Electronically Excited States of Sodium-Water-Clusters
Claus Peter Schulz∗ and Christiana Bobbert
Max-Born-Institut, Max-Born-Str. 2a, D-12489 Berlin, Germany
Taku Shimosato, Kota Daigoku, Nobuaki Miura, and Kenro Hashimoto†
Computer Center and Department of Chemistry, Tokyo Metropolitan University/ACT-JST,
1-1 Minami-Ohsawa, Hachioji-shi, Tokyo 192-0397, Japan
(Dated: Received 27 February 2003; accepted 17 September 2003)
The lowest electronically excited state of small Na(H2 O)n clusters have been investigated experimentally and theoretically. The excitation energy as determined by the depletion spectroscopy
method drops from 16950 cm−1 for the sodium atom down to 9670 cm−1 when only three water
molecules are attached to the Na atom. For larger clusters the absorption band shifts back towards
higher energies and reaches 10880 cm−1 for n = 12. The experimental data are compared to quantum
chemical calculations at the MP2 and MRSDCI levels. We found that the observed size-dependence
of the transition energy is well reproduced by the interior structure where the sodium atom is surrounded by water molecules. The analysis of the radial charge distribution of the unpaired electron
in these interior structures gives a new insight into the formation of the “solvated” electron.
I.
INTRODUCTION
Complexes containing alkali-atoms surrounded by polar solvent molecules are interesting species to study the
behavior of loosely bound metal valence electrons in polar environment. Since the first report of the solvation
of alkali metal in liquid ammonia by Weyl in 1864,1 the
solvation phenomena of metals in liquid ammonia and
other polar solvents has been studied extensively.2 It has
been found that in liquids solvated electrons are formed
when alkali metal is dissolved. These solvated electrons
are completely detached from the alkali metal ions and
exhibit spectroscopic properties of their own. On the
other hand, in very small complexes with only a few solvent molecules attached to the alkali atom the valence
electron is still bound in the coulomb field of the metal
ion core. Thus, research on the formation of solvated
electrons in model systems is of fundamental interest for
the understanding of elementary chemical processes such
as solvation, electron transfer, charge induced reactivity
and the structure of the liquid phase in general.
A variety of clusters have been studied to gain
information at the solvation process on a molecular
−
−
level. Negatively charged (H2 O)n and (NH3 )n clusters were one of the first species, where photo detachment energies have been determined.3 Later absorption
−
spectra of (H2 O)n have been reported.4 Sr+ (H2 O)n ,
+
Ca (H2 O)n and Mg+ (H2 O)n clusters have been studied
and photo dissociation spectroscopy to understand the
ion solvation.5–8 Fuke et al. reported photoelectron spec−
−
tra of the anion complexes Li(H2 O)n and Na(H2 O)n .9,10
On theoretical side the energetics and structures
−
−
of negatively charged (H2 O)n and (NH3 )n clusters
have been studied by applying a modified continuum
model.11,12 Quantum mechanical calculations were carried out to elucidate the geometric and electronic
structures of Ca+ (H2 O)n and Mg+ (H2 O)n clusters.13,14
The geometries and detachment energies of Li− (H2 O)n ,
Li− (NH3 )n , Na− (H2 O)n , and Na− (NH3 )n have been ex-
plored by ab initio MO method.15–18
In the past years, our group has investigated the spectroscopic properties of size selected neutral sodium-water
and sodium-ammonia clusters. Ionization potentials (IP)
for Na(H2 O)n and Na(NH3 )n clusters up to n ≈ 20 have
been determined by one photon ionization.19–22 The measurements have shown that the IP decreases with increasing number of solvent molecules. While for Na(NH3 )n
clusters the ionization potential decreases smoothly with
increasing cluster size towards the bulk value of 1.5 eV,
the IP of Na(H2 O)n stays constant for n ≥ 4 at 3.2 eV. A
similar behavior was found when sodium was replaced by
cesium or lithium.10,23 The interesting difference in the
size dependence of IPs in M(H2 O)n and M(NH3 )n , especially the unusual behavior of the IPs in the former clusters has motivated many theoretical studies.24–30 Among
them, Hashimoto and Kamimoto have found for the first
time the excess electron localization and the formation
of the two center state in Li(H2 O)n (n ≥ 4).28
Another important spectroscopic quantity is the energy of the lowest electronically excited state, which
asymptotically corresponds to the 3p ← 3s transition in
the bare sodium atom. The energetics of this transition
can give information on the charge density of the sodium
valence electron. The first electronically excited state
of Na(NH3 )n complexes has been studied extensively in
our group.31–33 As a general result of these studies, it
has been found that the 3p ← 3s excitation energy decreases from its atomic value (16950 cm−1 ) with increasing number of ammonia molecules bound to the sodium
atom and reaches 6000 cm−1 when the first solvation shell
is filled at n = 4. This size dependence of the transition energy resembles that of the separation between
the 32 S(Na) ← 31 S(Na− ) and 32 P(Na) ← 31 S(Na− )
type bands in the photoelectron spectra for the negatively charged Na(NH3 )n . The decrease of 2 S–2 P energy
separations with increasing cluster size has been theoretically attributed to the growth of the Rydberg-like
wavefunction of the valence electron in the clusters.17 For
2
FIG. 1: Scheme of the experimental setup. The excitation
laser enters the apparatus counterpropagating to the cluster
beam, while the ionization laser in perpendicular to the cluster beam.
