Ab initio Evaluation of a Consistent Set of Various Properties of the Benzene Ar Cluster

3.5.1.1 Potential Energy Surface

Five different structures of the benzene-■-Ar complex (cf. Fig. 3) were studied at the ab initio HF level with inclusion of correlation energy [57]. The respective stabilization energies and optimal structures are summarized in Table 4. Clearly, the Cgv structure A

Figure 3. Structures of various benzene---Ar complexes; the individual structural types are labelled by A, B, C, D, E (cf. Table 4).

is the moststable; displacing Ar from the C6 axis (structures B, C) leads to destabiliza-tion. Passing from sandwich structures to a planar structure results in an important stabilization energy decrease. The intermolecular distance found for structure A (3.534 À) agrees well with the corresponding experimental value (3.584 î).

Table 4. Optimal interaction energies and distances for five different structures of the benzene--Ar cluster.

Structure® .

A

B

C

D

E

R(k)

3.5

3.7

3.7

5.2

6.0

- AE (cm-1)

351

261

245

147

a Cf. Figure 3

3.5.1.2 More Accurate Calculations for the Global Minimum

The C6v structure of the benzene Ar---complex was studied athigher theoretical levels [58]. It was found that the MP2 calculation with small basis sets gives a very good estimate of the stabilization energy; this is, however, due to compensation of errors. Only very large basis sets in combination with the CC method give satisfactory results, which converge to the experimental value.

3.5.1.3 Preparation of the Empirical Potential

Two types of empirical potential were fitted [59] to the benzene---Ar ab initio P.E.S. The first potential, the global potential (Eq. 2) is generally applicable while the second one, basically the Morse-type potential (Eq. 3) is, due to its complexity, limited to the evaluation of the vibrational spectrum

The individual terms have the following meanings: rn and rCi are the H---Ar and C---Ar distances, ah a2, a3 and a4 are constants and Ch C2, C3, C4, N and M were fitted to the ah initio P.E.S. (for details see [59]).

The Morse-type potential was fitted only to the sandwich structures of the cluster; for details, see [59].

3.5.1.4 Vibrational Frequencies

Intermolecular frequencies were obtained [59] numerically by solving the vibrational Schrodinger equation utilizing empirical potentials obtained in the preceeding step. Both available experimental studies yield vibrational bands at about 40 and 31 cm"1 and assign them to intermolecular stretching and the first overtone of the intermolecular bending vibration. The third band at about 64 cm"1 was assigned either to the third overtone of the intermolecular bending or the first overtone of the intermolecular stretching vibration. The theoretical Morse stretching (39 cm"1) agrees nicely with the experimental findings. The other theoretical Morse frequencies, bending (29 cm"1), first overtone of bending (57 cm"1) and combination modes (62 and 63 cm"1) differ from the experimental assignment. On the basis of our [59] and van der Avoird's [60] theoretical studies, a new assignment of the experimental peaks was made, which fully agrees with that suggested theoretically, i.e., the bands at 31 and 63 cm"1 correspond to the fundamental of bending and a combination mode.

3.5.1.5 Molecular Dynamics Simulations

The number of structures of higher benzene---Ar„ clusters increases rapidly with increasing n and equals about 300 for n = 7; localization of these minima is very tedious. It must be mentioned here that the use of "chemical intuition" for these purposes is rather limited and some more objective method, like the quenching method, should be utilized. Experimental techniques have become very sophisticated over the past few years and allow us to detect various structures of the cluster. It is, however, not easy to specify the particular structure and the theory should assist in this direction. Because experiments are carried out at non-zero temperature it is essential to include the entropy term. The only feasible way of doing this, i.e., to determine the relative AG term for various structures of the cluster, is to perform molecular dynamics (MD) simulations.

As an example, the MD study of the isomers of the benzene-••Ar2 cluster [61] will be described. Two isomers of the cluster exist. The global minimum corresponds to the (1/1) isomer (having the argon atoms on the opposite sides of benzene; D6h), while the (2/0) isomer (both argons are localized on one side of benzene) has higher energy. For benzene---Ar, the "global" potential described above was utilized, while the Ar---Ar interaction was described by the empirical 6-12 Lennard-Jones potential. To find the relative abundances of the two isomers, very long MD runs (100-400 ns) must be performed. At low temperatures (below 27 K) the population of the (1/1) isomer is

V= kjdi + cfy) + kxxzw(dx + cfy) + kzy - Dt Here De is the dissociation energy of the complex, and w = 1 - exp(- adz) kxx, kzz and kxxz are parameters.

100 %. At higher temperatures, the relative population of the (2/0) isomer becomes higher than that of the (1/1) isomer (64 % :36 %) and in the temperature interval studied (27-37 K) it is almost temperature-independent. Temperatures above 37 K could not be reached because the clusters dissociate. The preference of the energetically less favorable (2/0) structure is clearly a consequence of the entropy term. A similar study was performed in our laboratory for higher benzene- --Ar„ clusters (n = 3-7).

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