J. Chem. Phys.
Supplementary Material
Trapping a Proton in Argon: Spectroscopy and Theory of the
Proton-Bound Argon Dimer and Its Solvation
D. C. McDonald,1 D. T. Mauney,1 D. Leicht,2 J. H. Marks,1 J. A. Tan,3,4,5 J.-L. Kuo,3,4* and
M. A. Duncan1*
1
Department of Chemistry, University of Georgia, Athens, Georgia, 30602, U.S.A.
Lehrstuhl für Physikalische Chemie II, Ruhr-Universität Bochum, 44801 Bochum, Germany
3
Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei, 10617, Taiwan, R.O.C.
4
Molecular Science and Technology Program, Taiwan International Graduate Program,
Academia Sinica, Nangang, Taipei, 11529, Taiwan, R.O.C.
5
Department of Chemistry, National Tsing Hua University, Hsinchu, 30013, Taiwan, R.O.C.
2
Email: [email protected]
S1
Figure S1. The mass spectrum generated in the molecular beam machine using a pulsed
discharge source with a supersonic expansion of 10% H2 in argon.
S2
Figure S2. Expanded view of the mass spectrum from Figure S1 above, showing the H+Arn
peaks just below those for the corresponding H3+Arn masses.
S3
Table S1. The computed energy of H+Ar at the MP2/aug-cc-pVTZ level of theory.
Complex
H+Ar
Total Energy (Hartrees)
-527.1725848
Relative Energy (kcal/mol)
0.0
Figure S3. The optimized geometry of H+Ar.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar at the MP2/aug-cc-pVTZ level:
2737.6 (564)
S4
Table S2. The computed energy of H+Ar2 at the MP2/aug-cc-pVTZ level of theory.
Complex
H+Ar2
Total Energy (Hartrees)
-1054.2231193
Relative Energy (kcal/mol)
0.0
Figure S4. The optimized geometry of H+Ar2.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar2 at the MP2/aug-cc-pVTZ level:
324.4 (0), 705.6 (46) , 705.6 (46) , 1047.9 (4914)
S5
Table S3. The computed energy of H+Ar3 at the MP2/aug-cc-pVTZ level of theory.
Complex
H+Ar3
Total Energy (Hartrees)
Relative Energy (kcal/mol)
-1581.2508384
0.0
Figure S5. The optimized geometry of H+Ar3.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar3 at the MP2/aug-cc-pVTZ level:
46.2 (1), 59.6 (6) , 323.5 (0) , 682.9 (68) , 706.0 (40) , 1038.0 (4574)
S6
Table S4. The computed energy of H+Ar4 at the MP2/aug-cc-pVTZ level of theory.
Complex
H+Ar4
Total Energy (Hartrees)
-2108.2789742
Relative Energy (kcal/mol)
0.0
Figure S6. The optimized geometry of H+Ar4.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar4 at the MP2/aug-cc-pVTZ level:
25.5 (1), 39.2 (0) , 52.1 (1) , 56.8 (4) , 63.6 (6) , 322.4 (0) , 678.5 (68) , 689.8 (53) ,
1026.8 (4262)
S7
Table S5. The computed energy of H+Ar5 at the MP2/aug-cc-pVTZ level of theory.
Complex
Total Energy (Hartrees)
H+Ar5
-2635.3070583
Relative Energy (kcal/mol)
0.0
Figure S7. The optimized geometry of H+Ar5.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar5 at the MP2/aug-cc-pVTZ level:
20.7 (2), 27.4 (1) , 31.2 (1) , 48.6 (0) , 51.6 (1) , 53.9 (0) , 63.0 (5) , 67.1 (8) , 322.1 (0) ,
668.5 (77) , 682.2 (57) , 1019.9 (3971)
S8
Table S6. The computed energy of H+Ar6 at the MP2/aug-cc-pVTZ level of theory.
Complex
H+Ar6
Total Energy (Hartrees)
Relative Energy (kcal/mol)
-3162.3351147
0.0
Figure S8. The optimized geometry of H+Ar6.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar6 at the MP2/aug-cc-pVTZ level:
14.7 (2), 24.4 (0) , 26.1 (1) , 31.0 (0) , 37.2 (3) , 49.0 (0) , 51.0 (0) , 52.9 (1) , 61.0 (0) ,
65.6 (6) , 72.6 (9) , 322.2 (0) , 658.0 (87) , 676.9 (60) , 1015.1 (3695)
S9
Table S7. The computed energy of H+Ar7 at the MP2/aug-cc-pVTZ level of theory.
