The Hartree–Fock magnetizability of C

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Ž . Chemical Physics Letters 285 1998 205–209

The Hartree–Fock magnetizability of C

₆₀

a,1

˚

^a ^b ^b ^c

Kenneth Ruud , Hans Agren , Trygve Helgaker , Pal Dahle , Henrik Koch , ˚

Peter R. Taylor

^d,e

aDepartment of Physics and Measurement Technology, Linkoping UniÕersity, S-58183 Linkoping, Sweden¨ ¨

bDepartment of Chemistry, UniÕersity of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway

cDepartment of Chemistry, Odense UniÕersity, DK-5230 Odense M, Denmark

dDepartment of Chemistry and Biochemistry, UniÕersity of California, San Diego, CA 92186-9784, USA

eSan Diego Supercomputer Center, P.O. Box 85608, San Diego, CA 92186-9784, USA Received 5 December 1997; in final form 9 January 1998

Abstract

Using London atomic orbitals and a recent parallel implementation of our second-order ab initio property code, we have determined the Hartree–Fock limit for the magnetizability of C₆₀ to bey359"5 ppm cgs, in excellent agreement with earlier basis-set limit extrapolations. Using diamagnetic exaltation as a criterion for aromaticity, our calculations show that C₆₀ is an aromatic molecule with a relative diamagnetic exaltation greater than that of benzene.q1998 Elsevier Science B.V.

w x

Since the first observation of C₆₀ 1 , numerous papers have appeared related to the structure, properties and formation of this compound see the recentŽ

w x.

book by Cioslowski 2 . In particular, much atten- tion has been devoted to the magnetic properties of this molecule so as to understand and elucidate its potential aromatic character. The majority of these investigations have concentrated on the nuclear w x shieldings of endohedral nobel-gas atoms in C₆₀ 2 , although calculations of the magnetizability of this

w x compound have also been presented 3–6 .

Ab initio calculations of molecular magnetizabilities are a difficult task, unless special care is taken in constructing the basis set. In practice, for all but the very smallest molecules, it is virtually impossible to reach basis-set saturation for the magnetizability in a

1Permanent address: University of Oslo.

conventional basis. One of the most dramatic illustra- tions of the poor basis-set convergence of the magnetizability occurs for the PF molecule: at the polar-₃ ized double-zeta level, the magnetizability of PF is₃ too large by a factor of about 2.5 and, with more than 500 basis functions, the magnetizability is still

w x about 15% off the Hartree–Fock limit 7 .

One way of overcoming the problem of basis-set convergence in calculations of magnetizabilities is the use of distributed gauge origins. In particular,

w x

London atomic orbitals 8,9 have proven successful in the study of magnetizabilities and rotational g

w x

factors 10,11 . Using correlation-consistent basis sets, it has been demonstrated that the Hartree–Fock limit can be determined to within 2% for molecules containing first- and second-row atoms if London

w x atomic orbitals are employed 12 .

Recently, Lazzeretti and coworkers presented two methods for evaluating the molecular magnetizability

Ž .

PII S 0 0 0 9 - 2 6 1 4 9 8 0 0 0 4 2 - 6

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by a continuous transformation of the origin of the current density with a formal annihilation of the diamagnetic or paramagnetic contribution to the current density CTOCD-DZ and CTOCD-PZ, respec-Ž

. w x

tively 13,14 . Although no direct comparison was made with the London approach, the CTOCD- DZrPZ methods showed promising convergence with respect to extensions of the basis set. In the present Letter, we determine the Hartree–Fock limit of the C₆₀ molecule in a London atomic-orbital basis

w x using a recent direct and parallel version 15 of the

w x

second-order property code dalton 16 . In addition to establishing the Hartree–Fock limit of the magnetizability, we take the opportunity to compare the performance of the CTOCD-DZrPZ approaches with the London atomic-orbital approach.

w x

In a number of papers 12,11,7 , we have advocated the use of the augmented correlation-consistent

Ž .

polarized valence double-zeta aug-cc-pVDZ basis

w x

of Dunning and Woon 17,18 , which has been shown to give magnetizabilities that are less than 2% off the Hartree–Fock limit. However, we have noticed that, for large molecules, the need for diffuse functions is reduced. For coronene, for example, the aug-cc- pVDZ and cc-pVDZ sets give virtually the same results, y293.3 and y292.8 ppm cgs, respectively w x19 . It thus seems reasonable to anticipate that the difference between the Hartree–Fock limit and the results obtained with the cc-pVDZ London basis should be less than 2% for molecules the size of C .₆₀ For the Hartree–Fock calculations presented here, we have therefore chosen to use the cc-pVDZ basis set. For C₆₀ this basis set gives a total of 840 basis functions which, although a large number, is a fairly straightforward undertaking with our computer code.

