J. Chem. Phys. 150, 154304 (2019); https://doi.org/10.1063/1.5092566 150, 154304
© 2019 Author(s).
A combined experimental and theoretical
investigation of Cs
+
ions solvated in He
N
clusters
Cite as: J. Chem. Phys. 150, 154304 (2019); https://doi.org/10.1063/1.5092566
Submitted: 12 February 2019 . Accepted: 14 March 2019 . Published Online: 15 April 2019
Ricardo Pérez de Tudela , Paul Martini, Marcelo Goulart , Paul Scheier , Fernando Pirani ,
Javier Hernández-Rojas , José Bretón , Josu Ortiz de Zárate, Massimiliano Bartolomei , Tomás
González-Lezana , Marta I. Hernández , José Campos-Martínez , and Pablo Villarreal
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A combined experimental and theoretical
investigation of Cs
+
ions solvated in He
N
clusters
Cite as: J. Chem. Phys. 150, 154304 (2019); doi: 10.1063/1.5092566
Submitted: 12 February 2019 • Accepted: 14 March 2019 •
Published Online: 15 April 2019
Ricardo Pérez de Tudela,
1
Paul Martini,
2
Marcelo Goulart,
2
Paul Scheier,
2
Fernando Pirani,
3
Javier Hernández-Rojas,
4
José Bretón,
4
Josu Ortiz de Zárate,
5
Massimiliano Bartolomei,
5
Tomás González-Lezana,
5,a)
Marta I. Hernández,
5
José Campos-Martínez,
5
and Pablo Villarreal
5
AFFILIATIONS
1
Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, 44780 Bochum, Germany
2
Institut für Ionenphysik und Angewandte Physik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria
3
Dipartimento di Chimica, Biologia e Biotecnologie, Universitá di Perugia, 06123 Perugia, Italy
4
Departamento de Física and IUdEA, Universidad de La Laguna, La Laguna, 38205 Tenerife, Spain
5
Instituto de Física Fundamental, IFF-CSIC, Serrano 123, 28006 Madrid, Spain
a)
Electronic mail: t.gonzalez.lezana@csic.es
ABSTRACT
Solvation of Cs
+
ions inside helium droplets has been investigated both experimentally and theoretically. On the one hand, mass spectra
of doped helium clusters ionized with a crossed electron beam, He
N
Cs
+
, have been recorded for sizes up to N = 60. The analysis of the
ratio between the observed peaks for each size N reveals evidences of the closure of the first solvation shell when 17 He atoms surround
the alkali ion. On the other hand, we have obtained energies and geometrical structures of the title clusters by means of basin-hopping,
diffusion Monte Carlo (DMC), and path integral Monte Carlo (PIMC) methods. The analytical He–Cs
+
interaction potential employed in
our calculations is represented by the improved Lennard-Jones expression optimized on high level ab initio energies. The weakness of the
existing interaction between helium and Cs
+
in comparison with some other alkali ions such as Li
+
is found to play a crucial role. Our
theoretical findings confirm that the first solvation layer is completed at N = 17 and both evaporation and second difference energies
obtained with the PIMC calculation seem to reproduce a feature observed at N = 12 for the experimental ion abundance. The analy-
sis of the DMC probability distributions reveals the important contribution from the icosahedral structure to the overall configuration
for He
12
Cs
+
.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5092566
I. INTRODUCTION
Helium droplets are considered as an ideal environment for
spectroscopical investigations of different species.
1–7
Among all pos-
sible impurities to dope the He
N
clusters, alkali ions have received
special attention in recent studies.
8–24
From the experimental side,
for instance, Müller et al.
25
performed a systematic study of the
formation and stability of snowballs obtained by femtosecond pho-
toionization of small alkali clusters bound to helium nanodroplets.
