8622
J. Phys. Chem. 1994, 98, 8622-8626
How Unequivocally Do Ion Chromatography Experiments Determine Carbon Cluster Geometries? Douglas L. Strout, Lewis D. Book, John M. Millam, Chunhui Xu, and Gustavo E. Scuseria* Department of Chemistry and Rice Quantum Institute, Rice University, PO Box 1892, Houston, Texas 77251 -1892 Received: April 27, 1994; In Final Form: June 17, 1994”
Ion chromatography experiments on carbon clusters have provided a powerful tool for characterizing the products of the laser ablation of graphite. Using this technique, several families of carbon clusters have been observed, and their role in a plausible fullerene formation process has been hypothesized. In this work, we have examined the experimental mobility results from a theoretical perspective. Our most interesting finding is the existence of a family of three-dimensional 2+4 cycloaddition products whose members match the experimental mobilities of the so-called “ring 111” family over a range of cluster sizes, whereas previous studies have asserted that the “ring 111” clusters are planar. In agreement with previous research, we find that the “ring I” and “ring 11” families consist of monocyclic and bicyclic rings, respectively. However, these families should be broadly defined so as to include ring structures with carbon branches, because short carbon branches have only a negligible effect on cluster mobility. Introduction Carbon clusters have long been of scientific interest,l especially since the discovery2 and recent macroscopic synthesis3 of the carbon cages now known as fullerenes. Numerous attemptsM have been made to explain the processes that occur in fullerene formation experiments. One method for producing fullerenes is the laser vaporization of graphite,2 and a recently developed technique’ known as “ion chromatographyn8has made it possible to study in greater detail the products of graphite vaporization. In an ion chromatography experiment, carbon cluster ions from the vaporization of graphite are mass-selected and then injected into a tube containing an inert buffer gas. The various carbon species drift along the tube under the influence of an electric field and arrive at the other end a t different times, as a function of their cross sections for collision with the buffer gas. The result is an experimental “arrival time distribution,” and the arrival time for each molecule is used to derive its “mobility”, which is directly proportional to the molecule’s velocity in the tube. A computational method9 has been developed for assigning the peaks in the arrival time distribution to certain families of carbon species. This involves calculating the mobilities of candidate structures and comparing them to experimental values. First, an optimized geometry is obtained from electronic structure calculations for a carbon structure. A Monte Carlo integration technique is then applied to derive the candidate’s cross section, which is converted to its mobility. von Helden et al.9 reported the results of ion chromatography experiments in which, for carbon species with more than 30 atoms, they observed five families of molecules. The five families were named “ring I”, “ring 11”, “ring 111”, “3-D ring”, and “fullerene.” That study also included an interpretation of the five arrival time peaks by matching the experimental mobilities to the computed mobilities of structures optimized by the semiempirical PM3 method. In the interpretation of von Helden et al., ring I clusters have the designation ”monocyclic ring” because they have structures that consist primarily of a single large ring. Ring I1 clusters are “bicyclic rings” because they have structures containing two intermediate-sized rings, and in similar fashion, ring I11 molecules are denoted “tricyclic rings” because they are characterized by the presence of three intermediate-sized rings. Fullerene structures are cages of sp2-hybridized carbon atoms. ~~
@
Abstract published in Advance ACS Abstracts, August 1, 1994.
