Electric properties of N2 O and CO2 molecules
Ben T. Chang, Omololu Akin-Ojo, Robert Bukowski, and Krzysztof Szalewicz
Department of Physics and Astronomy, University of Delaware
Newark, DE 19716
(September 10, 2003)
Abstract
In this supplementary material we report accurate multipole moments and
dipole-dipole polarizabilities of N2 O and CO2 . Convergence of these propeties
with respect to basis set and theory level up to CCSD(T) has been carefully
analyzed. The quadrupole, octupole, and hexadecapole moments of N2 O have
been found significantly different from experimental values and are probably
more accurate than the latter.
1
I. INTRODUCTION
Our interest in accurate values of the electric properties of the N2 O and CO2 molecules
was motivated by the fact that these properties determine to some extent the character
of intermolecular interactions and are used in construction of analytic fits to interaction
potential surfaces. In addition, comparisons of the properties calculated in a given basis
with literature data provide an evaluation of the quality of this basis.
While initially our intention was only to tabulate a subset of the properties already
calculated analytically at the SCF and MBPT2 levels for the purpose of the fit using the
POLCOR suite of programs [1,2], after finding discrepancies with literature we have extended
this comparison by calculating these properties using higher levels of MBPT as well as the
CCSD(T) method and the finite field approach. In addition to our basis set, used in the
interaction energy calculations, we ahve also used augmented correlation-consistent basis
sets, denoted by aug-cc-pVXZ ≡ aXZ, of Dunning et al. [3] and properties-optimized basis
sets of Sadlej [4].
A set of multipole moments and polarizabilities for the systems of interest are listed
in Table S–I. Higher-rank multipole moments, polarizabilities, and dynamic polarizabilities
have been calculated as well, but no comparisons with literature are possible in these cases.
For linear molecules, the spherical multipole moments Ql0 provide the complete information
about the charge distribution. These are defined in the standard way (see, e.g., Ref. [19])
and are therefore equal to the Cartesian moments (µz , Θzz , Ωzzz , and Φzzzz for l = 1 − 4,
respectively). The dipole-dipole polarizability [19] tensor is diagonal for linear molecules
and can be represented by the average polarizability ᾱ = (2αxx + αzz )/3 and the anisotropy
of polarizability ∆α = αzz − αxx . The origin of coordinate system was fixed at the center
of mass of a given molecule with the z axis oriented along the molecular axis. In the
case of N2 O the oxygen atom is on the negative part of the z axis. We have used the
following conversion factors from atomic units: dipole moment 1 ea0 = 8.478 358 × 10−30
Cm2 = 2.541×10−18 esu; quadrupole moment 1 ea20 = 4.486 554×10−40 Cm2 = 1.345×10−26
2
esu; octupole moment 1 ea30 = 2.374 182 × 10−50 Cm3 = 0.7117 × 10−34 esu; hexadecapole
moment 1 ea40 = 1.256 363 × 10−60 Cm4 = 0.3766 × 10−42 esu; dipole-dipole polarizability 1
e2 a20 /Eh = 1.648778 × 10−41 C2 m2 /J.
All the quantities computed in the finite-field approach include the so-called response
or relaxation effects. The properties computed analytically can be obtained both with or
without the response, except for the MBPT2 polarizabilities. In the latter case, we add to
the SCF part of the polarizability computed with response (i.e., to the Coupled HartreeFock (CHF) value) the “true” second-order component calculated without response (the
second-order diagrams already present in the CHF part are, of course, excluded from this
component). This value of polarizability is relevant here since it is computed at the level of
theory equivalent to that used in SAPT. The strengths of the field used in the finite-field
method were: 1 × 10−4 a.u., 1 × 10−4 a.u., 2 × 10−5 a.u., 1 × 10−6 a.u., 5 × 10−3 a.u., and
5 × 10−3 a.u. for Q10 , Q20 , Q30 , Q40 , αzz , and αxx , respectively. The fields were created
by placing point charges along the the z axis far from the molecule [13]. For instance,
placing charges of −16Q, Q, −Q, and 16Q along the z axis at −2z0 , −z0 , z0 , and 2z0 ,
respectively, produces a (quasi)dipolar field of 6Q/z02 , i.e., 1 × 10−4 a.u. with z0 = 1000 bohr
and |Q| = 104 e/6. The energies were calculated with both Q = |Q| and Q = −|Q| and the
required moment was obtained as a derivative of the energy with respect to the applied field
using the central difference formula. The agreement between the finite field and analytic
results was at about 0.0001 a.u. for Q10 and Q20 , 0.0005 a.u. for Q30 , 0.005 a.u. for Q40 ,
0.01 a.u. for ᾱ, and 0.02 a.u. for ∆α in cases where comparisons could be made, showing
the high accuracy of numerical differentiations in the finite field approach for the moments.
