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Rigid and nonrigid benzene·Ar2 van der Waals heteroclusters
Volume 188, number I ,2
CHEMICAL PHYSICS LETTERS
3 January 1992
Rigid and nonrigid benzene-AT2 van der Waals heteroclusters Narda Ben-Horin,
Uzi Even and Joshua Jortner
School ofChematry, Tel-Aviv Universily, Tel-Aviv 69975.Israel
Received 27 September 1991
The nonrigidity of the one-sided (2 IO) isomer of benzene,Ar2 is manifested by the occurrence of a hierarchy of distinct isomerization processes and by the marked broadening of the photoionization efficiency curves, which were explored by constant energy molecular dynamics simulations.
M.A,, heteroclusters, consisting of an organic aromatic molecule (M) bound to rare-gas atoms (A), provide interesting information regarding the size dependence of energetic, structural, thermodynamic, dynamic and chemical properties of large finite systems [ l-4 1. In this context, various isomerization processes were identified by molecular dynamics (MD) and Monte Carlo computer simulations in carbazolesAr, [ 4-71, 9,l 0-dichloroanthracene*An (A=Ar, Kr) [8], tetracene.Ar, [9] and benzene.Ar,, [ lo,1 1] heteroclusters, which involve correlated surface motion (of A atoms on one side of M), surface melting (i.e. uncorrelated motion of A atoms on the microsurface of M), side crossing (of A atoms from one microsurface of A to the other), wetting-nonwetting (or two-dimensional (2D) to three-dimensional (3D) ) transition and a rigidnonrigid transition. These studies [4-91 established the hierarchical occurrence of several isomerization processes for a single heterocluster composition and the sequential occurrence of distinct isomerization processes with increasing the size of the M.A, heterocluster for a given M. This isomerization pattern is specified by the finite surface density p,=n/(M microsurface area) of the rare-gas atoms [8 1. Benzene.Ar,, heteroclusters, which are characterized by a large value ofp,, are expected to exhibit a hierarchy of isomerization processes even for small values of n. In addition, isomerization phenomena are governed by the initial isomer configuration, being distinct for interior and for surface states of M in or on the heterocluster [&lo]. In order to explore the M
and n size dependence and the isomer specificity of M,A,, heterocluster isomerization phenomena, we have conducted constant energy MD simulations of the benzene.Ar, isomers, which constitute the smallest heteroclustets of the M.A,, family. The recent spectroscopic studies of Schmidt, Mons and Le Calve? [ 11,12] on the spectral shifts and rotational counters of the SO+S, transition and the ionization potentials from So for benzeneeAr heteroclusters, provide compelling evidence for the existence of two structural isomers, i.e. the two-sided ( 1I 1) structure and the one-sided (2 IO) structure of this heterocluster. Our MD simulations of the nuclear motion in So of benzene.Arz isomers reveal that: (i) The (111) structure constitutes a rigid molecule over a broad temperature domain, until it converts to the (2 IO) structure at P 50 K. (ii) The (2 IO) structure constitutes a nonrigid molecule over all the relevant temperature domains (T< 70 K), with an onset of isomerization processes at T> 3 K. (iii) The (2 IO) benzene.Ar, heterocluster exhibits a hierarchy of isomerization processes with increasing temperature. In conjunction, we have performed MD simulations of the ionization potentials of the ( 111) and (2 (0) benzene. Arz heteroclusters. The simulated photoionization efficiency curves are confronted with the experimental data of Schmidt et al. [ 121, elucidating the effects of nuclear dynamics and the implications of molecular nonrigidity for the ionization processes of these heteroclusters. The constant energy MD simulations of benzene-A+ utilized a fifth-order predictor-corrector
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method for the integration of the equations of motion. The integration step was chosen as IO-” s at low temperatures (TG 10 K) and was gradually decreased with increasing temperature, being taken as 2 x lo-‘” s at T> 60 K. The first stage in the simulations involved an equilibration stage, with the initial random velocities being characterized by a Boltzmann distribution to give the desired temperature and with the total angular momentum being set to zero. A short simulation ( 10’integration steps) was then performed with the velocities being frequently scaled to achieve the desired temperature, followed by an uninterrupted short simulation to monitor the attainment of the final temperature. In some simulations we have used gradual heating of an equilibrated configuration for the initiation of the equilibration stage. Subsequently, constant energy trajectories were generated in the ground electronic state for a time of 1.O- 1.5 ns. Energy was conserved in these simulations to better than 1 part in 10’. The benzene molecule was taken to be rigid at its ground state equilibrium geometry. The potential surfaces of benzeneeAr, heteroclusters in the ground electronic state were specified using Lennard-Jones atom-atom potentials [ 13]. The potential parameters for the C-Ar and H-Ar interaction were taken from the prescription of Ondrechen et al. [ 131. Standard parameters were used for the Ar-Ar interaction [4]. We tested the role of electrostatic benzene-Ar chargeinduced dipole interactions within the ground state potential surface in order to ascertain their effects on the energetics and the relative stability of isomers. Calculations of the charge-induced dipole interactions between the atomic charge distribution of benzene (obtained using the MNDO method [ 141 in conjunction with Liiwdin’s charge density analysis [ 15,161) and the rare-gas atom, resulted in the stabilization energy of -6 cm-’ for the equilibrium nuclear configuration of benzenesArI, which constitutes 1.5% of the total ground state binding energy, and can be safely neglected. In each MD simulation the following observables were calculated by averaging over the entire trajectory: (i) The potential energy U. (ii) The standard deviation (7of the distances between the rare-gas atoms and the center of mass of benzene and the Cartesian components I?~(&x, y, z) of d. The molecule-fixed Cartesian coordinate axes were specified 14
3 January I992
by the origin at the center of mass of benzene, with x (in the molecular plane perpendicular to the C-C bond), y (in the molecular plane passing through two C atoms) and z (perpendicular to the molecular plane). We have also evaluated the normalized standard deviation u, where the two contributions of the Ar atoms are normalized by their distances from the center of mass of benzene, and the normalized three Cartesian components of (T,i.e. o,, aY and CJ=.(iii) The standard deviation 6 of the Ar-Ar distance normalized by the average Ar-Ar distance. The MD simulations of the structure, energetics and nuclear dynamics of the ( 111) and (2 ]0) isomers of benzene.Ar, were performed over a broad temperature domain (T= 3-70 K). Static structural data were inferred from short time (100ps) averaging of lowtemperature (T=3 K) simulations, yielding the equilibrium coordinates (x,, y,, z, ) and (x,, y,, zz) of the two Ar atoms (in A). The following information was obtained for the ( 1) I) isomer: ( 1) Its static low-temperature (3 K) equilibrium structure is (0, 0, 3.47) and (0, 0, -3.47), which corresponds to a D6,, symmetry. (2) With increasing the temperature in the range 3-50 K, this structure (expressed by the average coordinates) is preserved. (3) The potential energy U and the structural fluctuation parameters a, a,, u,,,u=and 6 exhibit a smooth temperature dependence (fig. 1). The increase of the structural parameters with increasing temperature is modest, e.g. (Tincreases from a=0.005 at 10 K to 0~0.06 at 50 K. (4) For T3 52 K the ( 1 ( 1) structure interconverts into the (2(O) structure. This nearly irreversible ( 1 I 1) + (2 IO) side crossing occurs already in the equilibration runs for the ( 11I ) isomer. Regarding the (2 ]0) isomer we observed that: (5 ) The low-temperature (3 IS) static equilibrium structure is (3.24, 0, 3.13) and (-0.4, 0, 3.49), which reflects a low-symmetry (C,) structure. One Ar atom is shifted (by 0.4 A along the x axis) from the benzene-Ar minimum and the second Ar atom is located above the midpoint between two H atoms, with the Arz dimer being slightly tilted relative to the benzene plane. Our structure of the (2 IO)isomer is in agreement with the recent results of Monte Carlo simulations for this system [ 111. This structure
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CHEMICAL PHYSICS LETTERS
I
’
’
0.2-
’
’ c’a
/
b 0.1 -
v#_,---
,*
0
IO
20 ““--“p30
40
60
70
80
TP’KI
Fig. 2. The temperature dependence of the standard deviation d (0 ) of the distances between the two Ar atoms and the center of mass of benzene and the Cartesian components (u) &, ( A ) 6,, and ( l ) & of a. Data for the (2 IO) isomer of benzene,Ar,.
