Supplemental Material for
The Spontaneous Synchronized Dance of Pairs of Water Molecules
Luiz F. Roncaratti, David Cappelletti*, Fernando Pirani
*To whom correspondence should be addressed; E-mail: david.cappelletti@unipg.it
This PDF includes:
Scattering experiments with water, ammonia and hydrogen sulphide
Model simulations
References
Figures S1, S2, S3, S4
Tables S1, S2
1
Scattering experiments with water, ammonia and hydrogen sulphide
As discussed in the paper, the observed phenomenology (a large increasing of the magnitude of the cross
sections) is promoted by the transformation, during each collision in gas phase, of free molecular rotations
into coupled pendular states. The involved interaction energies, at the relevant intermolecular distance of 1
nm, are of the order of 0.1 kJ/mol. The long-range anisotropy of the intermolecular energy, mostly
electrostatic and due to the interaction of the molecular permanent dipoles, is comparable with the molecular
rotation quantum and is the main driver of the phenomenon. These small energies are at the limit of accuracy
of any quantum mechanical calculation, but our experimental observable, the absolute magnitude of the
integral cross section, is very sensitive to these small effects at long-range and therefore can differentiate the
collision behavior of molecular systems with different permanent dipoles. Therefore, molecular beam (MB)
scattering experiments with a series of hydrogenated polar molecules have been performed in order to
investigate systematically, as a function of changing dipole moment products, the phenomenology described
in the paper, under the same conditions. All measurements have been performed with the same MB apparatus.
Integral cross section Q(v) values, for the D2O-D2O, D2O-ND3, D2O-H2S and ND3-H2S colliding pairs, are
reported in Fig.S1. For each pair of polar molecules the data of two reference systems, investigated with the
same experimental technique, have also been plotted for an important comparison. For the sake of simplicity,
only calculated best-fit cross sections are shown. The choice of the references was suggested by the
polarizability α of the involved partners, whose values1 are reported in Tab.S1, together with those the dipole
moment μ of hydrogenated molecules2.
It is useful to recall that α and μ are the fundamental physical properties determining strength and range of the
leading interaction components, such as the size repulsion (Vrep), dispersion (Vdisp) and induction (Vind)
attractions. Their combination provides a resulting interaction potential which is expected to be very similar
for the investigated pair and for the corresponding reference systems. Such potentials can be represented by
an Improved Lennard Jones (ILJ) function, as widely discussed in ref 5.
2
For the reference systems the comparison involves Q(v) calculated in the laboratory frame by using the ILJ
interaction potentials which provided the best fit of the experimental data and whose relevant parameters are
given Refs 3-9.
Obviously, there are small (even if clearly distinguishable) differences between the cross sections of each pair
of reference systems (of the order of 5-10% depending on the collision velocity, see blue and red curves in
each panel of S1). These differences are due, as expected, to changes in finer details of the interaction
potentials, mostly confined in the potential well at intermediate intermolecular distances. Indeed, looking to
these tiny differences is the main scope of our work (for a recent perspective see Ref. 8).
For the present case of polar pairs (D2O-D2O, D2O-ND3, …) the operative conditions of our experiments
(randomly oriented projectile and target molecules; broad rotational state distributions) were such that we
expected to average out the effect of electrostatic forces and to observe in the cross sections differences of the
same order of magnitude as for the red and blue corves in Fig. S1, but surprisingly this was not the case.
Figure S1 clearly indicates that each pair of hydrogenated polar molecules exhibits a cross section Q(v) much
larger, well outside the experimental uncertainty, than those of the corresponding reference systems. The
same figure shows also that while the cross sections of the references increase from the left to the right panel,
according to the increase of the long range attraction, defined by Vdisp+Vind, the Q(v) values, measured for the
scattering of each polar molecule pair, decrease in going from D2O-D2O to ND3-H2S. In particular, we
evaluated that the cross sections ratio of the polar pairs versus the corresponding reference systems (in
average) is ~ 2.5 for D2O-D2O, ~1.7 for D2O-ND3, ~ 1.4 for D2O-H2S and ~1.2 for ND3-H2S. This is a clear
indication that the polar molecules in our experiment are not freely rotating during the collisions but are
trapped in pendular states where they experience an interaction much stronger than the average (the average
Velec is zero). Moreover, this cross section ratio shows a clear linear dependence on the product of the dipole
moments, as plotted in Fig.S2. Interestingly, the observed linear dependence has an intercept Q/Qref=1 which
corresponds to a dipole moment product of about one square Debye. 2
3
For the polar pairs collisions the interaction potential V has been formulated as detailed in the text (VILJ+Velec)
and the results obtained from the fitting of the experimental data are reported in Fig. S3, where are shown
also the VILJ component of the interaction.