Complex
H+Ar7
Total Energy (Hartrees)
-3689.3635800
Relative Energy (kcal/mol)
0.0
Figure S9. The optimized geometry of H+Ar7.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar7 at the MP2/aug-cc-pVTZ level:
23.1 (0), 23.1 (0) , 28.1 (2) , 28.1 (2) , 36.2 (0) , 36.2 (0) , 41.9 (3) , 48.8 (0) , 48.8 (0) , 53.5 (0),
59.6 (0) , 59.6 (0) , 71.3 (8) , 71.3 (8) , 322.4 (0) , 658.9 (79), 658.9 (79) , 1014.4 (3459)
S10
Table S8. The computed energy of H+Ar8 at the MP2/aug-cc-pVTZ level of theory.
Complex
H+Ar8
Total Energy (Hartrees)
-4216.3906331
Relative Energy (kcal/mol)
0.0
Figure S10. The optimized geometry of H+Ar8.
The predicted unscaled frequencies (cm-1) and IR intensities (shown in parentheses, km/mol) for
H+Ar8 at the MP2/aug-cc-pVTZ level:
13.9 (0), 19.0 (0) , 23.6 (0) , 27.5 (1) , 30.0 (1) , 31.5 (0) , 36.4 (0) , 36.4 (0) , 40.8 (0) , 46.0 (3),
48.8 (0) , 49.3 (0) , 53.7 (0) , 59.8 (0) , 63.3 (0) , 71.6 (8) , 75.6 (9), 321.5 (4) , 655.6 (76) ,
658.9 (77) , 1022.4 (3527)
S11
Figure S11. Comparison of simulated spectra using unscaled harmonic frequencies from
MP2/aug-cc-pVTZ calculations for H+Arn , n = (28).
S12
Table S9. Structural parameters and harmonic normal mode analysis for H+Ar2 at the MP2 level.
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
Freq.
Inten.
Freq. Inten.
Freq.
Inten.
Freq.
Inten.
cm-1
km/mol
cm-1
km/mol
cm-1
km/mol
cm-1
km/mol
ν1 (σg+)
321
0
324
0
325
0
325
0
ν2a (πu)
687
50
706
46
687
46
681
46
ν2b (πu)
687
50
706
46
687
46
681
46
ν3 (σu+)
1041
4938
1048 4913
1074
4886
1053
4890
RAr-H+
1.5129 Å
1.5018 Å
1.5019 Å
1.5014 Å
RAr-Ar
3.0258 Å
3.0036 Å
3.0039 Å
3.0028 Å
<Ar-H+-Ar
180.00°
180.00°
180.00°
180.00°
Mode #
S13
Table S10. Anharmonic coupled frequencies for H+Ar2 using the PES obtained at the
CCSD(T)/aug-cc-pVXZ//MP2/aug-cc-pVQZ (X = D, T, and Q). The intensities were evaluated
using the DMS at MP2/ aug-cc-pVXZ//MP2/aug-cc-pVQZ (X = D, T, and Q).
Reference geometry and normal coordinates at MP2/aug-cc-pVQZ
PES at CCSD(T)/aug-cc-pVXZ//MP2/aug-cc-pVQZ
DMS at MP2/aug-cc-pVXZ//MP2/aug-cc-pVQZ
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
Freq.
Inten.
Freq.
Inten.
Freq.
Inten.
Fundamentals
cm-1
km/mol
cm-1
km/mol
cm-1
km/mol
ν1 (σg+)
289
0
292
0
293
0
ν2a (πu)
664
57
670
53
658
52
ν2b (πu)
664
57
670
53
658
52
ν3 (σu+)
984
2640
986
2706
1000
2728
ν1+ ν3
1231
1292
1238
1241
1253
1221
2ν1+ ν3
1476
362
1485
323
1500
307
3ν1+ ν3
1737
67
1735
59
1750
53
4ν1+ ν3
2029
5
2004
6
2017
5
2121
56
2135
59
2133
65
573
0
579
0
581
0
1335
0
1345
0
1323
0
1349
0
1357
0
1339
0
Combination
bands
Interesting
Combination
bands of ν2
with ν3
Overtones
2ν1
2ν2a, 2ν2b
and
ν2a+ ν2b
S14
Figure S12. Calculated coupled and anharmonic stick spectrum for H+Ar2 with various basis sets.
For brevity, aug-cc-pVXZ is abbreviated as AVXZ. The reference geometry is the minimized
geometry at the MP2/AVQZ level. The potential energy surfaces were built at the
CCSD(T)/AVXZ//MP2/AVQZ level and are labeled in the subplots. The dipole moment
surfaces (DMS) were constructed from the MP2/AVXZ//MP2/AVQZ levels.
