Chemical Physics 433 (2014) 60–66
Contents lists available at ScienceDirect
Chemical Physics
journal homepage: www.elsevier.com/locate/chemphys
A theoretical study of temperature dependence of cluster formation
from sulfuric acid and ammonia
Nara Lee Chon a, Shan-Hu Lee b,⇑, Hai Lin a,⇑
a
b
Chemistry Department, University of Colorado – Denver, Denver, CO 80217, United States
College of Public Health, Kent State University, Kent, OH 44242, United States
a r t i c l e
i n f o
Article history:
Received 28 September 2013
In final form 20 January 2014
Available online 11 February 2014
Keywords:
Sulfuric acid
Ammonia
Cluster
Nucleation
Temperature dependence
a b s t r a c t
We have performed density functional theory (BL3YP) and ab initio (MP2) calculations to investigate the
energetics of the cluster formation for (NH3)m(H2SO4) and (NH3)(H2SO4)n (m, n = 1–6) in the atmospherically-relevant temperature range between 200 and 300 K. For (NH3)m(H2SO4) clusters, the binding
increases from m = 1 to 6 at 200 K, while the most stable complex at 300 K is the cluster with m = 2.
For (NH3)(H2SO4)n clusters, the binding is more stable for those with larger n. There is a strong temperature dependence for the (NH3)m(H2SO4) cluster formation; the lowest free energy shifts from m = 6 at
T = 200 K to m = 5 around T = 240 K and further to m = 2 at T P 280 K. The temperature effects on (NH3)
(H2SO4)n clusters are much less stronger, while there is still a similar trend which favors larger n in the
entire temperature range from 200 to 300 K.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
Atmospheric nucleation (formation of solid or liquid aerosols directly from gas phase species) is an important source of secondary
aerosols [1]. Nucleation contributes significantly to the global production of cloud condensation nuclei (CCN) [2]. Although a large of
number of observational studies have been made in the nucleation
field [3], aerosol nucleation mechanisms are not well understood,
especially at the molecular cluster level. Direct measurements of
chemical composition of newly nucleated clusters are very limited
[4] and the thermodynamics properties of formation of molecular
clusters have rarely been measured [5]. From this perspective, theoretical computational studies of nucleation clusters can provide
indirect yet valuable information on molecular structures and energetics of clusters formed at the very early stage of aerosol nucleation.
Atmospheric observations and laboratory studies have consistently pointed out that sulfuric acid (H2SO4) is the key aerosol
nucleation precursor [3]. But in the boundary layer conditions,
H2SO4 concentrations (105 107 cm3) are not sufficient to explain
the measured nucleation rates, indicating that other ternary species are also involved in nucleation [6]. Ammonia (NH3) is one of
key ternary species. NH3 is abundant in the atmosphere and is
the main base compound in the atmosphere and thus has a unique
capability to stabilize H2SO4 clusters via acid–base reactions [7].
⇑ Corresponding authors. Tel.: +1 330 672 3905; fax: +1 330 672 6505 (S. Lee).
E-mail addresses: [email protected] (S.H. Lee), [email protected] (H. Lin).
http://dx.doi.org/10.1016/j.chemphys.2014.01.010
0301-0104/Ó 2014 Elsevier B.V. All rights reserved.
However there are still controversies on the role of NH3 on
H2SO4 nucleation. Global modeling studies have shown that ternary homogeneous nucleation of H2SO4, NH3, and H2O is the dominant nucleation mechanism in the entire troposphere and lower
stratosphere [8], and this nucleation mechanism has been also
used to explain atmospheric observations of new particle formation in the Eastern United States [9]. However, laboratory studies
also showed that the enhancement of nucleation rates by NH3 is
moderate (only at the 10–1000 range), although tens of pptv to
tens ppbv level of NH3 was used in these studies [7a–e]. Another
important role of NH3 in nucleation is that it can form H2SO4–
NH3 clusters initially under higher concentrations of NH3 in the
atmosphere and then ammonium components are replaced by
lower concentrations of amines in the aerosol phases [10].