Therefore, only the results from finite-field calculations are reported in Table S–I except
for the MBPT2 polarizabilities. We have also agreed to virtually all digits published, i.e.,
to within 0.00001 a.u., on the quadrupole moment of CO2 with Coriani et al. [14] at the
SCF and MBPT levels and to about 0.0001 a.u. on the CCSD and CCSD(T) levels. Our
calculations were performed correlating all electrons. We have checked using the aDZ basis
that the differences between the multipole moments of N2 O computed in this way and using
3
the frozen-core approximation were smaller than 2%.
II. DIPOLE MOMENT OF NITROUS OXIDE
It has been pointed out by Frisch and Del Bene [5] many years ago that the dipole
moment of N2 O is even more pathological than the well-known case of CO. This is clearly
related to its very small value of only 0.0633 a.u. [6,7] (typical dipole moments of polar
molecules are of the order 1 a.u.). The smallness of the dipole moment results, of course,
from the similarity of N2 O to molecules of D∞h symmetry, like CO2 , which have zero dipole
moments. The pathological properties of the dipole moment of N2 O are well visible in
Table S–I. The SCF-level result is of the right sign [7] (with our choice of the coordinate
system, the positive sign of Q10 corresponds to the − ONN+ charge distribution) but four
times larger than the experimental value. The MBPT2 level of theory predicts a wrong sign
and the MBPT series appears to diverge. Only the CCSD(T) method agrees fairly well, to
within 10%, with experiment. This level of agreement is not surprising taking into account
the convergence patterns in theory level and also in basis set size. Within the range of basis
sets used by us, the convergence is nonuniform and one cannot exclude that the residual
basis set incompleteness error is of the order of 5% for the largest basis used by us. As
mentioned above, the rate of convergence in the theory level is even more nonuniform. In
fact, one could assume that the theory truncation error may account for the whole of the
discrepancy with experiment. On the other hand, the CCSD(T) method rarely gives errors
larger than a few percent for first-order properties of closed-shell systems so that 10% might
be an overestimation of this uncertainty. One more significant source of error could be the
vibrational correction. This correction is unknown for N2 O but for the similarly small dipole
moment of CO the vibrationally averaged value is 10% larger than the value corresponding
to the equilibrium geometry [20]. Since we use the vibrationally averaged rather than the
equilibrium geometry, the effects of vibrational averaging of the dipole moment should be
smaller in our case, but still a vibrationally averaged calculation may be needed to reach a
4
better agreement between theory and experiment for N2 O. Two out of the three sources of
uncertainty of theoretical result can be eliminated by performing calculations in still larger
basis sets and computing the vibrational correction. Unfortunately, it would be presently
not practical to perform calculations at a level of theory higher than CCSD(T) for a molecule
of the size of N2 O.
We have found only one previous calculation of the dipole moment of N2 O. The results of
Frisch and Del Bene [5] were obtained in a basis set of triple zeta quality and including diffuse
functions so the results should be comparable to those computed by us in the basis aTZ. The
agreement indeed is reasonably good as at identical levels of theory the discrepancies are of
the order of 0.01 a.u. Moreover, these discrepancies are mostly due to the differences in the
N2 O geometries used in the two calculations. We performed a calculation in the aDZ basis
at the same experimental geometry as used in Ref. [5]. The differences with respect to the
aDZ results listed in Table S–I were between 0.007 and 0.012 a.u., reproducing the pattern
of differences with the results of Ref. [5]. The highest level of theory used by Frisch and Del
Bene, the quadratic configuration interaction method with noniterative triples, QCISD(T),
is an approximation to CCSD(T). The QCISD(T) result from Ref. [5], given in the last
column of Table S–I, is significantly further from experiment than our CCSD(T) values.
