Anharmonicity effects in cluster isomerization

Anharmonicity V. Bernshtein

24 July 1992

CHEMICAL PHYSICS LETTERS

Volume 195, number 4

effects in cluster isomerization

and I. Oref

Department of Chemistry, Technion - Israel Institute of Technology, Ha$a 32000, Israel

Received 4 May 1992

The dynamics of Na,Cl, clusters is explored and various methods of calculating cluster reaction rates are compared. The Na.&l, cluster undergoes isomerization from a low temperature cube to ladder and ring forms. The potential surface is dotted with low barriers such that back and forth reactions take place and at equilibrium significant concentrations of the ladder and the ring forms are present. Molecular dynamics and statistical methods of calculating rate coefficients are compared and the effects of internal energies and anharmonicities on the rate coefficients are discussed. The effects of a non-steady state between an activated molecule and a transition state on the energy-dependent RRKM rate coefficient are evaluated.

1. Introduction Rare gas clusters were lately the subject of intensive investigation, and their physical characteristics as a function of the energetics and cluster size were explored [ 11. Recently, the salt clusters Na4C14[ 2,3 1, I&Cl4 [ 4 ] and &Cl5 [ 41 were investigated and their structures and dynamics explored. The low temperature Na4C14tetramer in a cube form isomerizes at high temperatures to the ladder and ring form [ 2,3 1, the reaction being depicted by the four rate coefficients scheme: cube*ladder*ring. The potential surface has a reaction coordinate which is shown in fig. 1 and has low barriers in the forward and backward isomerization directions. Thus, forward and backward dynamical calculations are required to elucidate the temporal behavior of an instantaneously excited cubic Na,Cl, cluster. Molecular dynamics (MD) calculations using instantaneous single mode or “statistical” excitation of the Na4C14cube show (a) fast intramolecular energy redistribution (IVR) compared to molecular isomerization lifetimes up to internal excitation energy of 20000 cm- ’ and (b ) back reactions take place and the equilibrium concentration of the intermediate ladder form increases with excitation energy and is Correspondence to: I. Oref, Department of Chemistry, Technion - Israel Institute of Technology, Haifa 32000, Israel.

CL* ‘2h

, r.* 4

I

Ladder

I Y Ring

5228 cm’

I

2554ctn-’

Fig. 1. The reaction coordinate for the isomerization of the Na4C14 clusters.

about equal to the concentration of the ring at high energies. The molecular dynamics calculations were interpreted by fitting the time-dependent results to a simple kinetic scheme which yields the four rate coefftcients: k, (cube+ladder), k2 (ladder-+cube), k3 (ring-tladder) and k4 (ladder-+ring). The high barrier cube-ladder path with rate coefficient k, is the rate determining step at low energies and it shows

0009-2614/92/% 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.

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the largest dependence on the initial excitation energy of the cube of the four rate coefficients of the system. The fact that IVR takes place at a much faster rate than does a chemical change implies that statistical calculations can be performed to obtain the four rate coefficients which define the kinetic scheme. Indeed, RRKM calculations were performed both on the NaCl and KC1 cluster systems. Each rate coefficient in the NaCl system [ 3 ] was calculated by the Marcus expression for k(E) using the frequencies of the molecules and transition states which were calculated from a potential consisting of a point Coulomb term and a Born-Mayer repulsion term [ 21. Thus, no adjustable parameters were used. In order to compare the MD rate coefficients values which did not include rotational effects with the RRKM values, the rotational partition functions and centrifugal corrections were omitted from the latter. It was found that the MD and RRKM rate coefficients were very similar as far as changes in magnitude as a function of the internal energy, but generally the RRKM values were larger, sometimes, by more than an order of magnitude. This discrepancy was attributed to two factors (a) lack of anharmonic corrections to the statistical calculations and (b ) the fact that classical MD calculations do not take into consideration the quantum nature of the system, namely the zero-point energy (ZPE) of molecules and transition states. The zero-point energy effects the dynamical calculations in two ways: (a) the barrier hight changes by the difference between the ZPE of molecule and transition state and (b) by ignoring the flow of ZPE from reactant to transition state and reaction coordinate mode. In salt clusters, where the barriers are very low, the ZPE is significant compared with the internal energy at low energies. Because of that reason, the difference in ZPE between reactants and transition states can be significant compared with barrier heights. It is the purpose of this work to estimate the effect of anharmonicity on the values of the RRKM rate coefficients and compare the latter with those calculated previously [ 2,3 ] by MD.

