Unimolecular Rate Constants versus Energy and Pressure as a

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Unimolecular Rate Constants versus Energy and Pressure as a Convolution of Unimolecular Lifetime and Collisional Deactivation Probabilities. Analyses of Intrinsic Non-RRKM Dynamics Shreyas Malpathak, and William L. Hase J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b00184 • Publication Date (Web): 22 Feb 2019 Downloaded from http://pubs.acs.org on February 23, 2019

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The Journal of Physical Chemistry

Unimolecular Rate Constants versus Energy and Pressure as a Convolution of Unimolecular Lifetime and Collisional Deactivation Probabilities. Analyses of Intrinsic Non-RRKM Dynamics Shreyas Malpathaka,b,c and William L. Hasea,* aDepartment

of Chemistry and Biochemistry

Texas Tech University Lubbock, Texas 79409 bDepartment

of Chemistry

Indian Institute of Science Education and Research, Pune, India 411008 cDepartment

of Chemistry and Chemical Biology Cornell University Ithaca, NY 14853

* [email protected] 1 ACS Paragon Plus Environment

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

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Abstract Following work by Slater and Bunker, the unimolecular rate constant versus collision frequency, kuni(ω,E), is expressed as a convolution of unimolecular lifetime and collisional deactivation probabilities. This allows incorporation of non-exponential, intrinsically non-RRKM, populations of dissociating molecules versus time, N(t)/N(0), in the expression for kuni(ω,E). Previous work using this approach is reviewed. In the work presented here, the bi-exponential f1exp(-k1t) + f2exp(-k2t) is used to represent N(t)/N(0), where f1k1 + f2k2 equals the RRKM rate constant k(E) and f1 + f2 = 1. With these two constraints, there are two adjustable parameters in the bi-exponential N(t)/N(0) to represent intrinsic non-RRKM dynamics. The rate constant k1 is larger than k(E) and k2 is smaller. This bi-exponential gives kuni(ω,E) rate constants which are lower than the RRKM prediction, except at the high and low pressure limits. The deviation from the RRKM prediction increases as f1 is made smaller and k1 made larger. Of considerable interest is the finding that, if the collision frequency ω for the RRKM plot of kuni(ω,E) versus ω is multiplied by an energy transfer efficiency factor βc, the RRKM kuni(ω,E) versus ω plot may be scaled to match those for the intrinsic non-RRKM, bi-exponential N(t)/N(0), plots. This analysis identifies the importance of determining accurate collisional intermolecular energy transfer (IET) efficiencies.

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I. Introduction Rice-Ramsperger-Kassel-Marcus (RRKM) theory is a fundamental and widely used model for interpreting the kinetics of unimolecular reactions; i.e. A* → products.1-3 The assumption of RRKM theory is rapid intramolecular vibrational energy redistribution (IVR), so that the intramolecular dynamics of A* is ergodic on the time-scale of the unimolecular reaction, and for fixed energy unimolecular reactions a microcanonical ensemble is maintained for A* as it dissociates.4-7 With this assumption the energy dependent rate constant for A* is time-independent and equals k(E). In the following a “brief” review is given of RRKM theory, including a procedure for incorporating intrinsic non-RRKM dynamics in expressions for the unimolecular kinetics. The Hinshelwood-Lindemann mechanism8,9 forms the basis for interpreting thermal unimolecular reactions; i.e.

A + M ↔ A∗ + M A ∗ →products

(𝑘1,𝑘 ―1)

(1a)

(𝑘2)

(1b)

Collisions, with a bath of inert gas phase molecules M, excite A and deactivate A* by rate constants k1 and k-1. For fixed energy E, with an initial microcanonical ensemble of states for A* and the RRKM assumption, k2 = k(E). Applying the steady-state approximation to the HinshelwoodLindemann mechanism for fixed E, results in the rate constant 𝜔 𝑘(𝐸) 𝜔