Clearly, the CCSD(T) value computed in the aQZ basis and equal to 0.0569 a.u. represents
the currently most accurate theoretical result for the dipole moment of N2 O.
III. QUADRUPOLE MOMENT OF NITROUS OXIDE
The convergence of the quadrupole moment with the level of theory is fast, in striking
contrast to the dipole moment: the difference between the SCF and CCSD(T) predictions is
less than 2%. Also the variations due to the improvements of the basis set are of the same
order. Therefore it is surprising that the prediction of the CCSD(T) method in the aQZ basis
still disagrees with the generally accepted experimental value of Reinartz et al. [6] by as much
as 10%. This disagreement is very large compared to the experimental error bar of Ref. [6]
5
amounting to 0.011 a.u. The disagreement with the experiment of Reinartz et al. is difficult
to explain since, unlike in the this case of the dipole moment, an increase of the level of theory,
basis set, and inclusion of vibrational effects would not be sufficient to significantly change
the theoretical predictions, as can be judged from the results of Table S–I and comparisons
to the quadrupole moment of CO2 . It is quite clear from the small differences between the
applied levels of theory seen in Table S–I that the theory-level truncation effect cannot be
much larger than about 2%. Similar size improvement can be realized by further basis set
extensions as indicated by the pattern of the basis set convergence (only 0.5% change between
aTZ and aQZ). A further evidence can be obtained by comparisons with CO2 . In a recent
paper Coriani et al. [14] estimated that the quadrupole moment of CO2 computed at the
CCSD(T) level using the aQZ basis is less than 1% different from the exact clamped-nuclei
value. The vibrational effects were found [14] to account for only 0.6% of the quadrupole
moment. Using the CCSD(T) level of theory, the a5Z basis set, and adding the vibrational
correction, these authors achieved agreement with experimental values [15,21,22] to about
0.1%. We can conservatively estimate that our best value of the quadrupole moment of N2 O
equal to -2.689 a.u. is accurate to about 2% or 0.05 a.u. This casts doubts on the accuracy
of the experimental result of Reinartz et al. [6] since the discrepancy between theory and
experiment is five times larger than the estimated theoretical uncertainty and two orders of
magnitude larger than the experimental error bars.
In view of the disagreement with the result of Reinartz et al. [6], we have looked in the
literature for other measurements of the quadrupole moment of N2 O. The value measured
by Buckingham et al. [8] agrees only slightly better with our prediction, although, because
of the large error bars, theory is close to the upper limit of the experimental range. One
should point out, however, that the comparison with the value of Buckingham et al. is
not fully meaningful since the experimental Q20 is not defined with respect to the center
of the mass of N2 O. Another experimental value was reported by Flygare et al. [9]. This
value agrees to within 1% with our prediction. The Flygare et al. value was quoted by
Reinartz et al. but the discrepancy was not discussed. The former value was measured for
6
the
15
N2 16 O isotope while the latter for
14
N2 16 O, however, this differences would effect only
the vibrational correction which, as discussed above, should be of the order of only 1%.
Thus, our results support the quadrupole moment measurement of Flygare et al. [9] over
the more recent value from Reinartz et al. [6].
IV. OCTUPOLE MOMENT OF NITROUS OXIDE
Since the octupole moment, similarly as the dipole moment, is zero for molecules of D∞h
symmetry, one would expect to encounter for N2 O similar convergence problems for both
moments. Indeed, as the results of Table S–I show, the octupole moment converges much
slower with the level of theory than the quadrupolar one, although the convergence is not
nearly as oscillatory as for the dipole moment. The sign of the octupole moment is the same
at each level of theory and the difference between the SCF and CCSD(T) values is 57%, a
very large correlation effect. The better convergence of Q30 compared Q10 is probably related
to the fact that the former moment, although not large (the magnitude of Q30 should be
a few times larger than that of Q20 to significantly contribute to the electric potential of a
molecule) is not that small as one might expect from near-symmetry arguments.
Comparison of the computed octupole moment with experiment reveals a strikingly large
discrepancy: the smallest in magnitude of the three experimental values of Q30 [10,11] is 2.5
times larger than the best theoretical result. The experimental values come from two different types of measurements and although no error bars are attached to either experiment, the
results differ by only 10%-25%, a small difference compared to the discrepancies with theory.