The geometry we have used is the same as that of

w x Ž .

Ref. 20 , with bond distances ofr CsC s139.15

Ž .

pm and r C–C s 145.55 pm. The dalton program can exploit only subgroups of D . In the case of_2h C , which has icosahedral symmetry I , the sub-₆₀ _h group used was C . All calculations were done on a_2h Cray-T3E in this point group, with the number of processors ranging from 8 to 96 for the various basis sets employed.

Although we do not use a larger basis set than w x that employed by Zanasi and coworkers 5,6 , the calculation of magnetizabilities using London atomic orbitals involves the evaluation of an expectation

value from two-electron integrals differentiated twice with respect to the external magnetic field – effec- tively increasing the angular quantum number of the differentiated orbitals by two – thereby significantly increasing the cost of the calculation. Indeed, the evaluation of the differentiated two-electron integrals takes as long as the full optimization of the wave- function or the iterative solution of the coupled Hartree–Fock equations. Thus, compared to a coupled Hartree–Fock calculation in the same basis without the use of London orbitals, our calculations in general require a 50% longer time. However, the evaluation of the differentiated two-electron integral expectation value is ideally suited for parallelization w x15 and it is also this ‘‘augmentation’’ of the standard basis set that gives us Hartree–Fock limit results even with moderately-sized basis sets.

Our results for the magnetizability of C₆₀ have been collected in Table 1, along with the results of

w x

Fowler and coworkers 3,5,6 . The result obtained

Ž .

with the cc-pVDZ basis set y359 ppm cgs is in remarkable agreement with the basis-set extrapola-

w xŽ .

tion of Fowler et al. 3 y357 ppm cgs , although this might be fortuitous.

In the table, we have listed separately the diamagnetic and paramagnetic contributions to the total magnetizabilities. For the London magnetizabilities, we have used the partitioning advocated by Gauss et

w x

al. 11 , where the diamagnetic term corresponds to the standard diamagnetic term obtained without the use of London orbitals. It is interesting to note that, in the London STO-3G calculations, the diamagnetic

Ž .

term which is a simple expectation value appears to be less well converged than the paramagnetic term.

We also note that, for the no-London magnetizabilities, the slow convergence occurs only in the paramagnetic contribution.

On the basis of previous experience with the convergence of magnetizabilities using London orbitals, we would estimate the cc-pVDZ result to be accurate to about 1.5%, possibly being slightly too

w x

paramagnetic 12 . It thus appears likely that the Hartree–Fock limit may fall between the result ob-

Ž .

tained here y359 ppm cgs and those obtained using the CTOCD-PZ and CTOCD-DZ approaches w x5,6 and that both methods thus appear to give good approximations to the Hartree–Fock limit. We note, however, that the discrepancy seen in Table 1 be-

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Table 1

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Hartree–Fock magnetizabilities in ppm cgs calculated with and without the use of London orbitals

a

dia para

Basis set Reference j j j

Ž .

STO-3G No London This work y13341 4525 y8817

Ž .

STO-3G London This work y13341 12821 y520

Ž .

cc-pVDZ London This work y13073 12714 y359

Ž . w x

STO-3G No London Fowler et al. 3 y13368 4815 y8554

Ž . w x

6-31G) No London Fowler et al. 3 y13430 9515 y3915

w x

Basis set extrapolation Fowler et al. 3 y357

w x

CTOCD-PZ Zanasi and Fowler 5 y365

w x

CTOCD-DZ2 Zanasi et al. 6 y381

w x

CTOCD-PZ2 Zanasi et al. 6 y391

w x a

Experiment Bandow et al. 21 y337

w x b

Bandow et al. 21 y166

w x

From Haddon et al. 22 y260"20

Ascribed to an hcp structure.b

Ascribed to an fcc structure.

tween the no-London STO-3G calculation of the w x present study and the same calculation of Ref. 3 indicate that differences in geometries may affect this comparison.

It is of some interest to investigate the stability of the London atomic-orbital approach for systems as large as C . For this purpose, we have included in₆₀ the table the results obtained with the minimal STO- 3G basis, for which we have calculated the magnetizability both with and without the use of London orbitals. Although the results are in neither case satisfactory, it is still quite remarkable that the Lon- don result is in error by a factor of only 1.45, whereas the no-London result is in error by a factor of 25. Although further calculations are needed to verify this relatively good behaviour of the London STO-3G basis, we anticipate that, for large molecular systems, STO-3G calculations may prove a viable and quick way of obtaining qualitative information about the molecular magnetizability.