0022-365419412098-8622%04.50/0
3-D rings are not conclusively identified in the von Helden et al. model, but they are suggested to have compact three-dimensional structures. In our research group, Book et al.1° have recently developed a mobility code from the von Helden et al. method9 and used it to explore the properties of computationally-derived carbon cluster mobilities. That study surveyed semiempirical and a b initio methods and presented twoimportant results. First, thecalculated mobilities were shown to be rather insensitive to the specific level of theory employed in the geometry optimization, meaning that fast, inexpensive methods are valid choices. Also, the mobilities were not significantly affected by molecular vibration at temperatures up to 500 K, that is, the mobilities were insensitive to the amount of molecular vibrational energy. Therefore, the use of equilibrium geometries for mobility calculations has been validated. In the present study, weemploy our recently developed schemelo to examine the results of the ion chromatography experiments. This computational study of the experimentally relevant clusters is carried out with the goal of elucidating the products of the laser vaporization and builds on the previous study by von Helden et a1.9
Mobility Studies Carbon molecules with 36,40,44, and 48 atoms are included in this study. These span a range of the most experimentally illustrative clusters. For larger species, the various ring families cannot be resolved in the experiment, and for smaller clusters, some of the five arrival time peaks are not observed.9 For each molecule, geometry optimization is carried out using the CerjanMiller optimization algorithm” on a recently developed tightbinding (TB) potential energy surface for carbon,12 including analytic second derivatives.13 TB is a semiempirical method that includes electronic structure contributions in the potential energy and has been used successfully in recent carbon-related applications.l4J5 The code of Book et al.”J is then used to compute mobilities that can be straightforwardly compared to experiment. In this work, monocyclic rings are studied as members of a larger class that includes slightly smaller rings with carbon branches, since short branches are not expected to significantly impact themobility of large rings. In theC36 case, a 36-membered ring is studied (Figure la), along with a 35-membered ring with a one-atom branch (Figure lb) and a 34-membered ring with a two-atom branch (Figure IC). Table 1 presents energy and 0 1994 American Chemical Society
Determining Carbon Cluster Geometries
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 8623
B
Figure 2. c36 2+2 cycloadducts (experimental mobility 3.30 cm2/V s): (a) cl8+cl8(mobility 3.29); (b) C16+c20 (mobility 3.26); (c) Cl,+C22 (mobility 3.25).
L
8
Figure 1. c36 monocyclic rings (experimental mobility 3.01 cmZ/V 9): (a) a 36-membered ring (mobility 2.89); (b) a 35-membered ring with a one-atom branch (mobility 2.89); (c) a 34-membered ring with a twoatom branch (mobility 2.90).
TABLE 1: Tight-Binding Energetics and Calculated Mobilities for CMMolecules figure label
experimental mobilitya
calculated mobility"
TB relative energy (eV)
monocyclic
la lb
3.01
2+2 adduct
2a 2b 2c 3a 3b 3c 4a 4b 4c Sa
2.89 2.89 2.90 3.29 3.26 3.25 3.64 3.62 3.65 4.48 4.58 4.44 6.00
18.8 21.2 19.5 20.0 20.6 20.1 18.6 19.0 19.3 12.5 11.9 12.2
molecule class
IC
2+4 adduct graphitic sheets fullerene 0
3.30 3.70 4.58 5.93
0.0
Mobility expressed in units of cm2/(V s).
TABLE 2 Calculated Mobility of C ~ Monocyclic S Rings (a 36 - D Membered Ring with an n-Membered Branch) n
mobility
0 1 2 3 4 8 12 16 20 24 expt9
2.89 2.89 2.90 2.89 2.89 2.88 2.85 2.82 2.78 2.72 3.01
mobility data for these three isomers. The calculated mobilities for the monocyclic rings are all within a few percent of each other and the experimental result. To fully illustrate that small branches have a negligible effect on mobility, Table 2 shows mobility data for other c36 monocyclic rings with longer branches than the isomers shown in Figure 1. These data indicate that structures with branches up to eight atoms long have essentially the same mobility. Therefore, these structures would be indistinguishable in the ion chromatography experiments, and all should be included with the family of monocyclic rings.
Figure 3. Two perpendicular views of three C36 2+4 cycloadducts (experimental mobility 3.70 cm2/V s): (a) Cl8+Cl8 (mobility 3.64); (b) c16+CzO7where C20 contributes four atoms to the cycloaddition (mobility 3.65); (c) c16+CzO, where CIScontributes four atoms to the cycloaddition (mobility 3.62).
Previously? the bicyclic ring peakin the arrival timedistribution has been attributed to 2+2 cycloaddition products. In Figure 2, we present examples of these molecules for c36. In this work, c36 2+2 cycloadducts of clE+Cl8, C16+C20, and C14+C22 are studied. Thedata in Table 1 illustrate that all thesestructures have similar mobilities and would be indistinguishable by ion chromatography techniques. The energy data demonstrate the influence of the stability of aromatic (Hiickel4n+2) rings, as illustrated by Table 1, in which cl8+cl8and C14+C2z are both lower in energy than c16+c20*
In the tricyclic ring family, we differ from the earlier interpretation9 that proposes planar structures as a cause of the ring I11 arrival time peak. As there is support for 2+2 cycloaddition products for bicyclic rings, we propose products of another bimolecular reaction, namely 2+4 cycloaddition, as candidates for ring 111. Figure 3 illustrates 2+4 cycloaddition products for C36, including CIE+C18 and two isomers of C16+C20
8624 The Journal of Physical Chemistry, Vol. 98, No. 35, 1994
Strout et al.
