Heats of formation of higher fullerenes from ab initio Hartree—Fock and correlation energy functional calculations

Volume 216, number 3,4,5,6

CHEMICAL PHYSICS LETTERS

31 December 1993

Heats of formation of higher fullerenes from ab initio Hartree-Fock and correlation energy functional calculations Jerzy Cioslowski Department of Chemistry and Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306-3006, USA Received 25 August 1993; in final form 25 October 1993

Standard enthalpies of formation of higher fullerenes are predicted from the corresponding HF/DZP and LYP/DZP total energies, calculated at the optimized MNDO geometries, using fullerene interconversion reactions. Without exception, the higher fullerenes are found to be more stable than CeOon a per carbon atom basis. Inclusion of electron correlation through the approximate LYP density functional additionally stabilizes the higher fullerenes, but has a very small effect on the relative energies of fullerenes of the same size. The calculated enthalpies of formation converge to the limit of 670-680 kcal/mol as the cluster sizes increase.

1. Introduction The C6,, and CT0 clusters are the main extractable constituents of the soot obtained by resistive heating of graphite. However, the soot extract also contains sizable amounts of higher fullerenes, such as CT6, C&, Cs2, and Cs4. These species can be separated by liquid chromatography. The evidence from NMR spectroscopy points to a single Dz isomer present in the CT6 fraction [ 11. In contrast, the CT8 fraction consistsoftwo (CZv+D3) [2] orthree (2xCZ,+ 1 xD3) [ 3 ] isomers. The CsZ and Cs4 fractions are mixtures of at least three (C2+CZv+D3) [ 31 and two ( D2 + DZd) [ 3,4 ] species, respectively. The standard enthalpy of formation of the C6,, cluster has been measured recently. Beckhaus et al. [ 5 ] obtained AH: = 545 f 1 kcal/mol at 298 K for the solid-state enthalpy of formation, whereas the combustion experiments of Steele and co-workers [ 6 ] yielded AH: = 579 f 3 kcal/mol. The latter value is in disagreement with the solid-state A@ of 543 + 4 kcal/mol measured by Kiyobayashi and Sakiyama [ 7 1. Taking into account the excellent agreement between the data from refs. [ 5,7], one can confidently assume that A@(&,, solid) = 544 kcal/mol. In order to calculate AH: (&,, gas), one needs the corresponding enthalpy of sublimation. The value of

AH,,,,=40? 1 kcal/mol at 707 K, obtained by Pan et al. [ 8 1, has been converted [ 61 to AHfub,,,= 56 f 5 kcal/mol. Therefore, one estimates A@( Ceo, gas) at about 600 kcal/mol. This value can be compared with the wide range of theoretical estimates. The MNDO method predicts A@(&,, gas) of 869 kcal/ mol [ 9,10 1, whereas the AM 1 and PM3 calculations yield 973 [ 91 and 811 kcal/mol [ 111, respectively. The value obtained with the MMP2 parameterization of molecular mechanics is 286 kcal/mol [ 91. When carbon group equivalents are used to convert the total energies computed with ab initio electronic structure methods, the resulting enthalpies are 582 [ 121 and 672 kcal/mol [ 131 at the HF/STO-3G and HF/6-3 1G* levels of theory, respectively. It is clear from the above data that the methods of the MNDO family badly overestimate AH: for C6,,, whereas the ab initio estimates fare much better. The solid-state enthalpy of formation of C,, has been also measured by Kiyobayashi and Sakiyama [ 7 1, who obtained AH! (C,, , solid) = 567 + 5 kcall mol at 298 K. Using the corresponding experimental enthalpy of sublimation AHsub, = 43 f 2 kcal/mol at 707 K [ 8 ] and converting it to mRbl = 60 f 5 kcall mol results in the estimate of about 627 kcal/mol for Iwp(C,,, gas). The standard enthalpy of the reaction

0009-2614/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved. SSDZ OOOOS-2614(93)E1325-B

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(70/6OFcio(g)-+Go(g)

PHYSICS

(1)

LETTERS

3 1 December

1993

pertinent molecular skeletons. The Hartree-Fock total energies, the approximate correlation energies, and their respective sums are listed in table 1.

is therefore equal to = -73 kcal/mol. At present, thermochemical data are not known for higher fullerenes. For this reason, in this Letter we report results of ab initio electronic structure calculations that are expected to provide reliable estimates of standard enthalpies of formation of several medium-size carbon clusters.