Molecular modeling studies with density functional theories
(DFT) and ab initio calculations have been performed for formation
of H2SO4 and NH3 clusters [11], but conclusions of these studies
are contradictory. Using DFT (B3LYP) calculations, Ianni and Bandy
[11h] studied structures, energetics, dipoles and rotational constants of NH3H2SO4(H2O)n (n = 0–5) and NH3(H2SO4)2H2O clusters, and concluded that the NH3(H2SO4)2H2O cluster does not
have enough free energy to initialize the atmospheric nucleation
and hence NH3 does not play roles in atmospheric aerosol nucleation. On the other hand, Kurten et al. [11d,11e] conducted ab initio
RI-CC2 calculations to derive total energies, free energies, binding
energies, and atmospheric cluster size distributions of NH3(H2SO4)m(H2O)n (m = 1, 2; n = 0–7) clusters, and showed that NH3 can
61
N.L. Chon et al. / Chemical Physics 433 (2014) 60–66
strengthen the formation of clusters containing 2 H2SO4 and 0–2
H2O molecules and so that NH3 favors thermodynamics of H2SO4
and NH3 binding via formation of ammonium bisulfate. Previous
studies also showed that the enhancing effect of NH3 is more pronounced for the formation of neutral clusters containing 2 H2SO4
molecules than larger clusters [11a,11d,11e,11i].
Recent RI-CC2 calculations by Kurten et al. [11c,12] further
showed that organic amine compounds can reduce Gibbs free
energy for the formation of H2SO4 clusters even more effectively
than NH3, so that the trace level of amines may play key roles in
aerosol nucleation [11c,12]. Another DFT (PW91PW91) study by
Nadykto et al. [11b] indicated that the thermochemical stability of
the H2SO4–amines–H2O complexes is indeed higher than that of
the H2SO4–NH3–H2O complexes, in a qualitative agreement with
Kurten et al. [11f]; but the enhancement in the stability due to
amines is not large enough (2–3.5 kcal/mol) to overcome the difference in typical atmospheric concentrations of NH3 and amines (two
to three orders of magnitude lower than NH3 [13]). It is possible that
the uncertainties associated with calculations at different levels of
the theory have caused different energetics in these studies
[11b,11c,12]. Calculations also showed that organic acids may be
important for the H2SO4 cluster formation, with or without NH3,
via hydrogen bonding between H2SO4 and carboxylic acids [11a,14].
For the case of the attachment of NH3 molecules to H2SO4 clusters, the presence of H2O may have small effects on the Gibbs free
energy of the H2SO4 and NH3 cluster formation [11a], although an
addition of H2O tends to slightly decrease the binding of NH3 to
H2SO4 clusters [11d,11e]. For example, the Gibbs formation free
energy of an unhydrated cluster containing one H2SO4 and one
NH3 derived from previous DFT calculations ranged from 4.5 to
8.7 kcal/mol, whereas the formation free energy of the hydrated
cluster ranged 3.6 to 7.6 kcal/mol [11a–h]. In contrast to these
cited studies, recent studies indicated that hydration can increase
the stability of H2SO4–amine clusters [3b,15].
Temperature dependence of thermodynamical properties is an
important parameter needed for understanding how a specific
nucleation mechanism plays roles at different altitudes (and hence
different temperatures) in the atmosphere. Quantum calculations
can provide such data, especially when experiments at lower temperature ranges are even more difficult to perform, because of the
increased wall losses of aerosol precursors (H2SO4 and NH3) at low
temperatures. But computational studies are still very limited on
the temperature dependence of Gibbs free energies for the cluster
formation involving H2SO4 and NH3 [11h,11i].
In the present study, we show MP2 and B3LYP calculations of
unhydrated (NH3)m(H2SO4) and (NH3)(H2SO4)n (m, n = 1–6) complexes at temperatures of 200, 220, 240, 260, 280, and 300 K, thus
within the entire temperature range from the ground level to the
lower stratosphere. We have optimized geometries of clusters
and computed binding energies, enthalpies, entropies, and Gibbs
free energies of the formation of these model clusters.
