THE JOURNAL OF CHEMICAL PHYSICS 130, 174313 共2009兲
H3+ as a trap for noble gases-3: Multiple trapping of neon, argon,
and krypton in XnH3+ „n = 1 – 3…
F. Pauzat,1,2,3,a兲 Y. Ellinger,1,2,3 J. Pilmé,2,4 and O. Mousis5,6
1
Laboratoire de Chimie Théorique, UMR 7616, UPMC University Paris 06, F-75005 Paris, France
Laboratoire de Chimie Théorique, CNRS, UMR 7616, F-75005 Paris, France
3
Almaden Research Center, 650 Harry Road, San Jose, California 95023-6099, USA
4
Faculté de Pharmacie, Université de Lyon, Université Lyon 1, F-69373 Lyon, France
5
Lunar and Planetary Laboratory, University of Arizona, 1629 E. University Blvd.,
Tucson, Arizona 85721, USA
6
Institut UTINAM, CNRS/INSU, UMR 6213, Université de Franche-Comté, 25030 Besançon Cedex, France
2
共Received 13 February 2009; accepted 8 April 2009; published online 7 May 2009兲
Recent studies on the formation of XH3+ noble gas complexes have shown strategic implications for
the composition of the atmospheres of the giant planets as well as for the composition of comets.
One crucial factor in the astrophysical process is the relative abundances of the noble gases versus
H3+. It is the context in which the possibility for clustering with more than one noble gas 共XnH3+ up
to n = 3兲 has been investigated for noble gases X ranging from neon to krypton. In order to assert our
results, a variety of methods have been used including ab initio coupled cluster CCSD and
CCSD共T兲, MP2, and density functional BH&HLYP levels of theory. All complexes with one, two,
and three noble gases are found to be stable in the Ne, Ar, and Kr families. These stable structures
are planar with the noble gases attached to the apices of the H3+ triangle. The binding energy of the
nth atom, defined as the XnH3+ → Xn−1H3+ + X reaction energy, increases slightly with n varying from
1 to 3 in the neon series, while it decreases in the argon series and shows a minimum for n = 2 in the
krypton series. The origin of this phenomenon is to be found in the variations in the respective
vibrational energies. A topological analysis of the electron localization function shows the
importance of the charge transfer from the noble gases toward H3+ as a driving force in the bonding
along the series. It is also consistent with the increase in the atomic polarizabilities from neon to
krypton. Rotational constants and harmonic frequencies are reported in order to provide a body of
data to be used for the detection in laboratory prior to space observations. This study strongly
suggests that the noble gases could be sequestered even in an environment where the H3+ abundance
is small. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3126777兴
I. INTRODUCTION
Noble gases are peculiar species whose ability to participate in molecular structures has been overlooked for many
years. The practical interest in the bonding capabilities of
these supposedly inert atoms started with the synthesis of the
X-halide family, with Xe+关PtF6兴− being the first example.1
More noble gas compounds followed, mainly based on xenon and, to a lesser extent, on krypton. A second breakthrough came recently with the synthesis of noble gas insertion compounds where the noble gas is part of molecules
traditionally used in current laboratory chemistry. Most of
them have been prepared by Räsänen and his co-workers by
photodissociation in the noble gas matrix, and some, such as
HXeH, HXeOH, HKrCN, HXeCN, and HXeNC, present a
clear astrophysical interest.2–5 In addition, theoretical studies
have been undertaken, showing the strength of the bonds
implied in this type of systems.6,7 At the other extremity of
the binding energy scale, laboratory and theoretical studies
on van der Waals complexes of noble gases with large sysa兲
Electronic mail: [email protected].
0021-9606/2009/130共17兲/174313/15/$25.00
tems such as carbazole8,9 or small molecules10–12 have been
carried out; however, the stability of these complexes, driven
by dispersion forces, is very weak.
In between these two extremes, we find another family
of noble gas compounds with binding energies in the range
of the hydrogen bonds:13,14 the ion-neutral complexes with
H3+. Examples of such molecular complexes are well
known in the laboratory15–18 in the form of 共H2兲nH3+. The
binding energy, hereafter taken as the absolute value of the
XnH3+ → Xn−1H3+ + X dissociation energy 共X = H2, Ne, Ar,
Kr兲, has been reported by several authors using various experimental techniques for n = 1 with values ranging between
5.8 and 9.6 kcal/mol. The thermochemical data allowed
Hiraoka18 to derive values of 6.9, 3.3, and 3.2 kcal/mol for
共H2兲nH3+ complexes with n = 1 – 3. For noble gases, much
less is known; one may cite the experimental work of
Hiraoka and Mori,19 who obtained thermochemical data on
ArnH3+ from which they derived binding energies of 6.69,
4.56, and 4.28 kcal/mol for n = 1 – 3 and an estimation of the
binding energy in NeH3+ at 0.4 kcal/mol.
Considering that H3+, ubiquitous in space, is a major
initiator of the ion-molecule chemistry, it is of considerable
interest to know its capability to form stable complexes. Up
130, 174313-1
© 2009 American Institute of Physics
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174313-2
J. Chem. Phys. 130, 174313 共2009兲
Pauzat et al.
possible with two electrons. All calculations were performed
using methods and basis sets as implemented in the
33
GAUSSIAN package. The computational procedure used in
this study has been presented in detail in a previous report21
and needs only to be summarized here.
FIG. 1. Stable structures of XnH3+.
Basis sets are correlation-consistent34–37 cc-pVTZ and
cc-pVQZ.
All structures were optimized at the BH&HLYP,
MP2, CCSD, and CCSD共T兲 levels with full correction
of the basis set superposition error 共BSSE兲 following
the counterpoise method,38 and the rotational constants were calculated accordingly.
The infrared spectra were obtained within the harmonic approximation. In order to be able to treat all
systems on an equal footing, compromise had to be
reached between accuracy and computational requirements and the level of theory had to be limited to
CCSD. No rovibrational corrections were evaluated.
共i兲
to now, attention has been focused on the complexes of H3+
with CO due to the large abundance of this molecule. These
complexes have been the object of several theoretical
studies20–23 with reference to the formation of HCO+ and
HOC+ since they were viewed as stabilized intermediates;
experimental data have been obtained on the H3+CO ion.24
A crucial factor with noble gases in space is their relative
abundances compared to H3+. The motivation here is to
probe whether several noble gases can be trapped by H3+. It
is an extension of our previous investigations on noble gas
XH3+ complexes 共X = Ne, Ar, Kr兲 that were originally limited
to single complexation.
共ii兲
共iii兲
II. COMPUTATIONAL BACKGROUND
From a theoretical point of view, H3+ based hydrogen
ionic clusters 共H2兲nH3+ have long been of interest to theoreticians. Directly pertinent to the present work are the systematic studies on the first terms of the series since these clusters
are the corresponding analogs of XnH3+. Examples that illustrate how difficult it is to describe these weakly bound structures are reported in literature.25–32 Based on the numerical
experiments provided by our previous studies,13,14 three levels of theory have been employed, namely, post-Hartree–
Fock 共post-HF兲 Møller–Plesset 共MP2兲, coupled cluster
关CCSD and CCSD共T兲兴, and DFT using the BH&LYP
formalism. It should be noted that CCSD共T兲 calculations are
irrelevant for H3+ in isolation since triple excitations are im-
III. STRUCTURAL RESULTS AND SPECTRAL
SIGNATURES
One stable minimum was identified for each complex
XnH3+ 共Fig. 1兲. This result was found for all noble gases,
including xenon, at all levels of theory whatever the extension of the basis set. The case of Xe, which showed significant structural differences from the other noble gases in our
previous study,14 will be discussed independently in a forthcoming report.
The geometries of the XH+ diatomics,39–41 of H2 and H3+
together with the IR frequencies are reported in Table I for
comparison.42–45 Since the general trends at any level of
theory hold whatever the basis set is, only the results ob-
TABLE I. Bond distances 共Å兲 and IR frequencies 共cm−1兲 for XH+ 共X = Ne– Kr兲 and hydrogenated fragments H2
and H3+. Note that ␯ refers to the harmonic vibrational frequencies, while ␯0 refers to the observed 共anharmonic兲 fundamentals.