The reasons for these discrepancies remain unclear. It is very unlikely that theoretical value
could increase by as much as a factor of 2.5. The difference between the MBPT2 value and
CCSD(T) value is 16%. The latter method represents the highest practically applicable level
of electron correlation and, as already mentioned, it is known to reproduce experimental data
often to within a few percent provided a large enough basis is used. Thus, an assumption
that the exact octupole moment is within 10% of the basis-set converged CCSD(T) value is
7
rather safe. The results in Table S–I show that the octupole moment is more sensitive to
the basis set than the quadrupole moment, as generally expected of higher-rank moments,
however, an assumption of 5% basis-set saturation error in the value computed using the
aQZ basis would be generous. Adding another 10% uncertainty due to vibrational effects,
we can estimate the error of theory to be less than 30%, grossly insufficient to explain the
discrepancies with experiment.
V. HEXADECAPOLE MOMENT OF NITROUS OXIDE
The disagreement with experiment is even more dramatic for the hexadecapole moment.
Unexpectedly, this moment converges slower with the theory level than the octupole moment,
the CCSD(T) value being 3 times larger in magnitude than the SCF value. Even the
difference between the MBPT4 and CCSD(T) moments is 27% of the latter. If the aDZ
results are disregarded due to the smallness of this basis, the basis set convergence is similar
to that observed for the quadrupole and octupole moments. While we cannot exclude that
our best value of Q40 equal to -3.904 a.u. may be a factor of two different from the exact
value, our value is about 10 times smaller in magnitude than the experimental result of Dagg
et al. [10]. While the disagreement is enormous, one should point out that the quoted values
of Q30 and Q40 were obtained in Ref. [10] as parameters in a fit to measured quantities and
the authors of Ref. [10] could obtain satisfactory fits also setting Q40 to zero. With this
choice, the value of Q30 changed to 13 a.u., increasing the discrepancy with our calculations.
VI. DIPOLE-DIPOLE POLARIZABILITY OF NITROUS OXIDE
Table S–I compares also the ab initio and experimental average static dipole-dipole polarizability, ᾱ, and the anisotropy of this polarizability, ∆α. The experimental values were
computed by us from the data published by Hohm [12]. There is a very good agreement, to
within about 2%, for the average polarizability of N2 O between calculations in our basis set
at the MBPT2 level without full response and the experiment. The discrepancy is larger,
8
about 12%, for the anisotropy of polarizability. This discrepancy is reduced to 5% if the
response is fully taken into account at the MBPT2 level. A larger aTZ basis and CCSD(T)
level of theory bring the ab initio values very close to the experimental ones: the discrepancies are only 0.4% for ᾱ and 2% for ∆α. Both quantities lie within the experimental error
bars.
VII. QUADRUPOLE MOMENT OF CARBON DIOXIDE
In contrast to N2 O, no significant discrepancies with experiment are observed for the
quadrupole moment of CO2 . Already with our basis set at the MBPT2 level an agreement
to within 3% has been reached and the computed value is within experimental error bars
[15]. A further increase in the theory level and in the size of the basis set leads to a nearly
prefect agreement with experiment as recently shown by Coriani et al. [14]. These authors
also computed the vibrational correction amounting to -0.016 a.u. and upon addition of this
correction reproduced experimental values [15] (see also Refs. [21,22]) to better than 0.1%.
Our results agree very well with the results of Ref. [14], as well as with the calculations of
Maroulis and Thakkar [13].
VIII. HEXADECAPOLE MOMENT OF CARBON DIOXIDE
For the hexadecapole moment of CO2 we found only one published theoretical result
[13], except for an earlier result from our group [23] (in the same basis as the present one
but with a slightly different geometry), and no experimental data. This quantity shows
exceptionally strong sensitivity to the basis set size, particularly at the SCF level of theory
where differences between some basis sets are a factor of 1.6 large. The agreement with the
calculations of Maroulis and Thakkar [13], the only published result, is fair. Only a small
part of the discrepancy is due to the slightly different geometries used in the two calculations:
we have recomputed Q40 with Maroulis and Thakkar’s basis [13] and our geometry obtaining
values -1.13, -1.36, and -0.95 a.u. at the SCF, MBPT2, and MBPT3 levels, respectively.