2. Computational methods
Geometries for (NH3)m(H2SO4) and (NH3)(H2SO4)n model complexes with m, n = 1–6 were optimized at the B3LYP [16] level with
the 6-31+G(d,p) [17] basis set using ORCA [18]. A larger number of
different initial geometries have been systematically constructed
in a stepwise manner, exploring the configurational space in an efficient way. For the (NH3)m(H2SO4) complex, a NH3 molecule was
manually added to the optimized (NH3)m1(H2SO4) complex of the
lowest energy at different positions with various orientations. The
same strategy was applied to the construction of (NH3)(H2SO4)n,
where H2SO4 was added one by one. While we have tried our best
to use the aufbau principle to grow clusters, one should be aware
of the limits of this method in exploring the configuration space.
In this perspective, a sampling scheme used by Xu and Zhang [15]
can be an alternative choice. Tight SCF convergence and a large grid
for integral evaluation were utilized for DFT calculations (see the detailed computational setup in Supporting Information). To refine the
interaction energies, single-point calculations were carried out at
the optimized geometries employing the RI-MP2 [19] (resolution
of the identity MP2) method with Dunning’s augmented
correlation-consistent basis sets (aug-cc-pVXZ, X = D, T, Q) [20]
implemented ORCA. The MP2 single-point energies were then
extrapolated to the complete basis set (CBS) limit using the following equation [21]:
EMP2
¼ EMP2
X
CBS þ
b
ð1Þ
X3
where EMP2
is the MP2/aug-cc-pVXZ (X = D, T, Q) energy, EMP2
X
CBS is the
extrapolated MP2/CBS energy, and b is a fitting parameter. Due to
the technical constraint, we were unable to converge MP2/aug-ccpVQZ calculations for (NH3)(H2SO4)6 clusters. Therefore, MP2/CBS
energies were not available for (NH3)(H2SO4)6. We note that, while
B3LYP- and MP2-geometry optimizations may lead to different
minima for hydrogen-bonded complexes, the optimized geometries
usually agree with each other [11g]. Moreover, variations in the
interaction energies due to small differences in the geometry are
likely marginal (1–2 kcal/mol) [22]. Therefore, we consider the
above MP2/CBS//B3LYP/6-31+G(d,p) strategy as a reasonable compromise between the accuracy and efficiency.
The binding energy (DU) associated with the complex formation, compared to free molecules, was computed by:
DU½ðNH3 Þm ðH2 SO4 Þ ¼ E½ðNH3 Þm ðH2 SO4 ÞÞ ½m EðNH3 Þ
þ EðH2 SO4 Þ
ð2Þ
DU½ðNH3 ÞðH2 SO4 Þn ¼ E½ðNH3 ÞðH2 SO4 Þn Þ ½EðNH3 Þ þ n
EðH2 SO4 Þ
ð3Þ
where E is the electronic energy. Normal-mode analysis was performed at the optimized geometries at the B3LYP/6-31+G(d,p) level.
Enthalpy changes DH and entropy changes DS were estimated in
the similar way. All thermodynamics quantities were estimated
assuming harmonic vibrations for temperatures T at 200, 220,
240, 260, 280, and 300 K. For example, the entropic contribution
Sv by a normal mode v was calculated based on [23]:
Sm ¼ Nk
~
bhcm
lnð1 ebhcm~ Þ
ebhcm~ 1
ð4Þ
1
where b = kT
, N is the Avogadro constant, k is the Boltzmann constant, T is temperature, h is the Planck constant, c is the speed of
~ is the vibrational wavenumber.
light, and m
The harmonic approximation likely overestimathe partition
functions for normal modes of very low frequencies, which are very
anharmonic. To reduce errors in the computation of entropic contributions for those normal modes, we have adopted an ansatz, where
we replaced the actual vibrational wavenumber by a cut-off value
~cutoff = 35 cm–1) when computing entropic contributions for a nor(m
mal mode, if the actual wavenumber was lower than the cut-off
value. Such a treatment has been successfully employed in our
previous study of kinetic isotope effects for methane reacting with
hydroxyl radical in the atmosphere [24]. Estimations of the entropy
of free energy without this treatment are denoted as Est1 in the present study, and those with this treatment as Est2. Both Est1 and Est2
free energies can be further refined by replacing B3LYP electronic
energies by more accurate MP2/CBS counterparts, and the corresponding results are indicated as Est1(MP2) and Est2(MP2),
respectively.