NeH+
Methoda
r
b
ArH+
␯
r
c
KrH+
␯
r
d
H 3+
␯
r
e,f,g
H2
␯共A1兲
␯共E兲
r
h
␯
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
0.989
0.987
2939 1.280
2991 1.280
2729 1.416
2739 1.413
2501
2562
CCSD/cc-pVQZ
CCSD/cc-pVTZ
0.986
0.985
2966 1.278
3015 1.278
2750 1.413
2759 1.411
2520 0.874
2578 0.875
3437
3432
2772 0.742
2763 0.742
4407
4414
MP2/cc-pVQZ
MP2/cc-pVTZ
0.992
0.990
2908 1.278
2957 1.279
2739 1.411
2752 1.409
2526 0.870
2593 0.871
3484
3481
2820 0.736
2813 0.737
4525
4531
BH&HLP/cc-pVQZ
BH&HLP/cc-pVTZ
0.990
0.990
2898 1.281
2913 1.282
2710 1.421
2715 1.421
2528 0.872
2534 0.873
3459
3456
2780 0.736
2778 0.737
4512
4516
Expt. 共␯兲
Expt. 共␯0兲
0.9912 2904 1.2804 2711 1.4212 2495 0.8734
3178
2521
a
With 2 electrons, triple excitations are not possible
共H2 and H3+兲.
The more that can be done is CCSD.
b
Ram et al. 共Ref. 39兲.
c
Johns 共Ref. 40兲.
0.7414 4401
4161
Warner et al. 共Ref. 41兲.
Cencek et al. 共Ref. 43兲.
f
Ketterle et al. 共Ref. 44兲.
g
Oka 共Ref. 45兲.
h
Huber and Herzberg 共Ref. 42兲.
d
e
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174313-3
J. Chem. Phys. 130, 174313 共2009兲
H3+ as a trap for noble gases
FIG. 2. Schematic normal modes for XnH3+ 共n = 1 – 3兲.
tained with the cc-pVQZ basis set will be discussed. In order
to provide some help in the identification of these complexes, we also propose best estimates of the spectral signatures at the end of each section.
For rotational constants, we have shown in our previous
study on ArH3+ 共the only complex observed to our best
knowledge兲 that the CCSD共T兲/cc-pVQZ level of theory provides the results closest to the experiment. We consider here
that this approach is also valid for the other noble gases and
is therefore able to provide a best estimate of the unknown
spectroscopic constants. Due to the close agreement with the
MP2/cc-pVQZ calculations, we found that it is worth reporting the values obtained at this much less expensive level of
theory.
For IR frequencies, we rely on the close agreement of
the CCSD共T兲/cc-pVQZ calculations of structures and
energy differences with the corresponding MP2 treatments.
The discussion on the IR signatures will mainly follow the
MP2/cc-pVQZ calculations, and the schematic normal coordinates are given in Fig. 2.
It is well known that applying proper scaling factors to
the high frequency vibrations provides best estimates of the
actual spectra. Here, one considers that the ratio of the observed to the calculated frequencies for H3+ in isolation 共see
Table I兲 can be applied to the frequencies of the H3+ fragment within the complexes. For that reason, the discussion
on the symmetric/antisymmetric bending vibrations will be
presented in terms of the frequency shifts with respect to
isolated H3+. No scaling procedure is applied at low wave
numbers in this case. It relies on the fact that the only known
experimental frequency for a noble gas H3+ complex, estimated from the centrifugal distortion of the microwave spec-
trum of the ArH3+, namely, ␯6 ⬃ 475 cm−1, is perfectly reproduced by the MP2/cc-pVQZ as it already was at the
CCSD共T兲 level.13,14
A. The Ne complexes
1. Structure of NenH3+ complexes
The structural parameters of the 共Ne兲nH3+ 共n = 1 – 3兲
clusters are given in Table II. With one Neon atom, the complex is clearly made of a neutral Ne atom interacting with
H3+. The optimized geometry, with Ne– HNe distances of
⬃1.72 Å 共DFT兲, 1.79 Å 共MP2兲, and 1.78 Å 关CCSD共T兲兴,
reveals a long-range interaction; this distance being close to
twice that of NeH+ 共0.99 Å兲 in isolation.39 The interatomic
distances in the H3+ part are close to the 0.874 Å calculated
at the CCSD level for the ion in isolation 共0.870 Å at MP2兲
and in excellent agreement with experiment; H – HNe is only
⬃0.008 Å longer and H–H is ⬃0.014 Å smaller.
With two Neon atoms, the Ne– HNe distance increases by
⬃0.05 Å for all theoretical approaches. The post-HF values
are systematically ⬃0.07 Å larger than in DFT. In this complex, the HNe – HNe distance is 0.013 Å larger than for H3+ in
isolation, whereas HNe – H is 0.007 Å shorter 关CCSD共T兲兴.
The net result is an overall shortening of the HNe – H bonds
by ⬃0.01 Å with respect to NeH3+.
With three Neon atoms, one observes an additional
lengthening of the noble gas to H3+ distance by ⬃0.03 Å in
the average for all levels of theory. The structure of the H3+
fragment is now the same as that in isolation.
The variation in the Ne– HNe distance with the level of
theory, i.e., the systematic shortening observed in DFT calculations, compared to MP2 and coupled cluster, simply il-
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174313-4
J. Chem. Phys. 130, 174313 共2009兲
Pauzat et al.
TABLE II. Optimized geometries 共Å兲 for NenH3+ after full correction of the
BSSE artifact.
Ne– HNe
HNe – H
NeH3+
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
1.780
1.807
1.804
1.835
1.787
1.812
1.723
1.723
0.882
0.882
0.881
0.881
0.878
0.878
0.884
0.885
Ne2H3+
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLP/cc-pVQZ
BH&HLYP/cc-pVTZ
1.824
1.853
1.846
1.877
1.832
1.858
1.773
1.778
0.867
0.870
0.868
0.870
0.865
0.866
0.866
0.867
Ne3H3+
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
1.855
1.886
1.878
1.909
1.866
1.892
1.811
1.817
HNe – HNe
H–H
0.860
0.862
0.861
0.863
0.856
0.858
0.855
0.855
0.887
0.887
0.885
0.885
0.883
0.883
0.890
0.890
0.874
0.875
0.874
0.874
0.871
0.871
0.874
0.874
lustrates the lack of an accurate description of the dispersion
effects in the first method. It should be mentioned that the
MP2 level of theory gives the closest values to CCSD共T兲.
2. Spectral signatures of the NenH3+ complexes
To our knowledge, there is no experimental information
available on these neon complexes. Our best estimates of the
rotational constants are the CCSD共T兲/cc-pVQZ values reported below. It is worth mentioning the quality of the
MP2/cc-pVQZ calculations that give the closest values
共within ⬃0.7% at most兲 of all the post-HF treatments that we
have considered in our extensive screening 共MP2 values in
parenthesis兲.
NeH3+:A = 1355共1368兲 GHz,
B = 35.56共35.41兲 GHz,
C = 34.66共34.52兲 GHz,
Ne2H3+:A = 114.8共114.2兲 GHz,
B = 3.147共3.132兲 GHz,
C = 3.063共3.049兲 GHz,
Ne3H3+:A = 3.018共2.996兲 GHz,
B = 3.018共2.996兲 GHz,
C = 1.509共1.498兲 GHz.
The dipole moments referred to the center of mass,
␮ ⬃ 7.8 D for NeH3+ and ␮ ⬃ 4.5 D for Ne2H3+, are large
enough to encourage laboratory experiments on these complexes. Ne3H3+ has no dipole moment for symmetry reasons
and will not be observed in microwave spectroscopy.
