aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Static dipole polarizability of C fullerene Rajendra R. Zope
Department of Physics, University of Texas at El Paso, El Paso 79968E-mail: [email protected]
Abstract.
The electronic and vibrational contributions to the static dipolepolarizability of C fullerene are determined using the finite-field method withinthe density functional formalism. Large polarized Gaussian basis sets augmentedwith diffuse functions are used and the exchange-correlation effects are describedwithin the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA).The calculated polarizability ( α ) of C is 103 ˚A , in excellent agreement withthe experimental value of 102 ˚A and is completely determined by the electronicpart, vibrational contribution being negligible. The ratio α ( C ) /α ( C ) is 1.26.The comparison of polarizability calculated with only local terms (LDA) in the PBEfunctional to that obtained with PBE-GGA shows that LDA is sufficient to determinethe static dipole polarizability of C .PACS numbers: Submitted to:
J. Phys. B: At. Mol. Opt. Phys. tatic dipole polarizability of C fullerene
1. Introduction C is perhaps the most studied carbon fullerene after the C fullerene. Several studieshave addressed electric response properties of C and its more famous cousin C [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Static dipole polarizability α of C ,which is the subject of present work, was recently measured in gas phase by Compagnonand coworkers[3]. Using the molecular beam deflection technique, they reported themean static dipole polarizability to be 102 ˚A with an error bar of ±
14 ˚A . Theoretically,polarizability of C has been subject of several investigations[4, 5, 6, 9, 15, 16]. Onlytwo of these studies are, however, at the ab initio level of theory. The first study isby Jonsson and coworkers[5], who using the self-consistent-field (SCF), and the multi-configuration self-consistent field (MCSCF) theories in combination with the 6-31++Gbasis set reported polarizability of C to be 89.8 ˚A . The second study is due to vanFassen et al. [16] who used time dependent density functional theory (TDDFT) andthree different basis sets to calculate α of C [16]. Using the largest basis (triplezeta with additional field induced polarization function TZVP+) they found α to be104.8 ˚A . These authors also computed α using the current-dependent Vignale-Kohnfunctional and concluded that polarizabilities calculated using the current-dependentfunctional gives good agreement with experimental values for C and C . Thegeometric structure of C was not optimized in both these works.The present article complements these earlier studies and reports the staticdipole polarizability of C calculated within density functional formalism, using largepolarized Gaussian basis sets augmented by diffuse functions. The calculations areperformed within the generalized gradient approximation using Perdew-Burke-Ernzerhofparametrization[18]. Unlike in previous works, we first determine the equilibriumstructure of the C fullerene at the same level of theory. We then compute thestatic dipole polarizability. We investigate the vibrational α vib contribution to thepolarizability α as well as the electronic α el contribution. The former, which has notyet been computed, is computationally significantly more expensive than the latter asit requires multiple optimization of C geometry in the presence of electric field or tatic dipole polarizability of C fullerene Figure 1.
The optimized geometry of C fullerene. The five inequivalent atomsin the D h structure are denoted by lower case letters. The bond distance betweenlabeled atoms are given in Table I. requires the calculation of full vibrational spectrum.
