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Probing molecular potentials with an optical centrifuge
A. A. Milner, A. Korobenko, J. W. Hepburn, and V. Milner
Citation: The Journal of Chemical Physics 147, 124202 (2017);
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Published by the American Institute of Physics
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Probing molecular potentials with an optical centrifuge
A. A. Milner,1 A. Korobenko,1 J. W. Hepburn,2 and V. Milner1,a)
1 Department
of Physics and Astronomy, The University of British Columbia, Vancouver,
British Columbia V6T 1Z1, Canada
2 Department of Chemistry, The University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
(Received 24 April 2017; accepted 15 September 2017; published online 29 September 2017)
We use an optical centrifuge to excite coherent rotational wave packets in N2 O, OCS, and CS2
molecules with rotational quantum numbers reaching up to J ≈ 465, 690, and 1186, respectively.
Time-resolved rotational spectroscopy at such ultra-high levels of rotational excitation can be used as
a sensitive tool to probe the molecular potential energy surface at internuclear distances far from their
equilibrium values. Significant bond stretching in the centrifuged molecules results in the growing
period of the rotational revivals, which are experimentally detected using coherent Raman scattering. We measure the revival period as a function of the centrifuge-induced rotational frequency and
compare it with the numerical calculations based on the known Morse-cosine potentials. Published
by AIP Publishing.
Potential energy surfaces (PES) are central to almost every
aspect of molecular dynamics as they govern both the motion
of atoms inside a single molecule1 and collisions and chemical reactions between several molecular reagents.2 Mapping
out PES of polyatomic molecules, possessing many degrees
of freedom, is a difficult task, which becomes especially challenging far from the equilibrium, where multiple vibrational
and rotational modes become strongly coupled. A common
approach consists of expressing the ro-vibrational energy of a
molecule as a function of internal molecular coordinates, calculating the frequencies of vibrational transitions, and adjusting the expansion coefficients so as to match the calculated
frequencies with experimental observations. The higher the
energies of the observed ro-vibrational transition, the wider
the region of the potential energy surface which can be probed
and accurately mapped out, the richer the molecular dynamics
available for quantitative studies.
A great deal of work has been done on triatomic
molecules, for which precise spectroscopic information about
highly excited vibrational transitions is readily available.3 For
the molecules studied here—N2 O, OCS and CS2 —vibrational
energies up to approximately 15 × 103 cm1 , 13 × 103 cm1 ,
and 14 × 103 cm1 , respectively, have been reported. In a
series of works, Zúñiga et al.4–6 developed optimal generalized internal coordinates to fit the ro-vibrational spectrum
of these molecules by approximating their potential energy
surface with a fourth-order Morse-cosine expansion,7
V (R1 , R2 , θ) = M2 y2 +
Mij yi yj +
Mijk yi yj yk
Mijkl yi yj yk yl ,
where R1 and R2 are the lengths of the two molecular bonds,
θ is the angle between them, M are the expansion coefficients,
y1 = 1 − e−a1 (R1 −R1,0 ) ,
y2 = cos θ − cos θ 0 ,
y3 = 1 − e
−a3 (R2 −R2,0 )
with the subscript “0” denoting the equilibrium value for the
corresponding coordinate in the ground ro-vibrational state.
Constants a1 and a3 determine the stiffness of the respective
In this work, we introduce a new experimental approach to
probing molecular potentials, which may assist in verifying the
above mentioned numerical models far away from the equilibrium bond lengths. The method is based on using high-energy
rotational, rather than vibrational, excitation and its utility is
demonstrated through the application to three molecular systems. To spin the molecules of interest to ultra-high states of
angular momentum J, we employ the technique of an optical
centrifuge.8,9 The total energy of a rotating molecule can be
expressed as
EJ (R1 , R2 , θ) = V (R1 , R2 , θ) +
~2 J(J + 1)
2I(R1 , R2 , θ)
where I is the molecular moment of inertia and ~ is the reduced
Planck’s constant. As the angular momentum increases, the
second term in the above expression pushes up the equilibrium
bond lengths R1,J and R2,J , corresponding to the minimum
of the total energy, to higher and higher values, eventually
resulting in dissociation. The effect is illustrated in Fig. 1,
where we plot the potential energy surfaces of N2 O at different levels of rotational excitation [J = 0, 250, 350, and 465
in plots (a)–(d), respectively], calculated with Eq. (3) and the
Morse-cosine PES coefficients from Ref. 4. While not much
distortion is happening below J ≈ 300 [Figs. 1(a) and 1(b)],
the shape of the PES changes rapidly at higher values of
the molecular angular momentum. Despite the potential rising above the dissociation energy, the centrifugal barrier due
147, 124202-1
Published by AIP Publishing.
