are the momentum, and coordinate of kth normal mode. λk = 4π2υk2, with υk is the frequency of kth normal mode. The normalized eigenvectors of the mass weighted force constant matrix is used to transform Cartesian qi’s and pi’s to normal mode Pk’s and Qk’s.50 In Figure 5, the energies of six inter-Bz-HFB normal modes with frequencies, 1.7 (υ1), 18.8 (υ2), 18.8 (υ2), 56.7 (υ3), 56.7 (υ3), and 67.7 (υ4) cm-1 are shown for eight trajectories from the simulation at 1500 K. Four left hand side trajectories, i.e., (a), (c), (e), and (g) are directly dissociating, whereas, right-hand side panels (b), (d), (f), and (h) display the trajectories where there are some Bz-HFB vibrations before the dissociation occurs. Similar figures for the simulations at 1000K (Figure S1) and 2000K (Figure S2) are presented in the 18 ACS Paragon Plus Environment

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supporting information. The dissociation times for all these trajectories at 1500 K as shown in Figure 5 are listed in Table 5. The highest time of the presented directly dissociation trajectories is ~2 ps which is trajectory (e). Other three trajectories of this category dissociate below 1 ps. For the late dissociating trajectories, the dissociation time ranges from ~5 ps to ~15 ps. Also shown in the same table are the total energies of the inter-Bz-HFB modes and intra-Bz-HFB modes of the complex obtained from random initial sampling for all those eight trajectories. It is observed and given in Table 5 that when the trajectories are dissociated directly the initial total energies in the inter-Bz-HFB modes (Einter) are quite large. For the four such trajectories presented here, this energy ranges from ~8 kcal/mol to ~22 kcal/mol. But the same Einter for the late dissociating trajectories is ~6 kcal/mol or less. The ZPE of six inter-Bz-HFB modes plus the dissociation energy of 5.93 kcal/mol is 6.24 kcal/mol. Trajectories, which are not dissociated directly, are obtained less than this energy in their inter-Bz-HFB modes from initial sampling and are waiting for the energy to flow from the intra-Bz-HFB modes for dissociation.

Table 5. Dissociation Time, Total Energy of Inter-Bz-HFB Modes, Einter, Total Energy of IntraBz-HFB Modes, Eintra, Energy of ν4 Mode, E(ν4), and the Highest Energy Inter-Bz-HFB Mode for Each of the Trajectories Shown in Figure 5. Dissociation Einter Eintra E (ν4) Highest Einter Trajectory time (ps)a (kcal/mol)b (kcal/mol) (kcal/mol) (kcal/mol) (mode) a 0.959 18.31 202.6 0.291 13.29 (ν2)

a.

b

9.181

6.026

214.6

0.678

3.407 (ν1)

c

0.609

22.29

198.7

6.678

6.678 (ν4)

d

14.90

6.051

214.7

1.258

1.880 (ν1)

e

1.938

8.136

213.0

0.290

3.644 (ν3)

f

5.079

4.532

216.4

0.871

1.332 (ν1)

g

0.631

15.92

204.9

1.065

6.397 (ν3)

h

8.498

6.017

214.6

0.678

1.645 (ν3)

Observed dissociation times for the trajectories presented

b. All energy values, namely, Einter, Eintra, E(ν4), Highest Einter are the randomly selected initial values for the trajectories presented