(d)
(C)
Figure 5. Fullerenes: (a) C36; (b) Ca; (c) CM;(d) Ca. @) W
U) (k) (1) Figure 4. Graphitic sheets: row (A) C36; row (B) Cw; row (C) Cu; row
(D)Cia. since either c16 or C20 can supply the four atoms to the 2+4 cycloaddition. Two views of each isomer are shown in Figure 3 in order to demonstrate the three-dimensional nature of these structures. Table 1 indicates that all these c36 2+4 adducts have mobilities consistent with ‘tricyclic ring” experimental mobility, and we propose these structures as candidates for the experimental “tricyclic ring” family, in sharp contrast to the model of planar tricyclic rings previously proposed.9 These 2+4 cycloaddition isomers are also energetically similar to one another, and therefore, any of these 2+4 cycloadducts could be responsible for the tricyclic ring mobility peak in the ion chromatography experiments. It has been suggested9 that graphitic structures have mobilities that are consistent with the “3-Dring” experimental family. In this work, graphitic structures are examined in detail to compare their mobilities to the experimental values. Figure 4 depicts three c 3 6 graphitic sheets, which is by no means an exhaustive list, and some of these molecules contain pentagons. Table 1 indicates that the mobilities of all these sheets are within a few percent of the experimental value, and the sheets have comparable energies. Therefore, graphitic structures need not consist solely of hexagons to be viable experimental candidates. A single fullerene isomer has also been studied in this investigation, simply toverify that the TB method yields a fullerene structure that has a mobility consistent with experiment. For c36, the fullerene isomer with D u symmetry is chosen, because it is known to be lowest in energy.13 This fullerene is pictured in Figure 5 , and its mobility is given in Table 1. The data show that the calculated mobility of this fullerene agrees well with the experimental value. The study of Cm is carried Out in analogous fashion to c36, and similar agreement between theory and experiment is seen. The C a results are shown in Table 3. The monocyclic ring, 2+2 cycloadduct and 2+4 cycloadduct isomers studied are completely analogous to c36 and are not pictured. ‘Monocyclic ring” includes a Cm ring, a C39 ring with a one-atom branch, and a C38 ring with a two-atom branch. The 2+2 cycloadducts are C20+C20, c18+c22, and c16+c24. The 2+4 cycloadducts include C20+C20 and two isomers of C18+C22. Several Cm graphitic sheets are shown in Figure 4, with energy and mobility results in Table 3. The most stable isomerI3 of Cm fullerene has 0 2 symmetry and is pictured in Figure 5 . Table 3 also contains mobility data for the fullerene. For C44, the monocyclic rings include a Cu ring, a C43 ring
TABLE 3: Tight-Binding Energetics and Calculated Mobilities for C s Molecules molecule class monocyclic 2+2 adduct 2+4 adduct graphitic sheets fullerene
isomer of experimental calculated TB relative figure label mobilitp mobilitr energy (eV) 40-ring 39-ring+l 38-ring+2 20+20 18+22 16+24 20+20 18+22b 18+22c 4d 4e 4f 5b
2.10 2.92 3.26 4.19 5.49
2.51 2.60 2.60 2.92 2.92 2.91 3.26 3.23 3.20 4.21 4.19 4.18 5.68
23.8 25.8 24.1 25.0 24.5 25.1 23.4 23.0 23.2 12.2 12.2 10.5 0.0
Mobility expressed in units of cmZ/(V s). The 22-membered ring contributes four atoms to the cycloaddition. The 18-membered ring contributes four atoms to the cycloaddition.
with a one-atom branch, and a c 4 2 ring with a two-atom branch. The 2+2 cycloaddition products studied as candidates for bicyclic rings are C22+C22, C2o+C24, and C I S + C ~Tricyclic ~. rings studied include the 2+4 cycloadducts of C22+C22 and two isomers of C2o+C24. Figure 4 illustrates Cu graphitic sheets, and Figure 5 shows the lowest energy fullerene isomer of C44.I3 The CM mobilities and energies are shown in Table 4, and consistent agreement is seen between our calculated mobilities and the experimental results.9 Good agreement between theoretical and experimental mobilities is also observed for the C48 clusters. A C48 ring, a C47 ring with a one-atom branch, and a c 4 6 ring with a two-atom branch are studied as monocyclic rings. Bicyclic rings candidates are the 2+2 cycloadducts for c24+c24, c22+c26and c20+c28. 2+4 cycloadducts for c 2 4 + c 2 4 and two isomers of C22+C26 are studied as tricyclic ring possibilities. Graphitic C48 sheets are shown in Figure 4, and the lowest-energy c 4 8 fullerene” is shown in Figure 6. The energy and mobility data for the C48 clusters are shown in Table 5.
