On the accuracy of collisional energy transfer parameters for reaction kinetics applications: detailed evaluation of data from direct experiments

It is shown that the spread among the various “direct” experimental 〈ΔE〉 data in the literature, so unsatisfactory for their application in chemical kinetics, can be removed consistently.

Underlying agreement within very small uncertainties is demonstrated for the case of the much studied collisional relaxation of highly vibrationally excited azulene.

Benchmark experimental data for the collisional energy transfer of highly vibrationally excited azulene obtained by the method of “kinetically controlled selective ionization (KCSI)” (U. Hold, T. Lenzer, K. Luther and A. C. Symonds, J. Chem. Phys., 2003, 119, 11 192) are used for a detailed comparison with earlier measurements employing time-resolved ultraviolet absorption (UVA) and infrared fluorescence (IRF).

The experimental UVA and IRF traces are simulated by convolution of the transient vibrational distributions g(E) during relaxation obtained from KCSI measurements with the respective calibration curves of the UVA and IRF experiments.

The differences between such simulations and the experimental curves are traced back to non-negligible contributions of azulene self-collisions in the UVA and IRF data.

Astonishing quantitative agreement is reached when azulene/bath gas mixing ratios of the corresponding UVA/IRF experiments are fully accounted for in the KCSI simulations.

The influence of self-collisions is thus quantitatively assessed as an important source of error in addition to the well-known problem of calibration curve uncertainties in UVA and IRF detection as discussed earlier (T. Lenzer, K. Luther, K. Reihs and A. C. Symonds, J. Chem. Phys., 2000, 112, 4090).


Collisional energy transfer (CET) is a key process in a large variety of chemical reaction systems.1,2

In cases like unimolecular or combination reactions, CET rates directly determine the total rates of the chemical reactions in the so-called “low pressure regime” and define the position of the fall-off range of diminishing pressure dependence at increasing densities.

The analysis of such pressure dependent reactions or the extrapolation of restricted experimental data in the fall-off range towards limiting low and high pressure values k0 and k is mostly done on the basis of well known analytical expressions from statistical rate theory.3,4

In this approach, CET information is considered only in the form of a single collision efficiency parameter βc.

An approximate analytical expression5 is mostly used to relate βc to 〈ΔE〉, the average amount of energy transferred per collision.

A more general and better alternative consists in a numerical solution of the energy specific master equation of the reaction system, which needs as input data specific rate constants k(E) for the reactive steps and correspondingly kcoll(E) for inelastic collisions, usually quoted as 〈ΔE(E)〉 = kcoll(E)Z–1, with Z being the gas kinetic collision number.

At chemically significant energies such CET data usually correspond to very large amounts of vibrational energy and densities of states.

Analysis of pressure dependent data on reaction rates or yields from measurements in the fall-off range or under chemical activation conditions was the first method to derive 〈ΔE〉 values under such conditions and is still used today.

However, the inherent disadvantages of this approach are well known: Due to the relation between βc and 〈ΔE〉, measurements of βc require a much higher precision to achieve the same level of accuracy as a corresponding direct determination of 〈ΔE〉.

But more important, such data evaluation unavoidably includes the absolute value for a reaction rate in the measured product expression with an unknown error limit or a tacit assumption of quasi-infinite theoretical accuracy.

The possibilities to clearly assign finer observed effects to either the energy transfer or the reactive properties are severely limited.

In principle this dilemma was overcome with the advent of direct experiments to measure 〈ΔE〉 at chemically relevant energies independent from any knowledge on chemical reaction rates.

Time-resolved infrared fluorescence (IRF)6 and “hot” ultraviolet absorption (UVA)7 became the most important techniques.

However, the agreement between various measurements turned out to be much less than might be expected.

This is very obvious for some frequently studied molecules, especially toluene and azulene, for which direct CET measurements with various techniques or even with the same have led to sometimes astonishing differences, see e.g. Figs. 1 and 2 and Fig. 22 in ref. 8 or Fig. 20 in .ref. 9

As all these methods directly rely on the quality of a calibration curve for data evaluation, arguments on e.g. deficiencies in the quality of these reference curves have been exchanged mutually but the issue could not be settled.

The considerable spread among reported “direct” 〈ΔE〉 values has led to a sort of common belief among users of 〈ΔE〉 data for chemical kinetics applications that the field of large molecule energy transfer at higher energies is still not really settled.

Typically then, averages of the available first moment data are taken in the hope to arrive at what is believed to be the best value.

And as a consequence 〈ΔE〉 values inferred from analysis of e.g. pressure dependent rate constants in the fall-off range are very generously called “reasonable”.

Thus the impact hoped from directly measured 〈ΔE〉 values has not yet been achieved, namely to define much closer boundaries for the CET part in pressure dependent chemical rate data, with improved resulting possibilities to identify more detail in disagreement between chemical reaction rate models and reality.