Milner et al.
J. Chem. Phys. 147, 124202 (2017)
FIG. 1. [(a)-(d)] Calculated potential
energy surfaces of N2 O at four values
of the molecular angular momentum J
= 0, 250, 350, and 465. The latter represents the largest value of J, for which the
PES still has a minimum. The lengths of
O–N and N–N bonds along the two horizontal axes are normalized to the corresponding equilibrium distances R1,0 and
R2,0 . Note the increasing energy scale.
(e) One-dimensional cross sections of
the displayed PES along the dissociation trajectories shown with solid and
dashed lines on the surfaces and contour plots, respectively. Black circles on
all plots mark the position of the energy
minimum (R1,J , R2,J ).
to the added angular momentum [clearly visible in Fig. 1(c)
for J = 350] prevents the molecule from falling apart. The dissociation eventually occurs at JD = 465 [Fig. 1(d)], when the
barrier disappears altogether, and with it disappears the potential minimum, needed to support bound vibrational states.
The process is illustrated in panel (e) of Fig. 1, where we
plot one-dimensional cross sections of the potential energy
surfaces, shown on the left, along the dissociation coordinate. Notably, before the molecule dissociates, its total energy
increases to about 105 cm1 , almost an order of magnitude
above the level typically accessible through pure vibrational
Rotational dissociation by an optical centrifuge has been
demonstrated experimentally with a diatomic chlorine,9 and
theoretically studied for diatomic (Cl2 , Refs. 8 and 11) and
triatomic (HCN, Ref. 12) molecules. In both cases, it was
predicted that the majority of molecules undergoing forced
accelerated rotation will be ejected from the centrifuge prior
to reaching the dissociation energy due to the Coriolis force.
The latter causes the molecular axis to turn away from the
laser field polarization and eventually fall behind the rotating trap. Hereafter we refer to the upper frequency limit,
above which the molecule can no longer follow the centrifuge as the Coriolis wall. In this work, we analyze the
molecular dynamics in the vicinity of the wall both numerically and experimentally, in several centrifuged triatomic
Figure 2(a) shows the calculated stretching of both molecular bonds in N2 O, OCS, and CS2 with the rotational quantum
number increasing from J = 0 to the calculated dissociation
limit JD = 465, 690 and 1186, respectively. The corresponding dissociation energies are 83 × 103 cm1 , 86 × 103 cm1 ,
and 132 × 103 cm1 . As anticipated for a linear molecule, the
equilibrium angle between the bonds remains zero regardless
of J. Although the internuclear distances cannot be measured
directly in our experiments, we extract the information about
the molecular bond stretching by investigating the periodic
dynamics of centrifuged molecules as a function of the rotational frequency, controlled by the centrifuge. Our analysis is
carried out as follows.
Given the rotational energy spectrum E(J), the classical
rotational frequency at any J can be calculated as
νJ =
E(J + 1) − E(J)
The results of these calculations in the simplest approximation
which neglects the zero-point vibrational energy, i.e., when
FIG. 2. (a) Stretching of bond #1 (O–N, S–C, S–C) and bond #2 (N–N, O–C,
S–C) in N2 O, OCS, and CS2 , respectively, with increasing angular momentum J. The bond lengths are plotted in dimensionless units, relative to the
corresponding equilibrium distances at J = 0, i.e., as R1,J /R1,0 and R2,J /R2,0 .