2. Computational details and Results
The geometry of C was fully optimized using the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm[19] using the NRLMOL suit of codes developed by Pedersonand coworkers[20, 21, 22]. The code employs an efficient scheme for numericalquadrature for the exchange-correlation integrals[23]. The D h symmetry[24] ofC fullerene was exploited to reduce the computational expenses during structureoptimization. The molecular orbitals in NRLMOL are expressed as a linear combinationof Gaussian orbitals. The Gaussian basis set for C consists of 5 s − , 3 p − , and 1 d − type Gaussians each contracted from 12 primitive functions. The total number of basisfunctions used for the structure optimization is 2450 and that used for polarizabilitycalculations is 2870. The exponents in the basis set are optimized iteratively byperforming a self-consistent calculation on isolated atoms[22]. More details about basisset and its construction can be found in Ref. [22, 25, 26]. The fully optimized structure tatic dipole polarizability of C fullerene is shown in Fig. 1. There are five inequivalent atoms in C . The optimizedpositions of these inequivalent atoms in atomic units are a ( -2.3127, 0.0000, -7.4352), b ( -4.5356, 0.0000, 6.0395 ), c ( 5.5080, 1.3063, 4.5799 ), d ( 5.9336, 2.6454, 2.2646), e ( 6.5245, 1.3720, 0.0000 ). The C fullerene geometry can be generated using the xyz coordinates of inequivalent atoms and symmetry operations of point group D h ,where the five-fold highest symmetry axis is the z − axis. In Table I, the calculatedbond distances are compared with some experimental and theoretical values reported inliterature. Agreement with experimental bond distances is quite good. The equatorialbond distance of 1.45 ˚A is smaller than the 1.538 ˚A measured in the gaseous electrondiffraction experiment. However, the smaller values comparable to the prediction of thepresent calculation, have been reported in other experiments. The distance between thepolar pentagons is 7.869 ˚A in good agreement with experimental value of 7.906 ˚A fromthe most recent experimental measurement[24].The elements of the static dipole polarizability tensor are α ij = − ∂ E∂F i ∂F j . (1)Here, E is the total molecular energy and F i is the i th component of the electric field.Eq. (1) is the coefficient of the second term in Taylor’s expansion of total energy E inthe presence of field: E ( F ) = E + X i (cid:16) ∂E∂F i (cid:17) F i + 12 X i,j (cid:16) ∂ E∂F i ∂F j (cid:17) F i F j + · · · (2)Alternatively, the polarizability tensor elements could also be obtained from the induceddipole moments. A number of methods have been formulated to obtain α ij and severalreview articles describing the details and applications of these methods exist[27, 28].In this work we compute α ij numerically using finite-difference formula. This requirescalculation of total energy of the molecule for various field values. The electroniccontribution to the polarizability α el is then obtained using a suitable approximationto compute the second derivative in Eq. 1. Alternatively, least-square fitting techniquecan also be used to extract the polarizability[29, 30, 10]. This way of calculating α el isknown as the finite-field (FF) method. The implementation of FF method is fairlystraight-forward. The Hamiltonian of the system is augmented by the term − ~F · ~r that tatic dipole polarizability of C fullerene ~F . The self-consistentsolution is performed to obtain total energy for a given value of electric field. Theself-consistent process takes into account the field-induced polarization or the so-called screening effects. The drawback is that addition of − ~F · ~r to the Hamiltonian, ingeneral, breaks the full point-group symmetry. Thus computational advantage of thefull point group symmetry is lost. In some cases, it may be still possible to use lowersymmetry for applied fields along a specific direction. The present calculations of α el ofC did not use any symmetry. The electric field step was chosen by ensuring that thecalculated polarizability is accurate and uncontaminated by higher polarizability. The α el obtained from induced dipole moments agrees with that obtained from energy within1%. The basis set used during structure optimization was augmented by a d − typediffuse function with 0.0772097 exponent. This methodology has been found to providea good description of α el of C and several molecules and clusters[10, 11, 31, 25, 26].For more details of methodology and application we refer reader to a recent review byPederson and Baruah[26]. The computational scheme used in this work has also beenapplied to C fullerene. In fact, one of the earliest calculation of polarizability ofC by Pederson and Quong used the same methodology[10, 11]. As mentioned earlier,the nuclear positions were assumed to be frozen during the calculation of α el . Therelaxation of nuclear positions in the presence of applied electric field also contribute tothe polarizability. This contribution is often called vibrational polarizability α vib andusually is the second largest contribution of the polarizability. In some cases, particularlyfor the system with ionic or hydrogen bonding it could be even larger than α el . Here, wecompute α