larger Na(NH3 )n clusters the excitation energy increases
slightly towards the absorption of the “solvated” electron
in liquid ammonia at 6300 cm−1 . As a second interesting result these spectroscopic studies revealed a strong
interaction between the electronic excitation of the unpaired electron with vibrations of the solvent molecules:
a strong absorption peak for all Na(NH3 )n clusters with
n > 3 was observed at 6600 cm−1 and assigned to the
first overtone of the symmetric and/or asymmetric N–H3
stretching vibration. The result was also confirmed by
pump-probe experiments in our lab, which showed a decreasing lifetime for the exited state with an increasing
number of solvent molecules.34,35
In this paper we present for the first time a spectroscopic study of sodium-water clusters. The absorption bands for the first electronically excited state of
Na(H2 O)n with n = 1 − 12 have been obtained experimentally by the depletion method as described in the
next section. The structures and the vertical transition
energies of these clusters with n up to 8 were investigated
by ab initio MO calculations at the correlated level. The
size and the structure dependence of the vertical transition will be presented in section III. In the final section
we will not only compare the experimental and theoretical results on the excitation energies of sodium-water
cluster but also discuss the relationship to our previous
study on sodium-ammonia clusters.
II.
oven and the nozzle are heated to temperatures 115◦ C
and 125◦ C, respectively, giving a water vapor pressure
of 1.4 bar. A skimmer collimates the supersonic jet and
the water cluster beam traverses to the sodium oven in a
separate chamber. The number of sodium atoms picked
up by the water clusters is controlled by the sodium vapor pressure. In this experiment an oven temperature of
200◦ C (pN a ≈ 1 · 10−4 mbar) gives the best condition to
form complexes containing only one sodium atom. The
formed mixed clusters enters the detection chamber further downstream. Here the clusters are excited and ionized with nanosecond laser pulses. The ions are mass
analyzed in a time-of-flight mass spectrometer and detected with MSP (MicroSpherePlates).
In our earlier spectroscopic study of sodium-ammonia
clusters it was found that the clusters fragment after excitation. We expect that sodium-water clusters exhibit
the same behavior. Therefore the method of depletion
spectroscopy have been used to determine the absorption band of Na(H2 O)n clusters. This method has been
described in detail in Ref. 33. Sodium-water clusters
are excited with a tunable laser beam counter propagating to the cluster beam. The laser radiation is generated by an optical parametric process. A commercial
laser system (Spectra Physics MOPO 730-10) was used,
which provides photon energies from 5500 to 24000 cm−1
with a small gap around the degeneration point of the
OPO between 13650 cm−1 and 14500 cm−1 . The clusters
are ionized 200 µs after excitation by photons of 4.66 eV
(266 nm) energy (4th harmonic of a Nd:Yag laser, Continuum Surelight).
For the evaluation of our data it is important that the
ionization does not introduce dissociation of the clusters
and thus an incorrect assignment of n. In our earlier
study of the ionization potentials22 we have found for
Na(NH3 )n that most of the excess energy is carried by
the photo electron. The remaining internal energy is
small and does not lead to fragmentation due to the high
binding energy between the ammonia molecules of about
0.3 eV.22 We can safely assume that this also holds for
Na(H2 O)n clusters especially because the binding energy
between the solvent molecules is even higher (0.5 eV).20
EXPERIMENTAL
In contrast to our earlier studies, where the mixed clusters were formed in a pulsed crossed beam type pick-up
source,20 in the experiments presented here a newly build
continuous pick-up source was used. The design is similar
to the setup used to dope large helium clusters with alkali
metal atoms.36 A brief overview will be adequate here. A
detailed description of the source and its properties can
be found in a separate publication.37
Figure 1 shows the experimental setup schematically.
Water clusters are prepared by expanding water vapor
through a 70 µm diameter nozzle into the vacuum. The
The raw ion signal taken with (Son ) and without (Soff )
excitation laser at each wavelength are converted to
the absorption cross section σ by the Beer-Lambert law
− ln(Son /Soff ) = σφ,38 where φ is the laser fluence also
measured at each point. As a typical example Figure 2
shows − ln(Son /Soff ) as a function of the laser fluence φ
for the Na(H2 O)3 cluster. The depletion signal increases
linearly with the laser fluence as predicted by LambertBeer law up to 2.5 mJ/cm2 and saturates for larger laser
fluence. The slope gives the photo absorption cross section σ. The all measurements presented in this paper
were taken in the linear part.