The harmonic frequencies in Table III show clearly two
types of vibrations, namely, those at high wave numbers corresponding to the internal vibrations of the H3+ fragment and
those at low wave numbers corresponding to the motions of
the H3+ entity in the field of the Ne atoms. The first set of
frequencies can be directly compared to the values obtained
for H3+ in isolation at the same level of theory 共see Table I兲.
For NeH3+, there is very little change between H3+ in the
complex and in isolation: +11 cm−1 at the MP2 level
共+9 cm−1 at CCSD; +12 cm−1 at BH&HLYP兲 for the ␯1
symmetric breathing vibration. The ␯2 symmetric stretching/
bending and ␯3 asymmetric stretching vibrations are no
longer degenerate in the complex and split by ⬃20 cm−1.
They are both shifted toward lower frequencies by ⫺40 and
−58 cm−1, respectively, at the MP2 level 共⫺30 and
−54 cm−1 at CCSD; ⫺60 and −81 cm−1 at BH&HLYP兲.
The lowest three frequencies are of two types: the ␯4 and ␯5
vibrations correspond to the out-of-plane and in-plane rotations of the H3+ fragment with ␯4 ⬎ ␯5, which is consistent
with the fact that the structure with Ne on top of the H3+
triangle is higher in energy than that where Ne is in the plane
of the triangle between two H atoms.14 The lowest frequency
␯6 represents the dissociation coordinate into Ne+ H3+.
For Ne2H3+, the ␯1 symmetric breathing vibration of the
+
H3 fragment remains close to that of the ion in isolation.
The symmetric stretching/bending and asymmetric stretching
vibrations swap positions with respect to NeH3+: the former
is shifted to lower frequencies by −100 cm−1 at the MP2
level 共−72 cm−1 at CCSD; −108 cm−1 at BH&HLYP兲,
whereas the latter is also shifted to lower frequencies, but to
a smaller extent 共⫺45, ⫺32, and −61 cm−1 at MP2, CCSD,
and BH&HLYP, respectively兲. Among the six remaining low
frequencies, the first five account for the displacements of the
H3+ triangle with respect to the frozen Ne atoms, the last one
being a very loose mode that can be assigned to the stretching vibration of Ne–Ne. The in- and out-of-plane rotations
are represented by ␯5, ␯4, and ␯7, respectively, whereas ␯8
and ␯9 describe the departure of H3+ from the Ne–Ne system.
For Ne3H3+, the three high frequencies are very similar
to those of H3+ in isolation with one A1 and two degenerate
E⬘ modes. The totally symmetric stretching is practically unaffected, whereas the E⬘ vibrations are shifted to lower frequencies by −71 cm−1 at the MP2 level 共−55 cm−1 at
CCSD; −88 cm−1 at BH&HLYP兲. The nine remaining low
frequencies do not involve any modification of the H3+ entity. One has the three rotations of the H3+ triangle inside the
Ne3 triangle 共in-plane ␯4 and out-of-plane ␯5 and ␯6兲, then
the in-plane degenerate displacements of H3+ inside the Ne3
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174313-5
J. Chem. Phys. 130, 174313 共2009兲
H3+ as a trap for noble gases
TABLE III. IR harmonic frequencies 共cm−1兲, absolute intensities 共km/mol兲, and ZPE corrections 共a.u.兲 for
NenH3+ at the CCSD 共top兲, MP2 共middle兲, and BH&HLYP 共bottom兲 levels of theory 共cc-pVQZ basis set兲.
NeH3+
Frequency
Ne2H3+
Ne3H3+
␯1
3446
3495
3471
共2兲
共1兲
共1兲
1A1
1A1
1A1
3445
3476
3453
共5兲
共5兲
共7兲
1A1
1A1
1A1
3447
3474
3443
共0兲
共0兲
共0兲
1A1
1A1
1A1
␯2
2742
2780
2720
共380兲
共421兲
共470兲
2A1
2A1
2A1
2740
2775
2719
共426兲
共481兲
共523兲
1B2
1B2
1B2
2717
2749
2692
共383兲
共422兲
共454兲
1E⬘
1E⬘
1E⬘
␯3
2718
2762
2699
共161兲
共175兲
共173兲
1B2
1B2
1B2
2700
2720
2672
共236兲
共258兲
共270兲
2A1
2A1
2A1
2717
2749
2692
共383兲
共422兲
共454兲
1E⬘
1E⬘
1E⬘
␯4
408
415
443
共5兲
共5兲
共6兲
1B1
1B1
1B1
449
487
485
共0兲
共0兲
共0兲
1A2
1A2
1A2
446
463
479
共0兲
共0兲
共0兲
1A⬘2
1A⬘2
1A⬘2
␯5
299
305
327
共12兲
共12兲
共13兲
2B2
2B2
2B2
390
417
431
共47兲
共35兲
共92兲
2B2
2B2
2B2
439
459
470
共0兲
共0兲
共0兲
1E⬙
1E⬙
1E⬙
␯6
297
301
349
共417兲
共424兲
共441兲
3A1
3A1
3A1
308
320
347
共422兲
共448兲
共395兲
3A1
3A1
3A1
439
459
470
共0兲
共0兲
共0兲
1E⬙
1E⬙
1E⬙
␯7
287
305
318
共18兲
共17兲
共16兲
1B1
1B1
1B1
318
323
352
共427兲
共435兲
共447兲
2E⬘
2E⬘
2E⬘
␯8
236
246
281
共280兲
共281兲
共291兲
3B2
3B2
3B2
318
323
352
共427兲
共435兲
共447兲
2E⬘
2E⬘
2E⬘
␯9
38
38
40
共30兲
共30兲
共27兲
4A1
4A1
4A1
107
109
120
共0兲
共0兲
共0兲
2A1
2A1
2A1
␯10
90
95
103
共226兲
共222兲
共216兲
1A⬙2
1A⬙2
1A⬙2
␯11
27
28
31
共8兲
共8兲
共7兲
3E⬘
3E⬘
3E⬘
␯12
27
28
31
共8兲
共8兲
共7兲
3E⬘
3E⬘
3E⬘
ZPE correction
0.022 58
0.022 96
0.022 80
triangle 共␯7 and ␯8兲; the out-of-plane movement of H3+
through the Ne3 triangle 共␯10兲 is very similar to ammonia
inversion. Finally, one has the three typical motions of the
Ne atoms at low frequencies 共␯9, ␯11, and ␯12兲.
Finally, the variation in the total zero point energy 共ZPE兲
shows a regular increase with the number of coordinated
0.024 13
0.024 57
0.024 48
0.025 27
0.025 65
0.025 59
noble gases which is mainly due to the associated increasing
number of vibrations. The technique used to produce these
transient species will most probably lead to a mixture of H3+
and the three complexes. The analysis of the infrared spectra
will be a difficult task since the infrared intensities are concentrated in the same frequency range for the three com-
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174313-6
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Pauzat et al.
plexes. At any rate, the stretching deformations of the H3+
fragment will be difficult to assign to a particular complex
since they are very close and of comparable intensities. In
such a situation, one could have thought that the Raman
spectra might be useful in distinguishing the vibrational
spectra of the NenH3+ clusters. Unfortunately, the Raman intensities are also very close and, consequently, cannot help in
solving the problem. The same difficulty will have to be
faced on the low frequency side since all the most intense
vibrations are around 300– 350 cm−1 in the three complexes.
IR frequencies 共cm−1兲 as well as relative intensities 共in parentheses兲 are reported below at the MP2/cc-pVQZ level
when larger than 10% of the most intense one is taken as
reference,
NeH3+:␯2 = 2485共0.9兲,
␯3 = 2469共0.4兲,
␯6 = 301共1.0兲,
Ne2H3+:␯2 = 2481共1.0兲,
␯5=417共0.1兲,
␯3 = 2432共0.5兲,
␯6=320共0.9兲,
Ne3H3 :␯2/3 = 2471共0.9兲,
+
␯8=246共0.6兲,
ArHAr
MP2/cc-pVQZ frequencies are in excellent agreement with
the corresponding CCSD共T兲 frequencies obtained in our previous study for the first term of the series, NeH3+,
␯3 = 2463 cm−1,
␯6 = 314 cm−1 .
