Full text of DOI:10.1007/BF02879812

Vol. 44 No. 4
SCIENCE IN CHINA (Series B)
August 2001
Theoretical study on cooperative and extra-additive
behavior of hydrogen-bonded clusters
TANG Junhui (唐俊辉), CHEN Penglei (陈鹏磊), ZHEN Zhen (甄 珍)
& LIU Xinhou(刘新厚)
Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100101, China
Correspondence should be addressed to Liu Xinhou (email: xhliu@ipc.ac.cn)
Received March 12, 2001
Abstract The stabilization energy ΔE(n) and four typical properties of hydrogen bond F—H…F
in chain-like and cyclic (HF)n clusters (n = 1—5) have been calculated using MP2 and three DF
levels of theory with the Gaussian 98 program, and 6-31++G** bases set. The results demonstrate
that the extra-additive or cooperative behavior in (HF)n clusters is very obvious. In addition, we
studied much larger chain-like (HF)n (n = 6, 9, 12, 18, 24) clusters using one of these DF methods.
Keywords: hydrogen-bonded, DF methods, non-additive or cooperative behavior.
Hydrogen-bonded clusters play an important role in many progresses of chemistry and bio-
chemistry[1,2]. In the past two decades, hydrogen-bonded clusters have received much attention
—
both theoretically and experimentally[3 5]. (HF)n cluster is the simplest hydrogen-bonded cluster
system. So that it is very useful to make the (HF)n clusters as the model clusters for the study of
the hydrogen-bonded clusters. Recently the non-additive or cooperative behavior in (HF)n clusters
is most interesting for all the investigators[2,6]. In general, the larger the (HF)n clusters, the more
obvious the non-additive behavior of the hydrogen-bonded clusters. However, because of the limit
of the computational capabilities, it is very difficult to compute larger (HF)n (n > 6) clusters with
the ab initio methods such as MP2. Fortunately, in recent years DF methods were confirmed very
—
9]
useful in the field of hydrogen-bonded clusters[7
.
In the present paper, we report computed results using ab initio/MP2 and three DF methods.
Through the use of all these four methods, we studied the extra-additive or cooperative behavior
in chain-like and ring-like (HF)n (n = 1—5) clusters. The results demonstrate that DF methods are
very useful in the field of hydrogen-bonded clusters. Then we computed much larger chain-like
(HF)n (n = 6, 9, 12, 18, 24) clusters using one of these DF methods.
1
Computational details
The three DF methods are (ⅰ) hybrid-B3LYP (B3 (Becke’s three-parameter) + LYP
(Lee-Yang-Parr)), (ⅱ) the BLYP(B (Becke’s 1988) + LYP) method, and (ⅲ) the hybrid-B3P86(B
+ P86(Perdew, 1986)). The ab initio calculations were performed at the MP2 levels of theory. The
Gaussian 98 program and 6-31++G** bases were used in each of these methods.

382
SCIENCE IN CHINA (Series B)
Vol. 44
2
Results and discussion
Generally, the formation of a hydrogen bond A—H…B is accompanied by the following
trends[2]: (ⅰ) The intermolecular distance R (A…B) is much shorter than the sum of the van der
Waals radii of A and B; (ⅱ) R (A-H) is longer in the complex than that in the monomer; (ⅲ) The
fundamental frequency ν (A-H) is smaller in the complex than that in the monomer; (ⅳ) The in-
frared intensity of ν (A-H) is larger in the complex than that in the monomer. In addition, proton
NMR chemical shifts are significantly smaller in the complex than that in the monomer.
The extent to which these trends are actually observable depends on the strength of the
hydrogen bond.
Before all, it must be indicated that in this paper the structures of chain-like (HF)n clusters
are zigzag lines. Fig. 1 shows the structures of chain-like and cyclic (HF)4 clusters.
Fig. 1. The structures of chain-like and cyclic (HF)4 clusters.
Because the averaged stabilization energy reflects the strength of the hydrogen bond directly, at
first we study the averaged stabilization energy. There are two methods to define the averaged
stabilization energy per hydrogen bond ΔE (n):
ΔEa (n) = E (n)−E (n −1) − E
(1)
or
ΔEb (n) = (E (n)−n*E (1))/m,
(2)
where E (n) is the total energy of the n-mer, m = n for cyclic clusters, and m = n −1 for linear clus-
ters. Here, the latter is adopted.
Fig. 2 shows the computed stabilization energy of cyclic and chain-like clusters obtained
with the four computational levels. All these results excellently agree with the previously pub-
lished theoretical studies[2]. The three DF variants deliver interaction energy that is in quite good
Fig. 2. Stabilization energy of chain-like and cyclic hydrogen fluoride clusters plotted versus n. ▼, MP2; ▲,
BLYP; ■, B3LYP; ●, B3P86.