0.0
TCKl Fig. 1.The temperature dependence of the internal energy Uand the normalized structural fluctuation parameten u and 6 for the ( 1I 1) isomer (0 ) and for the (2 (0) isomer ( n) of benzene-Arz.
characterizes the heterocluster only at low temperatures, while at finite temperatures (T> 5 K) large amplitude motion sets in. (6) The structural fluctuation parameters for the (2 IO) isomer exhibit a nonsmooth temperature dependence (figs. 1 and 2), which marks the onset of a hierarchy of the following isomerization processes: (a) Onset of correlated one-dimensional ( 1D) motion at Tz3 IL The nuclear motion of the Ar, dimer along the x axis is enhanced with raising the temperature, as manifested by the rise of u, (fig. 2). (b) Onset of correlated 2D surface motion at 10 K. A steep rise in gYand a slight rise in a= are exhibited (fig. 2), while a, and g have already attained a high value (figs. 1 and 2). Concurrently, 6 does not reveal a break. These features mark the onset of a correlated 2D surface rotational motion of the Ar, dimer parallel to the benzene surface. These MD data concur with recent Monte Carlo simulations, which reveal that the six equivalent H-C-C-H wells of the
(2 I0) isomer of benzene.Ar* are equally explored at 10K [ll]. (c) Onset of uncorrelated surface melting at 5254 K. Here u and 6 exhibit a simultaneous increase of slope. (d) Onset of side crossing at 47-54 K. Here a, exhibits a break followed by a sharp increase. The side crossing involves the motion of one Ar atom from (2 (0) to the other side of the microsurface and back. The lifetime of the ( 1 ) 1) structure is 5 5 ps. The distinction between processes (c) and (d), which occur in the same temperature domain, was inferred from analysis of MD trajectories, which also sets the temperature limits for the onset of these isomerizations. (e) Onset of 2D+ 3D transition at 62 K. MD trajectories revealed the transition from the (2 (0) structure to the ( I+ 1ZIO) structure, which contains one Ar atom in the first solvation layer (ZZ 3.5 A) and one Ar atom in the second solvation layer (2x7.0 A). The lifetime of the 3D (1+ 1210) structure was 3-5 ps. (7) At T2 73 K an eficient fragmentation process (2 IO) + ( 1I0) +Ar sets in. Characteristic time scales for fragmentation of a single atom are ~300 ps at this temperature. We are now in a position to compare the energetits and nuclear dynamics of the ( 111) and (2 IO) isomers. (8) The potential energy CJof the ( 111) structure is lower than that of the (210) structure over the 75
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temperature range 3-52 K (fig. 1). At the lowest temperature the ( 111) structure is energetically more stable than the (2 10) structure by AU= 55 cm-‘, in accord with the estimate of Schmidt et al. [ 121, with the energy difference being reduced to AU= 10 cm-’ at 50 K. The energetic stability of ( 11I ) does not preclude the simultaneous formation of ( 11I ) and (2 ]0) isomers in supersonic expansions of Ar seeded with benzene [ 121. (9) In the temperature range 3-52 K the ( 111) and (2 IO) isomers do not interconvert on a time scale of I ns. ( 10) At temperatures above the onset of side crossing T> 52 K, the (2 10) structure is dominant. The dominance of the (2 ]0) structure at high temperatures is reflected by the nearly irreversible (l]l)-r(2]0) side crossing of (111) at T>52 K (point (4) ) and the short 5 5 ps lifetime of the ( 1 I 1) structure originating from the (2]0)+ ( 1 11) side crossing (point 6 (d) ). Despite the energetic stability of the ( 1) 1) structure (point (7) ) the prevalence of (2 IO) is due to entropy contributions. (11) The structural fluctuation parameters are systematically higher for the (2 10) isomer than for the ( 111) isomer over the entire temperature coexistence domain (fig. 1). The high values of the structural fluctuation parameters, e.g. 0> 0.1 at T> 10 K, for the (2 IO) isomer (fig. 1 ), reflect the nonrigidity of this structure. According to the celebrated Lindeman criterion [ 171, which was applied to finite systems [ 17-201, the onset of a rigid+nonrigid isomerization in clusters is characterized by the normalized structural fluctuation parameters exceeding the value of 0.1. Accordingly, the onset of the nonrigid (2 IO) structure of benzene*Arz is specified by ui~ 0.1, being obtained at Tb 10 K. In contrast, for the (1) 1) benzene.Ar? we find that (T<0.1 and the structure is rigid over the entire temperature existence range of this isomer. This analysis demonstrates the occurrence of hierarchical isomerization in the “small” benzene.Ar, (2 IO) heterocluster. Some details of the analysis will have to be modified for the low temperature onsets (T< 5 K) where quantum effects [ 2 1,221 will be of some importance, but the general picture, which involves a proliferation of Ar, isomerization processes for benzene.Ar,, is quite clear. Correlated surface motion (CSM ) was previously simulated for (2 10) 76
