2027
J. Phys. Chem. 1992, 96, 2027-2029 the CCC bending angle, are fixed at those in the experimental geometry of the ground-stateallyl radical.’ An upper state angle of 1 17.5’, rotational temperature of 150 K, and line width of 1.0 cm-’ yield a close match with the rotational structure in Figure 2. The rotational constants used for the simulation are A” = 1.803, E N = 0.328, C N =0.278, A ‘ = 1.619, E’= 0.351, and C‘= 0.288 cm-I. Because allyl is so close to the prolate limit, only AK, = 0 subbands were included in the simulation. We note that a CCC bond angle of 117S0 is quite close to the 118’ predicted for allyl cation by ab initio calculation^^^ and is therefore consistent with our photoelectron results above. The large decrease in A displaces each successive subband origin to the red. In addition, the increase in both B and C causes each subband to shade to the blue. Each of the features to the red of the band maximum is the unresolved P-branch head for a subband with a single value for K,. The band maximum itself consists of several unresolved low-K, P-branch subbandheads. Because of the large change in A, the subband Q-branches do not overlap to produce a central maximum as is typical for asymmetric rotor type A bands. The R-branches produce a broad feature to the blue of the band maximum which, while much less prominent in the spectrum than would be expected from the simulation, is nevertheless present in all scans over the 248.15-nm band. We can speculate that the relative weakness of the R-branches with respect to the P-branches may be due to an electronic transition moment that is not constant over the band. Such behavior would not be surprising for a state that is heavily mixed with other nearby excited Etates. The vibrational structure of the C[22Bl] R[12A2]transition evident in Figure 1 affords no simple interpretation. The gross spacings of 1.390 cm-I can be attributed to excitation of the totally symmetric u7 CCC bending vibration in the excited state.I6 A long progression in that mode is consistent with the decrease in equilibrium CCC bond angle, upon electronic excitation, that comes out of the simulation of the rotational structure of the 248.15-nm band. At a more detailed level of analysis, the richness of the spectrum in Figure 1 is puzzling, especially because the absence of hot bands suggests that sequence bands should also be weak. We are currently considering two possibilities: (i) nonplanarity in the excited state would introduce more allowed bands and further complicate the spectrum with a double-well
-
(15) Wiberg, K. B. Private communication. Calculations were at the HF/6-31G* level. (16) This value for u, for the C[2*B excited state would be similar to that reported for the same vibration in the [12A1(3s)]excited state (ref 14) and the cation (ref 9).
J
potential problem, or (ii) vibronic coupling may allow levels of the 8[l2AI(3s)]state, the onephoton forbidden 3s RydbeJg state whose origin is only 249 cm-l to the red of the C[22Bl] X[12A2] origin, to borrow intensity from allowed transitions to levels in c[22Bl] and appear as extra bands. The coupling could be promoted by one quantum of the uI2CH2symmetric twisting mode of bl symmetry. Callear and Lee2 reported a strongly broadened series of absorpLions centered at 224.9 nm which they assigned to the C[22Bl] X[12A2] transition. None of the bands were rotationally resolved, and no vibronic assignments of individual bands were given. Interestingly, most of the bands in Figure 1 were reported i, that work as “weak bands, almost line-like features” but not as part of the ‘main system”. No regularities were reported in the spacings. We suggest a tentative interpretation for the far-UV spectrum of allyl radical. The C[22Bl] X[12A2]transition, whose origin band lies at 248.15 nm, shows an extended Franck-Condon envelope due, at least in part, to a substantial change in the equilibrium CCC bond angle upon excitation. Approximately 3500 cm-l above the origin, a fast radiationless process markedly shortens the excited-state lifetime, broadening the spectrum and introducing irregularities to the band spacings. This is consistent with our observation that the bands to the blue of 238 nm are progressively weaker and broader in 1 + 1 resonant MPI despite their increasing intensity in the absorption measurements by Callear and Lee. While the spectroscopicstudies presently favor no one radiationless process over another, ab initio calculation^^^ have suggested that a disrotatory closure of allyl radical to cyclopropyl radical is favorable for the C[22Bl]state. If so, time-delayed pumpprobe 1 1 resonant MPI spectroscopy of allyl radical may prove to be an interesting kinetic probe of excited-state dynamics. Further wGrk is undezway to identify and label the vibronic bands of the C[22Bl] X[12A2]transition using selective isotopic substitution, photoelectron spectroscopy, and ab initio calculations.