9
This large basis set sensitivity is peculiar to Q40 as the agreement of the two calculations
for Q20 is very good. The convergence of the hexadecapole moment with respect to the
size of the basis set is not only slow but also highly non-monotonic, as Table S–I shows.
The hexadecapole moment computed using the aDZ and the aQZ bases are quite different
from that obtained using the aTZ basis. Moreover, Maroulis [24] has recently obtained a
near basis set limit value of Q40 at the SCF and MBPT2 levels equal to -1.76 and -2.61
a.u., respectively. These values indicate that even the aQZ basis is far from saturation and,
actually, it is the present basis that is closest to these limits.
The hexadecapole moment of CO2 is fairly small (smaller in magnitude than the
quadrupole moment). Thus, the multipolar expansion of CO2 interactions with other species
will be will be very well described by approximating the electric potential of CO2 by the
quadrupole moment only. The smallness of the hexadecapole moment of CO2 may be of
relevance for computer simulations and interpretation of experiments measuring far-infrared
collision induced absorption spectra of gaseous CO2 . Recent computer simulations [25] assumed a value of -10.8 a.u. for this moment based on some old ab initio calculations, about
four times larger than our best estimate (-2.46 a.u.).
IX. DIPOLE-DIPOLE POLARIZABILITY OF CARBON DIOXIDE
The average static dipole-dipole polarizability of CO2 computed in our basis set at the
MBPT2 level of theory and using the finite-field approach agrees extremely well, to within
0.5%, with experiments [12,16,17]. This is clearly to some extent fortuitous since at the
CCSD(T) level the discrepancy is 2%. The latter agreement is still rather satisfactory.
The anisotropy of polarizability differs from experiment more significantly, by 10-11% at
the MBPT2 level and by 6-7% at the CCSD(T) level. This quantity may be somewhat
more difficult to measure accurately as discussed recently by Chrissanthopoulos et al. [18].
These authors recommend the value of 13.8 ± 1.6 a.u. for the anisotropy of (electronic)
static polarizability of CO2 as the best representation of experimental data. Our results are
10
within the experimental error bars at all correlated levels of theory.
The MBPT2 polarizability computed with a partial inclusion of the response effects is
17.97 and 16.52 a.u. [23] for ᾱ and ∆α, respectively. These values are almost 3 and 9%
larger than the respective finite-field MBPT2 results and further from experiments. This
untypically large response effect results mostly from the parallel component which is about
5% larger in the former than in the latter approach. The corresponding perpendicular component is only 0.9% larger, which leads to some enhancement of the effect for the anisotropy.
The effects of the complete inclusion of response at the MBPT2 level are similar for N2 O
and CO2 .
11
REFERENCES
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[2] P.E.S. Wormer and H. Hettema, POLCOR package, University of Nijmegen, The
Netherlands, 1992.
[3] T.H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989); R.A. Kendall, T.H. Dunning, Jr.,
and R.J. Harrison, ibid. 96, 6796 (1992); D.E. Woon and T.H. Dunning, Jr., ibid. 98,
1358 (1993); D.E. Woon and T.H. Dunning, Jr., ibid. 100, 2975 (1994).
[4] A. J. Sadlej, Coll. Czech. Chem. Comm. 53, 1995 (1988).
[5] M.J. Frisch and J.E. Del Bene, Int. J. Quantum Chem. Symp. 23, 363 (1989).
[6] J.M.L.J. Reinartz, W.L. Meerts, and A. Dymanus, Chem. Phys. 31, 19 (1978).
[7] H. Jalink, D.H. Parker, and S. Stolte, J. Mol. Spectr. 121, 236 (1987).
[8] A.D. Buckingham, C. Graham, and J.H. Williams, Mol. Phys. 49, 703 (1983).
[9] W.H. Flygare, R.L. Shoemaker, and W. Hüttner, J. Chem. Phys. 50, 2414 (1969).
[10] I.R. Dagg, A. Anderson, T.F. Gmach, C.G. Joslin, and W. Smith, Can. J. Phys. 68,
1440 (1990).