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structures of carbazole,Ar, [4,5], tetracene*Ar, [ 91 and 9,10-dichloroanthracene.Kr, [S]; however, the CSM constitutes the only isomerization process which was identified for these heteroclusters. For the benzene.Ar, heterocluster rich and diverse isomerization phenomena (processes 6(a)-6 (e) ) were recorded, which were previously simulated only for large heteroclusters, i.e. for 9, IO-dichloroanthracene*A, (A=Ar, Kr) side crossing was documented for n 3 5 while the 2D+ 3D transition was observed only for n> 10 [ 81. The present analysis demonstrates that the smallest heterocluster of the MeA,, family (as far as the size of the aromatic molecule and the coordination number are concerned), which exhibits distinct isomers, reveals a rich manifold of isomerization processes. This state of affairs bears a close analogy to the isomerization of small neat raregas clusters, where Ar, exhibits a rigid-+nonrigid transition [ 201. An attempt was made to elucidate some of the spectroscopic manifestations of the nonrigidity of benzene,Ar, (2 IO) by the simulation of the adiabatic ionization potentials and the photoionization efficiency (PIE) curves of the ( 1 I 1) and (2 IO) isomers, which were measured by Schmidt et al. [ 121. The shift 61 of the ionization potential of benzene*Ar2 at a fixed nuclear configuration relative to the bare molecule was calculated from the electrostatic stabilization energy of the ionic state of the heterocluster relative to its neutral ground state. The ionization potential shift at the ground state nuclear configuration {rA)=rl, r2 (where r, and r2 are the temporary coordinates of the two Ar atoms) is SV({r,})=S1/(+)({r,})-SV(“)({r,}), where SVc+)({rA}) and SV(“)({rJ) are the sum of the charge-induced dipole interactions and the induced dipole-induced dipole interactions for benzeneArZ and for benzene.Ar2, respectively. The details of the calculations of SF+) and 8Vco) are given elsewhere [ 8,231. The charge distribution of benzene and of benzene+ were calculated from o and R molecular orbitals obtained by the MNDO method [ 141 with the atomic charges being obtained using the Liiwdin procedure [ 15,161, The ionization potential shifts 6V( {rA)) for a fixed ground state nuclear configuration were summed over the entire trajectory. The distribution P(6V) of the values of 6V exhibits a continuous change from zero to a constant
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value, which is normalized to unity. P(6Y) represents the simulated PIE curve. The simulated ionization potential shift 61 was calculated by a linear extrapolation of P( 6 V) to zero. The width, W, of the photoionization PIE curve was somewhat arbitrarily evaluated from the difference between the upper and lower values of 6 Vwhere P( 6 V) starts deviating from a constant value (fig. 3). Fig. 4 displays the temperature dependence of 61 and W, for the two isomers which reveal the following features: ( 1) 61 is systematically lower for the ( 11I ) relative to the (2 IO) isomer, reflecting the enhanced electrostatic stabilization of the benzeneaAr,+ ( 1 11) structure. We note, however, that for both isomers Sl< 0. (2) I1511increases with increasing temperature for both isomers, manifesting the role of thermal nuclear motion on the decrease of the electrostatic stabilization. The overall change of the 62for both ( 1 I 1) and (2 IO) is small, i.e. x 20% over the temperature range studied herein. (3) W reveals a marked temperature deDendence
0.8
_ 0.6 > C.0 z 04
&km-1) Fig. 3. The PIE curves for the ( 11I ) isomer (-) at I7 and 25 K andforthe(2lO)isomer(---)atl6and23KThetempcralures are marked on the curves. The PIE curve widths W, and W,, which are marked by horizontal arrows, correspond to the width for the ( I 11) isomer at 17 K and to the width for the (2 10) isomer at 16 K, respectively.
-500 300 -
1
I
I
40 11°K)
I
1
60
1
80
Fig. 4. The temperature dependence of the ionization potential shifts SI and for the PIE curve widths W of the ( 11I ) isomer (Cl) and the (210) isomer (m)ofbenzene.Ar,
for both isomers, increasing from W= 20 cm- ’at 10 K to W=150 cm-’ at 50 K for (111) and from W=70 cm-’ at 10 K to W=270 cm-’ at 75 K for (210). (4 ) Over the experimentally relevant temperature domain T=3-40 K, the width of the PIE curve is considerably broader for (2 IO) than for ( Ill). As evident from fig. 3, the larger value of W for (210) reflects the nonrigid structure of this heterocluster. We can now confront the results of the simulations with the experimental data. Schmidt et al. [ 12] have observed a marked qualitative and quantitative difference between the PIE curves for the (1 I 1) and (2 IO) isomers of benzene.Ar,. For the ( 1 I 1) isomer a sharp PIE threshold W(exp.)=25 cm-’ with Gl(exp.) = - 334 cm-’ is observed, which was attributed to the adiabatic threshold [ 121. Our calculated values for the ( 1 I I ) isomer (fig. 4) are: &I=-445 cm-’ and W=25 cm-’ at 10 K, and 61= -428 cm-’ and W=35 cm-’ at 20 K, being in reasonable agreement with experiment (with the experimental data [ 12] presumably corresponding to the vibrational temperature domain of lo-20 K). The situation is drastically different for the (2 IO) 77
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isomer for which a broad ionization threshold was experimentally recorded [ 12 ] with W( exp. )x 100 cm-’ and an apparent ionization potential shift of GZ(exp.) = + 137 cm-‘. The calculated W values for the (210) isomer, W=70 cm-’ at 10 Kand W=90 cm-’ at 20 K, are in good agreement with the experimental value for W [121. On the other hand, the calculated value of SZfor (2]0), i.e. Sl= -365 cm-’ at 10 K and 6Z= - 350 cm-’ at 20 K differs both in sign and in magnitude from the experimental value of sZ( exp. ) = + 137 cm-’ [ 121. The positive experimental value of 61 for the (2 (0) isomer implies the apparent energetic destabilization of the ionic state relative to the ground neutral state of the heterocluster. This conclusion contradicts our results regarding the energetic stabilization of the ionic state as well as the ubiquity of experimental data for other heteroclusters, which always show that 61~ 0 [ 8,2327]. Schmidt et al. [ 121 have argued that the positive experimental value for 6Z(2 IO) corresponds to a