3. Results and discussion

2. Details of calculations

(N/6O)Go(g)-‘G(g)

HF/DZP total energies of eight fullerenes, namely CeO, C,,,, CT6, CT8 (2 isomers), Cg2 and CS4 (2 isomers), were calculated at the MNDO optimized geometries with the TURBOMOLE system of programs [ 141. In addition, approximate correlation energies were computed from the HF/DZP electron densities using the LYP functional [ 15 1. For the Go, C70, and C,, clusters, the well-known respective Ih, D5d, and Dz structures were assumed. For the C,*, C&, and CS4 species, the lowest-energy structures were considered. In particular, two C& isomers of C7*, called here CzV(A) and CzV(B ), were the subject of our calculations. These isomers correspond to the structures 5 and 4 in ref. [ 16 1. For Cs2, the C2 isomer, equivalent to the structure 3 in ref. [ 171, was selected. Finally, for the Cs4 cluster the low-energy D2 and DXd isomers, corresponding to the structures 22 and 23 in ref. [ 18 ] were considered. A detailed justification of the above selection has been previously published [ 19 1, together with drawings of the

by assuming negligible differences in the zero-point energies (ZPE) and the entropic contributions. Justification for this assumption is based upon the fact that such contributions are proportional to the number of carbon atoms and therefore cancel to a large extent for the reactions (2). The data presented in table 2 clearly show that, without exception, the higher fullerenes are more stable than C6,, on a per carbon atom basis. Approximate inclusion of electron correlations energy through the LYP functional makes this effect even more pronounced. Taking into account the large uncertainties in the experimental data, the calculated enthalpies of - 60.1 kcal/mol (HF/DZP) and -63.9 kcal/mol (LYP/ DZP) for the reaction ( 1) compare favorably with their experimental counterpart derived in section 1, the LYP/DZP value being in a better agreement. The same is true about the agreement between our values and those of - 69, - 64, and - 62 kcal/mol derived respectively from the published MNDO [ 9,201, AM 1 [ 91 and PM3 [ 111 enthalpies of formation. On the

Table 1 The calculated

‘) E( HF/DZP b, E(LYP/DZP ‘) E(LYP/DZP

390

Hartree-Fock,

correlation,

The calculated total energies can be converted into the standard enthalpies of fullerene interconversion reactions

and total energies of carbon clusters

Cluster

&F (au) a)

E,,

CSO GO CT6 ( DZ isomer) C,s ( Clv isomer A) CT8 ( Cfv isomer B ) Cg2 (Cr isomer) Cgq (Dz isomer) Cgq ( Dzd isomer)

-2271.618540 -2650.317406 -2877.515630 -2953.264385 -2953.258278 -3104.751805 -3180.517012 -3180.518325

- 12.669664 - 14.787261 - 16.057021 - 16.480822 - 16.481347 - 17.328700 - 17.752406 -17.751968

at MNDO geometry). at MNDO geometry) at MNDO geometry).

(2)

-E(HF/DZP

at MNDO geometry).

(au) ‘)

E,,, (au) ‘) -2284.288204 -2665.104667 -2893.572651 -2969.745207 -2969.739625 - 3122.080505 -3198.269418 -3198.270294

Volume 216, number 3,4,5,6

CHEMICAL PHYSICS LETTERS

Table 2 The calculated energies of the reaction (N/6O)C&g)

-C,(g)

Cluster

UHF (kcal/mol)

A&, (kcal/mol)

C60 C70 C,6 (D2 isomer) C,s (C, isomer A) C,s ( Czv isomer B ) Cg2 (C, isomer) Cs4 (D2 isomer) Cs., (DZdisomer)

0.0 -60.1 -82.9 - 100.6 -96.1 - 129.6 - 157.5 - 158.4

0.0 -63.9 -88.4 - 107.0 - 103.5 - 138.0 - 166.9 - 167.4

other

hand,

the MMP2

ular mechanics

yields

parameterization a very poor

estimate

of molecof - 11

[9]. The agreement is also very poor for the HF/3-2 1G value of - 44 kcal/mol [ 201. Finally, there is a large discrepancy between our values for C,6 and the estimate of - 111 kcal/mol calculated from the published [ 1 ] AM 1 enthalpy of formation. It is interesting to note that, despite the marked effect on the stabilities relative to C& the electron correlation effects have rather a small influence on the differences between the energies of isomers with the same number of carbon atoms. This fact is well illustrated for the &,(A)/C,,(B) pair of isomers of the CT8 cluster. The data compiled in table 3 show that the electron correlation contribution to the energy difference is only - 0.4 kcal/mol, much smaller than the error of at least 2.0 kcal/mol introduced by kcal/mol