𝑘𝑢𝑛𝑖(𝜔,𝐸) = 𝑘(𝐸) +

(2)

where ω is the collision frequency per molecule and, with the strong collision assumption, equals k-1[M]. The probability A* has energy in the range E → E + dE is P(E)dE and equals dk1/k-1. Integrating kuni(ω,E) over this probability gives the thermal unimolecular rate constant kuni(ω,T), ∞𝑘(𝐸)𝑃(𝐸)

𝑘𝑢𝑛𝑖(𝜔,𝑇) = 𝜔 ∫0 𝑘(𝐸) + 𝜔 𝑑𝐸

(3)

where the RRKM assumption has been made for k(E). A phenomenological collision-averaged rate constant is found from a Stern-Volmer analysis of chemical activation,10 radiationless transition,11 or overtone excitation experiments.12 3 ACS Paragon Plus Environment


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This rate constant is defined as13

𝑘(𝜔,𝐸) = 𝜔𝐷 𝑆

(4)

where D is the probability of forming the decomposition products, and S is the probability of forming collisionally stabilized reactant. With the RRKM model of a time-independent unimolecular rate constant for a monoenergetically excited reactant, k(ω,E) is pressure, i.e. ω, independent and equals the RRKM rate constant k(E). II. Unimolecular Rate Constants without the Steady-State Approximation and Random Lifetime Assumption A straightforward way to include possible non-RRKM effects for both kuni(ω,E) in Eq. (2) and k(ω,E) in Eq. (4) is to consider the unimolecular reactant’s lifetime distribution P(t)

𝑃(𝑡) = ―[𝑑𝑁(𝑡) 𝑑𝑡] 𝑁(0)

(5)

which was first introduced by Slater,14 further considered by Bunker,4,15 and reviewed by Robinson and Holbrook16 and Forst.17 Here N(t) is the number of reactant molecule’s remaining versus time. With an initial microcanonical ensemble of reactant states comprising N(0), any non-RRKM effects are referred to as intrinsic non-RRKM.5 For RRKM dynamics N(t) is exponential, i.e. the random lifetime assumption,4 but non-exponential for intrinsic non-RRKM. In the simplest analysis of using P(t) to formulate non-RRKM expressions for kuni(ω,E) and k(ω,E),4,19-21 one assumes that collisions between the inert gas phase bath molecules M and reactant A* are uncorrelated and that each collision results in stabilization of A*. (The latter may be modified to account for “weak” collisions16-18). The probability that A* avoids a collision for time t is W(t);14 i.e.

𝑊(𝑡) = 𝑒 ―𝜔𝑡

(6)

Since P(t)dt is the probability A* dissociates in the time interval t → t + dt, the total probability of dissociation is


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∞

𝐷 = ∫0 𝑊(𝑡)𝑃(𝑡)𝑑𝑡

(7)

The probability of stabilization S equals 1 - D, i.e.

∞

𝑆 = 1 ― ∫0 𝑊(𝑡)𝑃(𝑡)𝑑𝑡

(8)

The above equations for D and S yield the rate constant k(ω,E) in Eq. (4) as a function of ω. The rate constant kuni(ω,E) in Eq. (2) equals ωD. For RRKM dynamics, P(t) in Eq. (5) is exponential4-7 and given by

𝑃(𝑡) = 𝑘(𝐸)𝑒 ―𝑘(𝐸)𝑡

(9)

where k(E) is the RRKM rate constant. Both the P(0) intercept and the exponential decay of P(t) give k(E). Inserting Eq. (9) into Eqs. (7) and (8) give Eq. (2) for kuni(ω,E) and k(E) for k(ω,E) in Eq. (4). For intrinsic non-RRKM dynamics, P(t) is non-exponential and N(t)/N(0) may be written as4,19,20

𝑁(𝑡) 𝑁(0) = ∑ 𝑓𝑖𝑒 ― 𝑘𝑖𝑡 𝑖

(10)

where the sum of the fi equals unity. (Other functions may be used to represent the non-exponential character of N(t)21,22). P(t) becomes

𝑃(𝑡) = ∑𝑖𝑓𝑖𝑘𝑖𝑒 ― 𝑘𝑖𝑡

(11)