nonadiabatic ionization process, which is accompanied by a large intermolecular configurational change. This proposal is in variance with our calculations. Fig. 3 reveals that 6Z< 0 for all the nuclear configurations of (2 10) which contribute to the PIE curve. Our calculations thus provide a semi-quantitative description of the dramatic broadening of the PIE curve, but fail to account for the sign reversal of 61 for the (2 (0) isomer. One can argue that this discrepancy reflects the failure of the electrostatic model [ 8,231. We are, however, rather confident in the validity of our electrostatic model because of the following reasons: (i) It accounts well for the size dependence of 61 for other heteroclusters, i.e. 9,10dichloroanthracene. A,, (A= Ar, Kr; n = l-l 8 ) [ 8, 241. (ii) It accounts for the energetics of ionization of the ( 111) isomer of benzene.Ar,. (iii) It accounts for the linebroadening of the PIE curves of both the ( 1( 1) and (2 ]0) isomers, providing discrimination between the rigid ( 111) and the nonrigid (210) structures. In order to reconcile theory and experiment we propose the existence of a lower-energy photoionization threshold of the benzeneeAr, (2 IO) heterocluster, which was not yet experimentally observed. As evident from the PIE curves of bare benzene, benzene.Ar, and benzene.Arz ( 1 I 1 ), two ionization thresholds are exhibited, which originate from the 78
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Jahn-Teller components, i.e. the 0( l/2) ground state and the 6(3/2) excited vibrational state of benzene+ [ 121. For the case of bare benzene the lowest sharp threshold is rather weak [ 12 1. In contrast, the (2 (0) PIE exhibits only a single threshold [ 121. A lower energy, weak and broad PIE threshold of the nonrigid benzene. Ar, (2 IO) heteroclusters may exist, being smeared out by the large energetic spread of its threshold. An estimate of the energy of this “hidden” threshold is of interest. Schmidt et al. [ 121 have observed a single ionization threshold for the (210) isomer at SZ=+137 cm-‘. We propose that this threshold corresponds to a 6( 3/2) vibrationally excited benzene+, which is located at ~355 cm-’ above the vibrationless level. Accordingly, the vibrationless (adiabatic) ionization threshold of (2 10) is expected to be located at 61(2]0)= -218 cm-‘. Thus the splitting between the ( 1 I 1) and (2 10) ionization thresholds is inferred to be 66Z= 6Z( 1 I 1) 6Z(2 10) = - 116 cm- ‘. This possible experimental estimate is in reasonable agreement with our calculated value of S&Z=- 80 cm-’ (at T= lo-20 K). A search for an additional low-energy, broad ionization threshold of the (2 10) isomer will be of considerable interest. This proposal calls for further experimental investigation of the PIE curves of this nonrigid heterocluster. This research was supported by the Binational German-Israeli James Franck Program for lasermatter interaction.
References [ 1 ] A. Amirav, U. Even and J. Jortner, 1. Chem. Phys. 75 ( 198I ) 2489. [ 21 U. Even, A. Amirav, S. Leutwyler, M.O. Ondrechen, Z. Berkovitch-Yellin and J. Jortner, Faraday Discussions Chem. Sot. 73 (1982) 153. [ 31 S. Leutwyler and J. Jortner, J. Phys. Chem. 91 (1987) 5558. [ 41 S. Leutwyler and J. BSsiger, Chem. Rev. 90 (1990) 489. [ 51J. Btisiger and S. Leutwyler, Phys. Rev. Letters 59 (1987) 1895. 1611S. Leutwyler and J. Blisiger, in: Large finite systems, eds. J. Jortner, B. Pullman and A. Pullman (Reidel, Dordreeht, 1987) p. 153. I7‘1J. Biisiger, R. Knochenmuss and S. Leutwyler, Phys. Rev. Letters 62 (1989) 3058.
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[SIN. Ben-Horin, U. Even and J. Jortner, Spectroscopic Interrogation of Heterocluster Isomerization, submitted for publication. [9]N. Ben-Horin, U. Even, J. Jortner and S. Leutwyler, Spectroscopy and Nuclear Dynamics of Tetracene-Rare-Gas Heteroclusters, to be published. [ IO] J.E. Adamsand R.M. Stratt, J. Chem. Phys. 93 (1990) 1358; L.E. Fried and S. Mukamel, Phys. Rev. Letters, in press. [ 1 I ] M. Schmidt, M. Mom, J. Le Calve, P. Millie and C. CossartMagos, Chem. Phys. Letters 183 ( 1991) 69. [ 121 M. Schmidt, M. Mons and J. Le Calve, Chem. Phys. Letters 177 (1991) 371. [ 131 M.J. Ondrechen, Z. Berkovitch-Yellin and J. Jonner, J. Am. Chem. Sot. 103 ( 1981) 7686. [ 141 M.J.S. Dewar and W. Thiel, J. Am. Chem. Sot. 99 (1977) 4899. [IS] P.-O. L&din, J. Chem. Phys. 18 (1950) 365. [ 16] E.Q. Stout Jr. and P. Politzer, Theoret. Chim. Acta 12 (1968) 379. [ 171 P.R. Couchman and C.L. Ryan, Phil. Msg. 37A (1978) 369. [ 181J. Jortner, Ber. Bunsenges. Physik. Chem. 88 ( 1984) 188.
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[ 191J. Jortner, D. Scharf, N. Ben-Ho&, U. Even and U. Landman, in: Proceedings of the International School of Physics “Enrico Fermi”, Course 107, The chemical physics of atomic and molecular clusters, ed. G. Stoles (NorthHolland, Amsterdam, 1990) p. 43. [20] R.S. Berry, T.L. Beck, H.L. Davis and J. Jellinek, Advan. Chem. Phys. 70 ( 1988) 74. [21] G. Franke, E. Hilfand L. Polley, Z. PhysikD 9 (1988) 343. [22] T.L. Beck, J.D. Doll and D.L. Freeman, J. Chem. Phys. 90 (1989) 5651. [23] N. Ben-Horin, H. Sanderovitch, U. Kaldor, U. Even and J. Jortner, An Electrostatic Model for the Energetics of Large Heterocluster Ions, submitted for publication. [24] K.M. Fund, W.E. Henke,T.R. Hays, H.L. Selzle and E.W. Schlag, J. Phys. Chem. 85 ( 1981) 3560. [25] K. Rademan, B. Brutschy and H. Baumglrtel, Chem. Phys. Letters 80 ( 1983) 129. [26] P.D. Dao, S. Morgan and A.W. Castleman Jr., Chem. Phys. Letters113 (1985)219. [27] N. Ben-Horin, U. Even and J. Jortner, J. Chem. Phys. 91 (1989) 331.