S15
To support our assignments in Table S10 and Figure S12, we performed two low-dimensional
calculations. One of them includes only the ArAr stretch (𝑄𝑠 ) and the H+ stretch (𝑄𝑎𝑠 ), while
the other only includes the degenerate bends (𝑄2𝑎 , 𝑄2𝑏 ). We label their potentials as
𝑉 2𝐷 (𝑄𝑠 , 𝑄𝑎𝑠 ) and 𝑉 2𝐷 (𝑄2𝑎 , 𝑄2𝑏 ). Their vibrational Hamiltonians are
̂𝑄2𝐷,𝑄 =
𝐻
𝑠 𝑎𝑠
̂𝑄2𝐷 ,𝑄 =
𝐻
2𝑎 2𝑏
−ℏ2 1 𝜕 2
1 𝜕2
[
+
]
2
2 𝜇𝑠 𝜕𝑄𝑠2 𝜇𝑎𝑠 𝜕𝑄𝑎𝑠
−ℏ2 1 𝜕 2
1 𝜕2
[
2 +𝜇
2 ]
2 𝜇2𝑎 𝜕𝑄2𝑎
2𝑏 𝜕𝑄2𝑏
(1)
(2)
Their corresponding vibrational Schrödinger equations are
̂𝑄2𝐷,𝑄 |𝑛⟩ = 𝐸𝑛 |𝑛⟩
𝐻
𝑠 𝑎𝑠
(3)
̂𝑄2𝐷 ,𝑄 |𝑚⟩ = 𝜀𝑚 |𝑚⟩
𝐻
2𝑎 2𝑏
(4)
Table S11 below shows the solutions of the eigenvalue problem of Equations (3) and (4). The
direct products of |𝑛⟩ and |𝑚⟩ can be used as test states to unmask the identity of the calculated
2133 cm-1 peak. If we define the vibrational wave function of the 2133 cm-1 peak as
|Ψ2133 𝑐𝑚−1 ⟩, and the test states as |𝑛, 𝑚⟩ ≡ |𝑛⟩⨂|𝑚⟩, then the projection of |𝑛, 𝑚⟩ onto
2133 𝑐𝑚−1
|Ψ2133 𝑐𝑚−1 ⟩ represents the linear combination coefficient, 𝑐𝑛𝑚
for |𝑛, 𝑚⟩.
In other words, we used the test states |𝑛, 𝑚⟩ to expand the |Ψ2133 𝑐𝑚−1 ⟩.
−1
2133 𝑐𝑚 |𝑛,
|Ψ2133 𝑐𝑚−1 ⟩ = ∑ 𝑐𝑛𝑚
𝑚⟩
2133 𝑐𝑚
𝑐𝑛𝑚
−1
= ⟨𝑛, 𝑚|Ψ2133 𝑐𝑚−1 ⟩
S16
(5)
(6)
Equations (5) and (6) can be used in unmasking the identities of any vibrational state
|Ψ𝑘 𝑐𝑚−1 ⟩. Table S12 in the next page shows the squares of the linear combination coefficients
for a few test states. These squares of the linear combination coefficients can be interpreted as
the weight/contribution of the test state on |Ψ𝑘 𝑐𝑚−1 ⟩.
S17
Table S11. Transition frequencies (cm-1) and intensities (km/mol) for low-dimensional
calculations at the CCSD(T)/aug-cc-pVQZ//MP2/aug-cc-pVQZ level.
̂𝑄2𝐷 ,𝑄 |𝑚⟩ = 𝜀𝑚 |𝑚⟩
𝐻
2𝑎 2𝑏
̂𝑄2𝐷,𝑄 |𝑛⟩ = 𝐸𝑛 |𝑛⟩
𝐻
1 3
State
|𝑛⟩
Frequencies
(cm-1)
Intensity
(km/mol)
State
|𝑚⟩
Frequencies
Intensity
(cm-1)
(km/mol)
0
0
0
0
0
0
1
300
0
ν1
1
724
52
ν2a
2
597
0
2ν1
2
724
52
ν2b
3
890
0
3ν1
3
1465
0
4
1465
0
Assignment
Assignment
4
1096
2742
ν3
5
1178
0
4ν1
5
1483
0
6
1358
1332
ν1+ν3
6
2221
0
7
1463
0
5ν1
7
2221
0
8
1617
383
2ν1+ν3
8
2251
0
9
1754
0
6ν1
9
2251
0
10
1885
83
3ν1+ν3
10
2989
0
11
2069
0
7ν1
11
2989
0
12
2180
13
4ν1+ν3
12
3027
0
S18
First
overtones of
the H+ bends
−1
2
𝑘 𝑐𝑚
Table S12. Squares of the linear combination coefficients (𝑐𝑛𝑚
) for the peaks at 2133, 2367, and 2590 cm-1 of the anharmonic
vibrational calculations at the CCSD(T)/aug-cc-pVQZ//MP2/aug-cc-pVQZ level of theory and basis set.