HAr – H
HAr – HAr
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
ArH3+
1.818
0.937
1.808
0.942
1.844
0.930
1.835
0.934
1.823
0.931
1.820
0.934
1.795
0.946
1.791
0.948
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLP/cc-pVQZ
BH&HLYP/cc-pVTZ
Ar2H3+
1.996
0.871
1.993
0.871
2.014
0.869
2.011
0.870
1.990
0.867
1.991
0.868
1.990
0.867
1.984
0.868
0.932
0.936
0.928
0.931
0.929
0.931
0.939
0.941
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
Ar3H3+
2.082
2.083
2.099
2.099
2.079
2.081
2.088
2.082
0.887
0.888
0.885
0.886
0.884
0.885
0.886
0.887
␯7/8 = 318共1.0兲,
␯10 = 90共0.3兲.
␯2 = 2484 cm−1,
TABLE IV. Optimized geometries 共Å兲 for ArnH3+ after full correction of the
BSSE artifact.
H–H
0.826
0.825
0.832
0.827
0.823
0.823
0.818
0.818
B. The Ar complexes
1. Structure of ArnH3+ complexes
The structural parameters of the 共Ar兲nH3+ 共n = 1 – 3兲 clusters are given in Table IV. With one argon atom,14,46 the
complex shows a stronger interaction between the noble gas
and the positive ion, although the ArH3+ distance remains far
from the 1.28 Å in ArH+ in isolation.40 The longer distance
between Ar and HAr mainly reflects the larger size of Ar with
respect to Ne. Comparing the levels of theory, this Ar– HAr
distance of ⬃1.79 Å 共DFT兲 is smaller than ⬃1.82 Å in
post-HF methods 关either MP2 or CCSD共T兲兴. With respect to
the neon complex, the difference between the two types of
methods is smaller because the contribution of van der Waals
forces is relatively less important due to the increased role of
electronic charge transfer 共CT兲 共see Sec. IV, below兲. The H3+
part is more affected since H – HAr is now ⬃0.07 Å longer,
while H–H is ⬃0.05 Å smaller than for the ion in isolation.
With two argon atoms, the Ar– HAr distance increases by
⬃0.17 Å. At the same time, the H3+ fragment is modified so
that two of its bonds 共H – HAr兲 are close to the ion in isolation, which represents an overall shortening of the H – HAr
bonds by ⬃0.06 Å with respect to ArH3+, In this structure,
the 共HAr – HAr兲 distance remains ⬃0.6 Å larger; DFT and
post-HF methods give close geometries.
With three argon atoms, the preceding trends are confirmed. The Ar– HAr is further increased by ⬃0.10 Å and the
H3+ fragment is ⬃0.01 Å different from its structure in iso-
lation. One observes here again that the MP2 and CCSD共T兲
values are very close. The BH&HLYP calculations present
the same behavior, giving practically the same values.
2. Spectral signatures of the ArnH3+ complexes
Only a limited set of spectroscopic data is available for
ArH3+. Tentative values were first obtained by proper scaling
of the ArD3+ spectrum according to the H/D ratio of the
atomic masses. Further studies involving mixed H/D species
by Bogey et al.47 provided additional spectroscopic data that
were used as input to more elaborate rigid, semirigid, and
flexible models. Within these models, the rotational constants
were adjusted to reproduce the experimental spectra to the
best possible. Three sets of the so-called experimental values
were obtained, from which a set of experimental constants
can be estimated 共with a large uncertainty on the A constant兲,
A = 1490 GHz,
B = 30.8866 GHz,
C = 30.1464 GHz.
A complete discussion of the rotational spectrum has been
presented in a previous report.14 It was shown that the quality of the results of the quantum calculations is at least equal
to that of the experimental values deduced from the analysis
of spectroscopic data based on phenomenological models.
The CCSD共T兲/cc-pVQZ level of theory was found to be the
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174313-7
J. Chem. Phys. 130, 174313 共2009兲
H3+ as a trap for noble gases
best level to use for theoretical predictions; besides, the
present study confirms the remarkable quality of the MP2
calculations 共see values in parentheses兲,
ArH3+:A = 1469共1481兲 GHz,
B = 30.84共30.85兲 GHz,
C = 30.21共30.22兲 GHz,
Ar2H3+:A = 89.68共90.46兲 GHz,
B = 1.390共1.398兲 GHz,
C = 1.369共1.377兲 GHz,
Ar3H3+:A = 1.251共1.257兲 GHz,
B = 1.251共1.257兲 GHz,
C = 0.6257共0.6284兲 GHz.
Besides ArH3+ whose high dipole moment 共␮ ⬃ 7.2 D兲
should favor detection, Ar2H3+ seems also a reasonable target for microwave spectroscopy 共␮ ⬃ 3.9 D兲; Ar3H3+ will
not be observed for symmetry reasons.
The harmonic vibrational frequencies of the ArnH3+
clusters 共n = 1 – 3兲 are reported in Table V together with the
corresponding absolute intensities. As for Ne, the spectrum
of any of these complexes shows two well-separated domains corresponding to the internal vibrations of the H3+
fragment 共high frequencies兲 and to the movements with respect to the Ar atoms at low frequencies.
For ArH3+, the harmonic frequencies corresponding to
the H3+ fragment of the complex are significantly different
from those of the free ion. The symmetric stretching ␯1 is
shifted to higher frequencies by +149 cm−1 at the MP2 level
共+130 cm−1 at CCSD; +180 cm−1 at BH&HLYP兲. As for
NeH3+, the originally degenerate deformations of the H3+ ion
are shifted to lower wave numbers, i.e., −455 cm−1 at the
MP2 level 共−410 cm−1 at CCSD; ⫺497 at BH&HLYP兲 for
the symmetric stretching ␯2 and −664 cm−1 at the MP2 level
共578 cm−1 at CCSD; −697 cm−1 at BH&HLYP兲 for the ␯3
asymmetric stretching/bending. The lowest three frequencies
␯4 and ␯5 are the out-of-plane and in-plane rotations of the
H3+ fragment, with ␯4 ⬎ ␯5, which is consistent with the fact
that the structure with Ar on top of the H3+ triangle is higher
in energy than that where Ar is in the plane of the triangle
between two H atoms.14 The lowest frequency ␯6 represents
the dissociation coordinate into Ar+ H3+.
For Ar2H3+, the ␯1 frequency acquires a strong component on the weaker HAr – HAr bond with a shift by −127 cm−1
at the MP2 level 共−99 cm−1 at CCSD; ⫺112 at BH&HLYP兲
toward lower energy when compared to H3+ in isolation. The
asymmetric and symmetric stretchings ␯2 and ␯3 that involve
the shorter H – HAr bonds are still shifted to lower frequency
by ⫺260 and −459 cm−1, respectively, at the MP2 level,
which represents an increase by 195 and 205 cm−1 compared to that of ArH3+, in agreement with the corresponding
shortening of the bond lengths. At the CCSD and BH&HLYP
levels, the corresponding shifts for ␯2 are ⫺216 and
−252 cm−1; for ␯3 the shifts are ⫺409 and −494 cm−1. In
the remaining low frequencies, the H3+ fragment moves as a
whole with respect to the heavier argon atoms. In ␯4 and ␯5,
one finds the out-of-plane and in-plane rotations of the H3+
triangle, with ␯6 being the second out-of-plane rotation. The
last three vibrations can be described as the symmetric and
antisymmetric displacements of the Ar atoms with respect to
the H3+ triangle and the lowest frequency is mainly a weak
stretching of the Ar–Ar distance.
For Ar3H3+, one has an equilateral triangle of Ar atoms
surrounding the equilateral triangle of H3+. The three highest
frequencies are very similar to those of H3+ in isolation with
one A1⬘ and two degenerate E⬘ modes shifted by ⫺197 and
−296 cm−1 at the MP2 level, respectively 共⫺150 and
−254 cm−1 at CCSD; ⫺182 and −301 cm−1 at BH&HLYP兲.