62
N.L. Chon et al. / Chemical Physics 433 (2014) 60–66
(a) m=1, n=1
(b) m=2, n=1
(c) m=3, n=1
(d) m=4, n=1
(e) m=5, n=1
(f) m=6, n=1
(g) m=1, n=2
(h) m=1, n=3
(i) m=1, n=4
(j) m=1, n=5
(k) m=1, n=6
Fig. 1. Optimized (NH3)m(H2SO4) and (NH3)(H2SO4)n clusters. Distance in Å. Color code: red for oxygen, yellow for sulfur, blue for nitrogen, and white for hydrogen. (For
interpretation of the references to colors in this figure legend, the reader is referred to the web version of this paper.)
Although our calculations were solely based on the global
minima identified by geometry optimizations, we expect that
the associated errors in Gibbs Free Energies are insignificant
when considering only the global minima compared to the case
obtaining the Boltzmann average over all minima for a given clus-
ter. For example, Xu and Zhang [15] previously showed that the
difference in free energy is only 0.27 kcal/mol between the single-global-minimum estimation and the Boltzmann average over
all found minima for the cluster formed by 1 H2SO4 and 2 H2O
molecules.
63
N.L. Chon et al. / Chemical Physics 433 (2014) 60–66
3. Results and discussion
Fig. 1 shows the optimized cluster model structures. While
(NH3)(H2SO4) and (NH3)2(H2SO4) clusters are probably better classified as hydrogen-bonded complexes, the solvation by more NH3
molecules promotes the deprotonation of H2SO4. The (NH3)3(H2SO4) cluster is the intermediate state where a proton is shared by
NH3 and H2SO4. The (NH3)m(H2SO4) clusters with m > 3 showed
that ammonium has formed. For (NH3)(H2SO4)n, such proton transfer is completed as early as for n = 2. Interestingly, the ammonium
ion was found staying on the surface of (NH3)(H2SO4)n clusters
(above H2SO4 molecules) instead of being buried inside. Such
ammonium-on-surface solvation is probably due to the large number of hydrogen bonds formed between H2SO4 molecules, which
lowers the energy.
Binding energies, enthalpies, entropies, and Gibbs free energies
are summarized in Tables 1 and 2. For a brevity, we only show
the data for T = 200 K and 300 K in the main text, and the complete
set of data for other temperatures T (200, 220, 240, 260, 280, and
300 K) are given in Supporting Information. First, it is apparent that
the MP2 binding electronic energy is higher in magnitude than the
B3LYP counterpart for a given complex. As the size of the complex
increases, the difference between the B3LYP and MP2 electronic
energies becomes more prominent, changing from 0.2 kcal/mol
for (NH3)(H2SO4) to 2.7 kcal/mol for (NH3)6(H2SO4) and 10 kcal/
mol for (NH3)(H2SO4)5. We suspect that the leading factor that contributes to such a difference in the electronic energy is an underestimation of dispersion energies by the B3LYP density functional
model [25]. Because dispersion interactions are increasingly important when more atoms are involved, it is crucial to accurately take
into account the dispersion interactions as the size of complexes
grows. The larger dispersion interactions in (NH3)(H2SO4)n than in
(NH3)m(H2SO4) clusters are also partly due to the larger percentage
of heavy atoms in (NH3)(H2SO4)n complexes. The basis set superposition errors (BSSE) have not been not corrected in B3LYP calculations, which is another source of the inaccuracy. Because the BSSE
will lead to an overestimation of binding energies (in B3LYP calculations), dispersion interaction contributions are probably even
more significant than what have been revealed by the differences
in B3LYP and MP2 electronic energies discussed above.
