Stalking the Step-Size Distribution: A StatisticalâDynamical Theory for

Chapter 15

Downloaded by NORTH CAROLINA STATE UNIV on January 10, 2013 | http://pubs.acs.org Publication Date: June 10, 1997 | doi: 10.1021/bk-1997-0678.ch015

Stalking the Step-Size Distribution: A Statistical-Dynamical Theory for Large-Molecule Collisional Energy Transfer John R. Barker Department of Atmospheric, Oceanic, and Space Sciences and Department of Chemistry, University of Michigan, Ann Arbor, MI 48109-2143 A theory of large molecule energy transfer based on state-to-state transition probabilities is derived and a demonstration calculation is presented for cyclopropane deactivation by helium. The calculations are shown to be practical even for energies where the vibrational state densities exceed 10 states/cm . For demonstration purposes the state-to-state transition probabilities are calculated using SSH(T) Theory, and thus the present calculations are expected to provide only general trends (in future work, more accurate state-to-state theories will be employed). The predicted collision step size distributions resemble the exponential model, but with strong fluctuations about the mean. The fluctuations arise from energy transfer propensities, which persist even at high energies. The predicted average energy transferred in deactivation collisions ( ) depends on vibrational energy, due to the changing fraction of inelastic collisions; the shape of the distribution function remains nearly unchanged. 10

-1

down

Energy transfer is of fundamental importance in unimolecular reactions (1-3), but theory has lagged seriously behind experiment for energy transfer involving highly excited large molecules (4-6). For small molecules, the theory has advanced to the point that quantitative predictions are possible (7-9). Beginning with the LandauTeller Theory (10), several simple theories have been developed which are distinguished by their relative complexity and accuracy (4,8). One of the most basic of these is the Schwartz, Slawsky, and Herzfeld Theory (11,12), as implemented by Tanczos (75): SSH(T) Theory. Other theories of varying complexity include the impulsive called "ITFITS" theory (14) and its "DECENT" (15-17) and "INDECENT" (18) extensions. The co-linear atom+diatomic SSH and ITFITS models have been extended to three dimensions and to polyatomic systems by utilizing the "breathing sphere" approximation (13,19). These theories describe 220

© 1997 American Chemical Society In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.


15. BARKER

Stalking the Step-Size Distribution

221

energy transfer with varying degrees of fidelity, but their predictive capabilities are limited. Classical trajectory calculations have been used extensively to investigate small molecule energy transfer in recent years (7,20,21). More recently, quantum mechanical close coupling calculations using realistic potential energy surfaces have been used to investigate energy transfer involving small and medium-sized molecules at low vibrational energies (9,22) and model systems at high excitation energies (23). Classical trajectory calculations currently provide the most generally useful theoretical approach for studying large molecule energy transfer (24,25). They have become practical for large molecule energy transfer only recently, however, because of their great demands on computer time (25,26). These studies provide qualitative insight and even semi-quantitative agreement with experiments. However, it is not clear whether the agreement with experiments is due to an accidental cancellation of errors in the potential energy surface and those inherent in the use of classical mechanics. Classical trajectory calculations may suffer from several deficiencies, due to differences between classical and quantum mechanics, but the quantitative importance of these deficiencies is not yet known (27,28). The most obvious deficiency is that zero point energy is not conserved in classical mechanics. Nonconserved zero point energy may enhance energy transfer and it can even contribute to bond dissociation at moderate vibrational energies which are by themselves insufficient to break a bond. A more subtle deficiency is that, unlike classical systems, quantum systems do not exhibit equipartition of energy (29). Therefore, quantum statistics are significantly different from classical statistics and it is not clear that classical trajectories are capable of properly reproducing effects which depend on the statistical amount of vibrational energy available in a given vibrational mode. It is often argued that the quasicontinuum of states in a large molecule at high vibrational energy validates the use of classical mechanics (25). Even at high densities of vibrational states, however, the average excitation per mode may be quite small and the zero point energy may be an important fraction of the total classical vibrational energy, conditions which tend to invalidate the use of classical mechanics. Quantitative comparisons between classical and quantum mechanics are currently being carried out on prototype energy transfer systems to determine the quantitative accuracy of classical mechanics (28). In the present paper, we take a new approach, which has been described elsewhere in detail (30). The aim of this work is to develop a single semi quantitative theory which can reliably describe both small and large molecule energy transfer over a very wide range of excitation energies. Such a theory is necessarily approximate and it may have only limited predictive capabilities, but it will provide an alternative to classical trajectory calculations. It also may provide insight into the important factors which govern large molecule energy transfer. In this paper, we derive and demonstrate a statistical dynamical theory which conserves zero point energy, obeys detailed balance, and employs state-tostate energy transfer theories for separable degrees of freedom. The theory is modular and the state-to-state theoretical module can be selected according to the required accuracy and computational efficiency. For the purpose of demonstrating

In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

222

HIGHLY EXCITED MOLECULES

the statistical-dynamical theory, we have used SSH(T) breathing sphere theory, but work is currently underway in our laboratory to use ITFITS (14), which is more accurate.