As for Ne3H3+, the nine remaining low frequencies do not
involve any modification of the H3+ entity. One has three
rotations of the three-hydrogen fragment 共␯4, ␯5, and ␯6兲
tumbling in the space defined by the three-argon triangle.
Then one finds the three translations in 共␯7 and ␯8兲 and
through 共␯9兲 the argon’s plane. Finally, the lowest three frequencies are the low frequency images of the highest three
with one symmetric A1⬘ and two degenerate E⬘ modes implying only the argon atoms.
Finally, the variation in the total ZPE shows a little increase between ArH3+ and Ar2H3+ compared to the same
variation for the Ne complexes that is related to the shortening of the two H – HAr bonds. The addition of the third noble
gas has the same consequences for the neon and argon series.
As for the Ne complexes, the infrared spectra will most
probably contain a mixture of H3+ and the three complexes.
However, its analysis should be less difficult since the infrared intensities at high frequencies are now well separated. On
the low frequency side, the ArH3+ ␯6 vibration should be
easily identified since it is ⬃100 cm−1 apart from the most
intense vibrations of Ar2H3+ and Ar3H3+ around
⬃350 cm−1. IR frequencies 共cm−1兲 and relative intensities
共in parentheses兲 when larger than 10% of the most intense
one taken as reference are reported below at the MP2/ccpVQZ level,
ArH3+:␯2 = 2112共0.1兲,
Ar2H3+:␯2 = 2289共1.0兲,
␯7 = 371共0.2兲,
␯3 = 1927共1.0兲,
␯6 = 478共0.6兲,
␯3 = 2111共0.3兲,
␯8 = 360共0.5兲,
Ar3H3+:␯2/3 = 2256共1.0兲,
␯7/8 = 388共0.6兲,
␯10 = 128共0.1兲.
For information, the corresponding CCSD共T兲 frequencies
obtained in our previous study for the first term of the series,
ArH3+ are
␯2 = 2104 cm−1,
␯3 = 1925 cm−1,
␯6 = 478 cm−1 .
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174313-8
J. Chem. Phys. 130, 174313 共2009兲
Pauzat et al.
TABLE V. IR harmonic frequencies, absolute intensities 共km/mol兲, and ZPE corrections 共a.u.兲 at the CCSD
共top兲, MP2 共middle兲, and BH&HLYP 共bottom兲 levels of theory 共cc-pVQZ basis set兲.
ArH3+
Frequency
Ar2H3+
Ar3H3+
␯1
3567
3633
3639
共25兲
共38兲
共47兲
1A1
1A1
1A1
3338
3357
3347
共29兲
共31兲
共27兲
1A1
1A1
1A1
3287
3287
3277
共0兲
共0兲
共0兲
1A1
1A1
1A1
␯2
2362
2365
2283
共106兲
共110兲
共104兲
2A1
2A1
2A1
2556
2560
2528
共1602兲
共1746兲
共1847兲
1B2
1B2
1B2
2518
2524
2479
共1109兲
共1205兲
共1252兲
1E⬘
1E⬘
1E⬘
␯3
2194
2156
2083
共1630兲
共1728兲
共1741兲
1B2
1B2
1B2
2367
2361
2286
共426兲
共463兲
共454兲
2A1
2A1
2A1
2518
2524
2479
共1109兲
共1205兲
共1252兲
1E⬘
1E⬘
1E⬘
␯4
782
809
800
共2兲
共2兲
共3兲
1B1
1B1
1B1
731
756
741
共0兲
共0兲
共0兲
1A2
1A2
1A2
667
691
669
共0兲
共0兲
共0兲
1E⬙
1E⬙
1E⬙
␯5
597
617
615
共8兲
共9兲
共10兲
2B2
2B2
2B2
629
651
639
共21兲
共24兲
共15兲
2B2
2B2
2B2
667
691
669
共0兲
共0兲
共0兲
1E⬙
1E⬙
1E⬙
␯6
455
478
490
共959兲
共1003兲
共989兲
3A1
3A1
3A1
490
515
498
共4兲
共4兲
共5兲
1B1
1B1
1B1
656
674
663
共0兲
共0兲
共0兲
1A⬘2
1A⬘2
1A⬘2
␯7
352
371
358
共888兲
共307兲
共286兲
3A1
3A1
3A1
377
388
383
共651兲
共671兲
共666兲
2E⬘
2E⬘
2E⬘
␯8
352
360
346
共298兲
共907兲
共945兲
3B2
3B2
3B2
377
388
383
共651兲
共671兲
共666兲
2E⬘
2E⬘
2E⬘
␯9
40
41
42
共8兲
共8兲
共8兲
4A1
4A1
4A1
123
128
126
共122兲
共117兲
共117兲
1A⬙2
1A⬙2
1A⬙2
␯10
106
109
109
共0兲
共0兲
共0兲
2A1
2A1
2A1
␯11
25
25
28
共1兲
共1兲
共1兲
3E⬘
3E⬘
3E⬘
␯12
25
25
28
共1兲
共1兲
共1兲
3E⬘
3E⬘
3E⬘
ZPE correction
0.022 69
0.022 91
0.022 57
C. The Kr complexes
1. Structure of KrnH3+ complexes
The structural parameters of the 共Kr兲nH3+ clusters
共n = 1 – 3兲 are given in Table VI. With one krypton atom, the
complex cannot be described any longer by a clear-cut interaction between two well-defined fragments. The H3+ frag-
0.024 73
0.024 99
0.024 57
0.025 85
0.026 10
0.025 72
ment is strongly distorted and no longer close to an equilateral triangle: the H – HKr distance of ⬃1.04 Å at the
CCSD共T兲/cc-pVQZ level 共1.01 Å at BH&HLYP兲 is
⬃0.20 Å longer than in H3+ and the H–H distance is only
⬃0.05 Å larger than for isolated H2. At the same time, the
Kr– HKr distance of ⬃1.72 Å in post-HF methods or
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174313-9
J. Chem. Phys. 130, 174313 共2009兲
H3+ as a trap for noble gases
TABLE VI. Optimized geometries 共Å兲 for KrnH3+ after full correction of the
BSSE artifact.
KrnH3+
HKr – H
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
KrH3+
1.748
1.681
1.773
1.707
1.761
1.731
1.801
1.799
1.041
1.097
1.023
1.075
1.024
1.049
1.012
1.019
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLP/cc-pVQZ
BH&HLYP/cc-pVTZ
Kr2H3+
2.091
2.096
2.117
2.064
2.092
2.097
2.097
2.093
0.874
0.874
0.872
0.873
0.871
0.871
0.870
0.871
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
Kr3H3+
2.203
2.204
2.222
2.222
2.202
2.202
2.215
2.211
HKr – HKr
H–H
0.798
0.789
0.801
0.792
0.796
0.792
0.799
0.798
0.957
0.964
0.954
0.967
0.952
0.956
0.966
0.969
0.894
0.895
0.891
0.892
0.890
0.891
0.892
0.893
⬃1.80 Å at BH&HLYP is closer to the interatomic distance
of 1.42 Å in the KrH+ diatomic.41 As discussed previously,14
the situation is intermediate between the two X ¯ H3+ and
XH+ ¯ H2 limit structures.
With two krypton atoms, the Kr– HKr distance increases
by ⬃0.30 Å. The situation is much closer to the usual finding of a noble gas interacting with H3+; two of its bonds
共H – HKr兲 are close to the ion in isolation, whereas the third
one 共HKr – HKr兲 is still ⬃0.08 Å larger; DFT and post-HF
methods give close geometries. The important point is the
drastic shortening of the two H – HKr bond lengths by
⬃0.15 Å 共one order of magnitude larger compared to the
corresponding variation in the neon series兲.
With three krypton atoms, one observes a further increase in the Kr– HKr distance by ⬃0.10 Å, whereas the H3+
fragment is ⬃0.02 Å apart from its structure in isolation.
Once more, the MP2 and CCSD共T兲 values are very close.
