Helium induced pressure broadening and shifting of HCN hyperfine transitions between 1.3 and 20 K

Article abstract of DOI:10.1063/1.1895905

We have measured the helium induced pressure broadening and shifting of the distinct hyperfine components of the j=1←0 and j=2←1 transitions of HC N14 at temperatures between 1.3 and 20 K. The HCN molecules were cooled to these temperatures using the collisional cooling technique. As a test of this cooling technique we measured the Doppler contribution to the spectral lines, and these measurements confirm that the molecules are at the same temperature as the walls of the spectroscopic cell. We observed that the hyperfine components of the 2←1 transition have distinct broadening coefficients that differ from one another by as much as 5%. The measured differences are in reasonable agreement with theoretical predictions. We have also performed molecular scattering calculations on three He-HCN potential energy surfaces in order to compare our results with theoretical expectations. At the lowest temperatures these calculations predict broadening coefficients that are considerably larger than the measured coefficients. We have previously found a similar discrepancy for two other molecules at these low temperatures, and we discuss possible experimental and theoretical origins for this persistent discrepancy.

Full text of DOI:10.1063/1.1895905

Helium induced pressure broadening and shifting of HCN hyperfine transitions
between 1.3 and 20 K
Theodore J. Ronningen and Frank C. De Lucia
Citation: The Journal of Chemical Physics 122, 184319 (2005); doi: 10.1063/1.1895905
View online: http://dx.doi.org/10.1063/1.1895905
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/18?ver=pdfcov
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THE JOURNAL OF CHEMICAL PHYSICS 122, 184319 ͑2005͒
Helium induced pressure broadening and shifting of HCN hyperfine
transitions between 1.3 and 20 K
Theodore J. Ronningen and Frank C. De Luciaa͒
Department of Physics, The Ohio State University, Columbus, Ohio 43210
͑Received 13 October 2004; accepted 28 February 2005; published online 11 May 2005͒
We have measured the helium induced pressure broadening and shifting of the distinct hyperfine
components of the j=1 0 and j=2 1 transitions of HC14N at temperatures between 1.3 and 20
K. The HCN molecules were cooled to these temperatures using the collisional cooling technique.
As a test of this cooling technique we measured the Doppler contribution to the spectral lines, and
these measurements confirm that the molecules are at the same temperature as the walls of the
spectroscopic cell. We observed that the hyperfine components of the 2 1 transition have distinct
broadening coefficients that differ from one another by as much as 5%. The measured differences
are in reasonable agreement with theoretical predictions. We have also performed molecular
scattering calculations on three He–HCN potential energy surfaces in order to compare our results
with theoretical expectations. At the lowest temperatures these calculations predict broadening
coefficients that are considerably larger than the measured coefficients. We have previously found
a similar discrepancy for two other molecules at these low temperatures, and we discuss possible
experimental and theoretical origins for this persistent discrepancy. © 2005 American Institute of
Physics. ͓DOI: 10.1063/1.1895905͔
I. INTRODUCTION
have used this technique to cool a range of small molecules
in order to study their millimeter-wave pressure broadening
and shifting4,5 and their rotational relaxation using time-
resolved double resonance measurements.6 Collisional cool-
ing has also been adapted by other laboratories for infrared
spectroscopic measurements,7 infrared line shape
measurements,8 and loading atoms and molecules into a
magnetic trap.9 In this paper we expand on this body of data
by reporting the helium induced pressure broadening and
shifting of the two lowest energy rotational transitions of
HCN between 1.3 and 20 K.
Over the last two decades there has also been a marked
increase in the accuracy of the potential energy surfaces de-
scribing the energetics of small molecule interactions. Mod-
ern ab initio techniques can generate surfaces that predict the
bound state energies of small van der Waals complexes to
within 0.1 cm−1.10 Given that the molecular scattering theory
needed to predict pressure broadening parameters is consid-
ered to be well established, it seems that the quality of the
potential energy surface is most likely to limit the accuracy
of pressure broadening predictions at low temperatures.
However, advances in the quality of surfaces have not
brought about good agreement between our line shape mea-
surements and theoretical predictions. To the contrary, for the
two systems that have been compared in greatest detail the
theoretical predictions diverge from our measurements below
30 K.4,5 The present study has once again found a substantial
disagreement between theory and experiment.
The gas phase dynamics of small molecules at tempera-
tures below 30 K has been important in the astronomical
community for several decades. Two recent advances in mo-
lecular physics, helium droplet solvation1 and the generation
of ultracold molecules from Bose–Einstein condensates,2
have expanded further the importance of understanding low
energy molecular interactions. This temperature regime is
also of inherent interest since a molecule’s kinetic energy is
comparable to its rotational energy, and the collision dynam-
ics are therefore dominated by resonant energy transfer be-
tween translation and rotation. The theoretical framework for
understanding these interactions is considered to be well es-
tablished, but experimental studies on gas phase molecular
dynamics at particularly low temperatures are relatively rare.
This is due in large measure to the fact that all molecules
except molecular hydrogen and carbon monoxide have a
vanishingly small vapor pressure below 30 K and therefore
cannot be studied under static equilibrium conditions.
For the last two decades3 our laboratory has employed a
collisional cooling technique to perform gas phase dynamics
experiments at temperatures as low as 1.25 K. In this tech-
nique a small amount of warm molecular vapor is injected
into a reservoir of cold buffer gas. The warm molecules col-
lide with the buffer gas, cool to the temperature of the res-
ervoir, and remain in the gas phase until they reach the res-
ervoir wall. By continuously flowing the vapor into the cell,
a quasiequilibrium sample of cold vapor is generated. We
The He–HCN system has several characteristics that
make it a good candidate for clarifying the source of this
discrepancy. From an experimental perspective, the presence
a͒
Electronic mail: fcd@mps.ohio-state.edu
0021-9606/2005/122͑18͒/184319/12/$22.50
122, 184319-1
© 2005 American Institute of Physics
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184319-2
T. J. Ronningen and F. C. De Lucia
J. Chem. Phys. 122, 184319 ͑2005͒
of closely spaced but fully resolved nuclear hyperfine struc-
ture in the HCN rotational transitions is useful in assessing
the precision of our measurements and analysis. For in-
stance, we can verify the frequency calibration by measuring
the distinct line centers across the frequency sweep, and in
the analysis of the data the presence of multiple lines allows
for a clean separation of the absorption line shapes from
background power variations. The presence of multiple hy-
perfine lines also allows us to measure multiple line shape
parameters in each spectrum that can be compared to one
another.
From a theoretical perspective, the potential energy sur-
face for He and HCN is fairly well established by a combi-
nation of ab initio calculations and empirical refinements
based on spectroscopic measurements of the bound
complex.11–13 The most recent of these surfaces was modi-
fied so that its bound state predictions reproduce the experi-
mental spectrum within about 100 kHz. Both bound state
energies and low temperature dynamics are strongly depen-
dent on the attractive region of the potential energy surface,
and so it seems reasonable to expect a surface that predicted
one type of data well to also predict the other with reason-
able accuracy. Also, comparing the dynamics predictions
from several potential energy surfaces may indicate how sen-
sitive these predictions are to the details of the surface.