Theory Statistical-Dynamical Application of State-to-State Models. The derivation (30) is presented in this section. Our objective is to calculate the probability that a large molecule with internal energy Ε undergoes a transition to new energy in the range E' to E'+dE. For present purposes, we will treat this process as Vibration-toTranslation (V-T) energy transfer, but the theory can be extended easily to include Vibration-to-Vibration (V-V) energy transfer. For a system with s separable degrees of freedom, the states cc and β can be designated with sets of s quantum numbers (vi, 1 < i < s). A transition can be written in terms of (Xg and β , or in terms of a and A , where the last represents the set of changes in quantum numbers: s

8

s

s

a = {vi,v ,... } s

8

2

(1)

s

Ps={vi',V2\...}s

(2)

Δ ={(νι'-ν ),(ν ·-ν ),...} 8

1

2

2

(3)

8

Equation (3) is convenient for considering transition probabilities, because it allows the quantum numbers to be partitioned into two groups: those which change during the transition and those which do not: Δ = {(vi'-vO, (Vj'-Vj), . . . } + {0, 0, . . . } „ 8

(4)

n

The states a and β can also be partitioned: s

8

CC

=

S

y

{ i> Yj, . . . } + {V , V , ... } -n = 0C + a . n

p

q

s

n

s

(5)

n

β = βη + β -η 5

(6)

δ

where η of the quantum numbers change during the transition and the rest do not. The identities of the specific degrees of freedom (i, j , ...) in each subset are retained in sets η and η _ (i.e. the two sets record which modes change and which ones do not change during a transition). For separable degrees of freedom, the internal energy is also partitioned: Ε = E + E . . For a system with separable degrees of freedom, we assume that the stateto-state transition probabilities from state a to state β depend only on a and β : η

8

η

n

8

n

s

8


n

η

15. BARKER

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223

Ρ(β ,Os) = Ρ(η ;βη,α ) = Ρ ( η ; Δ , α ) . 8

η

η

η

η

(7)

η

Thus, every state with the same a (and η ) has the same transition probability. We are interested in the transition probability averaged over a small energy increment. The total number of states in an energy increment dE is ρ (η ;Ε) dE, where ρ (η ;Ε) is the total density of states for all s degrees of freedom at energy E. The functional form and magnitude of the density of states is determined by the identities, retained in η , of the s degrees of freedom. The average transition probability from a state in the energy range Ε to E+dE to all states in the energy range E' to E'+δΕ is equal to the sum of all such transition probabilities divided by the number of initial states. Thus, the average transition probability can be expressed n

η

8

8

8

8


8

j

= ^ δΕ

/P (x,E)dx ,

C

j = Ps

(10a)

c

E

s

= — Χ

£

C

„ ι^

where

Λ

Χ

α (0)

n=l ,

τ-, τ-,

χ

7(η . ;Ε,Ε . ) = 8

η

8

7(η -η;Ε,Ε .η)Ρ(ηη;βη,αη) 8

(Ha)

δ

β(Ε')

η

π

Ps-nC^s-n^Es-n)

——— . Ρ8(η ;Ε)

(lib)

δ

When the summation is truncated at η < s, the energy differential dE does not appear explicitly in the final expression. It is convenient to write Equation (11) as a sum of terms which are indexed by the number of quantum numbers which change during a transition:

over the thermal distribution of initial relative speeds (57). The thermal energy transfer rate constant kc is obtained from η

η

C

2

k(T) = k
, c

C

(13)

C

where is the average relative speed and σ is the Lennard-Jones collision radius. For a V - T energy transfer process involving a polyatomic, the SSH(T) Theory transition probability (32,33) depends sensitively on three sets of parameters: steric factors Ρο(βη>α ), the characteristic length for the repulsive potential λ, and the squares of the vibrational amplitude coefficients A i . For predictive calculations, the vibrational amplitude coefficients can be obtained from a normal coordinate analysis and the exponential repulsion characteristic length (λ) can be estimated on the basis of Lennard-Jones parameters (12,13). The steric factors (Ρ (βη»α η)) are usually taken as equal to 2/3 for each vibrational quantum number which changes during a transition (55) : Ρ (βη,α η) = (2/3) . For the purpose of fitting experimental data, or the results of higher order theories, the amplitude coefficients, steric factors, and length parameter may be adjusted. In the present work, anharmonicities have been introduced on an ad hoc basis by using anharmonic transition energies in place of the harmonic values. η

2

0

n

0


226

HIGHLY EXCITED MOLECULES

Implementation m

For computational purposes, note that each Ρ(η ;βη»θί ) Equation (7) is approximately a delta function at E'-E = ΔΕβ , the energy difference in the transition. By using a finite δΕ, one obtains a "binned" average transition probability
0. Operationally, one can test for convergence of the sum over η after evaluating each P term in Equation (12) and the summation can be truncated when satisfactory convergence is achieved. Since the summations over cc and β have many more terms for large n, truncation saves considerable computer time. It is also convenient for computational purposes to calculate all possible transitions in a relatively large energy range and then "bin" the results according to ΔΕβ . In this way,

Stalking the Step-Size Distribution: A StatisticalâDynamical Theory for

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