LETTERS

24 July 1992

2. Theory The RRKM rate coefficient, rotational effects excluded, is given by the expression k(E)=

1 W(E+) /-I p(E*) ’

(1)

where Wand p are the number and density of states respectively and + represents a transition state and * an excited molecule. Anharmonicities effect the value of k(E) through the number and density of states. It can be seen immediately, that there is a cancellation effect in eq. ( 1) and even though Wand p each may individually increase substantially, the ratio will be effected to a lesser extent. For cyclopropane the changes between harmonic and anharmanic sums of states vary between 0.2% at 10 kcal/ mol to a factor of 2 at 100 kcal/mol [ 5 1. This is to be compared with only 30% change in the rate coefficient for the isomerization of cyclopropane when anharmonicity was introduced into the expression of k(E) [ 6 1. This relatively small change in the value of k(E) is due to the above-mentioned cancellation effect. The anharmonicities of the Na,Cl, clusters and their transition states are not known and cannot be extracted from any procedure which calculates the normal modes of a molecule. Therefore, we have assigned Morse potentials and anharmonicities to the normal modes and calculated densities and number of states by the direct-count method, which utilizes the Stein-Rabinovitch algorithm [ 5 1. Using the frequencies of the clusters and the transition states reported before [ 3 ] and barrier heights with and without corrections for ZPE differences, we have calculated the ratio kj( anhar) /ki( har) for a series of anharmonicity constants xti, i indicates the species and j the mode to which this anharmonicity was assigned. The results of the calculations are reported in section 3.

3. Results and discussion For a hydrocarbon molecule where the frequencies span the range 300-3000 cm-‘. Stein andRabinovitch [ 51 have, for cyclopropane, chosen x=0.025 for the 3000 cm-’ C-H stretching fre-

418

quencies and 0.0073-0.00845 for lower frequency values of the normal modes. Tou and Liu [ 71 have chosen x=0.025, 0.01 and 0.007 for a contrived molecule with normal mode frequencies of 3000, 2000 and 1000 cm-’ respectively while Schlag et al. [ 81 have, for cyclopropane, chosen anharmonicities in the range x0.05-0.01 5. We have chosen, therefore, x in the range of 0.02-0.001 and calculated the ratio k( anhar) /k( har) at various values of x. The 18 frequencies of the Na,Cl, clusters are almost identical with an average value of z 150-200 cm- ’ depending on the species [ 2,3 1. Therefore, one value only of x was assigned to each species. The frequencies are given in refs. [ 2,3 ] and the parameters for the RRKM calculations are given in table 1. Fig. 2 shows values of the ratio k(anhar) /k( har) as a function of energy for various values of x when x is assigned the same value for excited molecule and activated complex. At the largest value of x, k, (C-+L) (anhar) is smaller than k( har) by a factor of 3 while for other values of x it is smaller by a factor of 2 or less. k2 (L-C ) (anhar ) is smaller than k(har) at all values of x at low energies ( < 12000 cm-‘) and larger than k(har) for x=0.01 and 0.02 at high energies. k3 (R-L) shows a similar behavior high energies and x=0.02, to k2. At k,(anhar)>k(har).Thecaseofk,(L+R)istheabberation, since, unlike other k, for low values of x, k(anhar)> k(har) while for large values of x, k( anhar) < k(har). The fact that in some cases k( anhar) > k( har) indicates that for high values of anharmonicity constants the number of states increases faster than the densities and at high energies it causes k(anhar) to be larger than k(har). To check whether anharmonicity corrections can

ring CL LR

CUBE-

LADDER

1.0.

0.001

;.-

L LADDER

2.0

NCUBE

, o,02

F

LADDER-

RING

-b

2.0 I

RING-

LADDER

.51

o.om 2000

2oooo

loo00 (cm-‘)

Fig. 2. Ratio of anharmonic to harmonic rate coefficient versus excitation of the Na4C14cluster.

E,“) (cm-‘)

Xb’

L"

5228 (5114) (C-L) 1702 (1729) (L-C) 496 (427) (L-R) 1468 (1438) (R-L)

0.13 0.13 0.13 0.09 0.13 0.11

6 1

‘) Barrier height (ZPE corrections in parentheses). b, Anharmonicity constants. ‘) Path degeneracy.

1.5-

ENERGY

Table 1 The parameters used in the RRKM calculations

cube ladder

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account for the total deviation of k(RRKM) from k( MD) we have calculated the four rate coefficients with values of x chosen to give the largest deviation of k( anhar) from k(har). The anharmonicities of the transition states were identical for the forward and backward reactions. E.g. the values of x for the cube to ladder transition state were identical to those of the ladder to cube reaction. To resemble the MD calculations as much as possible, the results pre419

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sented in figs. 3 and 4 are for RRKM calculations without ZPE corrections. As can be seen from fig. 3, k( anhar) values are closer to the MD values than the k(har) values. In no case is it possible to obtain complete agreement nor could one be expected. An-

24 July i 992

CUBE--

LADDER

12

f\

_______-__-------

.f LADDER-

12 -

LADDER -

CUBE

CUBE ______----

RING -

LADDER

LADDER-

RING

G s RING -

B

LADDER

-f

9LADDER-

RING

ENERGY Fig. 4. Ratios of k(anhar)/k(MD) (-_) versus excitation energy.