[11] C. Boulet, N. Lacombe, P. Isnard, Can. J. Phys. 51, 605 (1973).
[12] U. Hohm, Chem. Phys. 179, 553 (1994).
[13] G. Maroulis and A.J. Thakkar, J. Chem. Phys. 93, 4164 (1990).
[14] S. Coriani, A. Halkier, A. Rizzo, and K. Ruud, Chem. Phys. Lett. 326, 269 (2000).
[15] C. Graham, D.A. Imrie, and R.E. Raab, Mol. Phys. 93, 49 (1998).
[16] W.Q. Cai, T.E. Gough, X.J. Gu, N.R. Isenor, and G. Scoles, Phys. Rev. A 36, 4722
(1987).
12
[17] M. P. Bogaard, A. D. Buckingham, R. K. Pierens, and A. H. White, J. Chem. Soc.
Faraday Trans. I 74, 3008 (1974).
[18] A. Chrissanthopoulos, U. Hohm, and U. Wachsmuth, J. Mol. Struct. 526, 323 (2000).
[19] A.J. Stone The Theory of Intermolecular Forces, Clarendon, Oxford, 1996.
[20] J.S. Muenter, J. Mol. Spectr. 55, 490 (1975).
[21] J.N. Watson, I.E. Craven, and G.L.D. Ritchie, Chem. Phys. Lett. 274, 1 (1997).
[22] M.R. Battaglia, A.D. Buckingham, D. Neumark, R.K. Pierens, and J.H. Williams, Mol.
Phys. 43, 1015 (1981).
[23] R. Bukowski, J. Sadlej, B. Jeziorski, P. Jankowski, K. Szalewicz, S.A. Kucharski, H.L.
Williams, and B.M. Rice, J. Chem. Phys. 110, 3785 (1999).
[24] G. Maroulis, private communication (2002).
[25] M. Gruszka and A. Borysow, Mol. Phys. 93, 1007 (1998).
13
TABLES
TABLE S–I. Multipole moments and polarizabilities of N2 O and CO2 . All quantities are
computed with response effects (except as noted) and are given in atomic units. Bases denoted by
aXZ are aug-cc-pVXZ bases of Dunning et al. [3]. The Sadlej’s basis is from Ref. [4].
basis (N )
SCF
MBPT2
MBPT3
MBPT4
CCSD(T)
N2 O: Q10
present (93)
0.259
-0.0114
0.156
-0.0491
0.0596
Sadlej (72)
0.257
-0.0149
0.151
-0.0478
0.0559
aDZ (69)
0.257
-0.0125
0.153
-0.0461
0.0567
aTZ (138)
0.255
-0.0123
0.154
-0.0590
0.0537
aQZ (240)
0.254
-0.0099
0.158
-0.0585
0.0569
literaturea
0.242
-0.0230
0.142
-0.0673
0.0341b
0.0633314 ±0.000009c
experiment
N2 O: Q20
present
-2.812
-2.773
-2.789
-2.685
-2.746
Sadlej
-2.744
-2.734
-2.741
-2.642
-2.699
aDZ
-2.774
-2.785
-2.794
-2.689
-2.748
aTZ
-2.811
-2.735
-2.750
-2.647
-2.703
aQZ
-2.808
-2.720
-2.737
-2.633
-2.689
-2.451±0.011d
experiment
-2.50 ±0.14e
-2.71 ±0.19f
N2 O: Q30
present
-1.881
-3.750
-2.851
-3.493
-3.229
Sadlej
-2.191
-3.964
-3.112
-3.640
-3.431
aDZ
-2.177
-3.817
-2.996
-3.552
-3.326
aTZ
-1.904
-3.623
-2.726
-3.422
-3.114
aQZ
-1.970
-3.681
-2.769
-3.480
-3.161
-10.7g
experiment
-9.3g
-8.4h
N2 O: Q40
present
-0.828
-3.134
-1.785
-3.859
-2.945
Sadlej
-0.730
-4.025
-2.050
-4.570
-3.435
aDZ
0.250
-2.735