In Sec. II we briefly review the collisional cooling tech-
nique and other experimental methods. We also introduce a
new data analysis method that fits multiple spectra simulta-
neously in order to improve the precision of the retrieved
parameters. We have carried out scattering calculations to
determine the pressure broadening and shifting predictions
from three potential energy surfaces, and the details of these
calculations are presented in Sec. III. Section IV compares
these predictions to the experimental measurements and ex-
amines possible explanations for the discrepancy between
them.
surement is discussed further in Sec. IV. The details of the
apparatus used to carry out the collisional cooling have been
published previously.4
Considerable effort was made to understand and, if pos-
sible, eliminate all contributions to the line shape other than
Doppler broadening and pressure broadening. Most other
possible distortions are related to the millimeter-wave
source. Our source, a phase-locked klystron followed by a
diode multiplier, has a roughly Gaussian profile with a half-
width at half-maximum ͑HWHM͒ of about 1 kHz, and the
nearest sidebands are 3 kHz away and 10 dB down. This
profile is narrow enough that it does not significantly affect
the absorption line shape, which, even at its narrowest Dop-
pler limit, has a HWHM of 8 kHz. During data collection the
source is typically swept through a 10 MHz range in 10 ms,
and the transmitted power is recorded directly without any
modulation applied. This typical sweep rate, though, is too
high for the narrowest lines. The molecular polarization of an
absorbing sample relaxes to a steady state with a half-life of
approximately 2/͑␲ HWHM͒. For the narrowest lines the
polarization does not relax to a steady state within the time it
interacts with the radiation unless the sweep rate is de-
creased. We did observe asymmetric distortions to the line
shapes in some low pressure data, and so these data were
collected again at a reduced sweep rate. The detected signal
is also passed through electronic amplifiers and bandpass
filters that are selected to avoid distorting the line shape.
The most tenacious source of unwanted broadening for
HCN is power broadening. A line is power broadened when
the rate of radiative population excitation exceeds the rate of
collisional relaxation. This is also sometimes referred to as
saturation.14 Power broadening is a particular problem with
HCN because the radiative transition rate is increased by its
large dipole moment, 2.985 D as compared to 0.109 D for
CO. Keeping the source power low enough to avoid power
broadening considerably reduces signal to noise, particularly
for the j=2 1 transition, and also sets a lower limit on the
helium pressure since the rate of collisional relaxation is di-
rectly proportional to the pressure. The presence of power
broadening was most easily identified by its impact on the
relative intensities of the hyperfine components. The most
intense components power broaden, and therefore have a re-
duced peak absorption, at lower powers than the less intense
components. This effect was also used as a signature to iden-
tify power broadened data sets so that they could be ex-
cluded from the final data analysis.
II. EXPERIMENTAL METHODS
A. Data collection
The collisional cooling technique used to cool the HCN
has been described elsewhere3 and will only be briefly re-
viewed here. A small amount of warm HCN is injected into
the cold cell approximately 2.5 cm away from the radiation
interaction region. As the molecules traverse this distance
they collide with cold He atoms and are brought into thermal
equilibrium. Even with the He pressure as low as 5 mTorr the
HCN molecules undergo several thousand collisions before
reaching the radiation beam. From a classical viewpoint, no
more than a few hundred collisions are necessary to remove
the excess kinetic energy from the HCN, and the order of
magnitude separation between these numbers gives us confi-
dence that the HCN molecules are thermalized before they
interact with the radiation. In this study we have also verified
that the kinetic temperature of the HCN molecules is the
same as the temperature of the cold cell by measuring the
Doppler contribution to the line shape. This Doppler mea-
The HCN sample was synthesized in our lab by reacting
KCN with H2SO4 and trapping the HCN vapor in a cold trap.
The purchased He gas had a stated purity of 99.995%. Fur-
ther purification was not necessary for either sample because
the HCN rotational transitions are well separated from any
possible contaminant transitions and the low experimental
temperature causes any impurities in the He to condense on
the cell walls. The HCN/He dilution ratio was kept below
1/1000 so that self-broadening can be ignored. The helium
pressure was measured with a Baratron style pressure gauge
with a range of 0.01–1000 mTorr. The gauge accuracy is
quoted by the manufacturer as ±0.12% of the reading over
the full range. Since the pressure was measured at room tem-
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184319-3
Helium induced pressure broadening
J. Chem. Phys. 122, 184319 ͑2005͒
by power broadening were excluded from the final fit. A
typical fit included ten spectra at helium pressures between 2
and 50 mTorr.
Our fitting routine uses nonlinear least-squares fitting to
determine the best fit for a range of physical parameters, but
the line shape parameters of greatest interest are the Doppler
width, line center at zero pressure ␯0, pressure broadening ␥,
and pressure shifting ␩. In a fitting procedure that fit each
spectrum separately, each of these parameters would be de-
termined multiple times, and the distinct values would then
have to be averaged using an appropriate weighting scheme.
A multispectrum fit determines the single value for each pa-
rameter that is most consistent with all of the data. Multi-
spectrum fitting is now regularly applied to analyze both
laboratory and field measurements in the infrared region, and
there is no reason it cannot be applied in the microwave and
millimeter-wave regions as well. We found this fitting
method to be particularly useful in identifying and diagnos-
ing conflicts between our data and the assumed line shape
model. For instance, distortions caused by power broadening
at low pressures were quite evident when fit simultaneously
with high pressure data that were not power broadened. If
each spectrum were fit separately, the distorted lines would
still fit to a Voigt model reasonably well, but the broadening
and Doppler parameters retrieved would be substantially dis-
torted. In contrast, with multispectrum fitting the width pa-
rameters are still accurate and the residual error in the power
broadened data is evident.
FIG. 1. Typical recorded spectra of the two HCN rotational transitions in-
vestigated in this study. The solid curve overlaying the data points is the best
fit, and the points below the data are the fit residual. A slowly varying
baseline, fit to a fourth-order polynomial, has been subtracted from both the
fit and the data.
perature it was necessary to apply a transpiration correction
in order to determine the pressure in the cold sample cell. In
our system the warm and cold regions are connected by a
stainless steel tube, and appropriate correction factors for this
setup were determined by Chen et al.15 based on measure-
ments by McConville.16 The cell temperature was measured
with a germanium resistor type thermometer with a manu-
facturer’s stated accuracy of ±5 mK. A typical data collec-
tion cycle, which averages several thousand frequency
sweeps, required 2–5 min. Over this time period the stability
of the temperature and pressure was consistently better than
1%, and any data sets that did include variations larger than
this were discarded prior to analysis.
In our model all of the hyperfine components are con-
strained to share a single Doppler width, but the other three
line shape parameters are allowed to vary for each compo-
nent. Forcing the components to share a Doppler width is
reasonable because there is no reason to expect the distinct
hyperfine states to have distinct velocity distributions. There
was no indication in fitting the data that this assumption was
violated. In the initial fits of our data we fit for the best
values of ␯ for each component. Our fits agreed with previ-
0
ous beam maser measurements18 but had less precision. We
B. Data analysis
tested the effect of holding ␯ fixed during the fit and found
that this did improve the precision of the retrieved shifting
parameters. All of the broadening and shifting parameters
0
A typical recorded spectrum is shown in Fig. 1. Each
spectrum is recorded at a particular temperature and pres-
sure, but in order to improve the accuracy of the fit and
remove some of the correlation among fit parameters we
used a multispectrum fitting procedure modeled on the work
of Benner et al.17 Our multispectrum fit combines all of the
spectra collected at a particular temperature and fits them
simultaneously. The fitting routine allows for variation in the
absorption line shape due to changes in helium pressure,
sample concentration, source power, sweep time, sweep
width, and electronic signal processing. The routine also al-
lows for the presence of a variable number of hyperfine lines,
each at a distinct frequency but with fixed relative intensities.