The angular momentum is increasing along each curve from J = 0 (lower
left corner) to the dissociation limit JD , labeled with circles. (b) Classical
rotational frequency of N2 O, OCS, and CS2 as a function of the molecular
angular momentum J, normalized by its value at dissociation (JD ). In both
panels and insets, crosses mark the location of the Coriolis wall, which limits
the adiabatic excitation by the centrifuge (see text for details).
Milner et al.
J. Chem. Phys. 147, 124202 (2017)
E(J) is taken as the minimum energy of the corresponding PES
at its equilibrium point, E(J) ≡ EJ (R1,J , R2,J , 0), are shown
in Fig. 2(b). The growth of ν J with the molecular angular
momentum is sub-linear, indicating strong centrifugal distortion. Similarly to the previously discussed8,12 behavior of Cl2
and HCN, the rotational frequencies of N2 O and OCS fall
off above the critical values of JC = 460 (green solid) and
JC = 650 (dashed blue), respectively. The corresponding frequency maxima of ν C = 9.4 THz and 6.2 THz, marked with
crosses in Fig. 2(b), represent the Coriolis wall on the way of
accelerated molecular rotation. Beyond this wall, the distance
between consecutive rotational levels starts decreasing and the
molecules can no longer adiabatically follow the centrifuge
Interestingly, the rotational frequency of CS2 increases
monotonically (dashed red line), indicating that the molecule
can climb the rotational ladder all the way to the dissociation
frequency ν D = 5.5 THz without escaping the centrifuge. This
conclusion can be interpreted as follows. In asymmetric linear
triatomic molecules, the centrifugal force stretches predominantly one of the two bonds (O–N, S–C and N–C in N2 O,
OCS and HCN, respectively), whereas the other one extends
to a much lesser degree and even starts shrinking with the
increasing rotational frequency [see Fig. 2(a)]. The quickly
stretching bond is responsible for the strong Coriolis force,
which makes the molecule lag behind, and eventually fall off,
the centrifuge without dissociating.
The symmetry of CS2 , on the other hand, effectively stiffens the molecule with respect to the centrifugal pull. The
lengths of both S–C bonds increase slower than in the asymmetric case, which results in the weaker Coriolis force and
better stability in the centrifuge. The difference between the
symmetric and asymmetric molecules can be better appreciated by comparing the respective potential energy surfaces
near dissociation. While the Gaussian curvature (the product of two principal curvatures) of the potential well of N2 O
remains positive with increasing angular momentum (Fig. 1),
its sign changes abruptly when the PES of CS2 develops saddle geometry at J = 1186 (Fig. 3). The subsequent sudden
elongation of one of the S–C bonds, even if accompanied by
strong Coriolis force, leads to an inevitable dissociation of
the molecule. Although the above classical picture does not
account for the quantum tunneling out of the well, the latter will
only expedite the dissociation process rather than preventing
Since the calculated critical frequency values for the Coriolis wall and the centrifuge-induced dissociation (ν C and ν D )
are both within the reach of our optical centrifuge (≤10 THz),
we were able to study the described behavior experimentally. After the molecules are spun up and released from the
centrifuge, we monitor their dynamics by means of coherent Raman scattering.13 An excited rotational wave packet
undergoes periodic revivals, separated in time by14,15
Trev =
d 2 E(J)/dJ 2
E(J + 1) − 2E(J) + E(J − 1)
Fractional revivals may also occur (as is the case here), depending on the particular structure of the wave packet.16 By comparing the experimentally found dependence of the revival time
FIG. 3. Calculated potential energy surfaces of CS2 at J = 1000, 1150 and
1250, around the dissociation point of JD = 1186. The lengths of both S–C
bonds along the two horizontal axes are normalized to the equilibrium distance
R1,0 = R2,0 = 1.55 Å. Black crosses in the two bottom panels mark the position
of the energy minimum. Note the increasing energy scale and the appearance
of two centrifugal barriers, clearly seen in the middle panel.
on the rotational frequency, Trev (νJ ), with the one calculated
using the theoretical energy spectrum E(J), the validity of the
latter can be readily assessed.