The authors concluded that the size of the doped helium clus-
ter depends on the mass of the alkali atom: whereas for Na
+
and
K
+
ions, clusters between 3 and 10 He atoms were observed, the
heavier alkali Rb
+
and Cs
+
led to the formation of snowballs con-
taining up to 41 He atoms. Differential mass spectra obtained as
abundance mass ratios of neighboring snowball intensities, I
N
+1
/I
N
,
as a function of the number of He atoms, N, were analyzed in order
to characterize the structure of the cluster around the ionic impu-
rity. In particular, indications of shell closures of the solvating atoms
were inferred from dips observed in such relative intensities. The
drop found in the spectra for He
N
Cs
+
at N = 16 is slightly shifted
with respect to the theoretical prediction reported by Rossi et al.,
26
who calculated N = 17.5 by numerical integration of the radial
density profile obtained in a shadow Monte Carlo (MC) calcula-
tion. The authors suggested the existence of another relatively stable
J. Chem. Phys. 150, 154304 (2019); doi: 10.1063/1.5092566 150, 154304-1
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structure at He
12
Cs
+
to explain the additional dip featured by the
differential intensity at N = 12
25
and referred to the classical lowest
energy configuration seen for He
12
K
+
in the calculations reported
in Ref. 27. The immersion of Cs
+
into the He
N
nanodroplets was
also experimentally investigated by Theisen et al.,
20
and the only
deviations from a smooth dependence of the photoion yield with
respect to N were observed around N = 17–18 and N = 50. Both fea-
tures were associated, in principle, to shell closures (first and second
shell, respectively) although, due to its non rigid character, the exact
number filling a second layer surrounding the ionic impurity could
fluctuate.
Previous theoretical investigations of He
N
Cs
+
in the literature
found a structure consisting of the ion localized at the center of a
droplet formed by surrounding He atoms in a snowball along well
defined shells.
26,28,29
Galli et al.
29
performed path integral Monte
Carlo (PIMC) calculations of these systems at T = 1 K in comparison
with
4
He nanodroplets doped with some other alkali and alkali-earth
ions such as Na
+
, K
+
, Be
+
, and Mg
+
. Density profiles of helium atoms
around the Cs
+
ion were calculated for different sizes N. The num-
ber of He atoms in the first shell changed from 17 for He
20
Cs
+
up to
18 when the calculation was performed for He
64
Cs
+
and He
128
Cs
+
.
The radial probability densities displayed different peaks associated
with the filling of shells around the cesium ion as more helium atoms
are added to the cluster. In particular, the authors of Ref. 29 saw
that the second maximum at ∼6 Å is characterized by He atoms with
high radial mobility which can participate in position changes with
atoms of the first shell, in agreement with conclusions reported in
Ref. 28.
In a recent experiment,
30
helium nanodroplets doped with
cesium in the presence of molecular hydrogen were subsequently
ionized with a crossed electron beam. The obtained mass spectra
were analyzed in order to study mainly the abundance of (H
2
)
N
Cs
+
.
The most prominent mass peaks were due, however, to the exis-
tence of He
N
Cs
+
clusters. The corresponding ion abundance curve
as a function of the number of He atoms was shown for sizes
smaller than N = 30, revealing just a suggested feature around
N = 15.
30
In this work, we present results for a similar experiment per-
formed only with cesium atoms in the pickup chamber in an attempt
to obtain much clearer conditions for a proper inspection of the
relative abundances of the different Cs
+
ions solvated with helium
atoms. In addition to this and in order to search for the most sta-
ble configurations, we have carried out extensive theoretical calcu-
lations by means of both classical and quantum mechanical (QM)
approaches. In particular, basin-hopping (BH), diffusion Monte
Carlo (DMC), and PIMC calculations have been performed in com-
parison with the new experimental data. Analogously as in previ-
ous theoretical investigations of similar clusters (see, for instance,
Ref. 31 for the case of He
N
Li
+
), our study includes the explicit
ab initio calculation of the He–Cs
+
interaction which has been used
to optimize an analytical representation according to an improved
Lennard-Jones (ILJ) expression.
The structure of the paper is the following: Details of the exper-
imental setup are given in Sec. II; the potential energy surface (PES)
is described in Sec. III; details of the theoretical methods employed
in our calculations are shown in Sec. IV. Results are shown and dis-
cussed in Sec. V and Sec. VI, respectively, and finally, conclusions
are given in Sec. VII.
II. EXPERIMENTAL DETAILS
Helium nanodroplets were produced by expanding helium
(Linde, purity 99.9999%) at a stagnation pressure of about 2.6 MPa
through a 5 µm nozzle, cooled by a closed-cycle cryostat (Sumitomo
Heavy Industries Ltd., model RDK-415D), into vacuum. The noz-
zle temperature was set to 9.9 K, resulting in an average droplet size
of about 5 ×10
5
He atoms.