3
have also corrected the basis set super position error for
the total binding energies by the counter-poise correction (CPC).42 From now on, we designate the ∆E with
only CPC as ∆Ecpc and that with both CPC and ZPC
as ∆Ezpc . The vertical transition energies (VTEs) of
Na(H2 O)n were calculated by the multi reference single
and double excitation configuration interaction (MRSDCI) method preceded by the complete active space selfconsistent-filed (CASSCF) calculations.43–47 The active
space for the CASSCF consists of five molecular orbitals
(MOs) corresponding to the 3s, 3p and 4s orbitals of Na,
and the ground and four low-lying excited states were
averaged with equal weight. The natural orbitals (NOs)
obtained by this method were used as one-particle functions in the MRSDCI calculations with the CASSCF reference. All single and double excitations from the five active orbitals and n high-lying occupied MOs corresponding to lone pairs of H2 O molecules were included in the
MRSDCI. We used the MOLPRO-2000 for the MRSDCI
calculations.48
FIG. 2: Depletion signal − ln(Son /Soff ) as a function of the
laser fluence φ for the Na(H2 O)3 cluster at 10000 cm−1 . The
signal saturates for φ > 2.5 mJ/cm2 . The absorption cross
section σ can be deduced from the slope of the linear part of
the signal.
III.
THEORETICAL
Molecular structures of Na(H2 O)n (n = 1 − 6 and
8) were optimized at MP2 level with the usual frozen
core approximation. The basis sets used were the 631++G(d,p).39 Since the number of potential minimum
configurations becomes very large for n ≥ 4, we have
optimized most n ≥ 4 clusters by referring to the lowenergy structures of Na(H2 O)n and Li(H2 O)n reported
in the previous ab initio studies.26,28,30 We also referred
to the anionic geometries of Na(H2 O)4 .17 The vibrational
analyses were carried out at each optimized geometry
to confirm the minima on the potential energy surfaces.
Harmonic frequencies of all n ≤ 3 clusters and a few
n = 4 structures were evaluated by computing the second derivative matrices analytically, while those of other
isomers were calculated by differentiating numerically
the first derivatives along the nuclear coordinates. The
program used for the geometry optimization and vibrational analyses was Gaussian-98.40 Total binding energies, ∆E(n), of the minimum structures were evaluated
by the following formula.
−∆E(n) = E [Na(H2 O)n ] − E [Na] − n · E [H2 O]
(1)
We have taken the zero point vibrational correction
(ZPC) into account by using the scaled harmonic frequencies. The scale factor, 0.956, was obtained from
the average ratio of the experimental fundamental41 and
the calculated harmonic frequencies of a free H2 O. We
IV.
A.
RESULTS
Spectroscopic Properties of Na(H2 O)n
The photoabsorption of the Na(H2 O)n clusters has
been investigated in a broad spectral range from the
3p ← 3s transition of the bare sodium atom (16950 cm−1 )
to the infrared region (6000 cm−1 ) except for a small gap
between 13650 cm−1 and 14500 cm−1 . Figure 3 shows the
spectra of Na(H2 O)n with n = 1 − 6. The circle symbols
present the experimental data. To guide the eye the full
line has been added, which was obtained by a polynomial
fit to the experimental data. Two vertical dotted lines
give the limit of the laser gap.
For the Na(H2 O) cluster (1:1 complex) only a weak
absorption is observed at the low energy side of the
scanned spectrum. The absorption increases noticeable at 13400 cm−1 close to the degeneration gap of the
laser. From the decreasing absorption towards higher
energies we assume that the main absorption band for
Na(H2 O) clusters is in the laser gap around 14000 cm−1 .
This would be a considerable red shift in comparison
with the 3p ← 3s transition of the free sodium at
16950 cm−1 , which is similar to our earlier observation
for the Na(NH3 ) complex.31 However, in contrast to the
sodium-ammonia complex no sharp vibrational lines were
resolved for Na(H2 O) even when the scan was repeated
with higher spectral resolution. Apart from this main
feature two very weak absorption bands are observed
around 8850 cm−1 and 12150 cm−1 . These bands coincides with transition energies to highly excited vibrational and rotational states of the water molecules, which
have been observed in atmospheric water vapor.49,50
The absorption for larger Na(H2 O)n complexes with
n ≥ 2 are much stronger. The absorption bands are
broad, asymmetric without any distinct structures. For
4
n Emax / cm−1 Eonset / cm−1
0
16960
1
14000
13450
2
10090
7300
3
9470
6180
4
9580
6140
5
9860
7330
6
10200
7390
7
10540
7820
8
10470
7830
9
11130
8280
10
10540
8090
11
10370
8030
12
10570
7570
∞
14100a
-
FIG. 3: Absorption spectra of Na(H2 O)n clusters. The open
are the experimental data while the full eye guiding line results from polynomial fit. The full vertical line at 9450 cm−1
denotes maximum for n = 3.
n = 2 the onset of the absorption is at 7300 cm−1 . The
maximum at 10085 cm−1 is shifted strongly to the near
infra red spectral region compared to the Na(H2 O) cluster. The maximum of the absorption for n = 3 and 4 are
shifted further to the red spectral range and have more
or less the same position. For easier comparison the full
vertical line in Fig. 3 denotes the position of the absorption maximum for Na(H2 O)3 at 9680 cm−1 . In the
right column of Fig. 3 it is clearly visible that the absorption maxima for larger clusters than n = 4 slowly
shifts to higher energies. This trend continues for even
larger Na(H2 O)n clusters with n = 6– 12 while the general shape of the absorption spectrum remains the same.