The experimental setup, shown in Fig. 4, is similar to that
used in our previous work.13 A beam of femtosecond pulses
FIG. 4. Experimental set up. BS: beam splitter, DM: dichroic mirror, CP/CA:
circular polarizer/analyzer, DL: delay line, L: lens. The gas cell is filled with
either N2 O, OCS, or CS2 under room temperature and pressure of 30 kPa, 10
kPa, and 5 kPa, respectively. An optical centrifuge field is illustrated above
the centrifuge shaper with k being the propagation direction and E the vector
of linear polarization undergoing an accelerated rotation.
Milner et al.
from a regenerative Ti:Sapphire amplifier (800 nm, 1 KHz
repetition rate, 30 nm full width at half maximum) is split into
two parts. One part is sent to a “centrifuge shaper,” implemented according to the original recipe of Karczmarek et al.,8
which converts the input laser field into the field of an optical
centrifuge, schematically illustrated in the inset. The shaper
is followed by a home built Ti:Sapphire multi-pass amplifier
(10 Hz repetition rate), which boosts the energy of centrifuge
pulses up to 30 mJ/pulse. The pulses are about 100 ps long
and their linear polarization undergoes an accelerated rotation, reaching the angular frequency of 10 THz by the end of
the pulse. The second (probe) beam is frequency doubled in
a nonlinear BaB2 O4 (BBO) crystal, time delayed by means
of a controllable translation stage and combined with the centrifuge beam on a dichroic mirror. Both beams are focused into
a cell, filled with either N2 O, OCS, or CS2 gas. Loose focusing
with a 1 m focal length lens down to a beam diameter of 90
µm, which limits the peak intensity of the excitation field to
below 2 × 1012 W/cm2 , is used to avoid strong-field effects,
such as molecular ionization, beam filamentation, and plasma
We detect the dynamics of the centrifuged molecules
using time-resolved coherent Raman spectroscopy. The
centrifuge-induced coherence between the rotational states
| J, M = J i and | J + 2, M = J + 2i (where M is the projection of J on the propagation direction of the centrifuge field)
results in the Raman frequency shift of the probe field. From
the selection rules ∆M = ±2 and the conservation of angular
momentum, it follows that the Raman sideband of a circularly polarized probe is also circularly polarized, but with an
opposite handedness. Due to this change of polarization, the
strong background of the input probe light can be efficiently
suppressed by means of a circular analyzer, orthogonal to the
input circular polarizer (CA and CP, respectively, in Fig. 4).
Raman spectra of the probe pulses scattered from the rotating
molecules have been measured as a function of the probe delay
relative to the centrifuge with a f /4.8 spectrometer equipped
with a 2400 grooves/mm grating.
FIG. 5. Raman spectrum of oxygen gas, excited by the same centrifuge field
as the one applied to N2 O, OCS, and CS2 molecules and used for retrieving
their rotational frequency from the experimentally recorded wavelength shift.
Horizontal frequency scale, shown at the top of the plot, represents the spectral range, in which every oxygen peak has been successfully identified and
assigned a well-known transition frequency. The rightmost peak at 398.2 nm,
marked with “λPr ,” corresponds to the input spectrum of probe pulses.
