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Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 415–423 www.elsevier.com/locate/jqsrt
Computational modelling for the clustering degree in the saturated steam and the water-containing complexes in the atmosphere Zdeneˇk Slaninaa,, Filip Uhlı´ kb, Shyi-Long Leec, Shigeru Nagasea a
Department of Theoretical Molecular Science, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Aichi, Japan b School of Science, Charles University, 128 43 Prague 2, Czech Republic c Department of Chemistry, National Chung-Cheng University, Ming-Hsiung, Chia-Yi 621, Taiwan Received 7 September 2004; accepted 2 May 2005
Abstract Recent computational findings of temperature increase of clustering degree in several quite different saturated vapors are analyzed further. A thermodynamic proof is presented, showing that this event should be rather common, if not general. Illustrations are based on the saturated steam and consequences for the atmosphere are discussed with both homo- and hetero-clustering. MP2 ¼ FC=6-311G results for the H2 O N2 (the best stabilization energy 1:62 kcal=mol, which corresponds to 1:44 kcal=mol with the G3 theory) and H2 O O2 (the best stabilization energy 0:96 kcal=mol) hetero-dimers, and the G1-theory ðO2 Þ2 data (the best stabilization energy 0:62 kcal=mol) are reported. The water-dimer thermodynamics is recomputed in an anharmonic regime and a remarkable agreement with experiment is found. The results have some significance for the atmospheric greenhouse effect. r 2005 Elsevier Ltd. All rights reserved. Keywords: Atmospheric complexes; Water clusters; Hydrogen-bonded heteroclusters; Greenhouse effect; Atmospheric cluster populations; Anharmonic water-dimer thermodynamics
Corresponding author.
E-mail address:
[email protected] (Z. Slanina). 0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2005.05.065
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1. Introduction Cluster studies have developed into a considerably broad field (see, e.g., [1–5]), covering aggregates of very different types. Although their experimental generation is frequently rather far from thermodynamic equilibrium, computational modelling of clusters has mostly worked— owing to obvious reasons—with dimers under the equilibrium conditions. Water clusters have been studied very intensively [6–13], including water hetero-clusters [3,14–19]. In order to estimate the possible effects of water clusters of any dimension, a thermodynamic extrapolation scheme has been applied [20,21]. Relative populations of water clusters in saturated steam (over ice or liquid water) have also been computed [22–25] based on the observed saturation pressure. It has been found in the model studies that the clustering degree increases with increasing temperature. It may be a surprising result but in fact it can be easily rationalized. While the equilibrium constants for cluster formation decrease with temperature, the saturated pressure increases. It is just the competition between these two terms which decides the final temperature behavior. As several computational approximations have to be used with the water clusters, some simple systems have also been investigated like dimerization in the argon-saturated vapor [26] with the dimerization equilibrium constant from advanced evaluations [27]. A temperature increase of the dimeric fraction has again been found. Similar results were obtained for Mg [28] and carbon [29] vapors. In this report, this interesting computational finding is further analyzed without any reference to a particular potential function or approximation, i.e., from a thermodynamic point of view. A simple thermodynamic criterion for the temperature enhancement of clustering degree is suggested and relation to the atmosphere is discussed ((H2 OÞ2 , H2 O N2 , and H2 O O2 ).
2. Clusters in saturated vapor A substance A exists in condensed (liquid or solid) and gas phases and they are in thermodynamic gas–liquid or gas–solid equilibrium. Moreover, in the gas phase clusters Ai are present as follows: iAðgÞ ¼ Ai ðgÞ
ði ¼ 2; . . .Þ,
(1)
described by the equilibrium constants: K p;i ¼
pAi piA
ði ¼ 2; . . .Þ.
(2)
The temperature dependency of K p;i is given by the van’t Hoff equation: d ln K p;i DH 0i ¼ dT RT 2
ði ¼ 2; . . .Þ.
(3)
where DH 0i denotes the (negative) standard change of enthalpy upon the respective i-merization.
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The total pressure P is the sum of all the partial pressures derived from Eqs. (2): P ¼ pA þ
1 X
piA K p;i .
(4)
i¼2
Eq. (4) can also be expressed using the monomeric molar fraction x1 : 1 ¼ x1 þ
1 X
xi1 Pi1 K p;i .
(5)
i¼2
Generally speaking, this algebraic equation cannot be solved in a closed form. However, we do not need the algebraic solution—it is just enough for our purpose to express the temperature derivative dx1 =dT. Moreover, we are not interested in a general pressure but in the saturated pressure. Let us suppose that the saturated pressure obeys the Clausius–Clapeyron equation: d ln P DH vap , ¼ dT RT 2
(6)
where DH vap is the (positive) enthalpy of vaporization. The temperature derivative dx1 =dT can be expressed as ! 1 1 X X dx1 DH 0i þ ði 1ÞDH vap ixi ¼ x1 xi (7) 1þ dT RT 2 i¼2 i¼2 where the xi terms are given as i1 xi ¼ xi1 K p;i . 1 P
(8)
Eq. (7) suggests a simple sufficient (although unnecessary) condition for the temperature decrease of x1 in the saturated vapor (i.e., for temperature increase of the clustering degree 1 x1 ). As long as DH vap 4jDH 0i j=ði 1Þ for every iX2, the temperature derivative dx1 =dT is negative and thus, the clustering degree increases with temperature in the saturated vapor. Table 1 illustrates the rule on the water dimer and other water oligomers as steam is a pertinent system [32,33] for such observations.