Graphitic Sheets: With or without Pentagons The present study demonstrates that, over the range of clusters studied, there exists a widearray of graphiticsheets with mobilities that are consistent with ”3-Dring” experimental results. It has been shown that sheets with pentagons have mobilities and energies comparable to sheets consisting only of hexagons. However, when pentagons are incorporated into graphitic sheets in this study, the pentagons are always on the periphery of the sheet. Interior
Determining Carbon Cluster Geometries
TABLE 4 Tight-BindingEnergetics and Calculated Mobilities for CU Molecules molecule isomer or experimental calculated TB relative class figure label mobility$ mobility energy (eV) monocyclic 44-ring 2.5 2.32 28.5 43-ring+l 2.35 30.5 42-ring+2 2.36 28.8 2+2 adduct 22+22 2.7 2.64 29.2 20+24 2.63 29.6 18+26 2.63 29.2 2+4 adduct 22+22 2.95 2.87 27.6 20+24c 2.94 27.9 20+24d 2.92 28.0 graphitic sheets 4g 4.0 3.92 12.0 4h 3.94 11.0 4i 3.94 13.1 fullerene 5c 5.23 5.38 0.0 Mobility expressed in units of cm2/(Vs). All experimental mobilities except that of the fullerene are estimates based on extrapolationsof the behavior of smaller clusters9 (Figure 8). The 24-membered ring contributes four atoms to the cycloaddition. The 20-membered ring contributes four atoms to the cycloaddition.
Figure 6. Planar tricyclic rings: (a) c36;9 (b) (240;~(c) CU; (d) (248.
TABLE 5 Tight-Binding Energetics and Calculated Mobilities for C a Molecules molecule isomer or experimental calculated TB relative class figure label mobilitya+b mobilitya energy (eV) monocyclic 48-ring 2.2 2.17 32.0 2.15 34.6 47-ring+ 1 46-ring+2 2.16 32.8 2+2 adduct 24+24 2.45 2.39 32.0 22+26 2.40 33.1 20+28 2.39 33.5 2+4 adduct 24+24 2.1 2.66 31.8 22+26c 2.71 31.5 22+26d 2.63 31.6 graphitic sheets 4j 3.7 3.72 10.8 3.73 12.3 4k 41 3.80 10.5 5.18 5.13 0.0 fullerene 5d a Mobility expressed in units of cm*/(Vs). All experimental mobilities except that of the fullerene are estimates based on extrapolations of the behavior of smaller clusters9 (Figure 8 ) . The 26-membered ring contributes four atoms to the cycloaddition. The 22-membered ring contributes four atoms to the cycloaddition. pentagons impart significant curvature to graphitic sheets, and it has been shown9 that graphitic structures with interior pentagons have mobilities that are inconsistent with experimental results. 2+4 Cycloaddition Products: “Tricyclic Rings” That Are Neither Planar nor Tricyclic
The most significant result of this study is the consistent match at all cluster sizes between the mobilities of 2+4 cycloadducts
The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 8625 TABLE 6 Energetic Comparison between Planar Tricyclic Rings and 2+4 Cycloadducts (Relative Energies in electronvolts) cluster size isomer TB MNDO DZBLYP C36 planar (Figure 6a) 0.0 0.0 0.0 2+4 (18+18) 1.3 -0.7 -0.2 0.0 C ~ O planar (Figure 6b) 0.0 0.0 1.5 2+4 (18+22)b -0.1 0.1 Ca planar (Figure 6c) 0.0 0.0 0.0 1.8 2+4 (22+22) 0.4 0.6 c 4 8 planar (Figure 6d) 0.0 0.0 0.0 1.8 2+4 (22+26)’ 0.6 1.1 a Energies at TB-optimized geometries. b Isomer in which the 22membered ring contributes four atoms to the cycloaddition. e Isomer in which the 26-membered ring contributesfour atoms to the cycloaddition. and those of experimental “tricyclic ring” mobilities. The interpretation of the “tricyclic” mobility peak as representing a 2+4 cycloadduct not only provides a viable alternative to the previous planar structure but also has an important advantage over the planar model, namely, that the 2+4 cycloadduct can be rationalized chemically as a product of a bimolecular reaction of two monocyclic rings. Such a reaction (2+4 cycloaddition) would not only be symmetry-allowed in the singlet state under the Woodward-Hoffman rules but would also be carried out with reactants, monocyclic rings with 18-24 atoms, that are not only abundant experimentally9but also energetically favorable when examined by theoretical methods.16 However, in order to advance these 2+4 cycloadducts as a viable alternative to the previous planar structures, it is important to know how the cycloadducts compare