CCSD(T)/aug-cc-pVQZ//MP2/aug-cc-pVQZ
Frequencies
in cm-1
1000
1253
1500
1750
2017
2133
2367
2590
Intensities
in km/mol
2728
1221
307
53
5
65
26
5
−1
𝑘 𝑐𝑚
(𝑐𝑛𝑚
)
2
|𝑛, 𝑚⟩
Assignment
|Ψ1000 𝑐𝑚−1 ⟩|Ψ1253 𝑐𝑚−1 ⟩ |Ψ1500 𝑐𝑚−1 ⟩ |Ψ1750 𝑐𝑚−1 ⟩ |Ψ2017 𝑐𝑚−1 ⟩ |Ψ2133 𝑐𝑚−1 ⟩ |Ψ2367 𝑐𝑚−1 ⟩ |Ψ2590 𝑐𝑚−1 ⟩
|4,0⟩
ν3
94.96
2.64
0.02
0.00
0.00
1.67
0.09
0.00
|6,0⟩
ν3 + ν1
2.24
89.64
5.24
0.08
0.00
0.48
1.44
0.19
|8,0⟩
ν3 + 2ν1
0.19
4.00
84.73
7.66
0.13
0.26
0.64
1.29
|10,0⟩
ν3 + 3ν1
0.01
0.51
5.30
80.35
9.84
0.09
0.55
0.62
|12,0⟩
ν3 + 4ν1
0.00
0.04
0.80
6.18
75.76
0.23
0.06
1.04
|4,5⟩
ν3 + 2ν2
1.13
0.03
0.01
0.00
0.03
71.27
8.77
0.04
|6,5⟩
ν1 + ν3 + 2ν2
0.15
1.28
0.06
0.05
0.02
7.19
54.38
14.64
|8,5⟩
2ν1 + ν3 + 2ν2
0.03
0.30
1.48
0.09
0.13
1.29
9.28
41.74
ν1
ν2
v3
ArAr str
H+ bend
IHB str
Legend
Note: |𝑛, 𝑚⟩ = |𝑛⟩⨂|𝑚⟩. |𝑛⟩ and |𝑚⟩ are states defined in Table S11.
S19
The data in Table S12 demonstrate that the 1000 cm-1 peak is composed of 94.96% |4,0>.
Referring to Table S11, n = 4 corresponds to the bright proton stretch. The peaks at 1253, 1500,
1750, and 2017 cm-1 are dominated by |n,m> = |6,0>, |8,0>, 10,0>, and |12,0> respectively.
Again, referring to Table S11, the values of n = 6, 8, 10, and 12 correspond to ν3 + ν1, ν3 + 2ν1, ν3
+ 3ν1, and ν3 + 4ν1 respectively. Therefore, we assign the 2017 cm-1 band to be ν3 + 4ν1.
We are left with the 2133 cm-1 peak. From Table S12 it appears that this peak is
interesting. It is mainly composed of |4,5>. Table S11 tells us that an n = 4 corresponds to the
bright proton stretch (ν3), and an m = 5 corresponds to the first overtone of the H+ bend. The
2133 cm-1 peak agrees reasonably well with the 2124 cm-1 peak of the measured H+Ar3
spectrum. This feature may be explained by the H+ stretch and H+ bend interacting. Whereas the
H+ bends are degenerate in H+Ar2, the simulation only shows one peak (2133 cm-1 at the
CCSD(T)/aug-cc-pVQZ//MP2/aug-cc-pVQZ) level. In H+Ar3, the degeneracy of the H+ bend is
broken. According to the harmonic theory, the bend towards the third argon atom shifts to a
lower frequency, which likely explains the two observed features at 2041 and 2124 cm-1.
Inclusion of the third argon in further simulations will be needed to test this hypothesis.
Lastly, both the first overtones of the H+ bend and the H+ stretch (i.e. 2ν2 and 2ν3) were
not observed. These were expected to have frequencies in the range 12001300 cm-1 and
18502000 cm-1 respectively.
S20