2. Spectral signatures of the KrnH3+ complexes
There are no spectroscopic data available to compare
with the theoretical values. The rotational constants at the
CCSD共T兲 and MP2 levels 共in parentheses兲 are remarkably
close,
KrH3+:A = 1575共1582兲 GHz,
B = 29.258共29.260兲 GHz,
C = 28.724共28.730兲 GHz,
Kr2H3+:A = 75.63共76.60兲GHz,
B = 0.6332共0.6307兲 GHz,
C = 0.6279共0.6255兲 GHz,
Kr3H3+:A = 0.5428共0.5441兲 GHz,
B = 0.5428共0.5441兲 GHz,
C = 0.2714共0.2720兲 GHz.
The dipole moments 共referred to the center of mass兲, namely,
⬃6.8 and ⬃4.8 D for KrH3+ and Kr2H3+ are large enough to
encourage laboratory experiments.
The harmonic vibrational frequencies of the 共Kr兲nH3+
共n = 1 – 3兲 are reported in Table VII together with the corresponding absolute intensities. Contrary to Ne and Ar, the
spectra of these complexes do not show two well-separated
domains for all the complexes, which can be rationalized by
the different type of bonding in KrH3+. For a better comparison with preceding results, the discussion will also focus on
the MP2/cc-pVQZ calculations using the schematic normal
coordinates in Fig. 2.
For KrH3+, there is only one high frequency vibration,
␯1, corresponding to the stretching of the H2 fragment interacting with the KrH+ part of the complex. It is shifted by
+361 cm−1 at the MP2 level 共+348 and +345 cm−1 at CCSD
and BH&HLYP, respectively兲. The degenerate vibrations are
strongly shifted by ⫺934 and −1438 cm−1 共MP2兲 to low
frequencies compared to the ␯2 and ␯3 asymmetric and symmetric stretching of H3+. This trend is also reproduced at the
CCSD 共⫺907 and −1416 cm−1兲 and BH&HLYP 共⫺859 and
−1291 cm−1兲 levels of theory. In this complex, the H3+ fragment has lost a large part of its identity. The next two low
frequencies, ␯4 and ␯5 are the out-of-plane and in-plane rotations of the H3+ triangle, whereas ␯6 represents the dissociation coordinate into Kr+ H3+ 共with a strong coupling with
the Kr– HKr elongation兲.
For Kr2H3+, the ␯1 frequency is now a symmetric breathing vibration shifted by −213 cm−1 at the MP2 level toward
the lower energy when compared to H3+ in isolation 共⫺170
and −175 cm−1 at CCSD and BH&HLYP, respectively兲. The
asymmetric and symmetric stretchings ␯2 and ␯3 that involve
the HKr – H bonds are also shifted to lower frequency by
⫺421 and −706 cm−1 共MP2兲, respectively. However, it is a
dramatic increase by 513 and 732 cm−1 compared to that of
KrH3+ that illustrates the reconstruction of the H3+ fragment.
As with Ne and Ar complexes, the remaining low frequencies hardly imply H3+. In all of them, the three-hydrogen
triangle moves as a whole with respect to the heavier krypton
atoms. In ␯4 and ␯5, one finds the out-of-plane and in-plane
rotations of the H3+ fragment, with ␯6 being the second outof-plane rotation. The last three vibrations are the symmetric
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174313-10
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Pauzat et al.
TABLE VII. IR harmonic frequencies 共cm−1兲, absolute intensities 共km/mol兲, and ZPE corrections 共a.u.兲 for
KrnH3+ at the CCSD 共top兲, MP2 共middle兲, and BH&HLYP 共bottom兲 levels of theory 共cc-pVQZ basis set兲.
KrH3+
Frequency
Kr2H3+
Kr3H3+
␯1
3785
3845
3804
共118兲
共135兲
共134兲
1A1
1A1
1A1
3267
3271
3284
共51兲
共59兲
共39兲
1A1
1A1
1A1
3203
3186
3184
共0兲
共0兲
共0兲
1A1
1A1
1A1
␯2
1865
1886
1921
共56兲
共59兲
共66兲
1B2
1B2
1B2
2419
2399
2417
共2631兲
共2810兲
共2920兲
1B2
1B2
1B2
2401
2389
2363
共1662兲
共1794兲
共1790兲
1E⬘
1E⬘
1E⬘
␯3
1356
1382
1489
共2067兲
共2188兲
共2171兲
2A1
2A1
2A1
2145
2114
2081
共530兲
共572兲
共533兲
2A1
2A1
2A1
2401
2389
2363
共1662兲
共1793兲
共1790兲
1E⬘
1E⬘
1E⬘
␯4
917
925
880
共2兲
共2兲
共2兲
1B1
1B1
1B1
820
850
796
共0兲
共0兲
共0兲
1A2
1A2
1A2
758
794
716
共0兲
共0兲
共0兲
1E⬙
1E⬙
1E⬙
␯5
724
727
697
共10兲
共10兲
共10兲
2B2
2B2
2B2
703
730
685
共12兲
共12兲
共17兲
2B2
2B2
2B2
758
794
716
共0兲
共0兲
共0兲
1E⬙
1E⬙
1E⬙
␯6
565
625
518
共2307兲
共2091兲
共1744兲
3A1
3A1
3A1
574
612
551
共1兲
共1兲
共2兲
1B1
1B1
1B1
722
749
688
共0兲
共0兲
共0兲
1A⬘2
1A⬘2
1A⬘2
␯7
386
414
374
共292兲
共307兲
共260兲
3A1
3A1
3A1
380
394
336
共762兲
共794兲
共762兲
2E⬘
2E⬘
2E⬘
␯8
318
328
371
共1200兲
共1237兲
共1213兲
3B2
3B2
3B2
380
394
336
共762兲
共794兲
共762兲
2E⬘
2E⬘
2E⬘
␯9
30
30
32
共3兲
共3兲
共4兲
4A1
4A1
4A1
129
134
125
共88兲
共82兲
共87兲
1A⬙2
1A⬙2
1A⬙2
␯10
78
80
74
共0兲
共0兲
共0兲
2A1
2A1
2A1
␯11
18
18
19
共0兲
共0兲
共0兲
3E⬘
3E⬘
3E⬘
␯12
18
18
19
共0兲
共0兲
共0兲
3E⬘
3E⬘
3E⬘
ZPE correction
0.020 99
0.021 39
0.021 21
and antisymmetric displacements of the Kr atoms with respect to the H3+ triangle and the lowest frequency is mainly
a weak stretching of the Kr–Kr distance.
For Kr3H3+, one finds again the equilateral triangle of
+
H3 embedded in the equilateral triangle of the Kr atoms.
The displacements of the hydrogen atoms corresponding to
0.024 29
0.024 49
0.024 13
0.025 61
0.025 83
0.024 92
the three highest frequencies are very similar to those of H3+
in isolation. The ␯1 and two degenerate ␯2 and ␯3 modes are
shifted by ⫺298 and −431 cm−1 at the MP2 level, respectively 共⫺234 and −371 cm−1 at CCSD; ⫺275 and
−417 cm−1 at BH&HLYP兲. As for Ne3H3+ and Ar3H3+, the
nine remaining low frequencies do not involve any modifi-
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174313-11
J. Chem. Phys. 130, 174313 共2009兲
H3+ as a trap for noble gases
TABLE VIII. Dissociation energies 共values in parentheses are obtained with ZPE corrections at the CCSD
level兲 for XnH3+ → Xn−1H3+ + X including full BSSE corrections and ZPEs 共kcal/mol兲.