It is clear from the trajectories presented in Figure 5 that whether or not the trajectories are dissociating directly, it is the υ4 inter-Bz-HFB mode with frequency 67.7 cm-1 which gains the highest energy among all six inter-Bz-HFB modes. This is to note here that the model used in eq 9 is harmonic whereas, the inter-Bz-HFB modes are highly anharmonic in nature. Hence, the energy calculated from eq 9 is arbitrary. However, since the energies of all the inter-BzHFB modes are calculated using the same Hamiltonian, a relative comparison of their energies is done here. It can also be evident from the trajectories shown in Figure 5 and the time of dissociation of those trajectories listed in Table 5 that the energy, although arbitrary, of υ4 interBz-HFB mode is quite similar at the point of dissociation. Therefore, the normal mode υ4 is recognized as the most responsible mode for Bz-HFB dissociation. A detailed analysis of the trajectories calculated at 1500 K shows that approximately, more than 95% of the trajectories are dissociated via υ4 inter-Bz-HFB mode. Out of the total energy in the inter-Bz-HFB modes (Einter) as listed in Table 5 for eight sample trajectories, the energy in υ4 inter-Bz-HFB mode is obtained and also listed in the same table and given as E(υ4). Except one (trajectory c) of the direct dissociating trajectories, the energies in υ4 mode are much smaller than the required dissociation energy. For example, in trajectory (a), 13.3 kcal/mol is added to one of the υ2 modes and only 0.3 kcal/mol in the υ4 mode. The direct dissociation of this trajectory at 0.96 ps suggests rapid vibrational energy redistribution among the inter-Bz-HFB modes. Animation of υ4 vibrational mode shows that this is a sliding vibrational mode and it can be concluded that the mechanism of most of the dissociations is slipping of one molecule over another in Bz-HFB complex. In Bz-HFB complex, a slipped parallel geometry is almost equally favourable as evident from ab-initio calculation.17 Therefore, the low energy pathway for Bz-HFB dissociation may be the sliding motion as observed. Such dissociation pathway is also common in inorganic complexes,51 where a slipped geometry of the dimer complex has favourable interactions. In comparison,


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the Bz dimer has the stretching mode with almost similar frequency value and was assumed as the responsible mode for the dissociation.24 The equilibrium geometry of Bz2 is a tilted Tshaped, whereas Bz-HFB is parallel stacked (Figure 1).

Figure 5. The normal mode energies of six inter-Bz-HFB modes with frequencies 1.7, 18.8, 56.7, and 67.7 cm-1. (a), (c), (e), and (g) are directly dissociating trajectories, whereas (b), (d), (f) and (h) are late dissociating trajectories. The simulation is done at 1500 K.

E. Non Random Excitation of Bz-HFB Complex To check the IVR bottleneck and explore the bi-expinential behaviour of Bz-HFB dissociation, two additional simulations are also performed. In one, an amount of 113.4 kcal/mol energy is randomly sampled to the inter-Bz-HFB vibrational modes, i.e., six low 21 ACS Paragon Plus Environment


frequency modes formed by Bz + HFB association, and in another, an amount of 203.2 kcal/mol energy is used for the sixty intra-Bz-HFB vibrational modes of Bz and HFB. In both the simulations, ZPE is added to all the modes. The results of these two additional simulations are compared with the same obtained when all sixty six vibrational modes of Bz-HFB are randomly excited and described in Section III.A. The above mentioned energies, i.e., 113.4 and 203.2 kcal/mol are the average energy at T = 1500 K and obtained from eq 3 plus the ZPE of the other modes of the complex for inter-Bz-HFB and inta-Bz-HFB vibrational modes, respectively. When only inter-Bz-HFB modes are randomly excited, all trajectories are dissociated directly within 3.7 ps. In comparison, the simulation where all Bz-HFB vibrational modes are randomly excited and discussed in Section III.A, 95% of the trajectories are dissociated directly within 4 ps at a temperature of 1500 K. When only inter-Bz-HFB vibrational modes are excited, there is no requirement of IVR for the dissociation of Bz-HFB complex and all the trajectories are dissociated directly. As a result, the N(t-t0)/N(t0) is exponential with rate constant kinter = 5.9 ps-1 and t0 = 0.3 ps. On the other hand, when the intraBz-HFB vibrational modes of Bz-HFB complex are sampled, the rate of dissociation is much smaller and a much longer time lag is obtained. After shifting the initial time to 6 ps (tʹ), the N(t-tʹ)/N(tʹ) is exponential and the rate constant (kintra) is obtained as 0.15 ps-1. As there is no initial energy except the ZPE in six inter-Bz-HFB modes in the latter simulation, the dissociation occurs only when the energy required for dissociation flows from the intra-BzHFB to the inter-Bz-HFB vibrational modes. The quantity 1/ kintra which is equal to 6.67 ps can, therefore, be considered as the IVR time for the Bz-HFB complex at 1500 K temperature. In Figure 6, N(t-t0)/N(t0) is plotted when (a) the intra-Bz-HFB and (b) the inter-Bz-HFB vibrational modes are randomly excited, respectively for Bz-HFB complex. The fit to the exponential function is also shown in each of the plots. In comparison with Bz2 dissociation where only intra-Bz-Bz modes are randomly excited, a value of 34.2 ps of tʹ was observed with


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a IVR time of 20.7 ps at 1000 K. At this temperature, Bz-HFB complex would be expected to provide a smaller tʹ and IVR time than 34.2 ps and 20.7 ps, respectively. Hence, a faster IVR rate is observed with respect to Bz2 dissociation due to an enhanced coupling between intraBz-HFB and inter-Bz-HFB vibrational modes, which may be through the low frequency intraBz-HFB vibrational modes.