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Acknowledgment. The authors acknowledge helpful discussions with Professors P. B. Kelly (UC Davis) and J. C. Weisshaar (U. of Wisconsin, Madison). Support from the Department of Energy and the Exxon E d ~ c a t i oFoundation ~l is gratefully acknowledged. Funding from the National Science Foundation for the purchase of the lasers used in this work is also acknowledged. (17) Merlet, P.; Peyerimhoff,S. D.; Buenker, R. J.; Shih, S. J. Am. Chem. SOC.1974, 96, 959. Farnell, L.; Richards, W. G. J. Chem. Soc., Chem. Commun. 1973, 334.
Modeling the Orientational Ordering Transition in Solid Cs0 Michiel Sprik,* Zurich Research Laboratory, IBM Research Division, 8803 Riischlikon, Switzerland
Ailan Cheng, and Michael I;. Klein Department of Chemistry and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323 (Received: December 6, 1991) We propose a new intermolecular potential for Ca molecules that not only reproduces the correct low-temperature structure but also correlates a wide range of experimental properties including the molecular reorientational time in the room temperature rotator phase, the volume change at the orientational ordering transition, and the librational frequencies in the low-temperature phase. Recent structural studies have shown that at low temperatures The the molecules in crystalline Csoare orientationally observed Pa5 structure has been rationalized by noting that *Corresponding author. BITNET address: SPR at ZURLVMI.
0022-3654/92/2096-2027$03.00/0
electron-rich short interpentagon bonds face electron-poor pentagon centers of adjacent molecules.2 To date, static structure ( 1 ) Heiney, P. A.; Fischer, J. E.; McGhie, A. R.; Romanow, W. J.; Denenstein, A. M.; McCauley, J. P., Jr.; Smith, A. B., III; Cox, D. E. Phys. Rev. Leu. 1991.66, 291 1-2914.
0 1992 American Chemical Society
2028 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 optimization and molecular dynamics calculations based on atom-atom potentials4’ have been unable to explain the experimental low-temperaturePa3 structure. They predict a tetragonal rather than a cubic low-temperature phase. The discrepancy in the ordered phase is in contrast to the surprisingly accurate description of the structure and dynamics of the disordered hightemperature phasee5 In this Letter we propose a new intermolecular potential for Ca molecules that not only reproduces the correct low-temperature structure but also correlates a wide range of experimental properties including the molecular reorientational time in the room temperature rotator phase,+* the volume change at the orientat;onal ordering transition, and the librational frequencies in the low-temperature phase.g In addition, in agreement with experiment, our molecular dynamics simulationsusing the new potential model exhibit a sharp ordering transition on cooling below room temperature and on applying a modest pressure to the sample. To achieve this it is necessary to explicitly include interaction sites on the electron-rich short C-C bonds and an electrostatic contribution to the intermolecular potential. The discrimination between two types of bonds can be justified qualitativelyby experimental2and theoretical evidence” indicating that the conjugation of the r-bond network is incomplete. This is reflected, for example, by the fact that the length of a bond shared by two hexagons is shorter than bonds of the pentagons. The explicit representation of multiple bonds in the modeling of intermolecular interactionshas an important precedent in the study of solid nitrogen.12 For this system, it was concluded that the structure of the high-pressure y phase could not be rationalized in terms of atom-atom potentials alone. Instead, a prolate ellipsoid repulsive profile had to be assumed and the atom-atom potential was modified accordingly by placing a third interaction center in the middle of the (triple) bond.12 The situation for Ca at low pressure is similar to high-pressure nitrogen. The reason is that the carbon atoms at the sites where the Ca molecules make contact are pressed together by the large cohesive forces of all the other atoms. Thus, many C-C contacts in solid Ca are much closer than those in graphite. These contacts are optimized when a short C-C bond is wedged into the center of a pentagon. The result is that in the low-temperature cubic phase the distance between short bonds is maximized.2 The spurious tetragonal structure imposed by the simple atom-atom model, on the other hand, is characterized by a crossing of short bonds at minimum ~eparation.