XH3+ → X + H3+
X2H3+ → XH3+ + X
X 3H 3+ → X 2H 3+ + X
共1.12兲
共0.89兲
0.94
0.75
1.00
0.88
1.54
1.53
共1.21兲
共1.01兲
1.06
0.88
1.19
1.00
1.63
1.67
共4.14兲
共3.46兲
3.85
3.21
4.15
3.49
4.13
3.82
共3.73兲
共3.32兲
3.63
3.10
3.93
3.35
3.63
3.39
共3.03兲
共1.20兲
2.81
1.16
2.62
2.03
4.05
3.72
共4.62兲
共3.99兲
4.33
3.75
4.68
4.08
4.39
4.08
Neon
CCSD共T兲/cc-pVQZ
CCSD共T兲/cc-pVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
0.98共1.02兲
0.71共0.77兲
0.82
0.58
0.97
0.72
1.61
1.57
CCSD共T兲/ccpVQZ
CCSD共T兲/ccpVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
7.28共7.20兲
6.89共6.79兲
6.63
6.23
7.20
6.66
8.29
7.92
CCSD共T兲/ccpVQZ
CCSD共T兲/ccpVTZ
CCSD/cc-pVQZ
CCSD/cc-pVTZ
MP2/cc-pVQZ
MP2/cc-pVTZ
BH&HLYP/cc-pVQZ
BH&HLP/cc-pVTZ
13.03共12.91兲
13.44共13.40兲
12.06
12.52
12.71
12.46
12.91
12.46
Argon
Krypton
cation of the H3+ entity. One has three rotations of the threehydrogen fragment 共␯4, ␯5, and ␯6兲 within the three-krypton
triangle. Then one finds the three translations in 共␯7 and ␯8兲
and through 共␯9兲 the krypton plane. Finally, the lowest three
frequencies are the low frequency images of the highest three
with one symmetric A1⬘ and two degenerate E⬘ modes involving only the argon atoms.
Compared to the neon and argon series, there is a sharp
increase in the ZPE with the addition of the second noble
gas. It is related to the electronic change in the complex that
will be analyzed in Sec. IV
Assuming that the IR spectrum is also a mixture of the
three complexes, the signatures of each of them should be
more easily identified than previously, in view of their wellseparated positions and intensity patterns. IR frequencies
共cm−1兲 and relative intensities 共in parentheses兲 when larger
than 5% of the most intense one taken as reference are reported below at the MP2/cc-pVQZ level,
KrH3+:␯2 = 3507共0.05兲,
␯3 = 1382共1.0兲,
␯6 = 625共1.0兲,
␯7/8 = 394共0.4兲.
For information, the corresponding CCSD共T兲 frequencies
obtained in our previous study for the first term of the series,
KrH3+ are
␯1 = 3515共0.05兲,
␯3 = 1273共0.9兲,
␯6 = 618共1.0兲.
IV. THE NATURE OF THE BONDING IN THE XnH3+
COMPLEXES
The evolution of the binding energies corresponding to
the formal reaction
XnH3+ → X共n−1兲H3+ + X
is given in Table VIII. The most important conclusion, as to
the calculation procedures, is that all methods, post-HF and
DFT, give similar results for all the noble gases. At this
point, it is worth emphasizing that the most elaborate level of
theory, CCSD共T兲/cc-pVQZ, is best reproduced by the MP2
calculations using the same cc-pVQZ basis set.
A. Electronic versus vibrational effects
Kr2H3+:␯2 = 2145共1.0兲,
␯7 = 414共0.1兲,
Kr3H3+:␯2/3 = 2136共1.0兲,
␯3 = 1890共0.2兲,
␯8 = 328共0.4兲,
Looking at the variations in the binding energies within
each series shows that the three noble gases behave differently. The analysis is illustrated in Fig. 3 where the total
binding energy is progressively decomposed into its compo-
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174313-12
J. Chem. Phys. 130, 174313 共2009兲
Pauzat et al.
FIG. 3. Components of the binding energy.
nents. The total binding energies 共electronic+ ZPE+ BSSE
corrections兲 are given in Fig. 3共a兲; neglecting the BSSE corrections leads to Fig. 3共b兲 and the electronic-only contribution is reported in Fig. 3共c兲. All the values are taken at the
same CCSD/cc-pVQZ level of theory.
The fact that the same evolution is observed in post-HF
and in DFT calculations, for which the BSSE correction has
a negligible impact on the structure calculated for these
complexes,14 is a first indication that the origin of the phenomenon has very little, if any, reason to come from an artifact in the BSSE correction. It is confirmed by the calculations without implementation of the BSSE correction 关Fig.
3共b兲兴 that show strictly the same variations 共the fact that the
binding energy of the second Ar and Kr atoms are very close
is a pure coincidence兲.
In electronic-only calculations 关Fig. 3共c兲兴, all three noble
gases behave similarly with a regular decrease in the binding
energy from the first to the second and third atoms. This
decrease is consistent with the progressive lengthening of the
distance between the X atom and the H3+ fragment and the
associated dilution of the charge 关see the electron localization function 共ELF兲 analysis in Sec. IV B兴. The effect is
therefore located in the vibrational contribution. More precisely, it comes from a balance between the electronic and
vibrational effects.
For neon, there is an unexpected but systematic increase
of ⬃0.1 kcal/ mol in the binding energy from the first to the
second and third Ne atom in all post-HF methods 关density
functional theory 共DFT兲 calculations do not show a regular
trend but the variations are too small to be significant兴. Extending the basis set from cc-pVTZ to cc-pVQZ results in a
systematic increase in the binding energies by
⬃0.2 kcal/ mol. Increasing the level of correlation from
CCSD to CCSD共T兲 with the cc-pVQZ on this system shows
a further increase of ⬃0.2 kcal/ mol for each of the binding
energies 关note that the MP2 results are in perfect agreement
with CCSD共T兲兴. The explanation of this trend is that the
difference between the successive ZPE of the complexes is
smaller than the associated decrease in the differences in the
electronic binding energies. These effects being opposite, the
net result is a small increase in the total binding energies
with the increasing number of associated Ne atoms. Com-
pared to experiments, our best result is about double of the
0.4 kcal/mol estimated by Hiraoka and Mori19 from the thermochemical plots for the binding energy of a single Ne atom.
For argon, the binding energy decreases from the first to
the second and third Ar atoms, which is consistent with the
progressive lengthening of the distance between the argon
atoms and the H3+ fragment. There is almost a factor of 2
between the binding of ArH3+ and that of Ar2H3+ 共7.2 versus
4.1 kcal/mol兲 at the CCSD共T兲 level. The bonding of the third
Ar is slightly lower than that of the second one 共3.7 kcal/
mol兲. Comparison with the available experimental dissociations energies19 of 6.69, 4.56, and 4.28 kcal/mol for ArnH3+
共n = 1 – 3兲 shows a remarkable agreement between theoretical
and experimental values. Although the numerical values may
still be a matter of discussion, it is worth mentioning that the
general trend is the same as for the pure hydrogen clusters. In
this case, the difference between the successive ZPE is fairly
constant, which explains that the variation in the total binding energy parallels that of the electronic energy.
For krypton, the binding energy is minimum for the second Kr atom. One observes a large decrease, by
⬃10 kcal/ mol, in the binding energy of the second Kr atom,
followed by a slight increase with the addition of the third
atom. This unexpected behavior is found whatever the type
of post-HF or DFT calculation. With one krypton, it has been
shown that there is a significant contribution of the
KrH+ ¯ H2 structure that vanishes with the addition of the
second krypton. The second complexation restores a slightly
distorted H3+ and a long distance weak interaction. The
strong increase in the ZPE contribution aforementioned in
the analysis of the vibrational spectra is at the origin of the
sharp decrease in the binding energy. The fact that the binding of the third krypton is ⬃2 kcal/ mol larger is simply due
to the small variation in the ZPE that is about the same as for
the neon and argon series.