Therefore they are not shown here. The shift towards
the blue spectral range with increasing cluster size can
be extrapolated to the absorption band of the hydrated
electrons in liquids51 with a maximum at 14100 cm−1 as
it will be shown in Section V. Table I summarizes the onset and maxima of the absorption bands for all Na(H2 O)n
clusters accessible in our experiment.
A closer inspection of Fig. 3 reveals a second absorption maximum around 16000 cm−1 for Na(H2 O)n clusters with n ≥ 4. This absorption peak increases for
larger clusters. We tentatively assign this peak to the
next higher electronic state of the Na(H2 O)n complex.
For cluster larger than n = 4 the electronic states are no
longer connected asymptotically to the electronic states
of the sodium atom because the valence electron starts
to localize outside the first solvation shell as it will be
shown further below in the theoretical section.
For Na(H2 O)n (n = 2 − 6) we were able to determine
the absolute absorption cross section. Although these
values have a quite large error as can be seen in Fig. 2 we
TABLE I: Energy of the absorption maxima and onset for
Na(H2 O)n clusters. a: From Ref. 51
n f6000−16000 cm−1
0
0.998
2
0.6(4)
3
0.7(4)
4
0.6(4)
5
0.5(4)
6
0.6(4)
∞
0.75
Ref. 53
This work
This work
This work
This work
This work
Ref. 51
TABLE II: Estimated oscillator strength for Na(H2 O)n clusters
will use them to estimate the oscillator strength f of the
first electronically excited state between 6000 cm−1 and
16000 cm−1 . As shown in Table II the oscillator strength
decreases only by 30 to 50% compared to the 3p ← 3s
transition in the bare sodium atom and remains relatively
constant with the cluster size. Our values for neutral
Na(H2 O)n clusters are in good agreement with measured
oscillator strength f ≈ 0.77 of negatively charged water
cluster which is also nearly cluster size independent.4
B.
Optimized geometries and energetics
The optimized structures and the total binding energies of Na(H2 O)n (n = 1–8) have been calculated.
Some selected isomers are shown in Figure 4. A completes overview about all optimized structures and tables
of their frequencies are given in the electronic version
(EPAPS).54 The labels of the form p + q(+r) under each
structure denote the numbers of the water molecules in
the first (p), second (q), and third (r) shells.
Na(H2 O) has a C2v structure with a Na–O bond and
the ∆Ezpc is about 4 kcal/mol. 2+0 and 1+1 type isomers of Na(H2 O)2 are as stable as each other. Their
5
2.371
126.9 2.398
2.526
2.355
69.3 0.967
W2a 2+0 C1
[8.9/11.6/15.3]
2.523
2.164
W3b 3+0 C3
[14.2/17.0/22.7]
W3c 3+0 C3
[14.1/19.4/25.7]
2.371
86.4
2.327
1.951 2.367
91.9
1.849 2.446
1.959 2.491
2.513++
+,,+ ,,
+
1.886 +
1.776
,, +
+ ,,
!!!!!
!!!
2.482 ! ! !!
1.870
92.3
1.761
W6f 3+3 C3
[34.1/47.1/62.2]
2.341
,,,, ++
++
+ ,, + ,, +
,, ++
1.890
1.918 1.769
161.2
2.356
$$$$$$$$$$
#### ""
$$$$$$$$$$
##
""
$$##$#$$
1.839"" ### "2.029
" ##
#
#
#
"
"
"
"
" ##
"
"
#" ## "
""
"
"" """"
"" " ##
## 1.962 #### 1.798
" ""
" ""
#
##
" #
# #
""
2.076
70.7
2.435
2.580
W8a 4+4 C2
[53.4/70.2/91.4]
2.334
123.6 Na
W3a 3+0 C1
[14.6/18.2/24.4]
W4a 4+0 C2
[22.8/28.7/37.6]
--------------++
,,+ ,, +
,+ , +
2.328
2.422
90.6
W5c 3+2 C1
[27.0/37.8/50.1]
W5a 4+1 C1
[30.7/39.1/51.1]
W2b 1+1 Cs
[8.9/11.9/15.4]
120.0
2.346
67.0
71.8
2.149
121.0
1.841
2.177
W1a 1+0 C2v
[4.4/5.0/6.6]
O
H
****
********
)( ) ( 2.624
)) ( ) ) ( (
(( )(( ) ()( )
63.6
(( ) )
)) ((
1.952
W4d 4+0 C4
[20.0/27.4/36.5]
2.351
89.8
2.319
1.782 1.904
2.444
2.352
1.887
1.900 W6a 4+2 C1
[38.5/49.5/64.0]
''''
2.453
& %% ''%& '% ' 2.486
%% && 2.029
&
1.828
%% && %% %% &&
&%& % && %%%% &&&& && %%%% && 1.881
1.834& %%
%%
%%
&&
2.038 &%
& % 1.794&& 1.902
%% && %%
&& %% &
1.834
1.857
''''''''
2.494
W8b 3+3+2 C1
[46.8/66.0/87.8]
FIG. 4: Optimized structures of Na(H2 O)n (n = 1– 8) calculated at MP2/6-31++G(d,p)level. Geometrical parameters are
given in Å and degrees. Molecular symmetry and total binding energies (∆Ezpc /∆Ecpc /∆E) in kcal/mol are also given under
each structure.