No. 4
COOPERATIVE AND EXTRA-ADDITIVE BEHAVIOR
383
agreement with the MP2 data. The good agreement indicates that, as far as the interaction energy
is concerned, the much cheaper DF variants reproduce reliable values.
Compared with the chain-like (HF)n clusters, the Cnh-symmetric structures are more stable
for n = 4, 5. However, larger rings will tend to become less stable. Therefore, for eventual ex-
trapolations to the solid, the analysis of the chain-like structures appears to be more appropriate.
The increase in the stabilization energy of the clusters takes place upon enlarging the size of the
chain-like (HF)n clusters. This is always called non-additive behavior. Fig. 2 indicates that the av-
eraged stabilization energy per hydrogen bond increases straightly while the clusters become lar-
ger. In other words, the larger the cluster, the more obvious the non-additive behavior in chain-like
(HF)n clusters. We call this extra-additive behavior.
However, the increase in the averaged stabilization energy per hydrogen bond is not unlimited. In
fact, the increase becomes slower as the size of the chain-like (HF)n clusters becomes larger (from n =
2 to 5). In order to obtain the limit in the averaged stabilization energy per hydrogen bond, larger
chain-like (HF)n clusters have to be investigated. So that larger chain-like (HF)n clusters (n =
6,9,12,18,24) were investigated using one of the DF methods and the results obtained with this method
are illustrated in fig. 2. From fig. 2 we can see distinctly that the limit is around –34 kJ·mol−1.
The R (H-F) and R (F…F) distances are important structural parameters, and the characteris-
tic structural parameters of the shortest hydrogen bond in each of the clusters are depicted in fig. 3.
It can be obviously seen that the R (H-F) elongates while the R (F…F) reduces with enlarging the
size of (HF)n clusters. Results obtained with all the three DF methods are in qualitative agreement
with the previous SCF investigations applying the same basis set[6]. On the other hand, there are
some quantitative differences between MP2 and DF results, in particular for the R (F…H) elonga-
tion computed with all the three DF variants and to a lesser degree for the R (F…F) contractions.
However, we can see from fig. 3 that the results computed with all the four methods have the
same trend and converge slowly. So that larger chain-like (HF)n clusters (n = 6, 9, 12, 18, 24) are
computed with one of the DF methods and the results are shown in fig. 3. From fig. 3 we can see
that while the (HF)n clusters are larger than (HF)12, the computed results converge rapidly and the
results (the R (H-F) distance is 0.094 nm and the R (F…F) distance is 0.25 nm) of (HF)24 cluster
are very close to the limit. The R (H-F) distances are quite difficult to determine experimentally
and hence only rather approximate crystal values of 0.095—0.097 nm[10] are available. The ex-
perimentally derived R (F…H) distances of about 0.253 nm for (HF)6 from electron diffraction[11]
and of about 0.250 nm from neutron diffraction of solid HF appear more reliable.
The harmonic ν (H-F) vibrational frequencies were also computed. The frequency of the
most intense ν (H-F) infrared mode of each cluster is shown in fig. 4. The strongest infrared active
mode in the chains is always the lowest within the group of F-H stretching. In the rings, the cor-
responding in-phase mode is optically forbidden, and the strongly infrared active mode is always
the second lowest and doubly degenerates. Because of the too large elongations of R (H-F) com-

384
SCIENCE IN CHINA (Series B)
Vol. 44
puted with the DF methods, the corresponding shifts of the ν (H-F) vibrational frequencies are
considerably too large. This trend is already visible in case of the cyclic trimer. The experimental
shift of the vibrationally allowed, and doubly degenerated F-H stretching mode with respect to the
monomer[12] is –249 cm−1, being in excellent agreement with the MP2 shift of –246 cm−1. The
computed MP2 shifts for the strongly infrared active F-H stretching mode of cyclic tetramer, pen-
tamer and hexamer are also in excellent agreement with the experimental data[13]
.
Fig. 3. (a) and (b) Computed R (F-H) distances of chain-like and cyclic hydrogen fluoride clusters, (c) and (d) computed
R (F…F) distances of chain-like and cyclic hydrogen fluoride clusters. ▼, MP2; ▲, BLYP; ■, B3LYP; ●, B3P86.
Fig. 4. Calculated ν (H-F) vibrational frequencies in chain-like and cyclic hydrogen fluoride clusters plotted
versus n. ▼, MP2; ▲, BLYP; ■, B3LYP; ●, B3P86.