Having established the Hartree–Fock limit, it is of interest to compare our results with experiment, quoted in Table 1. Such a comparison is particularly interesting as the magnetizability has been shown to be surprisingly unaffected by electron correlation w23–25 , the change in the magnetizability due tox electron correlation being less than 2%, although

w x

some exceptions are known 26,27 . Furthermore, the magnetizability is also largely unaffected by rovibra-

w x tional effects 28 .

From Table 1, we note that our result is in good agreement with the experimental result ascribed to a hexagonal close-packed phase, for which the properties are assumed to resemble most closely the proper-

w x

ties of the isolated gas-phase molecule 21 . On the basis of a large number of theoretical calculations of the diamagnetic magnetizability of molecules, we have suggested that there is a paramagnetic shift of approximately 10% on going from the gas to the

w x

liquid phase 19,29 . This suggestion is corroborated by experiment in the cases where the magnetizability has been measured in both liquid and gas phases,

w x

with the latter corrected for a calibration error 30,12 . A 10% increase in the experimental number gives a

‘‘gas-phase’’ result of y371 ppm cgs, which is less than 4% from our calculated value and even less considering that we estimate our result to be slightly too paramagnetic. However, the large difference between the magnetizability of the hexagonal close-

Ž . Ž .

packed hcp and face-centered cubic fcc phases w x21 indicates that the use of such ‘‘liquid-phase’’

scalings may be dubious.

The use of the diamagnetic exaltation – that is, the difference between the observedrcalculated magnetizability and the magnetizability estimated on the basis of the sum of ‘‘atomic’’ magnetizabilities – has been advocated by Schleyer and Jiao as the only

w x

unique definition of aromaticity 31 . In light of the discussion in the previous paragraph, it is clear that the use of such an exaltation as a criterion for

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aromaticity has to be based on a consistent set of atomic parameters, compared with data obtained in the same manner as that used for deriving the atomic magnetizabilities. High-quality London atomic orbital results should be particularly suited for this purpose. Using our most recent fit of atomic magne-

w x

tizabilities 19 , from which we have obtained a carbon magnetizability ofy4.53 ppm cgs, we get a total magnetizability for C₆₀ of y272 ppm cgs.

Comparing this number with our estimated Hartree–

Fock limit result, we note that the diamagnetic exaltation in C₆₀ amounts to 32%, significantly larger than the corresponding exaltation of 24% in benzene

2. Thus, on the basis of this criterion alone, C₆₀ is undoubtedly an aromatic molecule with a strong diamagnetic exaltation. However, other properties of C , in particular its addition chemistry, do not₆₀

w x support such a conclusion 32 .

Encouraged by the surprisingly good quality of the C₆₀ magnetizability in the STO-3G London basis, we have calculated also the magnetizability of C , which we found to be₇₀ y672 ppm cgs – that is, about 17 times smaller in magnitude than the result

w x

obtained by Fowler et al. 4 in a conventional STO-3G basis. Assuming a constant ratio between the Hartree–Fock-limit and STO-3G magnetizabilities in a London basis, we obtain for C₇₀ a Hartree–

Fock-limit magnetizability of abouty463 ppm cgs.

Ž . Ž .

Our calculations thus give j C₇₀ rj C₆₀ s1.3, which is smaller than that predicted by Baker et al.

w x33 from no-London STO-3G calculations and sig-Ž . nificantly smaller than the experimental ratio of 1.9 w x34 . Our STO-3G results thus give some support to a model where the electrons may be considered as essentially delocalised giving rise to a free diamag- w x netic circulation. Using this model, Baker et al. 33 estimated the ratio between C₇₀ and C₆₀ to be 1.2.

However, even though our STO-3G result for the dia- and paramagnetic terms may be correct to about 2% to 4%, these errors are too large to enable us to draw any definite conclusions about possible similar-

2This exaltation is based on a hydrogen atomic magnetizability ofy3.53 ppm cgs, giving a total magnetizability ofy48.36 ppm cgs, which can be compared with the magnetizability calculated with an aug-cc-pVDZ basis set using London orbitalsy60.2 ppm

w x cgs 19 .

ities and differences in the origin of the diamagnetic magnetizability in these fullerenes.

Acknowledgements

This work has received support from NSC at the University of Linkoping and The Research Council¨

Ž .

of Norway Program for Supercomputing through grants of computer time. PRT was supported by NSF Cooperative Agreement No. DASC-8902825 and Grant Nos. CHE-9320718 and CHE-9700627.

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