~ It can be expected therefore, as in the case of solid N2,that allowing the electron-rich short bonds to interact will affect the relative stability of the two phases. In the first implementation of the interacting bond model, the (2) David, W. I. F.; Ibberson, R. M.; Matthewman, J. C.; Prassides, K.; Dennis, T. J. S.; Hare, J. P.; Kroto, H. W.; Taylor, R.; Walton, D. R. M. Nature 1991, 353, 147-149. (3) Sachidanandam, R.; Harris, A. B. Phys. Rev. Lett. 1991, 67, 1467 (Comments). Heiney, P. A.; Fischer, J. E.; McGhie, A. R.; Romanow. W. A.; Denenstein, A. M.; McCauley, J. P., Jr.; Smith, A. B., 111; Cox, D. E. Ibid. 1991, 67, 1467 (Reply) 1468. (4) Guo, Y.; Karasawa, N . ; Goddard, W. A. 111 Nature 1991, 351, 464-467. ( 5 ) Cheng, A.; Klein, M. L. J . Phys. Chem. 1991,95,6750-6751. Cheng, A.; Klein, M. L.; Phys. Rev. E 1992, 45, 1889. (6) Tycko, R.; Dabbagh, G.; Fleming, R. M.; Haddon, R. C.; Makhija, A. V.; Zahurak, S. M. Phys. Rev. Lett. 1991, 67,1886-1889. (7) Johnson, R. D.; Yannoni, C. S.; Dorn, H. C.; Salem, J. R.; Bethune, D. S.Science, in press. (8) Neumann, D. A.; Copley, J. R. D.; Cappelletti, R. L.; Kamitakahara, W. A.; Lindstrom, R. M.; Creegan, K. M.; Cox, D. M.; Romanow, W. J.; Coustel, N.; McCauley, J. P., Jr.; Maliszewskyj, N . C.; Fischer, J. E.; Smith, A. B., I11 Phys. Rev. Lett. 1991, 67,3808. (9) Heiney, P. A.; Vaughan, G. B. M.; Fischer, J. E.; Coustel, N.; Cox, D. E.; Copley, J. R. D.; Neumann, D. A.; Kamitakahara, W. A,; Creegan, K. M.; Cox, D. M.; McCauley, J. P., Jr.; Smith, A. B., 111 Phys. Rev. E , to be published. (10) Neumann, D. A.; Copley, J. R. D.; Kamitakahara, W. A,; Rush, J . J.; Cappelletti, R. L.; Coustel, N.; McCauley, J. P., Jr.; Fischer, J. E.; Smith, A. B., 111; Creegan, K. M.; Cox, D. M., submitted to J . Chem. Phys. (1 1) Feuston, B. P.; Andreoni, W.; Parrinello, M.;Clementi, E. Phys. Rev. B: Rapid Commun. 1991, 44, 4056. (12) Raich, J. C.; Mills, R. L. J. Chem. Phys. 1971, 55, 1811-1817.
Letters -180 r
. 3
-190
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I - ,
0
,
.
I
100
,
,
,
I
200 TIK
,
,
,
1
,
I
300
Figure 1. Energy (a) and lattice parameter (b) hysteresis loop. Solid curves are molecular dynamics averages evaluated with decreasing temperature and dashed curves with increasing temperature.
60 atomic 12-6 sites C of ref 5 (e = 28 K, ucc = 3.4 A) were supplemented with 30 similar 12-6 sites D located at the centers of the double bonds. The molecule was treated as a rigid framework of atoms and bonds with the geometry taken from ref 11. For the u rameter of the DD interaction we chose a value of uDD = 3.6 The parameter uCDwas set equal to 3.5 A and the coupling strength distributed equally over all interacting pairs; hence eDD = tCD = tCC = e = 12 K. Single bonds were not explicitly represented at this level of approximation. This seemingly minor modification was sufficient to destabilizethe tetragonal phaseSwith respect to the Pa3 structure described in refs 2 and 3. With this potential model we also found another energetically less favorable configuration which is very similar to the configuration proposed in a recent neutron scattering study.I3 Surprisingly, all properties in the disordered phase at 300 K are left virtually unchanged. The stabilization of the Pa3 structure by an interaction center on the electron-rich bonds demonstrates that the orientational ordering is indeed sensitive to fine details of the electronic structure. The ordering transition temperature T, of this first version of the model, however, turned out to be only 110 K, which is more than a factor of 2 lower than the experimental value of T,= 260 K.1s697J3J4We were unable to increase T, by more than 50% by adjusting the potential parameters. Again, experience with the modeling of solid nitrogen suggested a way to further refine the potential. The nitrogen dimer, as a consequence of the triple bond, carries an appreciable electrostatic quadrupole moment.’ The corresponding intermolecular coupling cannot be ignored in a description of the solid. Assuming that the interaction of the electron-rich bonds in Ca also has an electrostatic component, we assigned a negative bond charge q D to the D sites and a compensating positive charge qc = - 4 D / 2 to the C atoms. As a result, the pentagons acquired a total charge of 5qc. The D sites are attracted by this accumulation of positive charge, which in turn enhances the stability of the Pa3 structure. However, in the current formulation of the intermolecular potential, the electrostatic contribution is a secondary effect. The short-range repulsion on the D sites is the term in the potential that contributes most to the orientational ordering. With the effective diameters fixed (UCC = 3.4 A, ucD = 3.5 A, and uDD = 3.6 A) the parameters e and q D were adjusted to improve agreement with the transition t e m p e r a t ~ r e , l . ~ the *~J~J~ (13) David, W . I. F.; Ibberson, R. M.; Dennis, T. J. S . ; Hare, J . P.; Prassides, K., submitted for publication in Eur. Phys. Lett. (14) Taylor, R. Compt. Rend. 1991, 312, 979-982. (15) Stogryn, D. E.; Stogryn, A. P. Mol. Phys. 1966, I I , 371.