Table 1
Thermodynamic properties (in kcal/mol) of (NH3)m(H2SO4), m = 1–6, at temperatures of 200 and 300 K at 1 atm. aData for other temperatures (220, 240, 260, and 280 K) are
included in Supporting Information.
m
DU
DU(MP2)b
1
16.75
16.93
2
31.25
31.92
3
41.17
41.69
4
50.03
51.27
5
62.83
64.73
6
71.02
73.71
T (K)
200
300
200
300
200
300
200
300
200
300
200
300
DH
15.45
15.43
28.28
28.22
36.58
36.55
43.34
43.19
54.41
54.11
61.19
60.72
TDS
DG
Est1
Est2
Est1
Est2
Est1(MP2)c
Est2(MP2)c
5.94
8.89
11.95
17.86
17.95
26.89
25.02
37.36
30.60
45.56
36.91
54.82
5.94
8.89
12.02
17.97
18.81
28.18
25.44
37.99
31.61
47.07
37.62
55.89
9.51
6.54
16.33
10.36
18.63
9.65
18.32
5.83
23.81
8.55
24.28
5.90
9.51
6.54
16.26
10.26
17.77
8.36
17.90
5.20
22.80
7.03
23.57
4.83
9.69
6.72
17.00
11.03
19.15
10.18
19.56
7.07
25.71
10.45
26.97
8.59
9.69
6.72
16.93
10.92
18.29
8.89
19.14
6.44
24.70
8.94
26.26
7.52
a
Computed at the B3LYP/6-31+G(d,p) level, unless otherwise indicated. Harmonic vibrations assumed for calculations of thermodynamic properties. Est1 and Est2 denote
two treatments in the estimation of entropic contributions by a given vibrational mode: the actual vibrational frequency was used in Est1, while in Est2 a cutoff value of
35 cm–1 replaced the actual vibrational frequency if it was lower than 35 cm–1.
b
Extrapolated to the complete basis set (CBS) limit, based on MP2/aug-cc-pVXZ (X = D, T, Q) energies.
c
Corrected by DU(MP2).
Table 2
Thermodynamic properties (in kcal/mol) of (NH3)(H2SO4)n, n = 1–6, at temperatures of 200 and 300 K at 1 atm.
included in Supporting Information.
n
DU
DU(MP2)b
1
16.75
16.93
2
39.87
40.91
3
61.56
66.22
4
77.64
84.16
5
93.42
104.15
6
105.85
N/Ad
T (K)
200
300
200
300
200
300
200
300
200
300
200
300
DH
15.45
15.43
36.13
36.71
57.20
56.98
71.49
71.04
86.34
85.95
97.66
97.13
TDS
a
Data for other temperatures (220, 240, 260, and 280 K) are
DG
Est1
Est2
Est1
Est2
Es1(MP2)c
Est(MP2)c
5.94
8.89
13.66
20.48
21.49
31.97
28.03
41.50
37.72
56.11
44.16
65.61
5.94
8.89
13.88
20.81
21.63
32.17
28.93
42.86
38.36
57.08
46.11
68.55
9.51
6.54
22.47
16.23
35.71
25.01
43.46
29.54
48.62
29.84
53.50
31.52
9.51
6.54
22.25
15.90
35.57
24.80
42.56
28.18
47.97
28.86
51.54
28.58
9.69
6.72
23.51
17.27
40.37
29.67
49.98
36.06
59.35
40.57
N/Ad
N/Ad
9.69
6.72
23.29
16.94
40.23
29.47
49.08
34.70
58.71
39.60
N/Ad
N/Ad
a
Computed at the B3LYP/6-31+G(d,p) level unless otherwise indicated. Harmonic vibrations assumed for calculations of thermodynamic properties. Est1 and Est2 denote
two treatments in the estimation of the entropic contributions by a given vibrational mode: the actual vibrational frequency was used in Est1, while in Est2 a cutoff value of
35 cm–1 replaced the actual vibrational frequency if it was lower than 35 cm–1.
b
Extrapolated to the complete basis set (CBS) limit, based on MP2/aug-cc-pVXZ (X = D, T, Q) energies.
c
Corrected by DU(MP2).
d
Due to technical difficulty, the MP2/aug-cc-pVQZ calculations failed to converge.