The role of the Velec component is null in the references systems and expected to be negligible for randomly
oriented and fast rotating polar molecules. The present experimental findings clearly indicate that, although
rotationally “hot” projectiles and targets are used, exhibiting a broad Boltzmann distribution of rotational
levels, the Velec component can affect the collision dynamics already at large intermolecular distances R,
forcing the orientation of a fraction of the collision pairs into defined angular cones, centered along the
intermolecular electric field direction. The use of the semiclassical relation, R=(Q/2π)1/2,
5
helps to identify
for each system the range of R relevant for the phenomenon (R > 0.8 nm in all cases). The results confirm
that, under the conditions of the present investigations, this spontaneous polarization plays an effective role
when the average intermolecular interaction becomes of the order 0.03-0.1 kJ/mol. In absolute scale this is a
small value of energy (although comparable with the rotational quantum), but its strong dependence on the
relative molecular orientation is sufficient to drive the trapping of two independent and freely rotating
molecules into a pair of pendular states. Such coupling is expected to be more efficient for the fraction of
molecules in low rotational states and to emerge in the proximity of the distance of the closest approach. In
conclusion, these experiments suggest that such effect decreases its efficiency along the D2O-D2O, D2O-ND3,
D2O-H2S and ND3-H2S series of systems. We are planning other scattering measurements to extend further
the phenomenology to investigate the role of the rotational distributions of the molecules and the possible
effect of higher order permanent multipoles (for instance the quadrupole moment).
Model Simulations
As discussed in the paper in order to rationalize the above findings, we have carried out a dynamical
treatment by using well-known electrostatic equations to define the force field that each molecule feels within
the collisional complex. The used coordinate system and all quantities relevant for the simulations are
4
illustrated in the paper. Also, two examples of results have been presented in the paper. Herein, we present a
further example of collisions between molecules initially rotating in perpendicular planes, in Fig S4. Collision
events are visualized by plotting the local coordinates of the two dipole moments in successive time frames as
the collision event progresses. The results confirm that the molecular coupling occurs with a time scale of few
ps and affects mostly molecules in low lying rotational states. In particular the figure shows that the formation
of pendular state is well evident when both water molecules are in J=1 and reduces for the higher rotational
levels.
To summarize, the results of computations clearly indicate that the phenomenon arises at large R; it becomes
particularly evident when projectile and target molecules are both in low rotational states; it is less active
when the rotational energy increases, but the effect persists even after an averaging over rotational Boltzman
distributions; the coupling occurs in a time scale of picoseconds, being this time scale comparable both with
the period of low lying-rotational states and also with the collision time. As a further proof of consistency, we
found that the order of magnitude of the effective energy driving this process is on the scale of the interaction
obtained from the analysis of the experimental data. Finally, the phenomenon is expected to be more
prominent for the low rotational states therefore for molecules in cold or ultra-cold conditions.
5
References
(1) Olney, T.N. ; Cann, N.M. ; Cooper, G.; Brion, C.E. ; Chem. Phys. 223, 59 (1997).
(2) Radzig, A.A.; Smirnov, B.M. Reference Data on Atoms, Molecules and Ions, Springer-Verlag, Berlin
Heidelberg (1985).
(3) Cappelletti, D.; Aquilanti, V.; Cornicchi, E.; Moix Teixidor, M. and Pirani, F.; J. Chem. Phys., 123,
024302 (2005).
(4) Roncaratti, L.F.; Belpassi, L., Cappelletti, D.; Pirani F.; and Tarantelli, F.; J. Phys. Chem. A, 113, 15223
(2009).
(5) Pirani, F.; Brizi, S.; Roncaratti, L.F.; Casavecchia, P.; Cappelletti, D.; Vecchiocattivi, F.; Phys.
Chem.Chem. Phys., Feature Article, 10, 5489 (2008).
(6) Aquilanti, V.; Cappelletti, D.; Pirani, F.; Roncaratti, L.F.; Int. J. Mass Spectr. 280, 72 (2009).
(7) Pirani, F.; Roncaratti, L.F.; Belpassi, L.; Tarantelli, F.; Cappelletti, D.; J. Chem. Phys. 135, 194301
(2011).