The t = 0 rate constant, for an initial microcanonical ensemble, is the RRKM rate constant k(E) and equal to P(t) at t = 0. For P(t) in Eq. (11), k(E) is

𝑘(𝐸) = ∑𝑖𝑓𝑖𝑘𝑖

(12)



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Effects of intrinsic non-RRKM dynamics on kuni(ω,E) and k(ω,E) have been investigated in previous studies.19,20,23-31 Multi-exponential P(t) have been used to calculate k(ω,E) in Eq. (2) for model H-C-C → H + C=C dissociation19,20 to compare with chemical dynamics simulations and for Cl----CH3Cl → Cl- + CH3Cl dissociation26 to compare with chemical dynamics simulations and experiments.32,33 Effects of fluctuations in quantum mechanical state-specific rate constants, and their distributions, on the unimolecular rate constants kuni(ω,E) and k(ω,E) have been considered.23-25,27 An example is the relationship between quantum dynamical calculations of the highly mode-specific decomposition for HOCl → OH + Cl and the Lindemann-Hinshelwood unimolecular rate constant kuni(ω,T) in Eq. (3).29 In interpreting experimental measurements of kuni(ω,T) for HCO → H + CO dissociation, Hippler and co-workers30 found that a mode-specific isolated resonance model represented their experimental results. The intrinsic non-RRKM dynamics of the Cl----CH3Br SN2 intermediate were studied by trajectories on an analytic potential energy surface and by direct dynamics.31 Different models were used to represent the phase space dynamics and non-exponential P(t), including a multi-exponential function. III. Effect of Intrinsic Non-RRKM Dynamics on kuni(ω,E) There have not been extensive analyses of the effect of intrinsic non-RRKM dynamics, and multi-exponential decomposition, on the pressure dependent Hinshelwood-Lindemann unimolecular rate constant kuni(ω,E) and that is the topic addressed here. To consider these dynamics, P(t) is written as a bi-exponential as in Eq. (11):

𝑃(𝑡) = 𝑓1𝑘1𝑒 ― 𝑘1𝑡 + 𝑓2𝑘2𝑒 ― 𝑘2𝑡

(13)

The RRKM rate constant is k(E) = f1k1 + f2k2. The Hinshelwood-Lindemann rate constant kuni(ω,E) is given by ωD and is

𝑘𝑢𝑛𝑖(𝜔,𝐸) = 𝜔[𝑓1𝑘1 (𝜔 + 𝑘1) + 𝑓2𝑘2 (𝜔 + 𝑘2)]

(14)

for D in Eq. (7). For the high-pressure limit, ω → ∞, kuni(ω,E) = f1k1 + f2k2 = k(E) and for the lowpressure limit, ω → 0, kuni(ω,E) = 0. The effect of non-exponential decomposition on kuni(ω,E) was 6 ACS Paragon Plus Environment