The absorption coefficient of each line is modeled as a
Voigt function, and when lines overlap the Voigt profiles are
simply added without taking into account any possible line
mixing. Ignoring line mixing is reasonable at the particularly
low pressures used in this experiment ͑the helium pressure is
never higher than 80 mTorr͒. This model does not take into
account any broadening due to the radiation source described
in Sec. II. Any spectra that appeared to have been distorted
reported here are from fits with each ␯ fixed.
0
III. THEORETICAL CALCULATIONS
A. Molecular scattering calculations
Molecular pressure broadening predictions for systems
in which only binary scattering events are important can be
calculated from the binary scattering matrix.19 We employed
20
the MOLSCAT molecular scattering program to calculate
scattering matrices and pressure broadening predictions for
three recently published He–HCN potential energy surfaces.
For low temperature predictions it is possible to perform
close coupling calculations at all of the relevant collision
energies. Close coupling calculations solve the quantum me-
chanical scattering equations exactly for a given potential
energy surface.
The three potential energy surfaces used in these calcu-
lations will be referred to throughout the rest of the paper as
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184319-4
T. J. Ronningen and F. C. De Lucia
J. Chem. Phys. 122, 184319 ͑2005͒
TABLE I. Energy spacing between successive scattering calculations within
the specified range of relative collision energies.
Range ͑cm−1
͒
Spacing ͑cm−1
͒
0.01–0.25
0.25–3.05
3.05–5.00
5.00–7.00
7.00–10.20
10.20–20.20
20.20–30.10
0.0010
0.0025
0.0125
0.0200
0.0400
0.0650
0.1250
0.3000
4.5000
30.10–102.00
102.00–247.50
the Atkins1E8,11 Toczylowski,12 and Harada13 surfaces. The
Toczylowski surface is the only one of these three that is
purely ab initio. The other two used measured microwave
and millimeter-wave transitions of the He–HCN van der
Waals complex as fitting parameters to refine the surface. All
three surfaces treat the HCN geometry as frozen and there-
fore only depend on the radial and angular separation of He
and HCN. The Atkins1E8 surface is a parametrized surface.
The parameters were determined from a combination of
theory and least-squares fitting to a limited set of spectro-
scopic data. Atkins and Hutson considered two functional
forms, designated as 1E8 and 2E8 in their paper, and found
that both forms reproduced the spectroscopic data with simi-
lar accuracy. The 2E8 surface is extremely similar to the
Toczylowski surface, and so we chose to perform calcula-
tions only on the 1E8 surface and assume that the 2E8 pre-
dictions are similar to the Toczylowski predictions. The
Harada surface uses the Toczylowski surface as a starting
point and scales the well depth and radial separation to bring
its bound state predictions into agreement with the measured
spectroscopic transitions. The derivation of the Harada sur-
face included 20 measured transitions that were not included
in the derivation of the Atkins 1E8 surface, and so only the
Harada surface accurately predicts all of the spectroscopic
measurements.
Each surface was input into MOLSCAT using its built-in
VRTP routine. This routine projects the surface onto appropri-
ate angular functions, Legendre polynomials in this case, at
each radial distance needed in calculating the propagation
integrals. For these three surfaces test calculations indicated
that it was necessary to include Legendre polynomials up to
ninth order and to use an eleven point Gauss integration in
order to accurately represent the surface.
For all three surfaces we calculated the scattering as a
function of the relative collision energy between 0.01 and
215 cm−1. Since the scattering below about 30 cm−1 is domi-
nated by scattering resonances, it is necessary to perform
calculations at an increasingly dense grid of energies as the
collision energy is reduced. Table I lists the combinations of
energy range and grid spacing we found were necessary to
adequately represent the shape of all of the resonances. The
radial integrals were calculated with a hybrid log-derivative
propagator.21 We performed a series of test calculations at
energies between 0.425 and 2.425 cm−1 to determine the ap-
propriate control parameters for this propagator. We verified
FIG. 2. Predicted pressure broadening cross sections as a function of the
relative collision energy. The energy is plotted on a log scale to emphasize
that the scattering resonances become increasingly narrow as the energy
decreases. These cross sections were calculated from the Harada potential
energy surface using the MOLSCAT computational package as described in the
text. Note that the j=2 1 cross section plotted here is from a calculation
that does not take any hyperfine quantum numbers into account. Taking
these quantum numbers into account generates a cross section with the same
shape but a slightly larger magnitude.
that the calculations converged within 1% with respect to the
propagator step size, the maximum range of integration, and
the changeover point for the two propagators. The number of
rotational states included in the basis was increased with in-
creasing energy in order to include at least three closed chan-
nels at each energy. The maximum total angular momentum
included at each energy was determined on the fly by the
convergence of the elastic scattering cross sections to within
0.3 Å2 and the inelastic cross sections to within 0.005 Å2.
All calculations were performed on a Cray SV1ex main-
tained by the Ohio Supercomputer Center.
A portion of the calculated pressure broadening cross
section ␴ as a function of the relative collision energy E is
plotted in Fig. 2. The cross section as function of temperature
T is determined by averaging the cross section over the dis-
tribution of relative collision energies, and is calculated us-
ing
ϱ
2
1
E
␴͑T͒ =
dE exp −
E␴͑E͒,
͑1͒
ͩ ͪ
ͩ ͪ
͵
kBT
kBT
0
where kB is Boltzmann’s constant. To perform this integra-
tion we used a linear interpolation between the calculated
energies. The range of relative energies calculated allows for
the calculation of predictions from 1 to 200 K.
In order to compare these predictions with measure-
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184319-5
Helium induced pressure broadening
J. Chem. Phys. 122, 184319 ͑2005͒
FIG. 4. Predicted pressure broadening cross sections as a function of tem-
perature for the hyperfine components of the j=2 1 transition. These pre-
dictions were calculated using the Harada potential energy surface. The
heavy curve below the hyperfine predictions is the cross section calculated
without taking the hyperfine structure into account.
FIG. 3. Frequency offsets and relative intensities of the nuclear hyperfine
components of the two HC14N rotational transitions studied here. Each hy-
perfine component is labeled by its F quantum numbers. The horizontal bars
indicate the 4.0 K Doppler HWHM for each transition.
nents do typically have distinct line shape parameters. The
number of total angular momentum channels accessible in a
collision is increased by the coupling of I and j, and so the
magnitude of the broadening is a function of F. This effect
has been observed at room temperature for several rotational
transitions of CH3I.23 At low temperatures it is possible to
make exact predictions for the differences by incorporating
the hyperfine quantum numbers into the calculation of the
broadening cross sections from the scattering matrix.
ments it is necessary to convert between the cross section
and the measured Lorentzian half-widths and line shifts.
They are related by
2
n
͗ ͘Re͓␴͑T͔͒ = Pͱ
⌬␯
=
Re͓␴͑T͔͒ = P␥
v
HWHM
2␲
␲3kBT␮
͑2͒
͑3͒
We calculated the hyperfine line shape cross sections
using the scattering matrices generated in the MOLSCAT cal-
culations and Eq. ͑16͒ from Green’s paper.22 ͑Note that this
equation contains a typographical error. Using Green’s nota-
tion, a factor of ͓Ka,Kb͔ needs to be included in the inner
summation as it does in Eq. ͑14͒.͒ We implemented this
equation by modifying MOLSCAT’s pressure broadening rou-
tines to incorporate the additional angular momentum cou-
pling coefficients and the increased summation ranges.
As Green points out, the j=1 0 transition of a linear
molecule is unique in that its hyperfine components all have
exactly equal broadening and shifting parameters. The HCN
j=2 1 transition is more typical in that the hyperfine line
shape parameters are predicted to differ from one another.