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2029
Letters
0
10
30
20
v I cm-' klgm 2. Librational spectrum (Fourier transform of the angular velocity autocorrelation) in the ordered phase at 100 K.
cohesive energy,16 the reorientational relaxation time in the disordered phaseI6v7and the librational spectrum in the ordered phase.I0 With a value of qD = -0.35e and e = 15 K we obtained reasonable overall agreement among all these four properties. Our calculations where performed using a standard constant pressure-constant temperature molecular dynamics algorithmI7and a system of 32 molecules replicated by periodic boundary conditions. The length of a run at a single state point was typically between 50 and 100 ps. The temperature dependence of the total energy at zero pressure is shown in Figure 1 The phase transition is an abrupt event occurring around 215 K. This temperatureis still somewhat lower than the experimental value of 260 K. The transformation is clearly first order with a hysteresis of 30 K. The total binding energy of our model can be compared to the enthalpy of formation AH = 171 kJ mol-' at 707 K reported in ref 16. Extrapolation of the data in Figure 1 to this high temperature yields U = 170 kJ mol-'. The first-order character of the transition is also evident in a plot of the cubic lattice parameter as a function of temperature (Figure 1). The calculated discontinuity agrees with recent measurement^.^ We have also investigated the effect of pressure at 300 K. An increase of only 9 kbar is required to order the system at this temperature. When the pressure is isothermally released again, the reverse transition is observed at 5 kbar. AvI
~
~~
(16) Pan, C.; Sampson, M. P.; Chai, Y.; Hauge, R. H.; Margrave, J. L. J . Phys. Chem. 1991, 95, 2944-2946. (17) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquid Clarendon: Oxford, U.K., 1987.
eraging these two pressures we obtain a point in the P,T diagram at 300 K and 7 kbar. Combined with the point at zero pressure, this gives a rough estimate of the slope of the transition curve. The calculated value of dT,/dP = 12 f 4 K/kbar is in accordance with the experimentally observed coefficient.18 Figure 2 shows the spectral density of the librational dynamics in the ordered phase at 100 K. Three separate modes can be distinguished; the sparse spectrum is likely related to the small system size. The characteristic librational frequency was recently measured experimentally to be 20 cm-l,loin fair agreement with the present model. The same degree of correspondence is also found for the relaxation time 7 of orientational correlations in the disordered phase. In the NMR studies of ref 6 and 7 the value of 7 at 300 K is determined as 12 and 8 ps, respectively. The result of our calculations is 9 f 2 ps. Evaluating the performance of our proposed intermolecular potential model under these diverse conditions, we conclude that (i) the explicit representation of interaction sites on the short electron-rich bonds is essential for understanding the orientational ordering in Cm and (ii) electrostatic interaction is also necessary.2 With the current completely empirical intermolecular potential most experimental quantities could be reproduced within an acceptable margin. The discrepancies that remain indicate a tendency to underestimate the anisotropy of the molecules, a deficiency that can be corrected by refining the parameters. A severe test of the type of model proposed here would be the application to other fullerenes, in particular, a prediction of the low-temperature structure of C70. The crucial point is that the strong intermolecular binding causes the closest intermolecular C-C contacts to in fact be repulsive and thus surprisingly very sensitive to the charge distribution. In this respect our picture of the intermolecular interaction differs from the approach taken in a recent tight-binding calculation of the energy of various structures of the Cbos01id.I~ Acknowledgment. We are indebted to Michele Parrinello and Francois Gygi for many valuable suggestions and comments. We also thank P. A. Heiney, J. E. Fischer, and R. Messmer for valuable discussions. (18) Kriza, G.; Ameline, J.-C.; Jerome, D.; Dworkin, A.; Szwarc, H.; Fabre, C.; Schutz, D.; Rassat, A.; Bernier, P.; Zahab, A. J . Phys. 11991, I , 1361-1 364. (19) Gumarsson, 0.; Satpthy, S.;Jepsen, 0.; Andersen, 0. K. Phys. Rev. krr. 1991, 67, 3002.