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studied by plotting kuni(ω,E)/k(E) versus ω/k(E). At high-pressure kuni(ω,E)/k(E) equals unity and at low-pressure it equals zero. Different sets of parameters were chosen for f1, k1, f2, and k2. Since k(E) = f1k1 + f2k2 and f1 + f2 = 1, the values for k1 and k2 are constrained and there are only two independent parameters. The different parameter sets considered here are listed in Table 1, with k1 and k2 larger and smaller than k(E), respectively. Plots of kuni(ω,E)/k(E) versus ω/k(E) are given in Figure 1 for the different parameter sets. The curves for the non-exponential, intrinsic non-RRKM N(t)/N(0) are below that of RRKM theory, except for the high and low pressure limits. This result, for a non-exponential N(t)/N(0), was shown analytically in previous work.29 For some parameter sets there is a substantial disagreement with the RRKM plot. This disagreement increases with decrease in f1 and increase in k1. For f1 equal to 0.2 and 0.4 there are substantial disagreements with RRKM for the large k1. If k2 is small, approaching zero, so that f1k1 ≈ k(E), a small f1 results in a k1 much larger than the RRKM k(E). Such a situation results in a large deviation from the RRKM plot of kuni(ω,E)/k(E) versus ω/k(E). For the plots in Figure 1, calculations were performed for k2 = 0, which represent those for a very small k2, approaching zero. In interpreting and fitting experiments, an energy transfer efficiency factor βc is often used to scale the collision frequency so the effective collision frequency is βcω.34-37 The effect is to lower the RRKM curve of kuni(ω,E) versus ω.18 For the analyses made above, the kuni(ω,E)/k(E) versus ω/k(E) plots for the bi-exponential, intrinsic non-RRKM N(t)/N(0) are assumed to represent experiment. It was found that if ω in Eq. (2) is replaced by βcω, and βc varied, the RRKM kuni(ω,E)/k(E) versus ω/k(E) plot may be adjusted to accurately fit the intrinsic non-RRKM plots. Attempts were made to determine the value of βc for the scaling analytically, but they were not successful. For the calculations in Figure 1, the most severe disagreement between the intrinsic non-RRKM and RRKM plots is for f1 = 0.2. The RRKM plot may be adjusted to fit the intrinsic non-RRKM plots for this f1 may by setting βc equal to 0.85, 0.5, 0.3, and 0.2 for k1 equal to 2k(E), 3k(E), 4k(E), and 5k(E), respectively. The intrinsic non-RRKM plot for k1 = 5k(E) and RRKM plot for βc = 0.2 are compared in Figure 2, illustrating the excellent agreement between the two plots. The important finding from the above analysis is that the RRKM kuni(ω,E)/k(E) versus ω/k(E) plot may be adjusted to fit those for intrinsic non-RRKM, bi-exponential N(t)/N(0), by using an energy transfer efficiency factor βc.



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IV. Summary In the current study the bi-exponential unimolecular dissociation probability N(t)/N(0) = f1exp(-k1t) + f2exp(-k2t) is used to represent intrinsic non-RRKM unimolecular dynamics. For this N(t)/N(0) the unimolecular rate constant kuni(ω,E), for which the collision frequency ω is proportional to pressure, disagrees with that given by RRKM theory. An important finding from this study is that the intrinsic non-RRKM kuni(ω,E) may be adjusted to fit kuni(ω,E) of RRKM theory by multiplying ω by an energy transfer efficiency factor βc. This analysis points out the critical need to determine accurate collisional intermolecular energy transfer (IET) efficiencies.18,34-37 For the bi-exponential N(t)/N(0), the RRKM rate constant k(E) equals f1k1 + f2k2. With this constraint and f1 + f2 = 1, there are only two variable parameters in the bi-exponential. For the analyses made here, the rate constant k1 is larger than k(E) and k2 is smaller. For the bi-exponential probability N(t)/N(0), the deviation from the RRKM kuni(ω,E) increases as f1 is made smaller and k1 made larger. O. K. Rice38 and D. L. Bunker4,5 have suggested that, if a molecule’s unimolecular dynamics is intrinsically non-RRKM, there may be a false high-pressure limit in kuni(ω,E) as illustrated in Figure 3. The non-RRKM dynamics arises from inefficient intramolecular vibrational energy redistribution (IVR).39 Thus, as suggested by Rice,37 for the molecule’s initial dissociation and the true high-pressure limit, the molecule may behave as a smaller molecule with a rate constant larger than that of RRKM theory. The false limit, at a lower collision frequency, then represents an IVR rate constant.4,5 The bi-exponential N(t)/N(0) considered here represents such dynamics since k1 is larger than the RRKM rate constant k(E). However, there is not a false limit in kuni(ω,E) for the bi-exponential N(t)/N(0). However, it is possible that other functional representations21,22 of non-exponential N(t)/N(0) may give a false high-pressure limit in kuni(ω,E). The work presented here illustrates the need to determine N(t)/N(0) distributions for unimolecular dissociation, and establish the importance of non-exponential dissociation dynamics and the functional form of N(t)/N(0) for such dynamics. Additional functional forms, for nonexponential N(t)/N(0) which could be considered in future work, include power laws, stretched exponentials, rational fractions,21,22,40,41 and a tri-exponential. There are possible experimental and simulation approaches for determining N(t)/N(0), but both are challenging. The ideal experiment would be to prepare a N(0) microcanonical ensemble and then follow its population versus time, 8 ACS Paragon Plus Environment