(8) Cappelletti, D.; Ronca, E.; Belpassi, L.; Tarantelli, F.; Pirani, F.; Accounts Chem. Res., 45, 1571 (2012).
(9) Cappelletti, D. , Candori, P. , Roncaratti, L. F. & Pirani, F. Mol. Phys. 108, 2179 (2010).
(10) Maitland, G., Rigby, M., Smith, E. & Wakeham, W. Intermolecular Forces: Their origin and
determination (Clarendom Press, Oxford, 1987).
6
FIGURES
8
D2O-D2O
(a)
7
D2O-ND3
(b)
D2O-H2S
(c)
ND3-H2S
(d)
D2O-D2O(VILJ)
D2O-ND3(VILJ)
D2O-H2S(VILJ)
ND3-H2S(VILJ)
D2O-O2
Ar-ND3
Ar-H2S
Kr-H2S
D2O-Ar
D2O-Kr
D2O-Xe
ND3-Xe
Ar-Ar
Ar-Kr
Ar-Xe
Kr-Xe
2
Q (nm )
6
5
4
3
2
1.0
1.5
2.0
2.5
1.0
1.5
2.0
2.5
1.0
v (km/s)
1.5
2.0
2.5
1.0
1.5
2.0
2.5
Figure S1 – Total integral cross sections Q(v), as a function of the collision velocity v for the D2O-D2O, D2OND3, D2O-H2S and ND3-H2S collision pairs. As comparison, red and blue curves are the cross sections of
selected reference systems, best fitting the experimental observables3-9. The theoretical cross sections, dashed
and dotted lines, are calculated with the JWKB method10 using the ILJ model5 for the vdW and induction
contributions of the interaction, to which an electrostatic component has been added (for the polar pairs, see
text).
7
Figure S2 – Average ratios of the polar pairs cross sections versus the reference systems as a function of the
permanent dipole product of the polar pairs.
8
(a)
V (meV)
0
Ar-Ar
0
-10
-20
D2O-D2O
-1
-30
V (meV)
Ar-Kr
11
8
9
10
0
-10
-20
-1
D2O-ND3
-30
(c)
0
V (meV)
10
(b)
0
Ar-Xe
0
-10
-1
-20
D2O-H2S
-2
-30
8
9
8
9
(d)
0
V (meV)
9
Kr-Xe
-10
0
-20
-1
ND3-H2S
-30
2
4
6
r (Å)
-2
8
10
Figure S3 – Best fit potentials, V(solid) and VILJ (dashed) plotted as a function of the intermolecular distance
R. In the insets a blow up of the long-range distance region is reported. These are the R values that mainly
determine the absolute value of Q(v). The best parameters (see text) are reported in Table S2.
9
Fig S4 - Examples of perpendicular molecular collisions at relative velocity g=1.0 km/s and impact parameter
b=1.0 nm. Initial rotational states are (a) J1=1 and J2=1 and (b) J1=3 and J2=5.
10
TABLES
Table S1 – Polarizability and permanent dipole moments.
System
D2O
Ar
ND3
Kr
H2S
Xe
α (Å3)
1.47
1.64
2.16
2.49
3.78
4.04
μ (D)
1.85
1.47
0.98
-
TABLE S2: Potential parameters defining the VILJ and Velec interaction components (see text). The resulting
long-range dispersion plus induction attraction coefficients Ctot is given by εR6m [5].
System
ref
ε
meV
D2O-D2O present 14.5
D2O-O2 [9]
13.4
D2O-Ar [4]
14.4
Ar-Ar
[5]
12.4
Rm
c
d
nm
nm
0.360 0.159 0.08
0.372
0.363
0.376
R0
nm
1.05
D2O-ND3
Ar-ND3
D2O-Kr
Ar-Kr
present 14.0
[7]
13.4
[4]
17.1
[5]
14.3
0.385 0.158 0.05
0.383
0.375
0.391
0.93
D2O-H2S
Ar-H2S
D2O-Xe
Ar-Xe
present 17.4
[6]
15.1
[4]
20.2
[5]
16.1
0.405 0.150 0.05
0.405
0.393
0.410
0.92
ND3-H2S
Kr-H2S
ND3-Xe
Kr-Xe
present 21.0
[6]
18.8
[7]
21.5
[5]
20.0
0.410 0.130 0.05
0.415
0.411
0.420
0.90
11