6
∆Ezpc are 8.9 kcal/mol.
Three 3+0 structures (W3a-c) and two 2+1 isomers
(W3d-e) were found for Na(H2 O)3 . The former clusters whose ∆Ezpc values are 14.1 - 14.6 kcal/mol are only
slightly more stable than the latter. A water molecule is
bound to W2a by forming the third Na-O bond in W3a,
while Na is located near the center of three equivalent O
atoms in W3b which is similar to the cationic structure.
Na is situated on a cyclic (H2 O)3 in W3c; the O-NaO angles are about 70 degrees. W3a and W3b can be
regarded as the beginning of the so called interior structures in which Na is surrounded by the first-shell ligands
from O sides, while W3c is the smallest surface complex where Na is situated on the surface of the hydrogenbonded water clusters.
For n = 4– 6, we also found interior (W4a-c, W5ab and W6a-e) and surface (W4d-f, W5c and W6f g) clusters as well as other isomers having one of the
n = 3 structures in the first shell (W5d-f ). We could
find the 4+1 surface complex at the HF level, however,
the optimization starting from the HF-optimized surface
geometry converged to W5f by the MP2 method. The
∆Ezpc values are around 20-23, 27-31 and 33-39 kcal/mol
for n = 4– 6, respectively.
Although it is practically impossible to examine all
possible minimum configurations for n = 8 we have optimized thirteen equilibrium structures for this size in total. Fig. 4 shows only the lowest energy structure we have
found for n = 8 (W8a) and in addition a typical surface
isomer (W8b). All other optimized structures are given
in the EPAPS.54 As we have seen already for the smaller
clusters n = 2– 6, there are several close-energy isomers
for each n and the similar situation is naturally expected
for n = 8. Thus, several isomers are considered to coexist
in the molecular beam even for small n.
C.
Vertical transition energies
The calculated vertical transition energies (VTEs) and
oscillator strengths to the 32 P- and 42 S-like states of
Na(H2 O)n for n = 0– 6 and two isomers with n = 8 are
listed in Table III. The results of all other n = 8 isomers
are given in the EPAPS.54 The VTE for the transition
to the 32 P state of bare Na atom was calculated to be
2.00 eV, which agrees with the experiment within 0.1 eV.
The calculated VTE for the à ← X̃ transition decreases
by 0.46 eV in the 1:1 complex, being in good agreement
with the observed lowering of the absorption maximum
from n = 0 to n = 1 (0.364 eV). Although the two n = 2
isomers are equally stable, the VTE value for the transition to the first excited state in W2a is lower than that of
W2b. Since the W2a structure reproduces the observed
lowering of the absorption maximum better we can assign
the observed peak to the à ← X̃ transition in W2a. The
present calculations predict that the band for the B̃ ← X̃
transition in W2a and that of à ← X̃ transition in W2b
are expected to appear at around 1.39 eV (11200 cm−1 )
FIG. 5: Comparison of the calculated vertical transition energies (VTE) and the experimental values for the à ← X̃
transition of Na(H2 O)n as function of n. The experimental
values are taken from Table I and the calculated values corresponds to the structures W1a, W2a, W3a, W4a, W5a,
W6a and W8a from Table III.
and 1.42 eV (11500 cm−1 ), respectively. The tail in the
higher energy side of the observed band maximum may
correspond to these transitions.
For n = 3, several transitions of the different isomers
can contribute to the observed band because the VTEs of
the low-energy structures are close to one another. The
calculated VTEs of the 32 P-type transitions in W3a-c
whose ∆Ezpc values deviate within 0.5 kcal/mol range
from 0.88 eV ( 7100 cm−1 ) to 1.25 eV ( 10100 cm−1 ).
They are between the lowest and the highest edges of
the observed band, and the oscillator strengths of these
transitions (0.2–0.4) are almost independent of the structures. Nevertheless, the present calculations show that
VTEs for the à ← X̃ transition of these 3+0 clusters
decreases from the lowest VTE value for n = 2, which is
consistent with the experimental observation.
The calculations also suggest that some lowest energy
structures such as W4a-e can contribute to the observed
band. The situation for n ≥ 5 is similar to n = 3 and 4.
It is worth noticing that the transition energies to 42 Slike state in the low-energy structures are shifted down to
1.7–2.0 eV for n ≥ 4 although the oscillator strengths are
small due to its forbidden nature. This fact supports
the tentative assignments of the second band around
16000 cm−1 observed for n ≥ 4 to the 42 S-type transitions.