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N(t), and see if it is exponential and, if not, determine the functional form of N(t). Following dissociation of an initial vibrational state or non-random energy distribution provides information concerning IVR, but not details of N(t). Direct dynamics simulations42 are a general approach for studying unimolecular dissociation, but given the short time scale for the simulations, it is usually necessary to consider energies much higher than those studied experimentally. Approaches need to be developed to make direct dynamics simulations applicable to thermal unimolecular dissociations.

Acknowledgements The research reported here is based upon work supported by the Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-17-1-0119 and the Robert A. Welch Foundation under Grant No. D-0005. Shreyas Malpathak was supported by a S. N. Bose Fellowship from the S. N. Bose Scholars Program for Indian students.

References



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28. W. L. Hase, Some Recent Advances and Remaining Questions Regarding Unimolecular Rate Theory. Acc. Chem. Res. 1998, 31, 659-665. 29. Song, K.; Sun, L.; Hase, W. L.; Grebenshchikov, S. Yu.; Schinke, R. Relationship between Mode Specific and Thermal Unimolecular Rate Constants for HOCl → OH + Cl Dissociation. J. Phys. Chem. A 2002, 106, 8339-8344. 30. Hippler, H.; Krasteva, N.; Striebel, F. The Thermal Unimolecular Decomposition of HCO: Effects of State Specific Rate Constants on the Thermal Rate Constant. Phys. Chem. Chem. Phys. 2004, 6, 3383-3388. 31. Paranjothy, M.; Sun, R.; Paul, A. K.; Hase, W. L. Models for Intrinsic Non-RRKM Dynamics. Decomposition of the SN2 Intermediate Cl----CH3Br. Z. Phys. Chem. 2013, 227, 1361-1379. 32. Tonner, D. S.; McMahon, T. B., Non-Statistical Effects in the Gas Phase SN2 Reaction. J. Am. Chem. Soc. 2000, 122, 8783-8784. 33. Mikosch, J.; Otto, R.; Trippel, S.; Eichhorn, C.; Weidemüller, M.; Wester, R. Inverse Temperature Dependent Lifetimes of Transient SN2 Ion-Dipole Complexes. J. Phys. Chem. A 2008, 112, 10448-10452. 34. Tardy, D. C.; Rabinovitch, B. S. Intermolecular Vibrational Energy Transfer in Thermal Unimolecular Systems. Chem. Rev. 1977, 77, 369–408. 35. Gilbert, R. G.; Luther, K.; and Troe, J. Theory of Thermal Unimolecular Reactions in the Falloff Range. II. Weak Collision Rate Constants. Ber. Bunsenges. Phys. Chem. 1983, 87, 169-177. 36. Oref, I.; Tardy, D. C. Energy Transfer in Highly Excited Large Polyatomic Molecules. Chem. Rev. 1990, 90, 1407-1445. 37. Zhu, L.; Chen, W.; Hase, W. L.; Kaiser, E. W. Comparison of Models for Treating Angular Momentum in RRKM Calculations with Vibrator Transition States. Pressure and Temperature Dependence of Cl + C2H2 Association. J. Phys. Chem. 1993, 97, 311-322. 38. Rice, O. K. Z. Phys. Chem. B 1930, 7, 226 [quoted by L. S. Kassel, Kinetics of Homogeneous Reactions; Chemical Catalog: New York, 1932]. 39. Hase, W. L. Unimolecular and Intramolecular Dynamics: Relationship to Potential Energy Surface Properties. J. Phys. Chem. 1986, 90, 365-374. 40. Shojiguchi, A.; Li, C.-B.; Komatsuzaki, T.; Toda, M. Fractional Behavior in Nonergodic Reaction Processes of Isomerization. Phys. Rev. E 2007, 75, 035204.