N.L. Chon et al. / Chemical Physics 433 (2014) 60–66
0
ΔG [kcal/mol]
-5
(A) T = 200 K
Est1
Est1(MP2)
Est2
Est2(MP2)
-10
-10
(A) T = 200 K
Est1
Est1(MP2)
Est2
Est2(MP2)
-20
-30
-40
-50
-60
-70
0
1
2
3
n
4
5
(B) T = 300 K
6
Est1
Est1(MP2)
Est2
Est2(MP2)
-10
-20
-30
-40
-50
1
2
3
n
4
5
6
Fig. 3. Binding free energies DG for the (NH3)(H2SO4)n complex formation at
temperatures T at (A) 200 K and (B) 300 K at 1 atm. Est1 and Est2 were computed at
the B3LYP/6-31+G(d,p) level, and Est1(MP2) and Est2(MP2) were obtained by
applying MP2/CBS electronic-energies corrections to Est1 and Est2, respectively.
Entropic contributions were estimated by using the actual vibrational frequencies
in Est1, whereas a cutoff value of 35 cm–1 replaced the actual value of a low
(<35 cm1) frequency in Est2.
-15
-20
-25
-30
1
0
-2
ΔG [kcal/mol]
0
ΔG [kcal/mol]
Second, we compare that the entropy contributions by Est1 and
by Est2. For the smallest complex (NH3)(H2SO4), Est1 and Est2 are
identical because all vibrational frequencies are higher than the
cutoff 35 cm–1. Low-frequency modes emerged for larger
complexes, for which Est1 and Est2 differed from each other. For
(NH3)m(H2SO4), differences are rather small, usually less than
1 kcal/mol. For (NH3)(H2SO4)n clusters, discrepancies are larger,
especially at T = 300 K, e.g. 3 kcal/mol for (NH3)(H2SO4)6. An
accurate treatment for the vibrational anharmonicity is highly
desirable. Unfortunately, there is not a simple and generally applicable way to handle the vibrational anharmonicity. Highly accurate
treatments of anharmonicity have been reported [25b], but they
are too complicate and expensive and so not feasible for the present study. Sampling the configurational space is preferred but is
too costly computationally at the MP2 or even the B3LYP level
for our clusters system, so we relied on the assumption of harmonic vibrations. The ansatz adopted in this work is more approximate in terms of accuracy, but it still provides more realistic
entropy estimations (Est2) than those without any adjustments
(Est1). Because the largest vibrational entropic contribution comes
from those lowest frequencies, the difference between Est1 and
Est2 provides a quick but rough idea on how large the uncertainties in entropic contributions are related to the vibrational
anharmonicity.
Fig. 2 shows the binding free energies as a function of m in (NH3)m(H2SO4) complexes, and Fig. 3 as a function of n in the (NH3)(H2SO4)n complexes. The curves illustrate the interplay between
enthalpy that favors the complex formation and the entropy that
disfavors the complex formation, mainly due to the loss of transla-
ΔG [kcal/mol]
64
2
3
m
4
5
6
Est1
Est1(MP2)
Est2
Est2(MP2)
(B) T = 300 K
-4
-6
-8
-10
-12
1
2
3
m
4
5
6
Fig. 2. Binding free energies DG for the (NH3)m(H2SO4) cluster formation at
temperatures T at (A) 200 K and (B) 300 K. Est1 and Est2 were computed at the
B3LYP/6-31+G(d,p) level, and Est1(MP2)and Est2(MP2) were obtained by applying
MP2/CBS electronic-energies corrections to Est1 and Est2, respectively. Entropic
contributions were estimated by using the actual vibrational frequencies in Est1,
whereas a cutoff value of 35 cm–1 replaced the actual value of a low (<35 cm1)
frequency in Est2.
