Negative Ions of p-Nitroaniline: Photodetachment, Collisions, and Ab Initio
Calculations
Supplementary Information
Byron H. Smitha, Angela Buonauguriob, Jing Chenb, Evan Collinsb, Kit H. Bowenb, Robert N.
Comptona,c, Thomas Sommerfeldd
a
Department of Physics, Univ. of Tennessee, Knoxville, TN 37996
b
Department of Chemistry, Johns Hopkins Univ., Baltimore, Maryland 21218
c
Department of Chemistry, Univ. of Tennessee, Knoxville, TN 37996
d
Department of Chemistry & Physics, Southeastern Louisiana Univ., Hammond, LA, 70402
Modeling
The raw data is output from the Analyst software as a peak amplitude ascertained through the
integration of a time-averaged mass peak paired with the corresponding collision energy.
Because the target gas in the collision process is not stationary, the collision energy associated
with a peak must be corrected. This energy can done by following Nalley et al.1
′
∞
𝑣 → 𝑣 = ∫ 𝑣𝑅 𝐹𝑅 (𝑣𝑅 )𝑑𝑣𝑅 .
0
for
∞
(𝑣𝑅 − 𝑣0 )2
(𝑣𝑅 + 𝑣0 )2
𝑣𝑅 1
𝐹𝑅 (𝑣𝑅 ) = ∫
{exp [−
] − exp [−
]} 𝐹(𝑣)𝑑𝑣
𝑣02
𝑣02
−∞ 𝑣 √𝜋𝑣0
where 𝑣0 is the mean of the projectile velocity distribution, 𝐹(𝑣). The velocities of each
projectile are assumed to be normally distributed about the collision energy with a standard
deviation consistent with the uncertainty in the collision energy. The initial distribution of
projectile velocities has been taken to be a Normal distribution with a mean equal to the collision
energy. The variance of this distribution is chosen arbitrarily; because the Normal distribution is
symmetric about the mean, the resultant transformed x-axis is independent of the variance.
Similar results have been proven with alternative methods2 and have been used thoroughly in
collisional charge transfer experiments.3
The non-linear modeling of energy resolved CID data has been contentious.4 Although several
traditional methods exist to calculate the model parameters, a recent analysis by Narancic et al.5
has shown that non-linear regression using the Marquardt-Levenburg optimization may be
insufficient. The authors cite the numerical estimation of derivatives used in the optimization
procedure.
As an alternative to Marquardt-Levenburg optimization, a non-parametric optimization algorithm
such as the Genetic Algorithm may be implemented. We instead use the Metropolis Algorithm to
create a Bayesian posterior distribution of the model parameters. In this way we not only obtain
information regarding the parameter estimates, but the full variance structure of each parameter
and the correlation as well. To use this algorithm we first use a coarse grid search over the
parameter space for a minimum error solution. These initial values were then used as the mean of
a Normal prior probability distribution for each parameter. A comparison of our results is
performed with CRUNCH produced by the Armentrout et al. group (Figures 1 and 2).4,6-19
Parameter error estimates in regression can be analytically derived in the case of linear
regression. For non-linear regression parameter error estimates again arise from the numerical
estimates of the derivatives. In contrast, the Bayesian paradigm allows for a direct estimate of the
parameter error through the posterior probability distribution of the parameter estimates.
Furthermore, the energy offset of the dependent variable observed in the fitting of I3− dissociation
can be incorporated in the parameter error through repeated fitting of the data after the
introduction of random noise on the order of the offset or as a hyperparameter associated with
the independent variable.
It is important to note that this empirical model shows a high level of inter-parameter correlation.
This is not uncommon to non-linear models, but may play a role in the optimization procedure.
In particular the parameters 𝜎0 and 𝐸0 have a correlation around 0.9. In order to work around this
correlation, the variance of each prior probability distribution was modified until the correlation
was minimized. This correlation will affect the modeling procedure due to multiple solutions
(and thus a wide spread of threshold energies) giving equivalent curves with respect to error.
Due to the existence of contact and surface potentials, specified voltages cannot be assumed to
solely characterize the ion laboratory energy. Calibration of the ion energy as well as the energy
spread was thus performed using the CID of I3− , i.e.