23,32
The resulting supersonic beam was
confined by a 0.8 mm conical skimmer, located 8 mm downstream
from the nozzle, and passed through a 20 cm-long differentially
pumped chamber. The droplets crossed through a pick-up cell filled
with cesium vapor produced in a resistively heated oven. The tem-
perature of the metal oven was set to 327.5 K. The doped helium
nanodroplets passed through another differentially pumped vacuum
chamber where they were crossed with an electron beam of 85 eV
energy. The emission current was 46 µA. Ions were guided by elec-
trostatic lenses into the extraction region of a commercial orthogo-
nal time-of-flight (TOF) mass spectrometer equipped with a reflec-
tron (Tofwerk AG, model HTOF). The effective mass resolution was
m/∆m = 5400 (∆m = full-width-at-half-maximum). The ions were
detected by a micro-channel plate operated in single ion counting
mode and recorded via a time to digital converter. Additional exper-
imental details have been described elsewhere.
33
The mass spectrum
was evaluated by means of a custom-designed software.
34
The pro-
gram corrects for experimental artifacts such as background signal
levels, imperfect peak shapes, and mass drift in case of long-term
measurements.
III. POTENTIAL ENERGY SURFACE
Pairwise two-body (2B) functions have been employed to
describe the He–He and He–Cs
+
interactions. In particular, for
He–He, we have used the potential reported in Ref. 35, whereas
for the He–Cs
+
contribution, a new potential energy surface (PES)
optimized on accurate coupled-cluster with single and double
and perturbative triple excitations [CCSD(T)] interaction ener-
gies obtained with the d-aug-cc-pV6Z
36
and def2-AQVZPP
37
basis
sets for He and Cs
+
, respectively, has been developed. We have
checked that the adopted basis set is sufficiently large to guaran-
tee well converged interaction energies, which are found to devi-
ate from those carried out in the global minimum region with the
d-aug-cc-pV5Z/def2-AQVZPP set of less than 0.3 meV (about 1%).
The CCSD(T) computations have been performed using the Mol-
pro2012.1 package.
38
For the analytical representation of the force
field, the ILJ formulation
39
of the atom-atom interactions has been
chosen
V(r)=𝜖
m
n(r)−m
r
m
r
n(r)
−
n(r)
n(r)−m
r
m
r
m
. (1)
In the expression above, 𝜖 is the potential depth, r
m
is the minimum
potential position, and n(r) is defined as follows:
39
n(r)=β + 4
r
r
m
2
. (2)
Values for the parameters used in Eqs. (1) and (2) are shown in
Table I.
Both the ab initio points and the analytical representation for
the He–Cs
+
interaction potential are shown in Fig. 1 besides the
J. Chem. Phys. 150, 154304 (2019); doi: 10.1063/1.5092566 150, 154304-2
Published under license by AIP Publishing
The Journal
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TABLE I. Parameters for the ILJ potentials employed for the He–Cs
+
interaction
shown in Eqs. (1) and (2). r
m
is given in Angstrom, 𝜖 is given in milli electron volt;
m, and β are dimensionless parameters.
m r
m
𝜖 β
4 3.35 13.70 9.5
He–He potential taken from Ref. 35. The comparison with the
He–Li
+
case, a system formed with a much lighter alkali ion treated
in a previous investigation,
31
reveals the weak character of the exist-
ing interactions responsible for the stability of the Cs
+
doped helium
clusters. As shown in Fig. 1, the He–Li
+
interaction is about 5 times
deeper than the He–Cs
+
. The reason has to be found in the differ-
ent charge distribution of each ion. Due to the smaller size of Li
+
,
the charge of the ion is distributed in a shorter and more com-
pact region, so the He atoms are capable to penetrate closer to
the impurity. Cs
+
is, on the contrary, bigger, and the charged elec-
tronic cloud extends therefore at a further distance with respect to
the center. The rare gas atoms can not get close to the impurity
thus resulting in a weaker interaction shifted at larger interatomic
distances.
IV. THEORETICAL METHODS
A. Basin-hopping
Putative global energy minima of He
N
Cs
+
clusters with N ≤30
were located using the BH method
40
also known as the “Monte
Carlo plus energy minimization” approach of Li and Scheraga.
41
This unbiased technique has been particularly successful for the
global optimization of various atomic and molecular systems.