7
n=0
Naa
n=1
W1a
eV f
C2v
2
3 P 2.00 0.33 12 B2
22 A1
12 B 1
2
4 S 3.08 0.00 32 A1
eV
1.54
1.57
1.64
2.44
f
0.31
0.26
0.32
0.03
n=2
W2a
C1
22 A
32 A
42 A
52 A
eV
1.24
1.39
1.56
2.17
f
0.31
0.32
0.25
0.04
W2b
Cs
2 2 A0
12 A00
3 2 A0
4 2 A0
eV
1.42
1.57
1.70
2.27
f
0.29
0.31
0.24
0.05
n=3
W3a
C1
22 A
32 A
42 A
52 A
eV
1.02
1.04
1.15
1.80
f
0.37
0.37
0.28
0.02
n=4
W3b
W3c
W3d
W3e
W4a
C3 eV f
C3 eV f
Cs eV f
C1 eV f
C2 eV f
12 E 0.88 0.37 12 E 1.19 0.31 22 A0 1.01 0.31 22 A 1.20 0.31 12 B 1.03 0.36
12 A00 1.31 0.31 32 A 1.37 0.32 22 B 1.11 0.34
2
2
2 A 1.12 0.30 2 A 1.25 0.23 32 A0 1.52 0.22 42 A 1.58 0.22 22 B 1.11 0.34
32 A 1.70 0.02 32 A 1.95 0.06 42 A0 1.97 0.06 52 A 1.93 0.04 32 A 1.80 0.04
W4b
C1
22 A
32 A
42 A
52 A
W4g
C2
12 B
22 A
22 A
32 A
a only
eV
1.10
1.18
1.29
1.86
eV
1.53
1.65
1.65
2.01
f
0.35
0.25
0.32
0.04
W4c
C1
22 A
32 A
42 A
52 A
f
0.30
0.22
0.22
0.03
n=5
W5a
C1
22 A
32 A
42 A
52 A
eV
0.90
1.18
1.21
1.72
eV
0.98
1.06
1.23
1.72
W4d
W4e
W4f
f
C4 eV f
C1 eV f
C2 eV f
0.31 12 E 1.11 0.31 22 A 1.13 0.33 12 B 1.29 0.31
0.32
32 A 1.21 0.33 22 B 1.32 0.31
2
0.26 2 A 1.30 0.19 42 A 1.41 0.20 22 A 1.61 0.20
0.02 32 A 1.84 0.08 52 A 1.90 0.06 32 A 2.10 0.07
f
0.34
0.35
0.21
0.05
W5b
Cs
22 A’
12 A”
32 A’
42 A’
eV
1.10
1.11
1.32
1.75
eV
1.15
1.22
1.44
1.79
f
0.33
0.33
0.19
0.03
W5c
C1
22 A
32 A
42 A
52 A
eV
1.10
1.19
1.43
1.88
f
0.33
0.34
0.13
0.04
W6b
C2
12 B
22 B
22 A
32 A
eV
1.15
1.36
1.42
1.79
f
0.31
0.30
0.17
0.09
W5d
C1
22 A
32 A
42 A
52 A
eV
1.36
1.54
1.56
1.88
f
0.30
0.23
0.23
0.07
f
0.35
0.31
0.14
0.03
W6c
Cs
12 A”
22 A’
32 A’
42 A’
eV
1.13
1.15
1.42
1.83
f
0.33
0.35
0.16
0.07
W5e
C1
22 A
32 A
42 A
52 A
eV
1.21
1.22
1.33
1.77
f
0.28
0.25
0.26
0.03
W5f
C1
22 A
32 A
42 A
52 A
eV
1.25
1.40
1.48
1.87
f
0.31
0.17
0.28
0.04
n=6
W6a
C1
22 A
32 A
42 A
52 A
W6d
C1
22 A
32 A
42 A
52 A
eV
1.16
1.22
1.30
1.76
f
0.31
0.34
0.14
0.01
W6e
C2
12 B
22 B
22 A
32 A
eV
1.16
1.38
1.43
1.74
f
0.34
0.29
0.10
0.04
W6f
W6g
C3 eV f
C1 eV f
12 E 1.08 0.31 22 A 0.98 0.32
32 A 1.06 0.31
2
2 A 1.45 0.14 42 A 1.30 0.17
32 A 1.80 0.11 52 A 1.70 0.08
n=8
W8a
C2
12 B
22 A
22 B
32 A
eV
1.12
1.46
1.46
1.82
f
0.36
0.06
0.27
0.01
W8b
C1
22 A
32 A
42 A
52 A
eV
1.09
1.10
1.50
1.85
f
0.31
0.30
0.14
0.12
1s orbital was frozen.
TABLE III: Vertical transition energies and oscillator strength (f ) corresponding to transitions from ground state to four
low-lying excited states in Na(H2 O)n at the MRSDCI level.
8
V.
A.
DISCUSSION
Comparison of experimental and theoretical
results
Figure 5 shows the experimental and theoretical transition energies of the à ← X̃ transition as a function
of n. The experimental transition energies for the band
maximum and onset are taken from Table I. The figure visualizes the findings of section IV A: the strong red
shift of the absorption energy for small clusters up to
n = 3 and the slight shift towards higher absorption energies for larger clusters. Fig. 5 also shows the calculated
vertical transition energies of the à ← X̃ transition for
the lowest energy structures of Na(H2 O)n for each n.
Taking into account that the calculations tend to underestimate the VTEs, the agreement with the experimental data is remarkably good. The rapid decrease of
the excitation energy for the small clusters containing up
to three water molecules is reproduced reasonably well.