The broadening cross sections predicted with and without
taking the hyperfine quantum numbers into account are de-
picted in Fig. 4. The predicted order of the broadening cross
sections, from largest to smallest, is the same at all tempera-
tures and for all three potential energy surfaces. Note that
neither the difference nor the ratio of any two hyperfine pre-
dictions is constant at all temperatures and so this effect can-
not be reduced to a simple multiplier or offset. The shifting
cross sections for the hyperfine components also differ from
the prediction made without taking the hyperfine quantum
numbers into account. The shifting predictions also maintain
a consistent order at all temperatures, but the order does
differ from the broadening order.
and
n
␯
center − ␯0 = −
͗ ͘Im͓␴͑T͔͒ = P␩,
v
2␲
where ͗ ͘ is the average relative velocity at T, n is the per-
v
turber number density, P is the perturber pressure, and ␮ is
the reduced mass of the colliding species. The use of the
ideal gas law to relate pressure and number density is valid
for helium, even at the low temperatures in this experiment,
due to the low pressures. These equations also define the
pressure broadening parameter ␥ and shifting parameter ␩.
B. Hyperfine transitions and line shape
The hyperfine structure for the two measured HCN tran-
sitions is sketched in Fig. 3. Each line represents a transition
between two F levels where F= j+I is the total angular mo-
mentum obtained by coupling the molecule’s rotational an-
gular momentum j and the nuclear spin angular momentum
I. The components are only fully resolved from one another
at low temperatures and pressures, but the splitting is large
enough that even at room temperature the presence of the
hyperfine must be taken into account in order to model the
line shape.
Green used HCN as an example in addressing the ques-
tion of whether line shape parameters are a function of the
nuclear hyperfine quantum numbers.22 His analysis assumes
that the nuclear spin is merely a spectator in the collision,
i.e., its projection on a space-fixed axis remains constant.
Even with this assumption he shows that hyperfine compo-
In addition to the differences in the diagonal elements of
the line shape parameter matrix, which define the broadening
and shifting of each line, the off-diagonal elements that de-
scribe line mixing are nonzero except for the j=1 0 tran-
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184319-6
T. J. Ronningen and F. C. De Lucia
J. Chem. Phys. 122, 184319 ͑2005͒
cooled to the same temperature as the cell. For both rota-
tional transitions our analysis retrieved the expected Doppler
width within ±3%. This corresponds to measuring the mo-
lecular temperature within ±6% since the Doppler width is
proportional to the square root of the temperature. By fixing
the Doppler width based on the cell temperature we could
slightly increase the precision of the broadening parameters,
and so all of the broadening and shifting data reported here
were obtained with the Doppler contribution fixed.
The helium pressure broadening of the HCN j=1
0
transition was measured in two previous experiments.24,25
Both experiments made measurements near 300 K, and
Nerf24 also measured the line shape at 195 K. His measured
cross section of 30.3±4.7 Å2 at 195 K is in agreement with
the predictions from all three potential energy surfaces con-
sidered here. Nerf did consider the possibility that the three
hyperfine lines had distinct broadening parameters or mixed
together. He showed, prior to Green’s theoretical explana-
tion, that the broadening parameters are equivalent for each
of the three lines and that there is no mixing.
FIG. 5. Error in retrieved Doppler widths, defined as Doppler
HWHMmeasured−HWHMexpected, for all measured lines as a function of the
Doppler HWHM expected if the molecules are at the same temperature as
the cell. The scatter in the 1 0 error is noticeably smaller for two related
reasons: first, the collapse of the molecular population into the j=0 state
leads to the 1 0 transition having a larger absorption coefficient and this
improves the experimental signal to noise ratio; second, maintaining suffi-
cient signal to noise for the 2 1 transition requires keeping the source
power slightly higher and this increases the distortion caused by power
broadening in the low pressure data.
We measured the j=1 0 absorption lines between 1.3
and 6.2 K. The broadening and shifting parameters are
shown in Table II, and the corresponding cross sections are
compared with theoretical predictions in Fig. 6. Our data
supports the theoretical prediction that the line shape param-
eters do not vary for the three hyperfine components. We
believe the scatter in the values measured for the three com-
ponents is the best measure of our experimental precision.
We therefore assigned a standard deviation of ±2% to the
broadening parameters. The precision of the line center mea-
surement is determined by the width of the line, and there-
fore the deviation assigned to the shifting is ±2% of the
broadening parameter at the same temperature. Our data do
not agree with the broadening predicted by our scattering
calculations. At 6.2 K the predicted broadening cross section
is about 18 s.d. above the measured value, and the discrep-
ancy grows as the temperature decreases. The shifting mea-
surements are also several standard deviations from any of
the predictions except at 1.3 K when the Toczylowski sur-
face’s predictions do intersect the measurements.
sition. We simulated the predicted mixing among the j=2
1 components over our full experimental pressure range,
2–80 mTorr, and found that mixing resulted in no more than
a 3% peak deviation from a model that ignores mixing. This
effect is sufficiently small that we chose to ignore mixing in
our data analysis.
IV. RESULTS AND DISCUSSION
A. Results
We have, for the first time in a millimeter-wave colli-
sional cooling experiment, measured the Doppler contribu-
tion to the line shape by treating it as a free parameter in
fitting the data. The difference between the measured Dop-
pler width and the width expected if the molecules are at the
cell temperature is plotted in Fig. 5. The fact that this error is
quite small, even for the Doppler widths below 10 kHz, in-
dicates both that the fitting method can retrieve line shape
parameters with high precision and that the molecules are
TABLE II. Measured helium pressure broadening and shifting parameters for the HCN j=1 0 hyperfine lines.
The hyperfine lines are identified by their F quantum numbers. The number in parentheses is the estimated
standard deviation discussed in the text. In some cases spectra were recorded at the same temperature on
multiple days. In these cases the spectra from different days were fit separately as a check that the line shape
parameters were reproducible, and the results of both fits are included.
Temperature ͑K͒
Broadening ͑MHz Torr−1
͒
Shifting ͑MHz Torr−1
͒
1
1
2
1
0
1
1
1
2
1
0
1
1.35
1.36
1.44
1.49
2.65
3.02
3.51
4.19
4.20
6.18
27.4͑0.5͒
27.3͑0.5͒
27.9͑0.6͒
28.3͑0.6͒
24.2͑0.5͒
23.6͑0.5͒
22.8͑0.5͒
21.0͑0.4͒
20.8͑0.4͒
17.3͑0.3͒
27.8͑0.6͒
27.7͑0.6͒
28.7͑0.6͒
29.0͑0.6͒
24.8͑0.5͒
23.9͑0.5͒
23.3͑0.5͒
21.4͑0.4͒
21.0͑0.4͒
17.5͑0.3͒
28.0͑0.6͒
27.2͑0.5͒
28.0͑0.6͒
28.4͑0.6͒
24.6͑0.5͒
23.7͑0.5͒
23.0͑0.5͒
21.2͑0.4͒
20.6͑0.4͒
17.3͑0.3͒
−4.6͑0.5͒
−5.6͑0.5͒
−5.4͑0.6͒
−4.9͑0.6͒
−3.9͑0.5͒
−4.3͑0.5͒
−3.4͑0.5͒
−3.1͑0.4͒
−4.0͑0.4͒
−2.7͑0.3͒
−4.4͑0.6͒
−5.3͑0.6͒
−5.2͑0.6͒
−4.7͑0.6͒
−3.7͑0.5͒
−4.2͑0.5͒
−3.2͑0.5͒
−3.0͑0.4͒
−4.2͑0.4͒
−2.9͑0.3͒
−4.6͑0.6͒
−5.3͑0.5͒
−5.3͑0.6͒
−4.8͑0.6͒
−3.6͑0.5͒
−4.3͑0.5͒
−3.3͑0.5͒
−3.0͑0.4͒
−3.8͑0.4͒
−2.6͑0.3͒
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130.209.6.50 On: Mon, 22 Dec 2014 19:47:45

184319-7
Helium induced pressure broadening
J. Chem. Phys. 122, 184319 ͑2005͒
surements, are in agreement with theoretical predictions.