tional entropies. Entropic contributions have more prominent
effects on the formation of (NH3)m(H2SO4) than (NH3)(H2SO4)n
clusters. For (NH3)m(H2SO4), whereas the binding was predicted to
be increasingly stronger from m = 1 to 6 at T = 200 K, the most stable
complex at T = 300 K was found to be at m = 2. The vibrationalanharmonicity treatment led to larger entropic changes upon binding and thus further enhanced the trend. The dispersion corrections
by MP2 electronic energies added to enthalpic contributions and
therefore acted in the opposite direction. However, adding the dispersion corrections was still unable to overcome the entropic driving forces at high temperature. For (NH3)(H2SO4)n complexes,
enthalpic contributions dominated, and the binding is more stable
for the larger n. For both types of complexes, it seems that adding
corrections due to the dispersion interactions has more profound effects than treating low-frequency vibrational modes differently
(between Est1 and Est2).
The temperature dependence of the binding free energy is
shown in Fig. 4, where we displayed the most realistic free energies
DG(Est2, MP2) for both types of complexes for all computed temperatures. Approximately, the binding free energy reduces from
T = 200 to 300 K by 0.03m kcal/mol/K for (NH3)m(H2SO4) and by
0.04n kcal/mol/K for (NH3)(H2SO4)n. The temperature dependency
has a profound effect on (NH3)m(H2SO4) complexes, among which
the lowest free energy shifts from m = 6 at T = 200 K to m = 5
around T = 240 K and further to m = 2 at T P 280 K. The temperature effects on (NH3)(H2SO4)n complexes, on the other hand, are
much smaller, and show a qualitatively similar trend that favors
larger n for all temperatures within 200–300 K.
65
N.L. Chon et al. / Chemical Physics 433 (2014) 60–66
ΔG(Est2,MP2) [kcal/mol]
0
(A) (NH3)m(H2SO4)
-5
220 K
-10
260 K
240 K
T (K)
280 K
-15
300 K
-20
-25
-30
1
2
3
4
m
5
6
7
8
(B) (NH3)(H2SO4)n
-10
n=4
34.7
40.4
43.3
n=5
39.6
47.2
51.0
200 K
220 K
-20
240 K
-30
260 K
280 K
-40
300 K
-50
-60
-70
DG
This work (MP2)
n=1
n=2
n=3
300
6.7
16.9
29.5
260
7.9
19.6
33.8
240
8.5
21.0
35.9
Torpo et al. [11i] (RI-MP2/aug-cc-pV(T+d)Z)
n=1
n=2
n=3
298
7.3
22.3
29.9
265
8.2
24.7
33.8
242
8.9
26.4
36.4
4. Conclusions and atmospheric relevance
0
ΔG(Est2,MP2) [kcal/mol]
Table 4
Temperature dependences of Gibbs free energies for the formation of (NH3)(H2SO4)n
clusters, reported from this work and a previous study [11i].
200 K
1
2
3
4
n
5
6
7
Fig. 4. Binding free energies DG(Est2, MP2) for (A) (NH3)m(H2SO4) and (B) (NH3)(H2SO4)n complex formations at various temperature T from 200 to 300 K at 1 atm.
The free-energies were computed at the B3LYP/6-31+G(d,p) level with MP2/CBS
electronic-energies corrections. For entropic contributions estimations, a cutoff
value of 35 cm–1 replaced the actual value of a low (<35 cm–1) frequency.