-1.220
-3.355
-2.390
aTZ
-1.205
-3.970
-2.460
-4.760
-3.735
aQZ
-1.266
-4.157
-2.660
-4.950
-3.904
-44.9g
experiment
-39.8g
14
N2 O: ᾱ
present
18.45i
20.13j
present
18.44i
19.90
19.15
20.26
19.77
aDZ
18.20
19.48
18.80
19.85
19.40
aTZ
18.58
19.81
19.04
20.12
19.62
19.70±0.10k
experiment
N2 O: ∆α
present
18.29i
21.45j
present
18.31i
20.04
18.75
20.72
19.59
aDZ
18.77
20.69
19.37
21.38
20.26
aTZ
18.36
20.00
18.66
20.66
19.49
19.1±0.6k
experiment
CO2 : Q20
present
-3.837
-3.086
-3.332
literaturel
-3.813
-3.053
3.292
literaturem
-3.795
-3.061
-3.114
-3.196
-3.169
-3.18±0.14n
experiment
CO2 : Q40
present
-1.550
-2.615
-2.000
-3.255
-2.790
aDZ
-1.225
-2.785
-2.110
-3.325
-2.835
aTZ
-2.005
-3.040
-2.510
-3.535
-3.155
aQZ
-1.195
-2.350
-1.805
-2.855
-2.460
literaturel
-1.23
-1.48
-1.07
-1.74o
CO2 : ᾱ
present
15.45
17.43
16.68
literaturel
15.85
17.97
17.20
17.40
17.17
17.63o
experiment
17.50±0.09k
experiment
17.51p
CO2 : ∆α
a
present
12.28
15.20
13.96
literaturel
11.77
14.57
13.42
15.13
14.65
14.27o
experiment
13.7±0.4k
experiment
13.8p
experiment
13.8±1.6q
Calculations of Frisch and Del Bene [5] using an spdf basis set [6-311+G(3df ,3pd)] and an
experimental equilibrium geometry (RNN = 1.1282 Å and RNO = 1.1842 Å).
b
Computed in Ref. [5] using the QCISD(T) method.
c
Reinartz et al. [6] and Jalink et al. [7].
15
d
Reinartz et al. [6].
e
Buckingham et al. [8]. This value is not defined with respect to the center of mass but
rather with respect to an effective quadrupole center [8].
f
Flygare et al. [9].
g
Dagg et al. [10]. The experiments measure only the magnitude of Q30 and Q40 . The two
values correspond to two different temperatures.
h
Boulet et al. [11].
i
The upper and lower values have been obtained through analytic and numerical (finite-
field) differentiation, respectively, of the field-dependent SCF energy.
j
A hybrid approach result that does not include all response effects, see text.
k
Hohm [12].
l
Maroulis and Thakkar [13] using a 120-term spdf basis at RCO =1.160 Å.
m
Coriani et al. [14] in d-aug-cc-pV5Z basis set at experimental equilibrium geometry of
RCO =1.15979 Å.
n
Graham et al. [15].
o
MBPT4 method without triples contribution.
p
Cai et al. [16] based on data from Bogaard et al. [17].
q
Chrissanthopoulos et al. [18].
16
TABLE S–II. Comparison of the observed and computed infrared transitions (in cm−1 ) in the
He-N2 O complex. The experimental data are from J. Tang and A. R. W. McKellar, J. Chem.
Phys. 117, 2586 (2002).