3. H2 O N2 and H2 O O2 hetero-dimers The importance of molecular complexes for spectral records and for thermophysical properties of the atmosphere has clearly been established [34–36]. There is a particular sub-group in the set of molecular complexes relevant to the atmosphere—the complexes with water, both water clusters and hetero-clusters with water as one component. According to the previous section, the contents of molecular complexes in saturated (or nearly saturated vapors) increase with temperature. The atmosphere can also be close to its saturation with water. Hence, the contents of water dimers or even of water-containing hetero-dimers can actually increase with warming up of the atmosphere. Strictly speaking, the contents of water clusters should increase under near-saturation conditions—the case of hetero-clusters is not covered by the above treatment and should be given a special numerical simulation. Let us consider a second component, B, that is not in a
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Table 1 Comparison of the DH 0i =ði 1Þ and DH vap terms (kcal/mol) for water System
i
T (K)
DH 0i =ði 1Þ
DH vap a
H2 O
2
298.15 400 500 298.15
4.3b 3.8b 3.5b 3.5c 6.0c 7.5c 7.5c 7.7c
10.5 9.4 7.8 10.5 10.5 10.5 10.5 10.5
2 3 4 5 6 a
Refs. [30,31]. Ref. [25]. c The G3 values from Ref. [60]. b
contact with its condensed form. For the total pressure P, two simplified components can be then considered, P ¼ PA þ P0 , the saturated (water) pressure PA (heat of vaporization DH A;vap ) and the pressure P0 as a constant pressure of the B component. It can be shown that a more general sufficient (though not necessary) condition for the temperature increase of the clustering degree can again be formulated, if some further simplifications are accepted: DH A;vap 4
jDH 0Ai j P i 1 PA
for every iX2,
(9)
DH A;vap 4
jDH 0Bi j P i 1 PA
for every iX2,
(10)
and finally DH A;vap 4
jDH 0Ai Bj j P i þ j 1 PA
for every pair of iX1 and jX1
(11)
(where DH 0Ai , DH 0Bi , and DH 0Ai Bj have an analogous meaning as the DH 0i terms in Eq. (3)). As for the Earth’s atmosphere, the P=PA scaling factor makes the conditions less convenient than in the homo-cluster case and a full numerical modelling should be applied instead. Once the B species is eliminated, which also implies P ¼ PA , the new rule is reduced back to the rule derived for the one-component saturated vapor. Clearly enough, those water-containing homo- and hetero-dimers are also greenhouse-effect agents, somewhat different from free water. Hence, it make sense to clarify if the water-containing complexes themselves could, if increased, contribute to some additional atmospheric warming up. This hypothesis can only be studied computationally at present. Anyhow, it would be interesting to estimate what kind of contribution to the greenhouse effect can be expected from watercontaining homo- and hetero-dimers. There are two particularly interesting water-containing hetero-dimers, viz. H2 O N2 and H2 O O2 . In order to investigate their temperature development in a mixture of air and saturated
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steam, we have been performing ab initio computations with the Gaussian program package [37]. The computations are carried out at the second-order Møller–Plesset (MP2) perturbation treatment with the frozen-core option ðMP2 ¼ FCÞ in the standard 6-311G basis set, i.e., the MP2 ¼ FC=6-311G or UMP2 ¼ FC=6-311G treatment for the H2 O N2 and H2 O O2 species, respectively. In both cases, three stationary points have been located. If we, somewhat arbitrarily, define hydrogen bonds here simply as H–O or H–N long-range linkages shorter than 3 A˚, the stationary points can be described as exhibiting two or one such hydrogen bond. In the H2 O N2 case, the lowest-energy structure is non-planar with two hydrogen bonds 2.76 and 2.80 A˚ long. Its MP2 ¼ FC=6-311G dimerization potential energy is DE ¼ 1:62 kcal=mol. It is followed by a planar structure with one hydrogen bond of 2.49 A˚ and dimerization energy of DE ¼ 1:59 kcal=mol. The third lowest structure is a planar species with two hydrogen bonds of 2.68 and 2.87 A˚ and a stabilization energy slightly less than 1:59 kcal=mol. The decrease in the potential energy upon dimerization is by about 0.5 kcal/mol smaller for the H2 O O2 species. The lowest-energy structure is also non-planar in this case; it has two hydrogen bonds of 2.75 and 2.87 A˚, and the related UMP2 ¼ FC=6-311G dimerization potential energy is DE ¼ 0:96 kcal=mol. The second lowest isomer is a planar structure with two hydrogen bonds 2.65 and 2.68 A˚ long, stabilized by 0:91 kcal=mol. Finally, the third species is a C 2v planar form with two hydrogen bonds 2.72 A˚ long and the dimerization energy DE ¼ 0:81 kcal=mol. The listed dimerization potential energy terms are without the BSSE correction [38]. Although the physical nature of the counterpoise approach has