B. ELF topological analysis
An interpretation of the evolution of the electronic bonding properties within the XnH3+ complexes as a function of
the type and the number of noble gas atoms has been obtained from a topological analysis of the ELF48,49 via addi-
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174313-13
J. Chem. Phys. 130, 174313 共2009兲
H3+ as a trap for noble gases
FIG. 4. 共Color online兲 ELF localization domains 共ELF= 0.85兲 of the ArH3+,
Ar2H3+, and Ar3H3+ complexes.
tional calculations using the TOPMOD package.50 Localization
domains were displayed with the MOLEKEL software.51 For
several years, the ELF analysis has been extensively used for
interpreting chemical bonding.52 Indeed, ELF is a powerful
tool to describe the chemical bonds because the function is
currently seen as a signature of the electronic pair
distributions.53 Thus, this method makes the partitioning of
the physical space into intuitive electronic regions 共volumes
called basins兲 possible: core basins around nuclei and valence basins in the lone pair and bonding region. The core
basins, C共X兲, where X is the noble gas, contain electrons that
are not involved in the chemical bonds. The protonated basin
V共X, H兲 corresponds to the X-H covalent bond, while the
basin V共H兲 or V共X兲 contains valence electrons not involved
in the X-H bond. Integrated quantities such as the basin
population can be calculated by integrating the charge density over the basin volume. In contrast to the ELF analysis,
the quantum theory of atoms in molecules 共QTAIM兲54 gives
a set of basins localized around the atoms. In this framework,
the contribution of an atom to the ELF population can be
calculated. The ELF localization domains of the XnH3+ com-
plexes are illustrated in Fig. 4, and Table IX presents the
population analysis of the complexes.
Figure 4 reveals several valence basins: one basin V共X兲
for each noble gas and one tricentric protonated basins
V共H1 , H2 , H3兲 for the hydrogen atoms involved in the H3+
fragment. As mentioned in a previous study on the formation
of XH3+ complex,14 the weak interaction between the gas X
and the H3+ moiety is essentially electrostatic since no covalent bonding basin V共X, H兲 is observed. However, the ELF
population analysis given in Table IX exhibits a net CT from
the gas toward the H3+ fragment 共ionic contribution兲. This
CT is systematically observed but its value varies with the
type of complex from −0.03e 共NeH3+兲 to −0.26e 共Kr3H3+兲.
This CT quantity clearly depends of the nature of the noble
gas since the Kr complexes show large values 共from −0.20e
to −0.26e兲, whereas the Ne complexes show very small values 共from −0.03e to −0.06e兲. It is consistent with the relative
atomic polarizabilities of Ne,55 Ar,55 and Kr 共Ref. 56兲 共0.39,
1.59, and 2.49 Å3兲; the most polarizable noble gas 共Kr兲 has
the largest interaction energies and the greatest degree of
charge transfer with H3+.
The relative values of CT between X共n−1兲H3+ and XnH3+
共n = 1, 2, or 3兲 are shown in Fig. 5. The three gases behave
similarly with a regular decrease in the relative CT from the
first to the second and third atoms. This decrease is consistent with the progressive lengthening of the calculated
X ¯ H3+ distance and with the electronic binding energy
共noncorrected or BSSE corrected兲 previously calculated.
This result presented in Fig. 5 confirms that the stability of
the complexes is mainly related to the CT X → H3+. However, the CT evolution can only provide a rationalization of
the electronic component of the binding energy since the
ELF population is calculated from the Kohn–Sham density
without ZPE correction. The details of the evolution, for example, the singularity of the Kr2H3+ complex, can be explained only by the vibrational correction.
V. CONCLUDING REMARKS
In the light of this study, two levels of conclusions, i.e.,
quantum chemical and astrophysical, can be drawn from the
TABLE IX. Population analysis 共electron兲 of the protonated basins involved in XnH3+ complexes. CT is the net
charge transfer 共electron兲 calculated as CT= 2.0− Pop关V共H1 , H2 , H3兲兴. The numbering of the basins corresponds
to that in Fig. 4.
Complex
Pop关H1兴
a
Pop关H2兴
a
Pop关H3兴
a
Pop关V共H1 , H2 , H3兲兴
CT共X → H3+兲
H 3+
0.67
0.67
0.67
2.00
0.00
Ne– H3+
Ne2 – H3+
Ne3 – H3+
0.51
0.55
0.69
0.76
0.55
0.69
0.76
0.94
0.69
2.03
2.05
2.06
⫺0.03
⫺0.05
⫺0.06
Ar– H3+
Ar2 – H3+
Ar3 – H3+
0.45
0.58
0.73
0.84
0.58
0.73
0.84
0.99
0.73
2.13
2.16
2.18
⫺0.13
⫺0.16
⫺0.18
Kr– H3+
Kr2 – H3+
Kr3 – H3+
0.47
0.59
0.75
0.87
0.59
0.75
0.87
1.06
0.75
2.20
2.24
2.26
⫺0.20
⫺0.24
⫺0.26
Atomic contributions of hydrogen to the population of tricentric protonated basin V共H1 , H2 , H3兲.
a
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174313-14
J. Chem. Phys. 130, 174313 共2009兲
Pauzat et al.
FIG. 5. Relative CT 共X → H3+ for X = Ne, Ar, and Kr兲 of the noble gas
complexes 共in electrons兲 calculated as 关CT共XnH3+兲 − CT共X共n−1兲H3+兲兴 with
n = 1, 2, 3.
clustering of the noble gases around the H3+ ion. Concerning
the chemical aspect, we have shown that clustering up to
three noble gases was possible for neon, argon, and krypton
atoms. This result was obtained using post-HF 关CCSD and
CCSD共T兲兴 levels of wave functions as well as DFT
共BH&HLYP兲 coupled with correlation-consistent basis sets
of high flexibility. The BSSE artifact was corrected all along
the optimization process using the counterpoise method,
which is necessary to obtain reliable values. The point in
common is that all the complexes are planar, with the noble
gases linked to the apices of the H3+ triangular fragment. The
topological analysis of their ELF has shown that the electronic binding energies are directly linked to the magnitude
of the charge transfer from the noble gas to the H3+ ion. This
general behavior is related to the increasing polarizability of
the noble gases from Ne to Ar and Kr. In fact, both effects,
i.e., the CT X → H3+ and the polarizability of the noble gases,
govern the stability of these complexes.
From a strict computational point of view, we have
found that the cost effective MP2 calculations provide energetic values and rotational constants close to the expensive
CCSD共T兲 treatments when the flexible cc-pVQZ basis set
was employed in both approaches. Rotational constants have
been proposed that should be an incentive for new laboratory
experiments in view of the values of the dipole moments of
the neon, argon, and krypton complexes. Besides, theoretical
IR spectra show characteristic bands with strong intensities
that should be detectable even if these complexes are of low
abundance.
Concerning the astrophysical aspect, it should be remembered that the H3+ ion is ubiquitous in space, as illustrated by recent detections in a large variety of environments:
in star forming regions,57 in diffuse interstellar media,58 at
the poles of Jupiter59 and of the other giant planets;60 it has
also been detected under its deuterated form H2D+ in a prestellar core61 and in protoplanetary disks.62
The stability of XnH3+ complexes has important astrophysical implications because the sequestration of noble
gases by H3+ may greatly influence their abundances in bodies such as giant gaseous planets, icy satellites, or comets.
Indeed, such trapping, if effective in space, would imply that
noble gases stay in the gas phase of protoplanetary disks
instead of being incorporated into the planetesimals that ultimately take part in the formation of the planets. In particular, the fact that more than one Kr can be trapped may be
decisive for the sequestration of this atom in regions where
the radiation field leads to abundances of H3+ comparable to
that of this species. For example, an important fraction of
Kr may have been sequestered by H3+ in the solar nebula gas
phase, thus implying the formation of at least partly
Kr-impoverished planetesimals in the formation zones of Jupiter and Saturn.63–65 In the case of Titan, which is the second largest satellite in the solar system, the noble gas deficiency measured in the atmosphere by the Huygens probe
during its descent66 may also result from the sequestration of
at least Kr by H3+ in the solar nebula gas phase before the
formation of its building blocks.67 Moreover, the stability of
the XH3+ clusters may also be an important factor for discriminating the origin of comets according to their heliocentric distance of formation.63,64
ACKNOWLEDGMENTS
The authors would like to thank one of the referees for
his positive suggestions and careful reading of the manuscript. This research has been supported by an IBM joint
study under the hospitality of the Almaden Research Center
and by the CNRS “Programme National de Planétologie.”
The largest calculations have been performed at the CINES
supercomputing facility.
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