J. Chem. Phys. 147, 124202 (2017)
To calibrate the frequency scale for the measured Raman
shift, we recorded the rotational spectrum of oxygen gas with
centrifuge and probe pulses as well as the settings of our spectrometer, identical to those used with the triatomic molecule of
interest. An example of such O2 spectrum is shown in Fig. 5. In
contrast to N2 O, OCS, and CS2 , oxygen’s much lower moment
of inertia enables us to resolve individual rotational lines. In
this case, 60 anti-Stokes peaks can be assigned to the corresponding Raman transitions | J + 2i → | J i, with the rotational
quantum numbers (only odd for oxygen, due to its molecular symmetry and nuclear spin statistics) spanning from 1 to
119. According to our earlier study of centrifuged oxygen,13
FIG. 6. Time-resolved rotational Raman spectra of centrifuged (a) N2 O, (b)
OCS, and (c) CS2 molecules as a function of the time delay between the centrifuge and probe pulses. Bright tilted lines correspond to the Raman signal
from the coherently rotating molecules. Note identical horizontal, but different vertical scales. An inset in plot (a) is a one-dimensional cross section
of the spectrogram along the white dashed line. It shows an example of the
time-dependent Raman signal generated by N2 O molecules rotating with the
frequency of 1 THz. Dim vertical “shadow” traces originating from the regions
of strong Raman signal [e.g., between the frequencies of 1 and 3.5 THz, or
above 6 THz in plot (b)] are spectrometer’s artifacts. See text for more details.
Milner et al.
J. Chem. Phys. 147, 124202 (2017)
its rotational energies are well described by the second-order
Dunham expansion,
EJ = B0 J(J + 1) − D0 J 2 (J + 1)2 ,
with the rotational and centrifugal constants for the ground
vibrational state B0 = 1.43 cm1 and D0 = 4.84 × 10−6 cm1 ,
respectively.17 Given the proper line assignment for O2 , the
experimentally measured wavelength shift can be converted
to the absolute frequency scale, shown at the top of Fig. 5.
Given the dense grid of resolved oxygen peaks, separated by
≈0.17 THz, we estimate our frequency accuracy as better than
102 THz.
The experimentally measured Raman spectrograms of
N2 O, OCS, and CS2 are shown in Fig. 6. The vertical scale
has been translated from the measured Raman shift to the
frequency of the molecular rotation as described above. The
bright tilted line on the left side of each plot reflects the
accelerated rotation of molecules inside the centrifuge (in
all three cases, the angular acceleration is exactly the same,
whereas the tilt appears different due to the difference in vertical scales). While spinning up, the molecules are “leaking”
from the centrifuge, producing an oscillatory Raman signal
in the broad range of rotational frequencies. An example of
such oscillations is presented in the inset in panel (a). The
plot shows a cross section of the two-dimensional spectrogram along the white dashed line and illustrates the signal
generated by those N2 O molecules, whose escape from the
centrifuge occurred when the latter was rotating at 1 THz. The
observed periodic oscillations indicate coherent evolution of
the excited rotational wave packet, with each peak corresponding to the moment of either fractional or full quantum revival,
Trev .
Substituting E(J) from Eq. (6) into Eq. (5) and assuming the regime of negligible centrifugal distortion, one finds
Trev = 1/(2B0 c), where c is the speed of light in vacuum. For
N2 O, OCS, and CS2 , the revival periods are 39.8, 82.2, and 153
ps, respectively. In the first two cases, fractional revivals occur
at Trev /2, hence, the oscillation period of ≈20 ps in the inset of
Fig. 6(a), whereas for CS2 , they also appear at Trev /4 = 38.2 ps.