100

10000

_I

17000 ENERGY

24Qoo

31000

km-’ 1

Fig. 3. Log rate coefficient versus excitation energy (---) RRKMharmonic, (-) RRKM-anharmonic, (---) molecular dynamics.

420

(---) and k(har)/k(MD)

harmonic calculations reduce the value of k( RRKM ) by a factor of up to 3 but it is still larger than k( MD), especially at low energies. For k, (C+L) and energy 10600 cm-‘, the ratio k(anhar)/k(MD)z3 and k( har) /k( MD) x 10 a decrease by a factor of about

CHEMICAL PHYSICS LETTERS

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3, but the discrepancy is still a factor of =3. At the same energy, the decrease in the harmonic values of k3 and k4 by anharmonic corrections is by a factor of 0.75 but values of k(MD) are much lower. A comparison between k( anhar) and k( MD) and k(har) and k(MD) over the whole energy range studied is given in fig. 4. In all cases k(anhar) is smaller than k(har) and both are upper bounds to MD rate coefficients. The discrepancy is noticeable especially at lower energies up to 16000 cm-l. At energies above this value, anharmonic values are as close to MD values as can be expected, thus providing a reasonable upper bound to the latter. In addition to anharmonicities which must be considered in the calculations of the RRKM rate coefficients, IVR times impose an upper limit to the validity of the basic assumption of a steady state between excited molecule and activated complex. The present system with low barriers is an ideal candidate for such a failure. To check this point, we have relaxed the steady state constraint and obtained the concentration of the products as a function of time for various values of k(E) . The rate coefftcient for the decomposition of the transition state, k+ taken as lOI s-l, is much larger than k(E) at low energies. Steady state is obtained and the RRKM formulation applies without any reservations. However, at energies where k(E) -k+, the situation, depicted in fig. 5, is totally different. The ratio of the concentration of transition states to the concentration of excited molecules vary with time. It attains steady state only after x 50% of the products were formed after elapsed

1

I 1.0 -

N+/N"

24 July 1992

time of 0.2 ps. In “normal” reactive systems with larger barriers and lower excitation energies the problem does not exist since k(E) is small compared to k+. It is only here, where the barriers are very low and applied energies are high, that conventional RRKM calculations fail. This combination of constraints leads to what might be called a non-intrinsic non-RRKM case, whereby a normal RRKM molecule behaves in an esoteric way at high energies and short times. This is to be contrasted with an “intrinsic non-RRKM” [ 91 molecule which might have a bottleneck in phase space. Such a species, yet to be found in polyatomic molecules, shows non-statistical behavior at lower excitation energies and longer times. In conclusion, RRKM calculations do not apply at high energies where the steady state assumption, which is a basic tenet of RRKM theory, fails. The results and conclusions reported here are within the range of applicability of the statistical theory. There is no doubt that some “improvement” is obtained by using anharmonic densities and number of states in the calculations of k(RRKM), the latter being an upper bound to molecular dynamics calculations. However, there must be additional sources for the discrepancy between the two ks, possibly zero-point energy effects, which should be addressed before a complete understanding of molecular dynamics and statistical methods is obtained.

Acknowledgement This work was supported by the United StatesIsrael Binational Science Foundation, by the German-Israeli Foundation for Scientific Research and Development and by the Fund for Promotion of Research at the Technion.

References [ I] T.L. Beck and T.L. Marchioro, J. Chem. Phys. 93 (1990) 0.4

0.8 TIME (pa)

Fig. 5. Concentration versus time. The asteriks indicates the excited molecule and + indicates the transition state.

1341. [2] A. Heidenreich, J. Jortner and I. Oref, J. Chem. Phys., in press. [3] A. Heidenreich, I. Oref and J. Jortner, J. Phys. Chem., submitted for publication. 14) J.P. Rossand R.S. Berry, J. Chem. Phys. 96 (1992) 517.

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[5] S.E. Stein and B.S. Rabinovitch, J. Chem. Phys. 58 (1992) 2438. [6] P.J. Robinson and K.A. Holbrook, Unimolecular reactions ( Wiley-Interscience, New York, I972 ). [7] J.C.TouandS.H.Lin, J.Chem.Phys.49 (1968) 4187.

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[8] E.W. Schlag, R.A. Sandmark and W.G. Valance, J. Chem. Phys. 40 (1964) 1461. [9] I. Orefand B.S. Rabinovitch, Accounts Chem. Res. 12 ( 1979) 166.