energetically to the planar tricyclic rings. To determine whether there is theoretical support for the idea that 2+4 cycloadducts are energetically competitive with planar tricyclic rings, energies are calculated with a wide variety of theoretical methods. Using (236 and cqg as examples, planar “tricyclic rings” are taken from the literature? and these structures are shown in Figure 6. An energetic comparison of these molecules with the 2+4 cycloadducts is presented in Table 6. TB energies are computed, along with semiempirical MND017 energies using the Gaussian 92 package.ls Also in Table 6 are D Z BLYP energy results for TB optimized geometries. BLYP refers to the Beckei9-Lee-Yang-Parr20 hybrid of Hartree-Fock and density functional theory.2iJ2 Hartree-Fock (HF) is employed as the direct self-consistent field method23 in the TURBOMOLE package,24and BLYP has been implemented in a program25 that interfaces with TURBOMOLE. DZ denotes the van Duijneveldt (7s3p) basis set contracted to [4s2pIa26 Planar tricyclic structures for CU and C48 have not been shown in previous literature, but they can be constructed by analogy with c 3 6 and Ca,and they are shown in Figure 6. Energetic comparisons to the 2+4 cycloadducts are also included in Table 6. All results in Table 6 indicate that the 2+4 adduct is energetically comparable to the previous “tricyclic ring.” Therefore, 2+4 cycloadducts are a novel and plausible means of interpreting “tricyclic ring” in the ion chromatography experiments. This new interpretation (that is, 2+4 cycloaddition) is very natural because 2+2 cycloaddition products are supported by previous studies9 and this work, and 2+4 cycloaddition should occur preferentially over 2+2 cycloaddition. 2+2 cycloaddition of smaller monocyclic rings, under the Woodward-Hoffman rules, requires that one of the reactant rings have a triplet electronic state. Since the 2+4 cycloaddition can proceed with rings in the singlet state, which is more stable than the trip1et,Z7the reaction should occur more readily than 2+2 cycloaddition. If 2+2 cycloaddition occurs in the laser vaporization experiments, it should be expected that 2+4 cycloaddition occurs as well.
8626 The Journal of Physical Chemistry, Vol. 98, No. 35, 1994 Conclusions In this paper, we have demonstrated that (a) 2+4 cycloaddition products of monocyclic rings are a viable three-dimensional alternative interpretation of the ring I11 experimental family, (b) graphitic sheets either with or without pentagons are candidates for the 3-D ring experimental family, and (c) monocyclic rings with branches should be included in the ring I family. The existence of 2+4 cycloadducts as products of graphite vaporization has implications for fullerene formation processes. It has already been suggested that 2+2 cycloadducts play a significant role in the fullerene growth and on the basis of our results it is reasonable to assume that 2+4 cycloadducts may also be important in the assembly of carbon cages. Also, the inclusion of pentagons into the graphitic sheets in this study does not provide support for the "pentagon roadB4mechanism because this study does not demonstrate any evidence for the highly curved intermediates involved in the pentagon road. Ion chromatography is a powerful tool for elucidating the products of the laser vaporization of graphite. This technique provides ion mobility information which, in conjunction with theoretical methods, can be used to identify classes of molecules generated in an experiment. However, as shown in this work, in some instances even the identification of classes is greatly uncertain.
Acknowledgment. This research is supported by the National Science Foundation (CHE-9321297) and the Welch Foundation. D.L.S.is a Texaco Foundation Fellow. G.E.S.is a Camille and Henry Dreyfus Teacher-Scholar. References and Notes (1) Weltner, Jr.; W.; van Zee, R. J. Chem. Rev. 1989,89, 1713. (2) Kroto, H. W.; OBrien, S. C.; Heath, J. R.; Curl, R. F.; Smalley, R. E.Nature 1985, 318, 162. (3) KrBtschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Nature 1990, 347, 354. (4) Smalley, R. E. Acc. Chem. Res. 1992, 25, 98.