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Rigid and nonrigid benzene-AT2 van der Waals heteroclusters Narda Ben-Horin,
Uzi Even and Joshua Jortner
School ofChematry, Tel-Aviv Universily, Tel-Aviv 69975.Israel
Received 27 September 1991
The nonrigidity of the one-sided (2 IO) isomer of benzene,Ar2 is manifested by the occurrence of a hierarchy of distinct isomerization processes and by the marked broadening of the photoionization efficiency curves, which were explored by constant energy molecular dynamics simulations.
M.A,, heteroclusters, consisting of an organic aromatic molecule (M) bound to rare-gas atoms (A), provide interesting information regarding the size dependence of energetic, structural, thermodynamic, dynamic and chemical properties of large finite systems [ l-4 1. In this context, various isomerization processes were identified by molecular dynamics (MD) and Monte Carlo computer simulations in carbazolesAr, [ 4-71, 9,l 0-dichloroanthracene*An (A=Ar, Kr) [8], tetracene.Ar, [9] and benzene.Ar,, [ lo,1 1] heteroclusters, which involve correlated surface motion (of A atoms on one side of M), surface melting (i.e. uncorrelated motion of A atoms on the microsurface of M), side crossing (of A atoms from one microsurface of A to the other), wetting-nonwetting (or two-dimensional (2D) to three-dimensional (3D) ) transition and a rigidnonrigid transition. These studies [4-91 established the hierarchical occurrence of several isomerization processes for a single heterocluster composition and the sequential occurrence of distinct isomerization processes with increasing the size of the M.A, heterocluster for a given M. This isomerization pattern is specified by the finite surface density p,=n/(M microsurface area) of the rare-gas atoms [8 1. Benzene.Ar,, heteroclusters, which are characterized by a large value ofp,, are expected to exhibit a hierarchy of isomerization processes even for small values of n. In addition, isomerization phenomena are governed by the initial isomer configuration, being distinct for interior and for surface states of M in or on the heterocluster [&lo]. In order to explore the M
and n size dependence and the isomer specificity of M,A,, heterocluster isomerization phenomena, we have conducted constant energy MD simulations of the benzene.Ar, isomers, which constitute the smallest heteroclustets of the M.A,, family. The recent spectroscopic studies of Schmidt, Mons and Le Calve? [ 11,12] on the spectral shifts and rotational counters of the SO+S, transition and the ionization potentials from So for benzeneeAr heteroclusters, provide compelling evidence for the existence of two structural isomers, i.e. the two-sided ( 1I 1) structure and the one-sided (2 IO) structure of this heterocluster. Our MD simulations of the nuclear motion in So of benzene.Arz isomers reveal that: (i) The (111) structure constitutes a rigid molecule over a broad temperature domain, until it converts to the (2 IO) structure at P 50 K. (ii) The (2 IO) structure constitutes a nonrigid molecule over all the relevant temperature domains (T< 70 K), with an onset of isomerization processes at T> 3 K. (iii) The (2 IO) benzene.Ar, heterocluster exhibits a hierarchy of isomerization processes with increasing temperature. In conjunction, we have performed MD simulations of the ionization potentials of the ( 111) and (2 (0) benzene. Arz heteroclusters. The simulated photoionization efficiency curves are confronted with the experimental data of Schmidt et al. [ 121, elucidating the effects of nuclear dynamics and the implications of molecular nonrigidity for the ionization processes of these heteroclusters. The constant energy MD simulations of benzene-A+ utilized a fifth-order predictor-corrector
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method for the integration of the equations of motion. The integration step was chosen as IO-” s at low temperatures (TG 10 K) and was gradually decreased with increasing temperature, being taken as 2 x lo-‘” s at T> 60 K. The first stage in the simulations involved an equilibration stage, with the initial random velocities being characterized by a Boltzmann distribution to give the desired temperature and with the total angular momentum being set to zero. A short simulation ( 10’integration steps) was then performed with the velocities being frequently scaled to achieve the desired temperature, followed by an uninterrupted short simulation to monitor the attainment of the final temperature. In some simulations we have used gradual heating of an equilibrated configuration for the initiation of the equilibration stage. Subsequently, constant energy trajectories were generated in the ground electronic state for a time of 1.O- 1.5 ns. Energy was conserved in these simulations to better than 1 part in 10’. The benzene molecule was taken to be rigid at its ground state equilibrium geometry. The potential surfaces of benzeneeAr, heteroclusters in the ground electronic state were specified using Lennard-Jones atom-atom potentials [ 13]. The potential parameters for the C-Ar and H-Ar interaction were taken from the prescription of Ondrechen et al. [ 131. Standard parameters were used for the Ar-Ar interaction [4]. We tested the role of electrostatic benzene-Ar chargeinduced dipole interactions within the ground state potential surface in order to ascertain their effects on the energetics and the relative stability of isomers. Calculations of the charge-induced dipole interactions between the atomic charge distribution of benzene (obtained using the MNDO method [ 141 in conjunction with Liiwdin’s charge density analysis [ 15,161) and the rare-gas atom, resulted in the stabilization energy of -6 cm-’ for the equilibrium nuclear configuration of benzenesArI, which constitutes 1.5% of the total ground state binding energy, and can be safely neglected. In each MD simulation the following observables were calculated by averaging over the entire trajectory: (i) The potential energy U. (ii) The standard deviation (7of the distances between the rare-gas atoms and the center of mass of benzene and the Cartesian components I?~(&x, y, z) of d. The molecule-fixed Cartesian coordinate axes were specified 14
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by the origin at the center of mass of benzene, with x (in the molecular plane perpendicular to the C-C bond), y (in the molecular plane passing through two C atoms) and z (perpendicular to the molecular plane). We have also evaluated the normalized standard deviation u, where the two contributions of the Ar atoms are normalized by their distances from the center of mass of benzene, and the normalized three Cartesian components of (T,i.e. o,, aY and CJ=.(iii) The standard deviation 6 of the Ar-Ar distance normalized by the average Ar-Ar distance. The MD simulations of the structure, energetics and nuclear dynamics of the ( 111) and (2 ]0) isomers of benzene.Ar, were performed over a broad temperature domain (T= 3-70 K). Static structural data were inferred from short time (100ps) averaging of lowtemperature (T=3 K) simulations, yielding the equilibrium coordinates (x,, y,, z, ) and (x,, y,, zz) of the two Ar atoms (in A). The following information was obtained for the ( 1) I) isomer: ( 1) Its static low-temperature (3 K) equilibrium structure is (0, 0, 3.47) and (0, 0, -3.47), which corresponds to a D6,, symmetry. (2) With increasing the temperature in the range 3-50 K, this structure (expressed by the average coordinates) is preserved. (3) The potential energy U and the structural fluctuation parameters a, a,, u,,,u=and 6 exhibit a smooth temperature dependence (fig. 1). The increase of the structural parameters with increasing temperature is modest, e.g. (Tincreases from a=0.005 at 10 K to 0~0.06 at 50 K. (4) For T3 52 K the ( 1 ( 1) structure interconverts into the (2(O) structure. This nearly irreversible ( 1 I 1) + (2 IO) side crossing occurs already in the equilibration runs for the ( 11I ) isomer. Regarding the (2 ]0) isomer we observed that: (5 ) The low-temperature (3 IS) static equilibrium structure is (3.24, 0, 3.13) and (-0.4, 0, 3.49), which reflects a low-symmetry (C,) structure. One Ar atom is shifted (by 0.4 A along the x axis) from the benzene-Ar minimum and the second Ar atom is located above the midpoint between two H atoms, with the Arz dimer being slightly tilted relative to the benzene plane. Our structure of the (2 IO)isomer is in agreement with the recent results of Monte Carlo simulations for this system [ 111. This structure
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I
’
’
0.2-
’
’ c’a
/
b 0.1 -
v#_,---
,*
0
IO
20 ““--“p30
40
60
70
80
TP’KI
Fig. 2. The temperature dependence of the standard deviation d (0 ) of the distances between the two Ar atoms and the center of mass of benzene and the Cartesian components (u) &, ( A ) 6,, and ( l ) & of a. Data for the (2 IO) isomer of benzene,Ar,.