42–50
The BH method relies on an extensive random search of the PES by
FIG. 1. Potential energy curves for He–Cs
+
(in circles ab initio points and in the
black solid line the ILJ analytical fit) and He–He
35
(in the blue dashed-dotted line)
interactions employed in this work. The He–Li
+
case (red dashed line) from Ref.
31 has been included for comparison in the bigger panel, while in the inset, the
He–He and He–Cs
+
cases are shown in a more reduced range.
large amplitude MC moves followed by systematic local optimiza-
tion. Suitable parameters for the present BH simulations were deter-
mined for all cluster sizes based on preliminary tests on He
12
Cs
+
.
These benchmarks consisted of 10
5
minimization steps and were
initiated from independent random geometries, varying the tem-
perature and the target acceptance ratios of the MC simulation.
Although our global minima remain putative, they were obtained
in all trajectories. This should ensure a reasonably high degree of
confidence.
The results below were obtained at a constant simulation tem-
perature of k
B
T = 2 meV (where k
B
is the Boltzmann constant)
and an acceptance ratio of 50%. A total of 4 runs of 5 ×10
5
BH
steps each were performed for all sizes. The quantum effects were
included through the zero-point energy (ZPE) function, in the har-
monic approximation.
31,51,52
To do it, we built a database of local
minima close to the global minimum for each cluster size. In some
cases, the geometry of the BH+ZPE global minimum differs from the
BH one.
B. Diffusion Monte Carlo
DMC
53,54
calculations are carried out by propagating in imagi-
nary time (τ = it) a time dependent Schrödinger equation, in such a
way that the general solution,
Ψ(τ)=
n
c
n
ψ
n
e
−τE
n
/
h
, (3)
will lead, in the long-time limit, to only one non-vanished term that
will correspond to E
0
, the ground state energy.
Depending on the cluster size and therefore the number of He
atoms, the number of replicas included are within the range 6000–
12 000, with typical time steps ∆τ = 40–80 a.u., over 4000–8000 time
steps and using descendant weighting with 6–9 generations.
C. Path integral Monte Carlo
The PIMC method employed in this work has been described
before several times,
31,51,52,55–58
so here we will only discuss the most
relevant details of the calculation. The density matrix of the system at
a temperature T is expressed as the product of densities at tempera-
ture T
′
= T ×M and is evaluated in a collection R
α
≡{r
α
1
, ..., r
α
N
}of
position vectors r
α
i
of the particles forming the cluster. The α index
runs over the M quantum beads or time slices.
The total energy of the Cs
+
doped helium clusters can be
obtained by means of the so-called virial estimator
59,60
expressed as
E(T)=
3 N
2β
−
1
2M
M
α=1
N
i=1
(r
α
i
−r
C
i
)⋅F
α
i
+
1
M
M
α=1
V(R
α
), (4)
where r
C
i
= M
−1
∑
M
α=1
r
α
i
defines the centroid of the M beads and
β = (k
B
T)
−1
. The first term in Eq. (4) corresponds to the classi-
cal kinetic energy after substracting the center of mass degrees of
freedom; the second one is a quantum correction where F
α
i
is the
force experienced by the i-particle on the α slice and the third term
describes the interaction between each pair of particles on that α
slice according with the pairwise interactions described in Sec. III.
The Cs
+
ion is located fixed at the origin, and it is not included
J. Chem. Phys. 150, 154304 (2019); doi: 10.1063/1.5092566 150, 154304-3
Published under license by AIP Publishing
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in the calculations, which are exclusively for the He atoms. This
is an acceptable approximation given its large mass in comparison
with He. The rigidity of the clusters is investigated by means of the
Lindemann index, defined as follows:
51,61
δ
i
=
1
M(N −1)
M
α=1
N
j≠i
r
α2
ij
−r
α
ij
2
r
α
ij
, (5)
where r
α
ij
=r
α
i
−r
α
j
is the He–He distance at the time slice α. More-
over, our analysis includes the calculation of the so-called radius of
gyration (ROG), obtained as follows:
(R
g
)
i
=
1
M
M
α=1
(r
α
i
−r
C
i
)
2
, (6)
which measures the quantum delocalization of each individual atom.
The PIMC calculation is performed at T = 2 K using M = 200 beads.