been clarified [39], its accuracy is not always satisfactory. In order to avoid the BSSE correction, one can consider an application of some version of the developing G-theories [40–42]. The G1 theory [40] applies the full fourth-order Møller–Plesset (MP4) treatment with the 6-311G , 6-311 þ G , and 6-311G(2df,p) basis sets. In addition, a quadratic configuration interaction with all singles and doubles (QCISD) and with a quasi-perturbative treatment of higher excitations (QCISD(T)) is applied with the 6-311G basis set. The effects of polarization space extension and diffuse functions are then assumed to be additive at the MP4 level, and a correction formula for further basis set incompleteness is applied. The G-theories are aiming [40–42] at a precise description of thermochemistry; however, they have not been tested for the description of weak molecular complexes. If the H2 O N2 case is recomputed at the G3 theory [42], we get the dimerization energy of 1:44 kcal=mol (the potential-energy change only, without the zero-point energy), i.e., not very different from the MP2 ¼ FC=6-311G term without the BSSE correction.
4. ðO2 Þ2 homo-dimers In order to observe the performance of the new tool, we have applied the G1-theory [40] to the three structures considered for ðO2 Þ2 in our previous evaluations [43]. The T-shape C 2v , rhomboid C 2h , and linear D1h exhibit the G1 dimerization energies of 0:62, 0:51, and 0:48 kcal=mol, respectively. In order to carry out through the computations, the frozen-core approach had to be applied, although not used in the original G1 scheme [40]. The potential-energy terms should, however, be further corrected for the zero-point vibrational energy and for temperature effects in order to be compared with the experiment [44].
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Table 2 The standarda Gibbs-energy change DG02 for the gas-phase water dimerization evaluatedb in an anharmonic approach and the related observed valuesc T (K)
372.4 373.0 423.0 573.15
DG02 (kcal/mol) Calculated
Observed
3.24 3.25 4.94 6.51
3.33 3.08 4.00 6.51
a
The standard state—ideal gas phase at 101 325 Pa pressure. The potential-energy change evaluated with the G3 theory and the anharmonic partition functions with the MP2 ¼ FC=6-311 þ þG approach, using the Gaussian 03 program package [54]. c Ref. [25] and therein quoted data. b
Orlando et al. studied [44] the temperature dependence of collision-induced absorption by oxygen over the temperature range 225–356 K. They obtained a heat of formation ðDH f Þ for ðO2 Þ2 as 1:1 0:5 kcal=mol. Long and Ewing reported [45,46] a DE f term of 0:53 kcal=mol near 100 K which corresponds [44] to a DH f of about 0:75 kcal=mol (for a wider, diversified perspective of tetraoxygen, cf. also papers [47–53]). Overall, the G1 results seem to be in an encouraging agreement with the observed values (interestingly enough, a much older treatment [47] is also in a reasonable agreement). However, from a practical point of view, the H2 O N2 system should be given preference in submission to still higher levels of theory (also because it is free of open-shell difficulties). As we are aiming at concentration terms, the quality of the partition functions is important, too. In order to have an insight into the computational reliability, the standard Gibbs-energy change DG02 for the gas-phase water dimerization is evaluated here in anharmonic rather then traditional harmonic approach (Table 2). While the potential-energy change is extracted from the G3 theory [42] computations, all the terms related to partition functions are evaluated within the MP2 ¼ FC=6-311 þ þG approach, using the anharmonic treatment included in the Gaussian 03 program package [54]. The agreement with the observed terms is considerably good, basically within possible experimental errors. The computational modelling of the atmospheric clustering will require further efforts before it can culminate with the simultaneous simulation of the temperature development of ðH2 OÞ2 , H2 O N2 , and H2 O O2 under the atmospheric conditions. Complexation with larger water clusters [55–62] can then represent another topic of a broader environmental interest.
Acknowledgements The reported research has been supported by a Grant-in-aid for NAREGI Nanoscience Project, and for Scientific Research on Priority Area (A) from the Ministry of Education, Culture, Sports, Science and Technology of Japan, by the Czech National Research Program ‘Information
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Society’ (Czech Acad. Sci. 1ET401110505), and by the National Science Council, Taiwan, Republic of China. Initial phase of the research line was supported by the Alexander von Humboldt-Stiftung and the Max-Planck-Institut fu¨r Chemie (Otto-Hahn-Institut). Last but not the least, the referee comments are highly appreciated, too.
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