I3− → I− + I2
(1)
Previous studies of this reaction place the threshold in the center of mass collision energy at
1.31±0.06 eV using photodissociation as well as collision induced dissociation.20-22 The observed
and uncalibrated threshold for reaction (1) above is shown in Figure 3. The tri-iodide anion
vibrational frequencies were estimated from a combination of calculation and experiment.21,23 A
dilute solution of iodine in acetonitrile was tested repeatedly for the presence of I3− by iteratively
increasing the concentration until enough ion signal was present to carry out a thorough analysis
of the dissociation threshold. The collision energy was varied in the lab frame from 4 to 27 eV in
steps of 0.5 eV.
Analysis of Error in CID
An assessment of the error in the modeling of CID data may be broken into two bases: Error
endemic to the modeling and that which results from experimental error. The former of these
may be addressed directly through the Bayesian MCMC by considering the 95% credible
intervals generated through the simulation. In this respect there is no inherent distribution
dependence (such as a Normal dependence in most parameter error estimates) because the
posterior distribution is generated numerically. For I3− this modeling error is 0.03 eV while it is
0.07 eV for each of the p-nitroaniline fragments. This estimate may be further corroborated
through repeated applications of the algorithm and analysis of the resultant fit parameters.
Furthermore, inclusion of a kinetic shift through the use of an RRKM reaction rate would result
in a decrease in the modeled threshold typically on the order of 0.2 to 0.3 eV.4-19 Absolute values
are not included here due to instability in the model resulting in unreasonable values for the fit
parameters. Nevertheless, we include a plot of the calculated reaction rates in CRUNCH (Figure
2). Note that the RRKM reaction rates corresponding to the loss of NO2- are relatively small
indicating a more dramatic change in the expected CID threshold.
A comprehensive analysis of experimental errors would require careful measurements of the
kinetic energy distribution of the projectile ions, the collision cell pressure, collision cell
effective length (the length over which collision occur), and the time of flight of fragment
anions. Each of these in turn may add to the error of the dissociation threshold. The zero of the
distribution can be assessed through a retardation analysis. Under the assumption of linearity in
the scale (dictated through uniform potential differences in the instrument), the mean of the
distribution may then be ascertained. A stronger analysis is to use a calibrating sample (in this
case I3− ) in order to identify the scale.
By utilizing the independent axis transformation described above, the effect of the FWHM of the
projectile distribution is minimized because it is integrated over. Therefore, the convolution
process is closer to Chantry’s single integral than to Tiernan’s method.2,3
Sources
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(1978).
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5. S. Narancic, A. Bach, and P. Chen, J. Phys. Chem. A 111, 7006 (2007).
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19. P. B. Armentrout, J. Chem. Phys. 126, 234302 (2007).
20. K. Do, T. P. Klein, C. A. Pommerening, and L. S. Sunderlin, J. Am. Soc. Mass Spectrom.
8, 688 (1997).
21. A. A. Hoops, J. R. Gascooke, A. E. Faulhaber, K. E. Kautzman, and D. M. Neumark, J.
Chem. Phys. 120, 7901 (2004).
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109, 9928 (1998).
Figure 1. Non-linear modeling of the cross-section of I − dissociation from I3− colliding with
argon.
8
6
log(Reaction Rate)
4
2
0
-2
0.00
1.00
2.00
3.00
4.00
5.00
6.00
5.00
6.00
-4
-6
-8
Center of Mass Energy (eV)
2
0
log(Reaction Rate)
0.00
1.00
2.00
3.00
4.00
-2
-4
-6
-8
-10
-12
Center of Mass Energy (eV)
Figure 2. RRKM reaction rates versus the center of mass collision energy for a) loss of NO and
b) loss of NO2− .
Figure 3. A comparison of the Bayesian MCMC fit (solid line) to CRUNCH’s MarquardtLevenburg non-linear regression (dashed line) for the loss of NO.
Figure 3. A comparison of the Bayesian MCMC fit (solid line) to CRUNCH’s MarquardtLevenburg non-linear regression (dashed line) for loss of NO−
2.
Figure 4. Transition state structures used in the CRUNCH calculations.
Figure 5. Potential energy curve resulting in the loss of NO−
2 calculated using a mod-redundant
algorithm at the B3LYP/6-31+G* level of theory. Note that the potential energy curve is strictly
uphill as expected.
Table 1. A comparison of the fit parameters resulting from the Bayesian MCMC used here and
CRUNCH’s fit.
Channel
Fit Method
σ0
E0
n
Bayesian MCMC
17.30
2.36
1.91
CRUNCH
9.711
2.16
2.40
Bayesian MCMC
11.43
3.82
2.14
CRUNCH
10.99
3.87
2.15
NO Loss
NO2¯ Loss