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Table 1. Values of k1 and k2 for Model Calculations of kuni(ω,E) f1a 13 ACS Paragon Plus Environment


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0.2

0.4

0.6

0.8

k1

2 k(E)

1.5 k(E)

1.25 k(E)

1.1 k(E)

k2

3/4 k(E)

2/3 k(E)

5/8 k(E)

3/5 k(E)

k1

3 k(E)

1.75 k(E)

4/3 k(E)

1.2 k(E)

k2

1/2 k(E)

1/2 k(E)

1/2 k(E)

1/5 k(E)

k1

4 k(E)

2 k(E)

5/3 k(E)

1.25 k(E)

k2

1/4 k(E)

1/3 k(E)

0

0

k1

5 k(E)

2.25 k(E)

k2

0

1/6 k(E)

k1

2.5 k(E)

k2

0

a. f1 is the fraction of the bi-exponential with k1. k(E) is the RRKM rate constant.

Figure Captions


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Figure 1. Plots for kuni(ω,E)/k(E) versus ω/k(E) for various values of f1 and k1. (a) k1 = k(E) : blue; 2k(E):red ; 3k(E): gray; 4 k(E): orange; 5 k(E): purple. (b) k1 = k(E) : blue; 1.5k(E):red ; 1.75k(E):green ; 2k(E): orange; 2.25k(E): purple ; 2.5k(E): green. (c) k1 = k(E) : blue; 1.25k(E):red ; 1.33k(E):green ; 1.66k(E): orange. (d) k1 = k(E) : blue; 1.1k(E): red ; 1.2k(E):green ; 1.25k(E): orange. Figure 2. Plots of kuni(βcω,E)/k(E) versus ω/k(E) for RRKM (black) and kuni(ω,E)/k(E) versus ω/k(E) for the bi-exponential N(t)/N(0) with f1 = 0.2 and k1 = 5k(E) (cyan), see Figure 1; βc = 0.2. Figure 3. Representative plot of kuni(ω,E) versus ω illustrating a false high-pressure limit as suggested by O. K. Rice37 and D. L. Bunker.4,5



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Figure 1.


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1.2 1

kuni/k(E)

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


0.8 0.6 0.4 0.2 0 0

20

40

ω/k(E)

60

80

100

Figure 2.



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Figure 3.


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Brief Biography of William L. Hase

William L. Hase is the Robert A. Welch Professor of Chemistry at Texas Tech University. He received his B.S. degree from the University of Missouri, Columbia (1967), and his Ph.D. from New Mexico State University (1970). He was a post-doctoral associate with Don Bunker, University of California, Irvine, before joining the faculty at Wayne State University (1973). He moved to Texas Tech University in 2004. Hase is a theoretical and computational chemist whose research interests include theories of chemical reaction dynamics and kinetics, algorithm and software development for chemical dynamics simulations, and applications of these simulations to a broad range of problems (hase-group.ttu.edu). He has a continual interest in intramolecular and unimolecular dynamics. His research group has written and maintains the chemical dynamics computer program VENUS (cdssim.chem.ttu.edu), which is interfaced with electronic structure computer programs for performing direct dynamics simulations.



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Brief Biography of Shreyas Malpathak

Shreyas Malpathak is a doctoral student at the Department of Chemistry and Chemical Biology, Cornell University. He graduated with a BS-MS degree from the Indian Institute of Science Education and Research (IISER), Pune, India in 2018. His master’s thesis focused on direct dynamics simulations of the unimolecular dissociation of 1,2-dioxetane, which was carried out at Texas Tech University with Prof. Hase. Having recently joined Prof. Nandini Ananth’s group at Cornell University, his current research interests include semi-classical dynamics in the condensed phase.


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TOC Graphic



338x190mm (96 x 96 DPI)

ACS Paragon Plus Environment

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182x97mm (96 x 96 DPI)



287x167mm (96 x 96 DPI)


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Unimolecular Rate Constants versus Energy and Pressure as a

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