Since all three potential energy surfaces predict similar dif-
ferences, this is a property that seems more sensitive to the
hyperfine quantum numbers than the details of the potential.
Our measurements therefore support the theoretical predic-
tions for the differences among the hyperfine broadening, but
the total magnitude of the broadening predicted from our
scattering calculations is consistently about 10 s.d. above the
measured broadening.
It is nearly impossible to draw any conclusions about the
variations of the shifting parameters of the j=2 1 compo-
nents because the uncertainties introduced by fit correlations
are large in comparison to the differences among the predic-
tions. Our measurements do place bounds on the magnitude
of the shifting, but the uncertainties are too large to allow for
a meaningful comparison among the predictions from the
three surfaces. Therefore the j=2 1 shifting results will be
excluded from the discussion in the next section.
FIG. 6. Measured and predicted cross sections for the broadening and shift-
ing of the hyperfine components of the j=1 0 transition as a function of
temperature. The three predictions plotted are calculated using the
Atkins1E8͑¯͒, Harada͑—͒, and Toczylowski͑- - -͒ potential energy sur-
faces. The three measured components are labeled with circles ͑F=1 1͒,
diamonds ͑2 1͒, and triangles ͑0 1͒. The capped bars illustrate the stan-
dard deviation assigned to the highest temperature measurement.
B. Discussion
The discrepancy between theory and experiment re-
ported here is not unique to HCN. Two previous studies of
CO4 and H2S5 line shape parameters at low temperatures
have both found that broadening predictions agree well with
measurements above about 30 K but are too large at tempera-
tures below that. This discrepancy has been critiqued from a
theoretical perspective for the case of CO.26 Given that three
cases of a similar discrepancy have now been observed, we
feel another critique of both the experimental methods and
the theory is in order.
No helium broadening studies have been reported previ-
ously for the j=2 1 transition. We studied this transition
between 1.3 and 18.5 K. The measured broadening and shift-
ing parameters are shown in Table III, and the corresponding
cross sections are compared with theoretical predictions in
Fig. 7. Parameters for the F=1 2 component were not
measured due to its small relative intensity, but it was in-
cluded in the line shape model with fixed parameters when
fitting the data. The standard deviation assigned to the broad-
ening is ±2% based on the precision established by the
1. Experimental procedure
These HCN measurements have provided experimental
confirmation of the precision and accuracy of several aspects
of our experiment. The measured line center for each of the
hyperfine components agrees with beam maser measure-
ments, and so our frequency calibration is accurate. We have
also measured line shape parameters for the individual hy-
perfine transitions that were the same within ±2% when
theory predicted they would be the same and that differed by
a few percent when theory predicted they would differ. This
precision in the line shape parameters is better than we have
been willing to assign to previous measurements with the
same apparatus.
1
0 measurements. The values determined for the shifting
display much greater deviations due to strong correlations
between the shifting parameters for overlapping lines in the
nonlinear least-squares fit. For example, the shifting of the
2
1 component is certainly unrealistically low and the 3
2 shifting is unrealistically high due to a negative correla-
tion between these two parameters. It therefore seems neces-
sary to assign a larger standard deviation of ±10% of the
broadening parameter in order to account for the errors in-
troduced by the fitting method.
One conclusion from these data is that the hyperfine
components do have distinct broadening parameters. The
broadening of the F=2 1 and 3 2 components consis-
tently differ by an amount larger than the experimental un-
certainty, and as predicted by theory the broadening of the
Cooling the molecules to the same temperature as the
cell is crucial to making meaningful comparisons with theo-
retical predictions. Assuming a higher temperature or su-
prathermal distribution of relative collision energies clearly
alters the cross sections derived from Eq. ͑1͒. Our measure-
ments of the Doppler widths of the HCN lines provide the
best confirmation to date that our implementation of the col-
lisional cooling method does bring the spectroscopic mol-
ecules into kinetic equilibrium with the buffer gas and that
the buffer gas is in equilibrium with the cell walls. Since the
HCN is thermalized solely by the He gas, its Doppler width
encodes both its own temperature and the temperature of the
He. One concern we raised in the past is that there may be a
thermal gradient across the cell due to the influx of warm
2
1 component is the larger of the two. The differences are
not as clear-cut for all of the components due to the noise in
the data, but we do believe our results for the differences
among the hyperfine components are in qualitative agree-
ment with theory for two reasons. First, a trend in the order-
ing does stand out from the noise, and this observed trend is
the same as the ordering predicted by theory. Second, the
magnitudes of the measured differences among the compo-
nents, taking into account the standard deviations in the mea-
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130.209.6.50 On: Mon, 22 Dec 2014 19:47:45

184319-8
T. J. Ronningen and F. C. De Lucia
J. Chem. Phys. 122, 184319 ͑2005͒
TABLE III. Measured helium pressure broadening and shifting parameters for the HCN j=2 1 hyperfine lines. The hyperfine lines are identified by their
F quantum numbers. The columns are ordered such that theory predicts the line shape parameters will increase when read from left to right. The number in
parentheses is the estimated standard deviation as discussed in the text.