31 December 1993

deleting polarization functions from the basis set. Taking into account that the ZPE contribution to the energy difference is estimated at = -0.3 kcal/mol [ 2 11, we find the CzV(B ) isomer to be less stable than its C*“(A) counterpart by ~3 kcal/mol. This estimate is much less than those obtained from tight binding [22] or MNDO [23] calculations, but significantly higher than the HF/STO-3G [ 2 11, HF/dz [21], and HF/3-21G [24] values. Inclusion of electron correlation also decreases the energy difference for the Dzd/D2 pair of the Cs4 isomers (table 4). In this case, augmentation of the basis set with polarization functions appears to be of less importance, presumably because of similar amount of strain present in both isomers. Finally, in table 5 we display standard enthalpies of formation of higher fullerenes predicted with the estimated value of mB(C&,, gas) =600 kcal/mol and the LYP/DZP data from table 2. With the increasing cluster sizes, the enthalpies of formation converge to a limit of 670-680 kcal/mol, which reflects the strain energy introduced by the presence of 12 five-membered rings.

4. Conclusions The conclusions reached from the present calculations are several. First of all, the LYP/DZP total energy calculations carried out at the MNDO opti-

Table 3 The energy difference between the C,,( B) and C,,(A) isomers of C,s Method

Ref.

AE (kcal/mol)

MM3 tight binding MNDO AMI PM3 MNDOC( SCF) MNDOC(BWEN) (at MNDOC(SCF) geometry) HF/STO-3G HF/3-21G (at MNDO geometry) HF/3-21G HF/dz (at HF/STO-3G geometry) HF/6-31G* (5d) (at HF/3-21Ggeometry) LDA HF/DZP (at MNDO geometry) LYP/DZP (at MNDO geometry)

L.211 [221

1.1 6.5 7.1 5.3 3.8 5.4 6.5 1.8 0.5 0.2 1.6 4.0 4.5 3.9 3.5

1231 [231 ~231 ~231 ~31 I-211 t231 ~241 [211 ~241 1251 this work this work

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Table 4 The energy difference

between the Dz., and D2 isomers of Cs4

Method

Ref.

MM3 tight binding MNDO AM1 PM3 sv7s4p sv7s4p (at MNDO geometry) LDA HF/DZP (at MNDO geometry) LYP/DZP (at MNDO geometry)

1261 1271

0.3

[26,28]

0.4 0.4

Table 5 The estimated

standard

enthalpies

0.8

-0.6

this work this work

0.9 0.5

Cluster

A@(gas)

C60 C70 C& C,s C,8 Csz Cs4 Cgq

600.0 a’ 636.1 671.6 673.0 676.5 682.0 673.1 672.6

(Dz isomer) (C,, isomer A) ( Clv isomer B ) (Cz isomer) (D2 isomer) ( DZd isomer)

AE (kcal/mol)

L.261 1261 1261 1261 I291

of formation

PHYSICS

0.4 0.4 1.4

of fullerenes

3 1 December

1993

and Dzd isomers of the Cs4 cluster and energetically almost degenerate, with the energy difference smaller than 1 kcal/mol.

Acknowledgement This work was partially supported by the National Science Foundation under the grant CHE-9224806, the Florida State University and NSCEE (Las Vegas) through time granted on their Cray Y-MP digital computers, the US DOE through its Supercomputer Computations Research Institute, and the donors of The Petroleum Research Fund administered by ACS (grant PRF 25076-G6).

(kcal/mol)

a) Assumed.

mized geometries are expected to provide reliable predictions of heats of formation for medium-size carbon clusters. The calculations show that, on a per carbon atom basis, higher fullerenes are significantly more stable than C6,,, confirming the trend previously observed in the MNDO data [ 9,20,23 1. Inclusion of electron correlation additionally lowers the standard enthalpies of the reactions (2) by 0.350.39 (N-60) kcal/mol. This amounts to a contribution of as much as -9 kcal/mol to the standard enthalpies of the (84/60) C,,(g) -C,,(g) reactions. Second, as expected from its approximately constant contribution per carbon atom, electron correlation energy has a rather small effect on relative stabilities of fullerenes of the same size. On the other hand, the relative energies, appear to be quite sensitive to the quality of basis sets, with the inclusion of polarization functions adding at least 2 kcal/mol to the energy difference between the two Czv isomers of c,s. Third, the present calculations confirm that the D2 392

LETTERS

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