0
00
JK
0 K 0 ← JK 0 K 0
a
c
a
Theory
Experiment
∆
c
Expt - Theo
101 ← 000
2224.6404
2224.6252
-0.0152
221 ← 330
2221.2174
2221.2437
0.0263
505 ← 606
2221.2980
2221.3042
0.0062
505 ← 616
2221.2974
2221.3042
0.0068
515 ← 606
2221.2988
2221.3042
0.0054
515 ← 616
2221.2982
2221.3042
0.0060
220 ← 331
2221.5832
2221.5415
-0.0417
404 ← 505
2221.7107
2221.7183
0.0076
414 ← 515
2221.7105
2221.7183
0.0078
101 ← 220
2222.0323
2222.0264
-0.0059
303 ← 404
2222.1258
2222.1335
0.0077
313 ← 414
2222.1235
2222.1335
0.0100
313 ← 404
2222.1242
2222.1335
0.0093
303 ← 414
2222.1251
2222.1335
0.0084
220 ← 321
2221.7032
2221.7684
0.0652
221 ← 322
2222.1320
2222.1425
0.0105
221 ← 312
2222.1035
2222.1876
0.0841
414 ← 423
2222.4196
2222.4044
-0.0152
404 ← 423
2222.4190
2222.4044
-0.0146
404 ← 413
2222.4095
2222.4210
0.0115
414 ← 413
2222.4101
2222.4210
0.0109
110 ← 221
2222.4681
2222.4475
-0.0206
202 ← 303
2222.5417
2222.5407
-0.0010
202 ← 313
2222.5435
2222.5407
-0.0028
212 ← 303
2222.5352
2222.5543
0.0191
212 ← 313
2222.5370
2222.5543
0.0173
110 ← 211
2222.5280
2222.5630
0.0350
101 ← 212
2222.9479
2222.9371
-0.0108
101 ← 202
2222.9415
2222.9472
0.0057
111 ← 212
2222.9671
2222.9773
0.0102
111 ← 202
2222.9606
2222.9873
0.0267
313 ← 322
2222.8793
2222.8455
-0.0338
313 ← 312
2222.8508
2222.8879
0.0371
303 ← 312
2222.8525
2222.8879
0.0354
202 ← 221
2223.2912
2223.2498
-0.0414
212 ← 221
2223.2848
2223.2611
-0.0237
000 ← 111
2223.3570
2223.3514
-0.0056
000 ← 101
2223.3774
2223.3906
0.0132
202 ← 211
2223.3511
2223.3656
0.0145
212 ← 221
2223.2848
2223.3760
0.0912
211 ← 220
2223.7450
2223.7065
-0.0385
101 ← 110
2223.7678
2223.7558
-0.0120
111 ← 110
2223.7870
2223.7959
0.0089
331 ← 330
2223.8165
2223.8520
0.0355
330 ← 331
2224.1548
2224.1069
-0.0479
221 ← 220
2223.8005
2223.8237
0.0232
220 ← 221
2224.1961
2224.1669
-0.0292
321 ← 322
2224.6216
2224.5244
-0.0972
321 ← 312
2224.5931
2224.5680
-0.0251
110 ← 111
2224.2259
2224.2148
-0.0111
211 ← 212
2224.6606
2224.6181
-0.0425
110 ← 101
2224.2464
2224.2541
0.0077
220 ← 211
2224.2561
2224.2821
0.0260
202 ← 111
2225.0491
2225.0174
-0.0317
212 ← 111
2225.0426
2225.0279
-0.0147
202 ← 101
2225.0696
2225.0566
-0.0130
111 ← 000
2224.6596
2224.6653
0.0057
221 ← 212
2224.7161
2224.7349
0.0188
221 ← 202
2224.7097
2224.7455
0.0358
312 ← 303
2225.1376
2225.0826
-0.0550
312 ← 313
2225.1394
2225.0826
-0.0568
322 ← 313
2225.1106
2225.1263
0.0157
212 ← 101
2225.0631
2225.0672
0.0041
211 ← 110
2225.4805
2225.4361
-0.0444
313 ← 212
2225.4634
2225.4361
-0.0273
303 ← 212
2225.4651
2225.4361
-0.0290
303 ← 202
2225.4586
2225.4478
-0.0108
313 ← 202
2225.4570
2225.4478
-0.0092
221 ← 110
2225.5360
2225.5534
0.0174
312 ← 221
2225.8872
2225.7902
-0.0970
312 ← 211
2225.9471
2225.9039
-0.0432
220 ← 111
2225.9540
2225.9339
-0.0201
414 ← 313
2225.8637
2225.8337
-0.0300
404 ← 313
2225.8630
2225.8337
-0.0293
414 ← 303
2225.8619
2225.8337
-0.0282
404 ← 303
2225.8612
2225.8337
-0.0275
322 ← 211
2225.9182
2225.9527
0.0345
220 ← 101
2225.9745
2225.9731
-0.0014
331 ← 220
2226.3996
2226.4321
0.0325
606 ← 505
2226.6535
2226.6016
-0.0519
606 ← 515
2226.6527
2226.6016
-0.0511
616 ← 505
2226.6541
2226.6016
-0.0525
616 ← 515
2226.6533
2226.6016
-0.0517
330 ← 221
2226.7678
2226.7306
-0.0372