Temperature ͑K͒
Broadening ͑MHz Torr−1
͒
Shifting ͑MHz Torr−1
͒
3
2
2
2
1
1
1
0
2
1
3
2
2
2
2
1
1
1
1
0
1.36
1.37
1.37
1.40
1.41
1.41
1.46
1.51
1.54
1.58
1.59
1.64
1.69
1.73
1.74
1.81
2.01
2.04
2.25
2.30
2.50
2.62
2.70
2.93
3.05
3.17
3.49
3.79
4.20
4.20
4.20
4.55
4.87
9.31
11.77
14.95
18.22
14.9͑0.3͒
15.4͑0.3͒
15.0͑0.3͒
14.8͑0.3͒
15.2͑0.3͒
15.0͑0.3͒
15.1͑0.3͒
15.3͑0.3͒
14.9͑0.3͒
14.6͑0.3͒
14.3͑0.3͒
15.2͑0.3͒
15.2͑0.3͒
15.0͑0.3͒
14.6͑0.3͒
14.3͑0.3͒
14.4͑0.3͒
14.7͑0.3͒
14.4͑0.3͒
14.2͑0.3͒
14.7͑0.3͒
14.2͑0.3͒
14.1͑0.3͒
14.5͑0.3͒
14.3͑0.3͒
14.5͑0.3͒
14.3͑0.3͒
13.9͑0.3͒
13.4͑0.3͒
13.7͑0.3͒
13.3͑0.3͒
13.2͑0.3͒
12.9͑0.3͒
10.3͑0.2͒
9.1͑0.2͒
14.9͑0.3͒
15.4͑0.3͒
15.0͑0.3͒
14.8͑0.3͒
15.2͑0.3͒
15.0͑0.3͒
15.1͑0.3͒
15.3͑0.3͒
14.9͑0.3͒
14.6͑0.3͒
14.3͑0.3͒
15.2͑0.3͒
15.2͑0.3͒
15.0͑0.3͒
14.6͑0.3͒
14.3͑0.3͒
14.4͑0.3͒
14.7͑0.3͒
14.4͑0.3͒
14.2͑0.3͒
14.7͑0.3͒
14.2͑0.3͒
14.1͑0.3͒
14.5͑0.3͒
14.3͑0.3͒
14.5͑0.3͒
14.3͑0.3͒
13.9͑0.3͒
13.4͑0.3͒
13.7͑0.3͒
13.3͑0.3͒
13.2͑0.3͒
12.9͑0.3͒
10.3͑0.2͒
9.1͑0.2͒
15.2͑0.3͒
15.4͑0.3͒
15.4͑0.3͒
15.2͑0.3͒
15.9͑0.3͒
15.2͑0.3͒
15.7͑0.3͒
15.4͑0.3͒
15.3͑0.3͒
15.0͑0.3͒
14.7͑0.3͒
15.2͑0.3͒
15.7͑0.3͒
15.0͑0.3͒
14.8͑0.3͒
14.7͑0.3͒
14.7͑0.3͒
14.9͑0.3͒
14.7͑0.3͒
14.7͑0.3͒
14.8͑0.3͒
14.2͑0.3͒
14.4͑0.3͒
14.4͑0.3͒
14.5͑0.3͒
14.3͑0.3͒
14.2͑0.3͒
14.0͑0.3͒
13.5͑0.3͒
13.8͑0.3͒
13.5͑0.3͒
13.5͑0.3͒
13.1͑0.3͒
10.5͑0.2͒
9.3͑0.2͒
15.9͑0.3͒
15.9͑0.3͒
15.9͑0.3͒
15.9͑0.3͒
16.5͑0.3͒
16.3͑0.3͒
16.2͑0.3͒
15.5͑0.3͒
16.1͑0.3͒
15.9͑0.3͒
15.9͑0.3͒
15.6͑0.3͒
16.3͑0.3͒
15.8͑0.3͒
16.0͑0.3͒
15.9͑0.3͒
15.9͑0.3͒
16.0͑0.3͒
15.9͑0.3͒
15.5͑0.3͒
15.6͑0.3͒
15.3͑0.3͒
15.6͑0.3͒
15.6͑0.3͒
15.4͑0.3͒
15.5͑0.3͒
15.2͑0.3͒
15.0͑0.3͒
14.4͑0.3͒
14.7͑0.3͒
14.5͑0.3͒
13.8͑0.3͒
13.7͑0.3͒
11.2͑0.2͒
9.9͑0.2͒
15.7͑0.3͒
15.9͑0.3͒
15.9͑0.3͒
15.8͑0.3͒
16.5͑0.3͒
16.1͑0.3͒
16.2͑0.3͒
16.0͑0.3͒
16.0͑0.3͒
15.6͑0.3͒
15.2͑0.3͒
15.8͑0.3͒
16.1͑0.3͒
15.6͑0.3͒
15.7͑0.3͒
15.4͑0.3͒
15.4͑0.3͒
15.9͑0.3͒
15.7͑0.3͒
15.4͑0.3͒
15.6͑0.3͒
15.3͑0.3͒
15.2͑0.3͒
15.4͑0.3͒
15.3͑0.3͒
15.6͑0.3͒
15.5͑0.3͒
15.0͑0.3͒
14.0͑0.3͒
14.7͑0.3͒
14.2͑0.3͒
14.1͑0.3͒
13.8͑0.3͒
11.3͑0.2͒
10.2͑0.2͒
8.7͑0.2͒
0.7͑1.5͒ 1.8͑1.5͒
0.8͑1.5͒ 2.0͑1.5͒
0.7͑1.5͒ 2.0͑1.5͒
0.6͑1.5͒ 1.9͑1.5͒
0.6͑1.5͒ 2.2͑1.6͒
0.7͑1.5͒ 1.6͑1.5͒
0.6͑1.5͒ 1.9͑1.6͒
0.8͑1.5͒ 1.6͑1.5͒
0.5͑1.5͒ 1.8͑1.5͒
0.4͑1.5͒ 1.5͑1.5͒
0.4͑1.4͒ 0.6͑1.5͒
0.8͑1.5͒ 1.6͑1.5͒
0.7͑1.5͒ 2.1͑1.6͒
0.6͑1.5͒ 1.7͑1.5͒
0.4͑1.5͒ 1.3͑1.5͒
0.4͑1.4͒ 0.8͑1.5͒
0.4͑1.4͒ 0.9͑1.5͒
0.6͑1.5͒ 1.8͑1.5͒
0.6͑1.4͒ 1.6͑1.5͒
0.6͑1.4͒ 1.5͑1.5͒
0.5͑1.5͒ 1.7͑1.5͒
0.6͑1.4͒ 1.4͑1.4͒
0.4͑1.4͒ 1.2͑1.4͒
0.2͑1.5͒ 1.6͑1.4͒
0.2͑1.4͒ 1.3͑1.5͒
0.2͑1.5͒ 1.6͑1.4͒
0.1͑1.4͒ 1.7͑1.4͒
0.0͑1.4͒ 1.4͑1.4͒
0.1͑1.3͒ 1.2͑1.4͒
−0.1͑1.4͒ 1.2͑1.4͒
0.0͑1.3͒ 0.9͑1.4͒
0.0͑1.3͒ 1.0͑1.3͒
−0.1͑1.3͒ 0.8͑1.3͒
−0.6͑1.0͒ 0.8͑1.1͒
−0.6͑0.9͒ 0.8͑0.9͒
−0.5͑0.8͒ 0.5͑0.8͒
−0.5͑0.7͒ 0.2͑0.7͒
1.7͑1.6͒
1.9͑1.6͒
1.8͑1.6͒
1.9͑1.6͒
2.1͑1.7͒
0.9͑1.6͒
1.7͑1.6͒
1.0͑1.6͒
1.4͑1.6͒
0.8͑1.6͒
−0.2͑1.6͒
1.0͑1.6͒
2.3͑1.6͒
1.3͑1.6͒
0.7͑1.6͒
0.1͑1.6͒
0.4͑1.6͒
0.9͑1.6͒
0.6͑1.6͒
0.3͑1.6͒
1.2͑1.6͒
0.4͑1.5͒
0.5͑1.6͒
1.6͑1.6͒
1.0͑1.5͒
1.6͑1.5͒
1.7͑1.5͒
1.4͑1.5͒
1.0͑1.4͒
1.3͑1.5͒
0.5͑1.4͒
0.5͑1.4͒
0.4͑1.4͒
0.9͑1.1͒
1.0͑1.0͒
0.8͑0.9͒
0.2͑0.8͒
1.1͑1.6͒
1.1͑1.6͒
1.1͑1.6͒
1.1͑1.6͒
1.0͑1.6͒
0.8͑1.6͒
0.9͑1.6͒
1.0͑1.6͒
0.9͑1.6͒
0.8͑1.6͒
0.4͑1.5͒
1.0͑1.6͒
1.2͑1.6͒
1.1͑1.6͒
0.6͑1.6͒
0.3͑1.5͒
0.5͑1.5͒
0.7͑1.6͒
0.5͑1.6͒
0.3͑1.5͒
0.6͑1.6͒
0.4͑1.5͒
0.3͑1.5͒
0.4͑1.5͒
0.4͑1.5͒
0.2͑1.6͒
0.0͑1.5͒
0.0͑1.5͒
0.7͑1.4͒
−0.1͑1.5͒
0.2͑1.4͒
0.2͑1.4͒
0.1͑1.4͒
−0.4͑1.1͒
−0.5͑1.0͒
−0.4͑0.9͒
−0.3͑0.8͒
1.1͑1.5͒
1.1͑1.6͒
1.0͑1.6͒
1.0͑1.5͒
1.1͑1.6͒
1.2͑1.6͒
1.0͑1.6͒
0.9͑1.6͒
1.0͑1.6͒
0.8͑1.6͒
1.0͑1.6͒
0.9͑1.6͒
1.2͑1.6͒
0.9͑1.6͒
1.1͑1.6͒
0.9͑1.6͒
1.0͑1.6͒
1.0͑1.5͒
1.0͑1.5͒
0.7͑1.5͒
0.9͑1.5͒
1.1͑1.5͒
0.8͑1.5͒
1.1͑1.5͒
0.7͑1.5͒
1.0͑1.5͒
0.7͑1.5͒
0.6͑1.5͒
0.5͑1.4͒
0.5͑1.4͒
0.4͑1.5͒
0.4͑1.4͒
0.4͑1.4͒
−0.1͑1.1͒
−0.2͑1.0͒
−0.2͑0.9͒
−0.1͑0.8͒
7.9͑0.2͒
6.9͑0.1͒
7.9͑0.2͒
6.9͑0.1͒
8.1͑0.2͒
7.0͑0.2͒
8.5͑0.2͒
7.7͑0.2͒
7.6͑0.2͒
gas. From the Doppler measurement, though, if there is a
gradient its peak temperature is no more than 6% above the
cell temperature. It is clear from Fig. 8 that assigning a gas
temperature 6% above the cell temperature does not account
for the discrepancy between theory and experiment. In order
for the measured broadening at a cell temperature of 1.35 K
to agree with the predictions from the Harada surface, the
gas temperature would have to be about 5.5 K.