The integration of the above terms shown in Eq. (4) is carried out
via a Metropolis MC algorithm averaging over a number of paths
{R
1
, R
2
, .. ., R
M
, R
M+1
}sampled by means of a staging procedure
62
moving about 10% (that is 20) of the beads. About 10
6
×N staging
moves for thermalization and about 10
7
×N for statistics were con-
sidered. A confinement procedure was employed to avoid the evapo-
ration of the He atoms beyond a cut-off radius defined by inspection
of radial probability density functions.
V. RESULTS
A. Experimental results
Integrated counts from the He
N
Cs
+
complexes have been
obtained as a function of the number of He atoms, N, as in pre-
vious investigations for similar systems.
10,30,31
The ion intensities,
shown in Fig. 2 for complexes up to N = 60, exhibit a mainly
structureless profile with the only exception of a drop after N = 12
and a soft shoulder between N = 14 and 18. In principle, one may
FIG. 2. Measured abundances of He
N
Cs
+
as a function of the number of He atoms,
N, up to N = 60. In the inset, present results (black squares) are compared for N ≤
28 with previous measurements (red circles) obtained in an experiment performed
with helium nanodroplets doped with cesium and molecular hydrogen.
30
relate both anomalies to a stable structure for He
12
Cs
+
, perhaps an
icosahedral arrangement as suggested for He
12
Kr
+
,
27
and to pos-
sible indications of complete filling of the first shell, respectively.
Much more smoother features are suggested at larger sizes, about
N = 33–35 and N = 53. In the inset of Fig. 2, the present ion mass
abundance is compared for N ≤28 with that reported in the recent
study by Kranabetter et al.
30
of helium nanodroplets doped not only
with Cs
+
but also with H
2
. The comparison between the curves
shown in Fig. 2 reveals that the small peak at N = 15 in the case
of the He
N
Cs
+
experiment (red circles in Fig. 2) is not seen in this
work, where no H
2
is introduced in the pick-up chamber besides
helium and cesium. The feature observed in Ref. 30 has its origin in
a double peak at m/z ∼193 assigned both to He
15
Cs
+
and a frag-
ment of a polydimethylsiloxane, S
3
C
3
H
9
O
+
4
, used as a lubricant for
turbo pumps. Once this additional contribution is removed after
extended baking, no intensity anomaly for He
15
Cs
+
is found and
the cluster abundance does not exhibit a suggested peak at N = 15
(see Fig. 2).
Two portions of the mass spectrum recorded after electron ion-
ization of the Cs doped helium nanodroplets are shown in Fig. 3
at the mass per charge range corresponding to the regions between
He
14
Cs
+
and He
16
Cs
+
and close to He
26
Cs
+
. Besides these con-
tributions from the Cs
+
doped clusters, peaks for He
+
N
with N =
47–49 and 59, respectively, are also observed. As already reported
in Ref. 23, all He
+
N
ions exhibit a narrow satellite peak (indicated
with vertical arrows in Fig. 3) next to their nominal mass, typically
0.1–0.3 amu upshifted. For the He
N
Cs
+
cluster, on the contrary,
this extra peak is seen only for He
26
Cs
+
but seems to be absent
for N = 14–16. As reported in Ref. 63, these extra features close
to the main peaks originate from metastable decay of cluster ions
on their way through the TOF mass spectrometer. They can be
understood as indications of unimolecular dissociation of the type
FIG. 3. Section of the mass spectrum obtained by electron ionization of Cs doped
Helium nanodroplets. Conditions: electron energy 85 eV, electron current 46 µA,
helium temperature 9.9 K, and He pressure 2.6 MPa. In this semilogarithmic plot,
all pristine helium cluster ions and the He
26
Cs
+
maximum peak exhibit a satellite
peak (indicated by arrows) upshifted by about 0.3 Thomson and about 5% of the
intensity of the main peak which can be assigned to fragmentation processes in
the ion extraction region of the TOF mass spectrometer.
J. Chem. Phys. 150, 154304 (2019); doi: 10.1063/1.5092566 150, 154304-4
Published under license by AIP Publishing
![FIG. 9. Evaporation energies as a function of N, the number of He atoms, ∆EN = − [EN − E(N−1)] for the HeNCs+ clusters as a function of the number of He atoms, N, obtained with the BH (black solid squares), PIMC (red circles), DMC (blue triangles), and BH+ZPE (empty squares). Units are milli electron volt.](/figures/fig-9-evaporation-energies-as-a-function-of-n-the-number-of-7z8qfgii.webp)