With the results from a second generation method, the kinetically controlled selective ionization (KCSI) this situation has changed.

As the only technique so far available, KCSI can determine complete distributions of CET transition probabilities P(E′,E) even at quasi-continuous densities of states.

This is done by monitoring the shape of the time dependent population distributions of highly vibrationally excited molecules, as they relax in the electronic ground state from a high initial energy.10

The resulting 〈ΔE〉 data are of unprecedented accuracy, typically on the order of 2–7%.

More important, KCSI data are widely and in favourable cases – like that of azulene treated in this paper – completely independent of external parameters of calibration data.

Thus they provide a benchmark for reevaluation of earlier direct data.

Already in the KCSI paper of toluene relaxation9 one example was shown, in which drastic discrepancies between an early UVA set of 〈ΔE〉 and new KCSI data was completely removed with a reconstruction of the experimental curve from time dependent KCSI population distributions and a very minor change in the most uncertain part of the calibration curve.

In this paper we apply a similar detailed analysis to various examples of the deactivation of highly vibrationally excited azulene, which has become a benchmark system for investigations of CET.

Time-resolved ultraviolet absorption has been applied in several studies.

The first experiments reported an almost energy independent 〈ΔE〉 for all bath gases, with the exception of excess energies below roughly 10 000 cm–1.7

Subsequently these studies were extended to other excitation energies.11,12

In contrast to the early study, an almost linear energy dependence of the first moment of energy transfer was found, though a leveling-off at the highest energies (about 50 000 cm–1) could not be ruled out.

The most recent measurements, carried out at temperatures ≥373 K for excitation energies <20 000 cm–1, show the highest signal-to-noise ratio obtained so far in an azulene UVA experiment.13

An almost linear energy dependence of 〈ΔE〉 was found over the whole energy range for all colliders, sometimes with substantial deviations from the earlier UVA data.

In addition, a variety of measurements employing time-resolved IR fluorescence exists.6,14–16

These data were later reanalyzed with an adjusted calibration curve.

This resulted in a substantial change of the original 〈ΔE〉 values by 30–50%, depending on the bath gas.17

We show that the deviations of all measurements in Figs. 1 and 2 from the KCSI curves can be removed within very narrow error limits, with the origins of the apparent deviations identified in agreement with experimental parameters originally given or very reasonable estimates of them.

It will become clear that the sometimes puzzlingly large differences can be traced back to two main effects: (1) contamination of the available data sets by – at first sight – small amounts of self-collisions and (2) uncertainties of the calibration curves needed for converting the raw UV absorption and IR fluorescence data.

By proper accounting for these two effects, we can show that all older measurements for azulene and toluene are in fact in complete agreement with the KCSI benchmark data.

Disagreements reported on CET for related systems are also highly likely due to self-collision and calibration issues.

Evaluation of available data and their simulation

The primary quantity obtained from UVA and IRF experiments is 〈〈ΔE(〈E〉)〉〉, which corresponds to a 〈ΔE(E)〉 “bulk averaged” over the normalized vibrational distribution g(E) in the ground electronic state with average energy 〈E〉: 〈〈ΔE(〈E〉)〉〉 = ∫∞0gE(E′)〈ΔE(E′)〉dE

Differences between microcanonical and the respective bulk averaged values of the first moments are however small, as long as one is sufficiently far away from thermal equilibrium.9

In the discussion of the first moment results, we will therefore only apply the notation 〈ΔE〉 for both quantities.

Comparisons for the bath gases argon and CO2 can be found in Figs. 1 and 2.

A similar plot for the helium case was already shown in Fig. 22 of ref. 8, which also contains the Lennard-Jones collision numbers required for converting the CET rate constants into the respective moments.

UVA experiments

The first CET experiments on azulene employing the UV absorption technique were carried out by Hippler et al.7

An almost energy independent 〈ΔE〉 was found for all bath gases, except for energies smaller than roughly 10 000 cm–1, where eventually 〈〈ΔE〉〉 = 0 has to be fulfilled in thermal equilibrium (〈Ethermal = 979 cm–1 for azulene).

Later, these experiments were extended to low11 and high excitation energies.12

It is immediately clear from Figs. 1 and 2 and also Fig. 22 of ref. 8 that the older UVA data lie considerably higher than the KCSI values over wide energy ranges.