Comparing calculations obtained in this work with experiments
is not straightforward, because of the complicated experimental
conditions and constraints that were missing in the model calculations. For example, one of such missing factors is H2O, which is easily evaporated from clusters before being detected in laboratory
studies [4b,4c]. The focus of the present study is on unhydrated
clusters, so we did not include any H2O molecules in the complex
models. Because of the small molecular size of H2O, we would expect that entropy plays a critical role in the hydrated-complex formation, like NH3, and temperature could have some impacts on
binding free energies [15,25b]. Although H2O molecules were not
included in the calculations, the present work still provides important information for an understanding of the cluster formation
processes from NH3 and H2SO4 molecules. Investigation of the
H2O effects is amongst our future research.
Table 3
A summary of Gibbs free energies for the formation of (NH3)(H2SO4)n clusters
compared to free molecules at 1 atmosphere, reported from different studies. This
study, 300 K; cited studies, 298 K.
This study
Est 1(MP2)/Est2(MP2)
Ianni and Brady [11h]
B3LYP/6-311++G(2d,2p)
Kurten et al. [11d]
PW91/DNP
Torpo et al. [11i]
RI-MP2/aug-cc-pV(T+d)Z
NadyktoandYu [11a]
PW91PW91/6-311++G(3df,3pd)
H2SO4NH3
2H2SO4NH3
3H2SO4NH3
6.7/6.7
17.3/16.9
29.7/29.5
7.3
22.3
29.9
7.77
19.42
4.54
8.7
We present for the first time the temperature dependence of
energetics for the formation of unhydrated H2SO4 and NH3 clusters
within the atmospherically-relevant temperature range (from 200
to 300 K). Our computational results are consistent with recent
CLOUD chamber experiments of homogeneous aerosol nucleation,
which showed a stepwise addition of NH3 molecules into H2SO4
clusters [4c,7e]. These laboratory studies showed that even for large
clusters containing 15 H2SO4 molecules formed under extremely
low NH3 concentrations (<50 pptv, lower than the typical
atmospheric NH3 concentrations, ppb or sub-ppb), the ratio of
NH3 to H2SO4 molecules clusters is still around 1 or even larger
(more basic than ammonium bisulfate) [4c,7e]. Sulfate components
in sub-micron and micron size aerosols are also partially or fully
neutralized by NH3 under most of the atmospheric conditions [26].
Our MP2/CBS-corrected Gibbs free energy (DG) for the H2SO4NH3 cluster is 6.7 kcal/mol at 300 K. In comparison, the previously
reported DG at 298 K is 4.54 kcal/mol (B3LYP/6-311++G(2d,2p))
by Ianni and Brady [11h], 8.7 kcal/mol(PW91/DNP) by Kurten
et al. [11d], 7.3 kcal/mol (B3LYP) by Torpo et al. [11i],
6.64 kcal/mol (RI-MP2) Kurten et al. [11c], and 7.77 kcal/mol
[PW91PW91/6-311++G(3df,3pd)] by Nadykto and Yu [11a]
(Table 3). Such differences are due to different uncertainties in different computational methods, as discussed in detail by Nadykto
et al. [11b].
The temperature dependence of energetics for the cluster formation of (NH3)(H2SO4)n is consistent with Torpo et al. [11i]
(Table 4). The temperature dependence shown in the present study
indicates that NH3 likely enhances atmospheric nucleation via formation of (NH3)2(H2SO4) clusters in the boundary layer conditions
and (NH3)(H2SO4)n (n = 1–6) complexes at a wide range of altitudes
where H2SO4 and NH3 are available. These results imply that NH3 is
critical for the formation of H2SO4 clusters under various atmospheric conditions. The relevant importance of NH3 in atmospheric
aerosol nucleation, compared to amines and organic acids, and the
synergetic effects of NH3 and amines [27], are needed to be investigated theoretically and experimentally.
Acknowledgments
This work was supported by National Science Foundation
(CHE0952337, AGS1137821, and AGS124198). N.C. was supported
by Undergraduate Research Opportunity Program (UROP) at University of Colorado, Denver. We thank Dr. Soroosh Pezeshki, Eun
Kim and Anatoly Khitrin for helpful discussions.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.chemphys.2014.
01.010.
66
N.L. Chon et al. / Chemical Physics 433 (2014) 60–66
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