0.0
TCKl Fig. 1.The temperature dependence of the internal energy Uand the normalized structural fluctuation parameten u and 6 for the ( 1I 1) isomer (0 ) and for the (2 (0) isomer ( n) of benzene-Arz.
characterizes the heterocluster only at low temperatures, while at finite temperatures (T> 5 K) large amplitude motion sets in. (6) The structural fluctuation parameters for the (2 IO) isomer exhibit a nonsmooth temperature dependence (figs. 1 and 2), which marks the onset of a hierarchy of the following isomerization processes: (a) Onset of correlated one-dimensional ( 1D) motion at Tz3 IL The nuclear motion of the Ar, dimer along the x axis is enhanced with raising the temperature, as manifested by the rise of u, (fig. 2). (b) Onset of correlated 2D surface motion at 10 K. A steep rise in gYand a slight rise in a= are exhibited (fig. 2), while a, and g have already attained a high value (figs. 1 and 2). Concurrently, 6 does not reveal a break. These features mark the onset of a correlated 2D surface rotational motion of the Ar, dimer parallel to the benzene surface. These MD data concur with recent Monte Carlo simulations, which reveal that the six equivalent H-C-C-H wells of the
(2 I0) isomer of benzene.Ar* are equally explored at 10K [ll]. (c) Onset of uncorrelated surface melting at 5254 K. Here u and 6 exhibit a simultaneous increase of slope. (d) Onset of side crossing at 47-54 K. Here a, exhibits a break followed by a sharp increase. The side crossing involves the motion of one Ar atom from (2 (0) to the other side of the microsurface and back. The lifetime of the ( 1 ) 1) structure is 5 5 ps. The distinction between processes (c) and (d), which occur in the same temperature domain, was inferred from analysis of MD trajectories, which also sets the temperature limits for the onset of these isomerizations. (e) Onset of 2D+ 3D transition at 62 K. MD trajectories revealed the transition from the (2 (0) structure to the ( I+ 1ZIO) structure, which contains one Ar atom in the first solvation layer (ZZ 3.5 A) and one Ar atom in the second solvation layer (2x7.0 A). The lifetime of the 3D (1+ 1210) structure was 3-5 ps. (7) At T2 73 K an eficient fragmentation process (2 IO) + ( 1I0) +Ar sets in. Characteristic time scales for fragmentation of a single atom are ~300 ps at this temperature. We are now in a position to compare the energetits and nuclear dynamics of the ( 111) and (2 IO) isomers. (8) The potential energy CJof the ( 111) structure is lower than that of the (210) structure over the 75
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temperature range 3-52 K (fig. 1). At the lowest temperature the ( 111) structure is energetically more stable than the (2 10) structure by AU= 55 cm-‘, in accord with the estimate of Schmidt et al. [ 121, with the energy difference being reduced to AU= 10 cm-’ at 50 K. The energetic stability of ( 11I ) does not preclude the simultaneous formation of ( 11I ) and (2 ]0) isomers in supersonic expansions of Ar seeded with benzene [ 121. (9) In the temperature range 3-52 K the ( 111) and (2 IO) isomers do not interconvert on a time scale of I ns. ( 10) At temperatures above the onset of side crossing T> 52 K, the (2 10) structure is dominant. The dominance of the (2 ]0) structure at high temperatures is reflected by the nearly irreversible (l]l)-r(2]0) side crossing of (111) at T>52 K (point (4) ) and the short 5 5 ps lifetime of the ( 1 I 1) structure originating from the (2]0)+ ( 1 11) side crossing (point 6 (d) ). Despite the energetic stability of the ( 1) 1) structure (point (7) ) the prevalence of (2 IO) is due to entropy contributions. (11) The structural fluctuation parameters are systematically higher for the (2 10) isomer than for the ( 111) isomer over the entire temperature coexistence domain (fig. 1). The high values of the structural fluctuation parameters, e.g. 0> 0.1 at T> 10 K, for the (2 IO) isomer (fig. 1 ), reflect the nonrigidity of this structure. According to the celebrated Lindeman criterion [ 171, which was applied to finite systems [ 17-201, the onset of a rigid+nonrigid isomerization in clusters is characterized by the normalized structural fluctuation parameters exceeding the value of 0.1. Accordingly, the onset of the nonrigid (2 IO) structure of benzene*Arz is specified by ui~ 0.1, being obtained at Tb 10 K. In contrast, for the (1) 1) benzene.Ar? we find that (T<0.1 and the structure is rigid over the entire temperature existence range of this isomer. This analysis demonstrates the occurrence of hierarchical isomerization in the “small” benzene.Ar, (2 IO) heterocluster. Some details of the analysis will have to be modified for the low temperature onsets (T< 5 K) where quantum effects [ 2 1,221 will be of some importance, but the general picture, which involves a proliferation of Ar, isomerization processes for benzene.Ar,, is quite clear. Correlated surface motion (CSM ) was previously simulated for (2 10) 76