One limitation in using the Doppler width as a tempera-
ture probe is that it only measures the velocity distribution
parallel to the radiation beam. In our system the radiation
passes through the cell perpendicular to the initial direction
of the HCN injection. If the HCN flowed as a beam the
velocity distribution parallel to the radiation would differ
from the distribution perpendicular to it. It is unlikely that
the HCN is flowing as a beam. The pressure in the HCN
reservoir was kept below 100 mTorr, and so we are far from
the pressure regime that would generate a jet expansion. The
best experimental evidence we have that the HCN molecules
are thermalized in all directions is that the absorption lines
broaden linearly with increasing pressure. If a suprathermal
velocity distribution were present at low pressures, the dis-
tribution at higher pressures would then be colder and closer
to a thermal distribution. A change in the velocity distribu-
tion would lead to a change in the broadening parameter, and
the line widths would then vary in a nonlinear fashion. This
is not observed.
One other possible source of systematic error is the tran-
spiration correction that must be applied to the measured
pressure. This correction is strongly nonlinear as a function
of the cell pressure. For our data the transpiration correction
factor, defined as Pmeter/Pcell, at 1.35 K ranges from 1.1 for
the highest helium pressure to 4.6 for the lowest. The conse-
quence of a substantial error in this correction factor would
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184319-9
Helium induced pressure broadening
J. Chem. Phys. 122, 184319 ͑2005͒
FIG. 7. Measured and predicted cross sections for the broadening and shifting of the hyperfine components of the j=2 1 transition as a function of
temperature. The five measured components are plotted separately and labeled by their F quantum numbers. The predictions were calculated using the
Atkins1E8͑¯͒, Harada͑—͒, and Toczylowski͑- - -͒ potential energy surfaces. The capped bars illustrate the standard deviation assigned to the highest
temperature measurements.
be to make the line broadening appear to be nonlinear as a
function of pressure. Since this is not observed, the correc-
tion factor appears valid within the precision of our measure-
ments.
Of the three potential energy surfaces considered in this
study, the Harada surface most accurately reproduces the
spectrum of the bound complex. Based on the measurements
presented here it is not clear that a particular surface consis-
tently produces the best predictions for the pressure broad-
ening and shifting. It is reasonable to say that the Atkins 1E8
surface produces the poorest predictions for these param-
eters. The Atkins 1E8 broadening predictions are consistently
larger than the other predictions, even as the temperature
increases, and its shifting predictions also diverge from the
measurements below 5 K. In discussing the two potentials
they calculated,11 Atkins and Hutson point out that the 1E8
potential does not use a “sophisticated” form for the repul-
sive wall. Their 2E8 potential addresses this shortcoming,
but based on the spectroscopic data available at that time
they could not claim that the 2E8 potential was genuinely
more accurate than the 1E8. Since the 2E8 potential is ex-
tremely similar to the Toczylowski potential, a comparison of
the predictions for the pressure broadening parameters does
confirm that the 2E8 repulsive wall is more accurate than the
1E8 as Atkins and Hutson expected.
2. Potential energy surfaces
The quality of a potential energy surface for two weakly
interacting species has been assessed primarily by its ability
to accurately predict the measured spectrum of the bound
complex. This measure of quality focuses attention on the
attractive region of the potential, including the long-range
attraction. As indicated in Fig. 2, the low energy broadening
cross sections are dominated by resonant features, and so low
temperature pressure broadening measurements are also sen-
sitive to the details of the attractive well. The higher energy
broadening is more dependent on the positive energy repul-
sive wall than either of these other two properties.
The broadening predictions from the Harada and Toczy-
lowski surfaces are similar enough, and their overall dis-
agreement with the measurements is large enough, that it is
not reasonable to judge between them based on these mea-
surements. The shifting predictions from these two surfaces
do diverge from one another below 5 K, and so the shifting
measurements may be a more sensitive indicator of an accu-
rate surface. The reason the shifting predictions diverge but
the broadening predictions remain similar is that shifting
originates solely from elastic scattering events while broad-
ening is influenced by both elastic and inelastic scattering.
The relative contribution from the two types of scattering
FIG. 8. Measured and predicted broadening parameter for the j=1 0,
F=2 1 component. The error bars on the measured points indicate the
±2% standard deviation in the broadening and the ±6% deviation in the
temperature determined from the Doppler width measurements.
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184319-10
T. J. Ronningen and F. C. De Lucia
J. Chem. Phys. 122, 184319 ͑2005͒
surements, and so it is possible that any error in the low
temperature line shape predictions is arising primarily from
inaccurate elastic scattering predictions.
If we assume the inelastic contributions predicted by the
Harada surface are already accurate, it is necessary that the
elastic contribution be reduced by about a factor of 2 in order
to bring the broadening and shifting predictions into agree-
ment with theory. The differences between the Atkins 1E8
and Harada surfaces reduced the elastic contribution to the
broadening by as much as 25%, and so fairly small changes
to the surface can produce substantially different elastic scat-
tering predictions. One possible source of residual error in
Toczylowski’s pure ab initio surface is the choice of mono-
mer geometry. The Toczylowski surface froze the HCN
monomer at its equilibrium geometry, but recent studies on
similar van der Waals complexes28 have indicated that a
more accurate surface is obtained by using the geometry of
the vibrationally averaged monomer. The empirical modifi-
cations made by Harada may have corrected to some degree
for any errors introduced by the choice of HCN geometry,
but the spectrum of the bound complex provides a limited
amount of information about a limited region of the poten-
tial. If the bound state energies and elastic scattering predic-
tions are not equally sensitive to the same features of the
potential energy surface, then a surface that accurately pre-
dicts one would not necessarily accurately predict the other.
Determining whether a single potential energy surface
can simultaneously predict the spectroscopic measurements
and the low temperature line shape measurements may re-
quire a potential energy surface inversion like that performed
by Geremia and Rabitz to determine the Ar–HCl potential.29
Their inversion does not assume a particular functional form
for the potential energy surface, but instead it maps out the
entire family of smooth surfaces that are capable of repro-
ducing both differential inelastic scattering measurements
and spectroscopic measurements of the bound complex.