Towards larger clusters the VTEs of these structures increase slightly in accordance with experimental data. Although it is difficult to assign the observed diffuse spectra definitely, the agreement between the experiment and
theory suggests that the lowest energy structures at the
present level are the major contributor to the spectra
for each n. For a rigorous comparison the finite temperature of the clusters has to be taken into account. In
earlier experiments a value of 200 K has been estimated
for the vibrational temperature of Na(NH3 )n clusters.52
We expect a similar value for Na(H2 O)n clusters. Since
a full Monte-Carlo simulations to determine the isomer
abundances is beyond the scope of this paper a simple
estimate using the Boltzmann distribution will be sufficient: At a temperature of 200 K the main contribution
to the observed spectra will come from isomers with an
energy of up to 0.4 kcal/mol above the most stable structure. Thus, the lowest energy isomers for each n will still
be the main contributors to the observed spectra since
the energy difference of most isomers is much larger.
B.
Electronic states
Although we could not assign the observed bands definitely due to the diffuse spectra, it is instructive to investigate the electronic character of the stable clusters which
should be an important contributor to the spectra. The
electronic states of M(H2 O)n (M = Li and Na) have been
studied by inspecting the contour surface of excess electron density or the space containing the half unpaired
electron.28,30 To get further insight about the electronic
nature, we have examined the radial distribution function
of the unpaired electron by dividing it into two components, ρ+ (r) and ρ− (r). They are contributions to the total radial distribution function ρ(r) from a half share with
z ≥ 0 and from that with z < 0, respectively, and evaluated using the singly occupied molecular orbital (SOMO)
label rρ− max ρ− max (r) rρ+ max ρ− max (r) N− (∞) N+ (∞)
Na
W1a
W2a
W3c
W4a
W4d
W6a
W6f
W8a
W8b
1.78
2.11
2.35
2.34
3.90
2.44
4.01
2.31
5.61
2.31
0.130
0.068
0.045
0.035
0.155
0.027
0.159
0.023
0.168
0.025
1.78
1.93
2.09
2.16
3.57
2.26
3.43
2.40
2.36
2.41
0.130
0.157
0.155
0.157
0.009
0.157
0.002
0.156
0.000
0.154
0.500
0.329
0.244
0.212
0.931
0.168
0.981
0.133
0.966
0.131
0.500
0.671
0.756
0.788
0.069
0.832
0.019
0.867
0.004
0.869
TABLE IV: Positions (rρ± max (Å)), values of peaks for
ρ± max (r), and N± (∞) at infinite separation from Na.
by unrestricted Hartree-Fock method with Na at the origin. We have taken the molecular symmetry axis as the
z axis except for W2a and W6a, with positive z values
in the direction opposite to the solvents. For W2a and
W6a, the z axis is set to a line passing through Na and
the point with the maximum SOMO density. We have
also calculated the number of electrons in the half spheres
with a radius of r, N+ (r) and N− (r), by integrating the
ρ+ (r) and ρ− (r) numerically. The results for small and
the most stable interior clusters for n ≥ 4 are shown in
the left hand side of Figure 6. We also present the results
of the typical surface clusters in the right column in Fig.
6 to examine the structure-dependence. The positions
(rρ+ max and rρ− max ) and values (ρ+ max and ρ− max ) of
the peaks for ρ+ (r) and ρ− (r) for each cluster are summarized in Table IV. N+ (r) and N− (r) at infinite r are
also given.
In Fig. 6 top left, we see the high ρ+ (r) and the low
ρ− (r) for W2a in comparison with the bare Na atom.
The water molecules bound to Na push the unpaired electron to the space in the direction opposite to the solvents.
In W4a-W8a, ρ− (r) is a dominant component of ρ(r).
The ρ+ (r) is very low in the whole r region for these clusters and that for W8a is actually invisible. The ρ− (r)’s
for W4a, W6a and W8a are peaked at 3.90 Å, 4.01 Å
and 5.61 Å with their values being 0.155, 0.159 and 0.168,
respectively. As shown in the middle left, the values of
the ρ+ (r) and ρ− (r) in the r ≤ 1.0 Å region for these
three are very small. The contribution of Na 3s orbital to
SOMO decreases and the unpaired electron is separated
from Na(+) as n grows. The bottom left panel shows
that more than 90 percents of the unpaired electrons are
distributed in space in the direction occupied by the water molecules. Note that the distance between the Na
atom and the outermost first-shell H atom in the negative z direction (3.029 Å (W4a), 3.052 Å (W6a) and
2.884 Å (W8a)) is shorter than the r giving the maximum ρ− (r). The excess electron is mostly distributed
outside the Na+ (H2 O)4 core in these clusters. The Na
is spontaneously ionized and the “solvated” electron is
excited in the interior clusters. In addition, the height-
9
0.20
ρ-(r)
W8a
ρ+(r)
0.15
0.15
ρ±(r)
ρ±(r)
W6a
0.20
Na
W2a
W4a
W6a
W8a
0.10
W4a
0.05
W2a
-8
-6
-4
-2
0
r/Å
2
4
6
8
0.00
-10
10
0.020
W1a
W6f
W8b
W3c
-8
-6
-4
ρ+(r)
-2
0
2
r(Å)
ρ-(r)
W2a
Na
4
6
8
10
0.010
W8a
W6a
ρ+(r)
0.015
ρ±(r)
0.015
W6f
W8b
W1a
0.010
W3c
W4a
0.005
0.005
0.000
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
r/Å
1.00
W2a
N-(r)
N+(r)
0.80
0.000
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
r/Å
1.00
N-(r)
N+(r)
0.80
W6f W3c
0.60
0.60
W8a
W6a
0.40
N±(r)
ρ±(r)
W4d
0.020
ρ-(r)
N±(r)
ρ+(r)
Na
0.05
0.00
-10
0.10
ρ-(r)
W1a
W3c
W4d
W6f
W8b
W4a
0.40
Na
W1a W4d W8b
0.20
0.00
-10
W4d
0.20
-8
-6
-4
-2
0
r/Å
2
4
6
8
10
0.00
-10
-8
-6
-4
-2
0
r/Å
2
4
6
8
10
FIG. 6: Components of radial distribution function of unpaired electron in the most stable interior(left) and representative
surface(right)clusters of Na(H2 O)n (n = 0– 8). Na is placed at the origin. ρ+ (r) and ρ− (r) are contributions from a half sphere
with the radius of r in space with z ≥ 0 and z < 0, respectively, (see text). Number of electrons in each half sphere, N+ (r) and
N− (r), are also presented in the bottom.