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structures of carbazole,Ar, [4,5], tetracene*Ar, [ 91 and 9,10-dichloroanthracene.Kr, [S]; however, the CSM constitutes the only isomerization process which was identified for these heteroclusters. For the benzene.Ar, heterocluster rich and diverse isomerization phenomena (processes 6(a)-6 (e) ) were recorded, which were previously simulated only for large heteroclusters, i.e. for 9, IO-dichloroanthracene*A, (A=Ar, Kr) side crossing was documented for n 3 5 while the 2D+ 3D transition was observed only for n> 10 [ 81. The present analysis demonstrates that the smallest heterocluster of the MeA,, family (as far as the size of the aromatic molecule and the coordination number are concerned), which exhibits distinct isomers, reveals a rich manifold of isomerization processes. This state of affairs bears a close analogy to the isomerization of small neat raregas clusters, where Ar, exhibits a rigid-+nonrigid transition [ 201. An attempt was made to elucidate some of the spectroscopic manifestations of the nonrigidity of benzene,Ar, (2 IO) by the simulation of the adiabatic ionization potentials and the photoionization efficiency (PIE) curves of the ( 1 I 1) and (2 IO) isomers, which were measured by Schmidt et al. [ 121. The shift 61 of the ionization potential of benzene*Ar2 at a fixed nuclear configuration relative to the bare molecule was calculated from the electrostatic stabilization energy of the ionic state of the heterocluster relative to its neutral ground state. The ionization potential shift at the ground state nuclear configuration {rA)=rl, r2 (where r, and r2 are the temporary coordinates of the two Ar atoms) is SV({r,})=S1/(+)({r,})-SV(“)({r,}), where SVc+)({rA}) and SV(“)({rJ) are the sum of the charge-induced dipole interactions and the induced dipole-induced dipole interactions for benzeneArZ and for benzene.Ar2, respectively. The details of the calculations of SF+) and 8Vco) are given elsewhere [ 8,231. The charge distribution of benzene and of benzene+ were calculated from o and R molecular orbitals obtained by the MNDO method [ 141 with the atomic charges being obtained using the Liiwdin procedure [ 15,161, The ionization potential shifts 6V( {rA)) for a fixed ground state nuclear configuration were summed over the entire trajectory. The distribution P(6V) of the values of 6V exhibits a continuous change from zero to a constant
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value, which is normalized to unity. P(6Y) represents the simulated PIE curve. The simulated ionization potential shift 61 was calculated by a linear extrapolation of P( 6 V) to zero. The width, W, of the photoionization PIE curve was somewhat arbitrarily evaluated from the difference between the upper and lower values of 6 Vwhere P( 6 V) starts deviating from a constant value (fig. 3). Fig. 4 displays the temperature dependence of 61 and W, for the two isomers which reveal the following features: ( 1) 61 is systematically lower for the ( 11I ) relative to the (2 IO) isomer, reflecting the enhanced electrostatic stabilization of the benzeneaAr,+ ( 1 11) structure. We note, however, that for both isomers Sl< 0. (2) I1511increases with increasing temperature for both isomers, manifesting the role of thermal nuclear motion on the decrease of the electrostatic stabilization. The overall change of the 62for both ( 1 I 1) and (2 IO) is small, i.e. x 20% over the temperature range studied herein. (3) W reveals a marked temperature deDendence
0.8
_ 0.6 > C.0 z 04
&km-1) Fig. 3. The PIE curves for the ( 11I ) isomer (-) at I7 and 25 K andforthe(2lO)isomer(---)atl6and23KThetempcralures are marked on the curves. The PIE curve widths W, and W,, which are marked by horizontal arrows, correspond to the width for the ( I 11) isomer at 17 K and to the width for the (2 10) isomer at 16 K, respectively.
-500 300 -
1
I
I
40 11°K)
I
1
60
1
80
Fig. 4. The temperature dependence of the ionization potential shifts SI and for the PIE curve widths W of the ( 11I ) isomer (Cl) and the (210) isomer (m)ofbenzene.Ar,
for both isomers, increasing from W= 20 cm- ’at 10 K to W=150 cm-’ at 50 K for (111) and from W=70 cm-’ at 10 K to W=270 cm-’ at 75 K for (210). (4 ) Over the experimentally relevant temperature domain T=3-40 K, the width of the PIE curve is considerably broader for (2 IO) than for ( Ill). As evident from fig. 3, the larger value of W for (210) reflects the nonrigid structure of this heterocluster. We can now confront the results of the simulations with the experimental data. Schmidt et al. [ 12] have observed a marked qualitative and quantitative difference between the PIE curves for the (1 I 1) and (2 IO) isomers of benzene.Ar,. For the ( 1 I 1) isomer a sharp PIE threshold W(exp.)=25 cm-’ with Gl(exp.) = - 334 cm-’ is observed, which was attributed to the adiabatic threshold [ 121. Our calculated values for the ( 1 I I ) isomer (fig. 4) are: &I=-445 cm-’ and W=25 cm-’ at 10 K, and 61= -428 cm-’ and W=35 cm-’ at 20 K, being in reasonable agreement with experiment (with the experimental data [ 12] presumably corresponding to the vibrational temperature domain of lo-20 K). The situation is drastically different for the (2 IO) 77
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isomer for which a broad ionization threshold was experimentally recorded [ 12 ] with W( exp. )x 100 cm-’ and an apparent ionization potential shift of GZ(exp.) = + 137 cm-‘. The calculated W values for the (210) isomer, W=70 cm-’ at 10 Kand W=90 cm-’ at 20 K, are in good agreement with the experimental value for W [121. On the other hand, the calculated value of SZfor (2]0), i.e. Sl= -365 cm-’ at 10 K and 6Z= - 350 cm-’ at 20 K differs both in sign and in magnitude from the experimental value of sZ( exp. ) = + 137 cm-’ [ 121. The positive experimental value of 61 for the (2 (0) isomer implies the apparent energetic destabilization of the ionic state relative to the ground neutral state of the heterocluster. This conclusion contradicts our results regarding the energetic stabilization of the ionic state as well as the ubiquity of experimental data for other heteroclusters, which always show that 61~ 0 [ 8,2327]. Schmidt et al. [ 121 have argued that the positive experimental value for 6Z(2 IO) corresponds to a nonadiabatic ionization process, which