Their family of surfaces differed from the best potential de-
rived from spectroscopic data alone30 in two important re-
spects: the steepness of the repulsive wall and the degree of
anisotropy. The scattering data were more sensitive to both
of these than the spectroscopic data were, and the same may
be true of the broadening data and account for the disagree-
ment between theory and experiment. Applying a similar in-
version to the He–HCN or He–CO system may be necessary
to conclusively determine whether modifications to the sur-
face can bring about good agreement with both the low tem-
perature line shape parameters and the spectroscopic transi-
tions.
FIG. 9. Predicted elastic and inelastic scattering contributions to the pres-
sure broadening cross section as a function of temperature. The definition of
each contribution is given by Eqs. ͑4͒ and ͑5͒. The three predictions plotted
are calculated using the Atkins1E8͑¯͒, Harada͑—͒, and Toczylowski͑- - -͒
potential energy surfaces. The sharp rise in the inelastic contribution from
j=2 below 5 K is dominated by relaxation to the j=0 state.
events can be calculated. The broadening cross section for an
isolated transition between two non-degenerate states ␣ and
␤ can be formulated as27
1
␴
=
͑␴␣,inelastic + ␴␤,inelastic + ␴
͒, ͑4͒
␣␤,elastic
␣␤,broadening
2
where ␴␣,inelastic is the sum of the total inelastic cross sections
for transitions out of ␣ to any other state, and the second
term is the corresponding sum for inelastic transitions out of
␤. The third term in Eq. ͑4͒, indicating the elastic scattering
contribution, is defined as
␴
=
d⍀͉f ͑⍀͒ − f ͑⍀͉͒2,
͑5͒
͵
␣␤,elastic
␣
␤
where f␣͑⍀͒ is the elastic scattering amplitude as a function
of the angle ⍀ between the precollision relative momentum
vector and the postcollision relative momentum vector.
Based on Eq. ͑4͒ and using the inelastic cross sections cal-
culated by MOLSCAT, the relative contribution from each of
the three terms can be determined. These contributions are
plotted as a function of temperature in Fig. 9.
The inelastic contributions to the Harada and Toczy-
lowski broadening predictions are quite similar, and so the
differences in the elastic predictions account for most of the
differences in their broadening predictions. Even the Atkins
1E8 inelastic predictions are close to the predictions from the
other two surfaces, implying that the inelastic contributions
are less sensitive to the details of the surface and that further
modifications to the surface are not likely to change them
substantially. For all three surfaces it is clear that the elastic
scattering contribution becomes equal to or larger than the
inelastic contributions as the temperature falls below 30 K.
This is the same temperature region where predicted line
shape parameters have been observed to diverge from mea-
3. Scattering formulation of pressure broadening
The connection between binary molecular scattering and
pressure broadening has been extensively developed and dis-
cussed over the last 40 years.31 A particularly thorough dis-
cussion of the assumptions employed in predicting pressure
broadening parameters is provided in
a review by
Ben-Reuven.32 In this paper he specifically addresses the
case of broadening that is dominated by scattering reso-
nances. These resonances should only cause a breakdown in
the calculation if the resonances cause the colliding species
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184319-11
Helium induced pressure broadening
J. Chem. Phys. 122, 184319 ͑2005͒
to stay close together for a period comparable to the time
between collision events. Under these conditions three-body
interactions need to be considered. For quasibound states of
van der Waals systems such as He–HCN, and at the pressures
used in this experiment, these two times are separated by
several orders of magnitude.26 Therefore a breakdown in this
assumption is highly unlikely.
lations, or the elastic scattering calculations specifically, that
modifying the potential energy surface does not address.
V. CONCLUSIONS
At the time our laboratory began making pressure broad-
ening measurements at low temperatures, ab initio potential
energy surfaces were fairly crude and not expected to pro-
duce particularly good agreement with measured parameters.
At this time many of the uncertainties with regards to both
the experiment and the potential energy surfaces have been
reduced, and so the gap between our measurements and the-
oretical predictions seems more substantial. By measuring
the Doppler width we have confirmed that the molecules are
cooled to the cell temperature. This measurement has largely
removed our principal concern about the accuracy of our
collisional cooling measurements. It is difficult to envision
other sources of systematic experimental error, but the per-
sistent lack of agreement between our results and theoretical
predictions leaves open that possibility. Inaccuracies in the
available surfaces may still be the source of the discrepancy.
It should now be possible to definitively address this possi-
bility by inverting a surface from all of the available data.
There is also a possibility that an error is present in the
theory or computation of low energy scattering. Experiments
that test molecular scattering predictions at these energies are
rare, and so it may be that some assumptions have simply
gone untested in this energy regime. Given the continuing
importance of these calculations for astronomy and their
emerging importance for understanding ultralow temperature
molecules, a shortcoming in either the theory or computa-
tional methods would be well worth correcting.
A second assumption that deserves consideration is the
assumed absence of any correlation between the spectro-
scopically active molecule and the bath molecules colliding
with it. At these low temperatures an inelastic collision event
increases a helium atom’s kinetic energy by an amount com-
parable to its initial energy, and this is not the typical case in
higher temperature measurements. The assumed lack of cor-
relation rests on the difference in the time scale of two
events: the relaxation of the molecular polarization that is
induced by the radiation field and the dissipation of the en-
ergy from the absorbed photon by the bath. If the two times
are well separated, then it can be assumed that the bath re-
tains a static energy distribution. The dilution ratio in this
experiment keeps these times well separated. A He atom ex-
cited by an inelastic collision has a much higher probability
of encountering other He atoms and dissipating its excess
energy rather than encountering another HCN molecule.
Also, both of these time scales would be altered by changes
in the He pressure, and since the only pressure dependent
effects observed were linear broadening and shifting there is
no experimental indication of a correlation.
It is clear from Fig. 9 that the predicted increase in the
pressure broadening cross section below 30 K is driven pri-
marily by the increase in the elastic scattering contribution.
The calculation of accurate elastic scattering cross sections
does require the inclusion of more total angular momentum
channels than are necessary for the calculation of inelastic
cross sections. In our case the elastic cross sections should be
well converged since the MOLSCAT calculations continue un-
til the elastic cross sections do not change by more than
0.3 Å2 with the inclusion of an additional angular momen-
tum channel. Also, in the energy regime where the elastic
contribution has the most significant impact, only a small
number of total angular momentum channels are energeti-
cally accessible.
While it is difficult to point to a particular problem with
the method or theory for calculating pressure broadening
cross sections, it is the case that there is only sparse experi-
mental evidence available to test theoretical predictions for
molecules at low temperatures. A recent experiment on in-
elastic rotational transition rates for CO colliding with He
found excellent agreement between predictions and measure-
ments at temperatures as low as 15 K.33 It seems, though,
that it is the elastic cross sections that are most important to
our measurements, and a recent prediction of an elastic scat-
tering diffusion cross section between 3He and CaH at 0.4 K
was larger than an experimental measurement.34 It is also
true that steady improvements in the purely ab initio poten-
tials have not brought about improvements in the agreement
between measurements and theoretical broadening predic-
tions; rather, the two continue to substantially disagree. This
may indicate a problem with the pressure broadening calcu-
ACKNOWLEDGMENTS
This research was funded by grants from the NSF and
ARO. Computing time was provided by a grant from the
Ohio Supercomputer Center. T.J.R. gratefully acknowledges
the support of the NSF Graduate Research Fellowship Pro-
gram.
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