Figure 6. N(t-t0)/N(t0) versus simulation time for (a) intra-Bz-HFB modes are excited, and (b) inter-Bz-HFB modes are excited. Fit are to the exponential function (see text)

Further analysis is made to obtain an idea of the unimolecular dissociation rate at 1500 K temperature for Bz-HFB complex using the models mentioned above. This is done by calculating the probability (f1ʹ) of six inter-Bz-HFB vibrational modes obtain an energy more or equal to the energy of 17.9 kcal/mol using eq 5 at T = 1500 K. This energy is the average vibrational energy of six inter-Bz-HFB modes at T = 1500 K. It is to note here that both classical (skBT) and quantum (eq 3) average vibrational energy expression provide same energy value 23 ACS Paragon Plus Environment


for the inter-Bz-HFB vibrational modes. The calculated probability is obtained as f1ʹ = 0.44. In other words, with a probability of 0.44 the inter-Bz-HFB modes will obtain the average vibrational energy or more at 1500 K and the dissociation will occur with a rate constant of 5.9 ps-1. It can be assumed that with a probability of (1 - f1ʹ) the rest vibrational energy may be sampled to the intra-Bz-HFB modes and the dissociation will take place with a rate constant of 0.15 ps-1. Therefore, an approximated unimolecular dissociation rate constant k(T) can be predicted as f1ʹkinter + (1 - f1ʹ)kintra and is equal to 2.68 ps-1. This value is quite similar to the k(t0,T) obtained from the simulation where all vibrational modes of Bz-HFB are randomly excited and reported in Table 3. The analysis presented here provides a justification of the biexponential behaviour of the Bz-HFB dissociation.

IV.

SUMMARY In this article, the detail study of unimolecular dissociation of Bz-HFB complex is carried out and compared with Bz2 dissociation. In Bz dimer, a weak coupling between six low frequency vibrational modes (inter-Bz-Bz), formed by Bz + Bz association, and sixty vibrational modes (intra-Bz-Bz) of two Bz molecules was observed which leads to a biexponential N(t-t0)/N(t0) versus time expression. The interest of this article is to check at what extent the coupling between those two class of modes can be increased, if at all possible. In case of Bz dimer the frequencies of intra-Bz-Bz vibrational modes are quite large with respect to the same of the inter-Bz-Bz modes. However, Bz-HFB complex is having smaller frequencies in some of the intra-Bz-HFB vibrational modes due to the larger mass of F atoms. In collisional energy transfer model,52 this lowering the frequency plays vital role in intramolecular vibrational energy transfer. It is observed in the work presented here that the same is the case for intramolecular vibrational energy transfer as well.


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The simulations are carried out at 1000, 1200, 1500, 1800 and 2000 K temperatures using the quasiclassical microcanonical normal mode sampling. The corresponding average quantum vibrational energies of Bz-HFB are lower than the same for Bz dimer for a particular temperature due to lower vibrational frequencies of the former. The relative dissociated trajectories versus time is found bi-exponential in Bz-HFB as also observed in Bz2 dissociation. The centre-of-mass separation between Bz and HFB is monitored for all trajectories throughout the dynamics. For 1500 K temperature, ~5% of the trajectories have a larger lifetime than 4 ps and do not dissociate directly. Therefore, the IVR bottleneck is not fully removed in Bz-HFB complex. However, there is an enhancement of the IVR rate is observed for this complex as compared to Bz dimer. With high simulation energies and large number of vibrational degrees of freedom, the microcanonical unimolecular rate constants at initial time are considered to be identical with the canonical ones and the Arrhenius parameters are obtained by fitting ln k(T) vurses 1/T. The activation energy is obtained similar to the binding energy of the complex and the pre-exponential factor is attained with an expected order of magnitude for weakly coupled aromatic complexes.24 In Table 4, the unimolecular rate constants of Bz-HFB dimer with initial average vibrational energies are compared with the same for Bz2 dissociation at temperatures where both the simulations are performed. The result shows that, instead of having lower initial vibrational and higher binding energies, the unimolecular rates are increased in Bz-HFB complex. The difference of rate constants become more prominent at higher temperatures. This observation suggests a stronger coupling between intra-Bz-HFB and inter-Bz-HFB vibrational modes in Bz-HFB complex with a better IVR efficiency between those modes with respect to the same in Bz dimer. The energies of six inter-Bz-HFB normal modes are monitored throughout the dynamics using a harmonic normal mode Hamiltonian. The results of eight sample trajectories are