Very recently, new high quality UVA results have been obtained for temperatures ≥373 K and excitation energies <20 000 cm–1.13

The UVA decay curves showing exceptionally low noise were almost monoexponential, which results in a linear energy dependence of 〈ΔE〉 over the whole energy range, due to the fact that the calibration curve in this energy region is practically linear (Fig. 3).18

For a direct comparison with the KCSI results of ref. 8 we calculated time-dependent UVA profiles from the KCSI data via: ε(t) = ∫∞0g(E,t)ε(〈E〉)dEi.e. a convolution of the g(E,t) distributions from a KCSI master equation analysis with a given UVA calibration curve ε(〈E〉) was used to reconstruct ε(t).

Such profiles can be directly confronted with the experimental UVA traces.

All UVA measurements for azulene rely on the empirical calibration data shown in Fig. 3.

The relationship between the absorption coefficient ε and the average internal energy 〈E〉 of the azulene molecules was established from thermal experiments in heated gas cells (open circles) or shocktubes (filled circles) and measurements after laser excitation (open squares).

For energies below 25 000 cm–1 the calibration curve is practically linear and can be well represented by the solid line in Fig. 3 using the expression:13

At higher energies, the calibration curve levels off and there is experimental evidence that it stays constant at least up to roughly 50 000 cm–1.12

To describe this behavior we modified the calibration curve given in ref. 19 by a factor of 2.2 to obtain agreement with the experimental points:

Above 31 790 cm–1, a constant value of ε = 17 650 l mol–1 cm–1 was assumed, so that the experimentally observed “saturation behavior” of the absorption coefficient is correctly described (dashed line in Fig. 3).

Fig. 4 shows a comparison of the most recent UVA data13 for the colliders helium, argon and CO2 with our master equation fits employing the P(E′,E) data from Table IV of .ref. 8

Note that the UVA data were recorded at slightly elevated temperatures (T = 373 K) so our fits had to be rescaled by the ratio of the respective KCSI and UVA Lennard-Jones collision numbers.

The overall fit to the UVA traces is very good.

Despite this, there are still visible deviations in the first moment plots, see, e.g., Fig. 2 (and especially also Fig. 22 of .ref. 8)

In Fig. 4(a), the decay of the helium UVA signal is slightly faster than in the KCSI fit.

In Fig. 4(c), the very slight differences in the decay of the experimental and simulated UVA signals for CO2 result in the different slopes of the energy dependence of 〈ΔE〉 in Fig. 2.

These deviations might be either due to a slight temperature dependence of the energy transfer or remaining uncertainties in the UVA calibration curve, as demonstrated in our earlier study.9

The minor differences for helium can probably be explained by a small contamination of efficient azulene self-collisions in the UVA experiments.

Note that the experimental UVA (and IRF) traces are most susceptible to self-collision effects in the case of inefficient colliders.

At this point we would like to comment on the clear deviations of 〈ΔE〉 found in older UVA measurements.

The argon data of Hippler and Troe in Fig. 1 exceed the KCSI values considerably, as it is also found for other colliders.

The even larger discrepancy for helium as a collider is documented as Fig. 22 in .ref. 8

In Fig. 5(a) the UVA trace for azulene* + argon is shown,11 which decays clearly faster than the simulated fit (dashed curve) using the KCSI parameters from Table IV of ref. 8 employing eqn. (3) as calibration curve.13

As in the helium case,8 the argon data seem to be contaminated by a visible amount of azulene self-collisions.

Such a contribution of collisions with a second bath gas partner in the mixture can be modeled by adding a second term to our general P(E′,E) expression with a parametric exponent in the argument [eqn. (11) in ref. 9]:

For the parameters C0, C1 and Y1 we use the optimized values from the KCSI analysis.8

The experimental conditions given in ref. 11 [P(azulene)/Ptotal = 0.6%] suggest a value x = 6 × 10–3 as a reasonable estimate for the contribution of efficient azulene self-collisions.

From the 〈ΔE〉 data in ref. 17 one can extrapolate that the behavior of azulene as collision partner should be very close to that of CHT.

We therefore approximated the azulene parameters B0, B1 and Y2 in the small second term of eqn. (5) by using the CHT values from Table IV of .ref. 8

The resulting simulation in Fig. 5(a) now shows good agreement with the original experimental trace.

Thus, it is strongly suggested that the UVA trace is contaminated by azulene self-collisions, which have not been considered in the extraction of the UVA 〈ΔE〉 values in .ref. 11

Fig. 5(b) shows another set of UVA measurements for azulene deactivation by the collider argon from Damm et al.12

Two sets of data are available for excitation at 193 and 248 nm.

The 248 nm curve was shifted so that the decay of both curves coincide.

Such a behavior is expected, because the UVA calibration curve appears to be essentially flat for energies above 30 000 cm–1 (Fig. 3).

Therefore the signal at 193 nm excitation shows an additional constant portion at early times before showing the same decay as the 248 nm signal.

Again we tried to fit the UVA traces using our KCSI parameters, but this time employing eqn. (4) (with the aforementioned constant extension at high energies), which should be valid up to excess energies around 50 000 cm–1.