is accompanied by a large intermolecular configurational change. This proposal is in variance with our calculations. Fig. 3 reveals that 6Z< 0 for all the nuclear configurations of (2 10) which contribute to the PIE curve. Our calculations thus provide a semi-quantitative description of the dramatic broadening of the PIE curve, but fail to account for the sign reversal of 61 for the (2 (0) isomer. One can argue that this discrepancy reflects the failure of the electrostatic model [ 8,231. We are, however, rather confident in the validity of our electrostatic model because of the following reasons: (i) It accounts well for the size dependence of 61 for other heteroclusters, i.e. 9,10dichloroanthracene. A,, (A= Ar, Kr; n = l-l 8 ) [ 8, 241. (ii) It accounts for the energetics of ionization of the ( 111) isomer of benzene.Ar,. (iii) It accounts for the linebroadening of the PIE curves of both the ( 1( 1) and (2 ]0) isomers, providing discrimination between the rigid ( 111) and the nonrigid (210) structures. In order to reconcile theory and experiment we propose the existence of a lower-energy photoionization threshold of the benzeneeAr, (2 IO) heterocluster, which was not yet experimentally observed. As evident from the PIE curves of bare benzene, benzene.Ar, and benzene.Arz ( 1 I 1 ), two ionization thresholds are exhibited, which originate from the 78
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Jahn-Teller components, i.e. the 0( l/2) ground state and the 6(3/2) excited vibrational state of benzene+ [ 121. For the case of bare benzene the lowest sharp threshold is rather weak [ 12 1. In contrast, the (2 (0) PIE exhibits only a single threshold [ 121. A lower energy, weak and broad PIE threshold of the nonrigid benzene. Ar, (2 IO) heteroclusters may exist, being smeared out by the large energetic spread of its threshold. An estimate of the energy of this “hidden” threshold is of interest. Schmidt et al. [ 121 have observed a single ionization threshold for the (210) isomer at SZ=+137 cm-‘. We propose that this threshold corresponds to a 6( 3/2) vibrationally excited benzene+, which is located at ~355 cm-’ above the vibrationless level. Accordingly, the vibrationless (adiabatic) ionization threshold of (2 10) is expected to be located at 61(2]0)= -218 cm-‘. Thus the splitting between the ( 1 I 1) and (2 10) ionization thresholds is inferred to be 66Z= 6Z( 1 I 1) 6Z(2 10) = - 116 cm- ‘. This possible experimental estimate is in reasonable agreement with our calculated value of S&Z=- 80 cm-’ (at T= lo-20 K). A search for an additional low-energy, broad ionization threshold of the (2 10) isomer will be of considerable interest. This proposal calls for further experimental investigation of the PIE curves of this nonrigid heterocluster. This research was supported by the Binational German-Israeli James Franck Program for lasermatter interaction.
References [ 1 ] A. Amirav, U. Even and J. Jortner, 1. Chem. Phys. 75 ( 198I ) 2489. [ 21 U. Even, A. Amirav, S. Leutwyler, M.O. Ondrechen, Z. Berkovitch-Yellin and J. Jortner, Faraday Discussions Chem. Sot. 73 (1982) 153. [ 31 S. Leutwyler and J. Jortner, J. Phys. Chem. 91 (1987) 5558. [ 41 S. Leutwyler and J. BSsiger, Chem. Rev. 90 (1990) 489. [ 51J. Btisiger and S. Leutwyler, Phys. Rev. Letters 59 (1987) 1895. 1611S. Leutwyler and J. Blisiger, in: Large finite systems, eds. J. Jortner, B. Pullman and A. Pullman (Reidel, Dordreeht, 1987) p. 153. I7‘1J. Biisiger, R. Knochenmuss and S. Leutwyler, Phys. Rev. Letters 62 (1989) 3058.
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[SIN. Ben-Horin, U. Even and J. Jortner, Spectroscopic Interrogation of Heterocluster Isomerization, submitted for publication. [9]N. Ben-Horin, U. Even, J. Jortner and S. Leutwyler, Spectroscopy and Nuclear Dynamics of Tetracene-Rare-Gas Heteroclusters, to be published. [ IO] J.E. Adamsand R.M. Stratt, J. Chem. Phys. 93 (1990) 1358; L.E. Fried and S. Mukamel, Phys. Rev. Letters, in press. [ 1 I ] M. Schmidt, M. Mom, J. Le Calve, P. Millie and C. CossartMagos, Chem. Phys. Letters 183 ( 1991) 69. [ 121 M. Schmidt, M. Mons and J. Le Calve, Chem. Phys. Letters 177 (1991) 371. [ 131 M.J. Ondrechen, Z. Berkovitch-Yellin and J. Jonner, J. Am. Chem. Sot. 103 ( 1981) 7686. [ 141 M.J.S. Dewar and W. Thiel, J. Am. Chem. Sot. 99 (1977) 4899. [IS] P.-O. L&din, J. Chem. Phys. 18 (1950) 365. [ 16] E.Q. Stout Jr. and P. Politzer, Theoret. Chim. Acta 12 (1968) 379. [ 171 P.R. Couchman and C.L. Ryan, Phil. Msg. 37A (1978) 369. [ 181J. Jortner, Ber. Bunsenges. Physik. Chem. 88 ( 1984) 188.
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[ 191J. Jortner, D. Scharf, N. Ben-Ho&, U. Even and U. Landman, in: Proceedings of the International School of Physics “Enrico Fermi”, Course 107, The chemical physics of atomic and molecular clusters, ed. G. Stoles (NorthHolland, Amsterdam, 1990) p. 43. [20] R.S. Berry, T.L. Beck, H.L. Davis and J. Jellinek, Advan. Chem. Phys. 70 ( 1988) 74. [21] G. Franke, E. Hilfand L. Polley, Z. PhysikD 9 (1988) 343. [22] T.L. Beck, J.D. Doll and D.L. Freeman, J. Chem. Phys. 90 (1989) 5651. [23] N. Ben-Horin, H. Sanderovitch, U. Kaldor, U. Even and J. Jortner, An Electrostatic Model for the Energetics of Large Heterocluster Ions, submitted for publication. [24] K.M. Fund, W.E. Henke,T.R. Hays, H.L. Selzle and E.W. Schlag, J. Phys. Chem. 85 ( 1981) 3560. [25] K. Rademan, B. Brutschy and H. Baumglrtel, Chem. Phys. Letters 80 ( 1983) 129. [26] P.D. Dao, S. Morgan and A.W. Castleman Jr., Chem. Phys. Letters113 (1985)219. [27] N. Ben-Horin, U. Even and J. Jortner, J. Chem. Phys. 91 (1989) 331.
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