presented at 1500 K temperature. Out of these eight, four trajectories dissociate directly and achieve 14 Å centre-of-mass separation between Bz and HFB within 2 ps, whereas, four trajectories have a larger lifetime and the same dissociation criteria is fulfilled within a range of 5-15 ps. The initial random excitation of vibrational modes sampled energy in excess of the lowest energy required for the dissociation among the inter-Bz-HFB modes in the directly dissociating trajectories, whereas the other four received lower in the same inter-Bz-HFB modes causing a larger dissociation time. This analysis also identifies the sliding inter-Bz-HFB mode with frequency 67.7 cm-1 as the responsible mode for the dissociation. It is also observed that although the initial sampling does not sample enough energy to this particular responsible mode for the directly dissociating trajectories, there is very fast IVR among the inter-Bz-HFB modes. Simulations results are also presented when the initial vibrational energies are sampled to the inter-Bz-HFB and intra-Bz-HFB vibrational modes, respectively. When the energy is sampled among the inter-Bz-HFB modes, all 1000 trajectories dissociate directly within 3.7 ps and N(t-t0)/N(t0) versus time is exponential with a rate constant as high as 5.9 ps-1. On the other hand, for simulation when initial vibrational energy is sampled to the intra-Bz-HFB modes, the same property become exponential with a rate constant of 0.15 ps-1 and with a consideration of larger time lag (tʹ) of 6 ps. The inverse of the latter rate constant gives a IVR time of 6.7 ps for Bz-HFB dimer. This IVR time is smaller than the same obtained for Bz2 dissociation suggesting a faster IVR rate between inter-Bz-HFB and intra-Bz-HFB modes. For the random initial quasiclassical microcanonical sampling of all the vibrational modes of Bz-HFB complex, the probability, to obtain the average vibrational energy or more in the inter-Bz-HFB modes, is calculated. With the help of this probability value and the rate constants mentioned above the unimolecular rate constant is assumed which is similar to the


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same obtained by the bi-exponential fit parameters of simulated [N(t0) - N(t-t0)]/N(t0) versus time. There are many future directives of the current investigation reported here. The association followed ensuing dissociation of Bz and HFB may be performed and the lifetime of the complex may be observed. Such simulation produces non-random excitation of the vibrational states and the energy redistribution between intra-Bz-HFB and inter-Bz-HFB modes become essential for dissociation to occur. It may also be interesting to see whether the ensuing dissociation of the complex is temperature dependent in this simulation. A simulation may be extended to the HFB2 dimer to check if the IVR bottle neck can be completely removed and exponential dissociation rate is observed. Furthermore, the dissociations may be performed in the bath model39,40,52 to see if the collisional vibrational energy transfer affects the dissociation rate. Since the lower frequency modes play leading role in the collisional energy transfer process, there may be a competition between the energy transfer pathway and the dissociation pathway in such dynamics.

AUTHOR INFORMATION Corresponding Author *Email: [email protected]

ACKNOWLEDGEMENTS The work presented here is supported by SERB-DST, file no. ECR/2017/001434. AKP likes to acknowledge the computation facility at NIT Meghalaya. HM and SSA like to acknowledge SERB-DST, file no. ECR/2017/001434, and NIT Meghalaya for their fellowships.



Page 28 of 41

Supporting Information Available: The normal mode analysis similar to Figure 5 is presented for 1000 K and 2000 K simulations.

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ln k(t0,T)

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E x 10 (kcal/mol)

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of Physical Chemistry 1.7The cmJournal -1 (a) (b) 18.8 cm -1 18.8 cm -1 56.7 cm -1 56.7 cm -1 67.7 cm

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A Better Understanding of the Unimolecular Dissociation Dynamics of

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