ening of the peak of ρ− (r) with increasing n and the
rapid rise of N− (r) are considered to be a sign of the
electron localization, which is similar to the formation of
the two-center state in Li-water clusters.28
In contrast, in Fig. 6 top right, we see the polarization
of the unpaired electron to the space free from waters
not only in the small but also in the n ≥ 4 surface complexes. The hydration shifts the peak gradually from
1.93 Å (W1a) to 2.41 Å (W8b) with keeping its height
almost unchanged (0.157 (W1a)) - 0.154 (W8b)). The
close examination of the functions in the r ≤ 1.0 Å region
indicates that both ρ+ (r) and ρ− (r) are zero at ∼ 0.10 Å
and at ∼ 0.55 Å reflecting the character of the Na 3s or-
bital (see the middle right). In the bottom right, around
0.17 electrons move to the free space in the 1:1 complex,
and about 80 percents or more unpaired electrons are distributed in the free space opposite to solvent molecules
for n ≥ 3. N+ (r) and N− (r) of the bare Na are nearly
constant for r ≥∼ 4 Å, while those of W1a-W8b become
insensitive to r at r ≥∼ 5 Å(W1a) - ∼ 7 Å(W8b). With
all right panels together, the spatial expansion of the Na
3s electron occurs gradually with stepwise hydration in
space opposite to the solvents in the these clusters. The
increasing Rydberg-like nature of the electron distribution results in the reduction of the 2 S–2 P energy separation, namely the red-shifts of the absorption bands.16–18
10
the solvent dependence of the electron localization mode.
One can assume that in ammonia the solvated electron
is more diffused and the excitation energies will be lower
than for Na(H2 O)n clusters. A full theoretical analysis
of the electronic wave functions in Na(NH3 )n clusters is
presently underway and will be given in a separate publication.
VI.
FIG. 7: Absorption energy as a function of (n + 1)−1/3 ∝ 1/r
for Na(H2 O)n and Na(NH3 )n clusters.
In addition, the positions and values of the peaks for
ρ+ (r) and ρ− (r) as well as N+ (r) and N− (r) do not depend very much on the cluster size for n ≥ 4, which is
considered to be responsible for the near constancy of the
VTEs in the surface clusters.
Finally, we present a comparison of the experimental
absorption energies for the two solvents. Figure 7 shows
the maximum of the absorption bands for Na(H2 O)n
along with our earlier data33 for Na(NH3 )n as a function of (n + 1)−1/3 , which is proportional to the inverse
of the cluster radius. For both solvents the initial absorption energy of the bare sodium atom (2.1 eV) decreases
by 1 eV with only three solvent molecules attached to the
alkali metal atom. Towards larger complexes the behavior of the water and ammonia solvent differs. While the
absorption energy for Na(H2 O)n increases for larger clusters, as discussed above, the energy decreases further to
approx. 0.7 eV for Na(NH3 )4 and then remains almost
constant at this value in the investigated mass range up
to n = 20. For both solvents the experimental gas phase
data can be extrapolated to the corresponding bulk values measured in liquid solvents. The different behavior
of the two solvents can be attributed to the differences
in the excess electron localization. Since the most stable forms of Na(NH3 )n are known to have the interior
structure,24,27 the analysis of their electronic states along
the above line is expected to give further insight about
∗
†
1
2
3
Electronic address: [email protected]
Electronic address: [email protected]
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Acknowledgments
The experimental work has been financially supported
by the Deutsche Forschungsgemeinschaft through SFB
450, Project A4. The theoretical work was supported in
part by the Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT).
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about all optimized structures and tables of their frequencies. A direct link to this document may be found
in the online article’s HTML reference section. The
document may also be reached via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from
ftp.aip.org in the directory /epaps. See the EPAPS homepage for more information.