Strout et al. (5) Heath, J. R. In Fullerenes: synthesis, properties and chemistry of large carbon clusters; Hammond, G. S., Kuck, V. J., a s . ; Washington, DC: American Chemical Society ACS Symposium Series No. 481; pp 1-23. (6) Wakabayashi, T.; Achiba, Y . Chem. Phys. Lett. 1992, 190, 465. (7) Kemper, P. R.; Bowers, M. T. J. Am. Soc. Mass Spectrom. 1990,
I, 197. ( 8 ) Kemper, P. R.; Bowers, M. T. J. Phys. Chem. 1991, 95, 5134. (9) von Helden, G.; Hsu, M.-T.; Gotts, N.; Bowers, M. T. J. Phys. Chem. 1993, 97, 8182. (10) Book, L. D.;Xu, C. H.;Scuseria, G. E. Chem. Phys. Lett. 1994,222, 281. (11) Cerjan, C. J.; Miller, W. H. J. Chem. Phys. 1981, 75, 2800. Baker, J. J . Compur. Chem. 1982, 3, 214. (12) Xu, C. H.; Wang, C. Z.; Chan, C. T.; Ho, K. M. J . Phys. Condens. Matter 1992, 4, 6047. (13) Xu, C. H.; Scuseria, G. E. unpublished results. (14) Strout, D. L.; Murry, R. L.; Xu, C. H.; Eckhoff, W. C.; Odom, G. K.; Scuseria, G. E. Chem. Phys. Lett. 1993, 214, 576. Xu, C. H.; Scuseria, G. E. Phys. Rev. Lett. 1994, 72, 669. (15) Zhang, B. L.; Wang, C. Z.; Ho, K. M.; Xu, C. H.; Chan, C. T. J . Chem. Phys. 1992,97,5007. Wang, C. Z.; Xu, C. H.; Chan, C. T.; Ho, K. M. J. Phys. Chem. 1992,96, 3563. (16) Raghavachari, K.;Strout, D. L.; Mom, G. K.;Scuseria, G. E.; Pople, J. A.;Johnson, B. G.; Gill, P. M. W. Chem. Phys. Lett. 1993, 214, 357.
Raghavachari, K.; Zhang, B. L.; Pople, J. A.; Johnson, B.G.; Gill, P. M. W. Chem. Phys. Lett. 1994, 220, 385. (17) Dewar, M. J. S.; Thiel, W. J . Am. Chem. SOC.1977, 99, 4899. (1 8) Gaussian 92, Revision C; Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Forsman, J. B.; Johnson, B. G.; Schlegel, H. B.;Robb, M.A.;Replogle,E.S.;,Gomperts,R.;Andres, J.L.;Raghavachari, K.; Binkley, J. S.;Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.;Pople, J. A. Gaussian, Inc.: Pittsburgh, PA, 1992. (19) Becke, A. D. Phys. Rev. 1988, ,438, 3098. (20). Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. 1988,837, 785. Mielich, B.; Savin, A.; Stoll, H.; Preuss, H. Chem. Phys. Lett. 1989, 157, 200. (21) Scuseria, G. E. J. Chem. Phys. 1992, 97, 7528. (22) Gill, P. M. W.; Johnson, B. G.; Pople, J. A,; Frisch, M. T. Int. J. Quantum Chem. Symp. 1992, 26, 319; Chem. Phys. Lett. 1992, 197, 499, (23) Almlbf, J.; Faegri, K.; Korsell, K. J. Comput. Chem. 1982, 3, 385. (24) Ahlrichs, R.; BBr, M.; HBser, M.; Horn, H.; K6lme1, C. Chem. Phys. Lett. 1989, 162, 165. (25) Odom, G. K.; Scuseria, G. E. to be published. (26) van Duijneveldt, F. B.;IBM Res. Rep. RJ 945, 1971. (27) Parasuk, V.; Almlof, J. Chem. Phys. Lett. 1991, 184, 187. (28) Hunter, J. M.; Fye, J. L.; Roskamp, J.; Jarrold, M. F. J . Phys. Chem. 1994, 98, 1810.