The difference stems from the absence of odd J’s in the rotational spectrum of CS2 due to the inversion symmetry of
the molecule and the nuclear spin statistics of its constituent
atoms. The rotational wave packets with all-odd and all-even
J values, separately created by the centrifuge in asymmetric N2 O and OCS, have an opposite sign contribution to the
Raman signal at Trev /4, thus suppressing the quarter-revival
One can see that the oscillation period remains relatively constant at lower rotation frequencies and then grows
quickly as the frequency approaches critical values, specific
to each particular molecule. Beyond this frequency, the rotational Raman signal disappears altogether, pointing to the fact
that a molecule can no longer follow the centrifuge field. Our
numerical analysis, described earlier in the text, suggests two
different mechanisms of the suppressed spinning: Coriolis
wall in the case of N2 O and OCS versus molecular dissociation of CS2 . Because of the inability of the Raman-based
technique to detect the dissociation directly, an alternative
method (e.g., based on ion detection9 ) will have to be implemented for verifying the nature of the observed escape from the
To carry out a quantitative study of the molecular bond
stretching, we analyze the dependence of the revival time on
the frequency of molecular rotation, Trev (νJ ). In order to do
that, we slice the two-dimensional spectrograms, recorded for
each molecule, at multiple values of ν J and find an oscillation period of the Raman signal from the corresponding
one-dimensional cross section, similar to that plotted in the
inset of Fig. 6(a). The results of this analysis are shown in
Fig. 7. At lower frequencies, the experimental revival periods follow closely the dependencies derived from Eq. (5) with
E(J) determined as the energy minimum of the corresponding Morse-cosine potential combined with the rotational term
[Eq. (3)]. As the rotational frequency increases towards the
upper frequency limits (dashed vertical lines), the experimentally measured revival periods begin to deviate from the calculated values. We attribute the disagreement to the zero-point
stretching and bending, which have been omitted in our calculations. At extremely high levels of rotational excitation, the
PES of a super-rotor becomes quite distorted and zero-point
vibrations have to be taken into account for the correct calculation of the equilibrium bond lengths. Detailed theoretical
investigation is needed to verify whether this is the case, and
whether Morse-cosine expansion of higher order is required in
order to explain the observed molecular dynamics at ultra-high
J values.
FIG. 7. Revival periods extracted from the experimental data shown in Fig. 6 (black diamonds with error bars), and theoretically calculated (solid lines) using
Eq. (5) with the total energy E J found from Morse-cosine potential energy surface, as explained in text. Dashed vertical lines mark the expected location of the
Coriolis wall for N2 O and OCS, and the calculated dissociation frequency of CS2 .
Milner et al.
In summary, we presented a method of probing molecular potential energy surfaces with an optical centrifuge. Our
approach is complementary to the one based on vibrational
spectroscopy in that it amounts to modifying the PES by
means of extreme rotational excitation and evaluating the ensuing effects of strong centrifugal distortion, which depend on
the properties of the molecular potential. The method has
been applied to three linear molecules, N2 O, OCS, and CS2 .
According to the numerical estimates based on the known
Morse-cosine potentials, centrifuge-induced ultra-fast rotation caused the molecular bonds to stretch by as much as
32%. Centrifugal bond stretching resulted in the experimentally measured lengthening of the rotational revival periods
and the eventual escape of molecules from the centrifuge.
In the case of CS2 , a simple classical model suggests that
molecular dissociation may have been reached at the critical
rotational frequency of 5.2 THz after the molecule climbed to
the high rotational level with a calculated angular momentum
JD = 1186.
On the other hand, we conclude that the experimentally
observed escape of N2 O and OCS from the centrifuge at the frequency of 8.9 and 5.9 THz, respectively, occurs due to the quick
stretching of the weakest molecular bond. The latter results in
the Coriolis force, which makes the molecular axis lag behind
the polarization axis of the rotating field until the two axes
fall out of alignment with one another and the molecule is lost
from the trap. In the quantum language, the experimentally
measured increase of Trev reflects the fact that the separation
of the rotational energy levels (and with it, the frequency of
the molecular rotation) ceases to grow with J. As a result,
the molecule can no longer follow the increasing rotational
frequency of the centrifuge. This interpretation of our experimental results is supported by the numerical calculations,
which show that both N2 O and OCS reach the Coriolis wall at
the respective critical values of angular momentum JC = 460
and JC = 650 before undergoing dissociation. Similar analysis
J. Chem. Phys. 147, 124202 (2017)
proved successful in describing the dynamics of centrifuged
nonlinear sulfur dioxide18 and can be applied to any molecule
amenable to centrifuge spinning, with an ultimate goal of
selective bond breaking.12
This research has been supported by the grants from CFI,
BCKDF, and NSERC and carried out under the auspices of the
UBC Center for Research on Ultra-Cold Systems (CRUCS).
We would like to thank Roman Krems for many stimulating
discussions of the observed phenomena.
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