The results are also included in Fig. 5(b) (dashed and dotted curves).

While the early part of the traces are very well reproduced, there are deviations at longer times.

The UVA traces decay faster.

This is consistent with the slightly higher –〈ΔE〉 at lower energies already seen in Fig. 1.

Again, an influence of self-collisions is very likely.

In these UVA measurements the self-collision contributions were smaller [P(azulene)/Ptotal = 0.12%].

Nevertheless, including a second term in our P(E′,E) to account for this (x = 1.2 × 10–3 and CHT parameters for azulene as above) leads to a simulation with better agreement (solid line).

One can therefore conclude that the existing body of UVA data is in full agreement with the KCSI results, and the differences observed are due to self-collisions in the older measurements and possibly remaining uncertainties in the UVA calibration curve.

The issue of calibration curves was already discussed at length in our earlier publication for toluene, where the seemingly very large discrepancies between the UVA and KCSI results could be quantitatively traced back to small uncertainties in the high energy portion of the UVA calibration curve9.

IRF experiments

An extended set of time-resolved IR fluorescence measurements for azulene was obtained by Barker and co-workers.6,14

These data were later reanalyzed by the same group employing an “improved calibration curve”.

Their reevaluation changed the original 〈ΔE〉 by up to 50% and yielded values at the energies 〈E〉 = 13 943 and 24 023 cm–1.17

The results for argon and CO2 of refs. 14 and 17 are included in Figs. 1 and 2 as dotted curves, including a linear interpolation for the reevaluated curve.

Similar to the KCSI results, the IRF measurements predict a linear energy dependence of 〈ΔE〉.

However, the IRF –〈ΔE〉 curves are much higher than the corresponding one from KCSI.

To analyze these differences, we again take the “direct” approach by comparing our master equation results with the IRF trace for azulene + argon given in .ref. 14

This is done in Fig. 6.

Experimental calibration points and a calculated calibration curve relating the IRF emission intensity and the energy E of the azulene molecules were given in refs. 17 and 20.

For our simulations we fitted the calibration curve by a fifth order polynomial:

The time-dependent IRF simulations employing KCSI g(E,t) can then be obtained via: I(t) = ∫∞0g(E,t)I(E)dE

Note that in the IRF experiments the fraction of azulene molecules was roughly 10%, and therefore much higher than in the UVA measurements discussed above.

For that reason, measurements at different argon partial pressures together with a “linear mixing rule” approach had to be applied in ref. 14 to extract values for pure argon conditions.

Not surprisingly, our KCSI master equation fit in Fig. 6 using eqn. (5) and the argon parameters from ref. 8 decays much slower (dashed line).

However – like for the UVA data in the preceeding section – perfect agreement is obtained when taking contributions of azulene-azulene collisions into account (solid line).

One can only speculate, why we obtain agreement between the IRF signal and our master equation simulation in Fig. 6, when at the same time in Fig. 1 the 〈ΔE〉 curves taken from refs. 14 and 17 show strong deviations.

One possible explanation might be that the “linear mixing rule” procedure employed in ref. 14 for different azulene/argon mixing ratios introduces considerable uncertainties when extrapolating to pure argon conditions, greatly affecting the accuracy of the 〈ΔE〉 curves.


We have demonstrated that the body of considerably varying data on 〈ΔE〉 in azulene reported in the literature is in an astonishing complete agreement with the more recent “self-calibrating” KCSI data8 if reanalyzed properly.

Analogous findings have been reported earlier for the CET of toluene.9

Together this indicates that a recommended preference of the precision 〈ΔE〉 data from KCSI marks a decisive step in defining strict, low error boundary conditions for the application in chemical reaction kinetics.

Deviations and spread in reported 〈ΔE〉 values from UVA and IRF reported in earlier studies (see Figs. 1 and 2 and ref. 8) are most probably due to the following two systematic errors in the UVA and IRF experiments, namely (1) contributions of efficient azulene self-collisions which were not correctly accounted for and (2) unavoidable uncertainties in the UVA and IRF calibration curves.

The second point can have drastic effects, as we have already shown in detail for toluene in .ref. 9

We believe that similar deviations between the results of different CET methods for other systems are mainly due to the same two effects.

The following advice for using large molecule CET data for kinetic applications can therefore be given: If available, KCSI data should be used, as they have the highest accuracy and can be – like in the azulene case presented here and in ref. 8 – independent of any external calibration.

Simply taking an average of several available measurements from different sources cannot be recommended in the light of the above discussion.

Experiments using high partial pressures of the parent molecule should be checked for the contribution of self-collisions, and corrected accordingly.

If possible, one should also assess the accuracy of the calibration curves used, as this can introduce a substantial source of error.