No. 4
COOPERATIVE AND EXTRA-ADDITIVE BEHAVIOR
385
Experimentally, the in-phase k = 0, F-H stretching frequency in the hydrogen fluoride crystal
is shifted by –896 cm−1 relative to the monomer frequency[14]. It is more difficult to extrapolate the
infinite chain limit in the case of hydrogen fluoride, because the optimal structure is not the fully
linear arrangement, and the non-additive contributions are of greater importance to the interaction
energy, which cause also a slower decay of the two edge effects at both sides of the chains. Hence,
chain-like clusters approach the bulk limit considerably slow. As shown in fig. 4, the lowest
ν (H-F) frequencies of chain-like clusters converge slowly, and actually larger clusters should be
treated. So that we computed larger chain-like (HF)n clusters (n = 6, 9, 12, 18, 24) using one of the
DF methods. The results obtained with the DF method are shown in fig. 4. From fig. 4, it is very
obvious that while the chain-like (HF)n clusters are larger than (HF)9, the computed results con-
verge rapidly and the ν (H-F) vibrational frequency in (HF)24 approaches the bulk limit (the F-H
stretching frequency is shifted by −1014 cm−1 relative to the monomer frequency).
The results of computed infrared intensities of the ν (H-F) modes are shown in fig. 5. The
methodical trends upon chain length elongation or upon increasing the ring size parallel those al-
ready observed for the frequencies, with the MP2 and B3P86 intensities as the limiting cases.
When the structural parameters and the vibrational frequencies are used, the computed infrared
intensities of the chain-like clusters allow, in combination with an analysis of the normal modes, a
nice separation of the convergence to edge and bulk properties, respectively. An indication for this
behavior can be found in fig. 5 by inspecting the intensities of the two highest-lying modes up to
the chain-like hexamer, corresponding to localized F-H stretches at non-hydrogen-bonded and
hydrogen-bonded edges, respectively. The results in larger chain-like (HF)n clusters (n = 6, 9, 12,
18, 24) obtained as one of the DF methods are illustrated in fig. 5, and the trends in the infrared
intensities of the ν (H-F) modes also parallel those already observed for the frequencies.
Fig. 5. Calculated infrared intensities A (F-H) of the ν (H-F) vibrational frequencies in chain-like and cyclic hydrogen
fluoride clusters plotted versus n. ▼, MP2; ▲, BLYP; ■, B3LYP; ●, B3P86.
3
Conclusion
In summary, we have investigated the stabilization energy ΔE (n) and four typical properties

386
SCIENCE IN CHINA (Series B)
Vol. 44
for hydrogen bond F—H…F in (HF)n clusters with MP2 and three DF methods. The good agree-
ment with the results of MP2 and DF methods confirm that DF methods is very useful in the field
of hydrogen-bonded clusters. In addition, we studied much larger chain-like (HF)n (n = 6, 9, 12,
18, 24) clusters using one of these DF methods, and the non-additive or cooperative behavior in
(HF)n clusters was obviously observed.
Acknowledegements This work was supported by the “Nineth-five” Important Project of the National Natural Science
Foundation of China(Grant No. 29992590) and the National Important Basic R & S Project (Grant No. G1999022501).
References
1. Aakeröy, C. B., Seddon, K. R., The hydrogen bond and crystal engineering, Chem. Soc. Rew., 1993, 22(6): 397—407.
2. Scheiner, Molecular Interactions, Chichester: John Wiley & Sons Ltd., 1997.
3. Sun, H., Watts, R. O., The infrared spectrum and structure of hydrogen fluoride clusters and the liquid: semiclassical and
classical studies, J. Chem. Phys., 1981, 96 (3): 1810—1812.
4. Lisy, J. M., Tramer, A., Vibrational predissociation spectra of (HF)n, n = 2—6, J. Chem. Phys., 1981, 75(9): 4733—4734.
5. Andrew, L., Fourier transform infrared spectra of HF complexes in solid argon, J. Phys. Chem., 1984, 88(13): 2940—
2949.
6. Karpfen, A., Cooperativity in hydrogen bonded clusters: an improved ab initio SCF study on the structure and energetics
of neutrual, protonated and deprotonated chains and cyclic hydrogen fluoride oligomer, J. Mol. Struct. (Theochem.), 1994,
314(1,2): 211—227.
7. Couronne, O., Ellinger, Y., An ab initio and DFT study of (N2)2 dimers, Chem. Phys. Lett., 1999, 306(1): 71—77.
8. Kovacs, A., Szabo, A., Comparison of ab initio and density functional methods for TeF6, Chem. Phys. Lett., 1999, 305(10):
458—464.
9. Gonzalez, L., Mo, O., Density functional theory study on enthanol dimers and cyclic enthanol trimers, J. Chem. Phys.,
1999, 111(9): 3855—3861.
10. Johnson, M. W., Sandor, E., The crystal structure of deuterium fluoride, Acta Cryst, 1975, B31(8): 1998—2003.
11. Habuda, S. B., Nuclear magnetic resonance data on proton positions in solid HF, Acta Cryst., 1971, B27 (8): 1677—1679.
12. Michael, D. W., Lisy, J. M., Vibrational predissociation spectroscopy of (HF)3, J. Chem. Phys., 1986, 85(5): 2528—2537.
13. Quach, M., Schmitt, U., Evidence for the (HF)5 complex in the HF stretching FTIR absorption spectra of pulsed and con-
tinuous supersonic jet expansions of hydrogen fluoride, Chem. Phys. Lett., 1993, 208 (5, 6): 446—452.
14. Kittelberger, J. S., Hornig, D. F., Vibrational spectrum of crystalline HF and DF, J. Chem. Phys., 1967, 46(8): 3099—
3108.