vib within double-harmonic approximation. A full account of the formulationof calculation of α vib used in this work can be found in Ref. [25].The mean or average polarizability α el , is one-third of the trace of the polarizabilitymatrix: α m = 13 ( α xx + α yy + α zz ) . (3)The calculated values of the polarizability components are α xx = α yy = 99˚A and α zz = 111˚A , where z − axis is along the five-fold axis. The mean polarizability α el is 103 ˚A . We have also calculated unscreened polarizability using the sum-over- tatic dipole polarizability of C fullerene fullerene[10, 11], the unscreened polarizability turns out to be 330 ˚A , roughly three times larger than the screened polarizability. In Table II calculated α el is compared with earlier published theoretical and experimental values. Recentmeasurements of gas-phase polarizability of C indicate α el to be 102 ˚A with arather larger error bar of about 14 ˚A . The predicted (PBE/NRLMOL) value of103 ˚A agrees well with the experimental value. As can be seen from the Table II,all ab-initio values fall within the error bar. The PBE/NRLMOL α el agree well withtime dependent density functional theory calculation with statistically averaged orbitalpotential (TDDFT/SAOP) polarizability[16]. But, it is larger than the predictionsby Hartree-Fock (HF) theory (HF/6-31+G)[32] and time dependent current densityfunctional theory (TDCDFT/VK)[16]models. The differences in the structure of C used in those calculations and that optimized in this work is one source of discrepancy.In particular, the higher order polarizabilities are often sensitive to the molecularstructure[33]. The other possible cause of differences is in the modeling of many bodyeffects. In our recent study that compared static dipole polarizabilities of 142 smallmolecules in the HF and PBE models, the HF polarizabilities were found to be smallerthan their PBE counterparts[34]. The larger value of α el in the PBE/NRLMOL than inthe HF model is consistent with this observation. The α el obtained within the time-dependent current density functional model (TDCFT/VK) falls intermediate betweenthe HF/6-31+G and PBE/NRLMOL predictions. We repeated calculations with onlylocal terms in the PBE functional, to estimate the effect of gradient correction to theexchange-correlation functional. This approximation gives α el to be 100 ˚A , indicatingthat local approximation is sufficient to determine the dipole polarizability of C .The vibrational contribution to the polarizability tensor within the double harmonicapproximation[25] is given as α vib,i,j = P µ Z i,µ ω − µ Z Tj,µ . Here, ω µ is the frequency of the µ th vibrational mode, Z i,µ is the effective charge tensor (See Ref. [25] for details). Thevibrational contribution along the five-fold axis is α vibzz is 0.43 ˚A while that alongthe transverse axis is 0.74 ˚A . Thus, in comparison with electronic polarizability,vibrational contribution to the dipole polarizability of C is negligible and the static tatic dipole polarizability of C fullerene Table 1.
The comparison of selected bond distances calculated in this work(PBE/NRLMOL) with those reported in literature. GED is the gaseuous electrondiffraction measurements[24]; Solid-State Electron diffraction (SED)[35]; Neutrondifftraction (ND) mesurments [36]; X-ray diffraction[37]; Hartree-Fock/Dobule ZetaBasis [38]; BP86/TZP[39]; PBE/NRLMOL (present).
Bond distance GED SED ND X-ray SCF/DZP BP86/TZP PBE/NRLMOLC1-C2 1.461 1.464 1.460 1.458 1.451 1.454 1.439C1-C6 1.388 1.37 1.382 1.380 1.375 1.401 1.389C6-C7 1.453 1.47 1.449 1.459 1.446 1.450 1.436C10-C12 1.386 1.37 1.396 1.370 1.361 1.395 1.383C7-C8 1.468 1.46 1.464 1.460 1.457 1.449 1.433C8-C9 1.425 1.47 1.420 1.430 1.415 1.441 1.426C9-C31 1.405 1.39 1.415 1.407 1.407 1.424 1.410C31-C32 1.538 1.41 1.477 1.476 1.475 1.471 1.452dipole polarizability of C is completely determined by the electronic polarizability.This result in consistent with the finding of Pederson and coworkers[25] for the C andis relevant for the dielectric response of carbon nanotubes.To conclude, static dipole polarizability of C is calculated within densityfunctional formalism using large polarized Gaussian basis sets. The calculated valuesof electronic ( α el = 102 . ) and vibrational ( α vib = 4 . ) polarizabilities indicatethat the vibrational contribution to the total polarizability of C is very small.This work is supported in part by the National Science Foundation through CRESTgrant, by the University of Texas at El Paso and by the Office of Naval Research.Authors acknowledge the computer time at the UTEP Cray acquired using ONR05PR07548-00 grant. references [1] Antoine R, Dugourd P, Rayane D, Benichou E, Broyer M, Chandezon F and Guet C 1999 Journalof Chemical Physics
Journal of Chemical Physics tatic dipole polarizability of C fullerene Table 2.
The comparison of calculated polarizability in (˚A ) with the experimentaland theoretical values reported in literature. Method C C C /C ReferenceGas phase 76.5 ± ±
14 1.33 Ref. [3]Ellipsometry 79.0 97.0 1.23 Ref. [40]EELS 83.0 103.5 1.25 Ref. [41]
Theory
Coupled Hartree-Fock/STO-3G 45.6 57.0 1.25 Ref. [42]Pople-Parr-Pariser model 49.4 63.8 1.29 Ref. [43]]Tight binding 77.0 91.6 1.19 Ref. [1]Bond polarizability model 89.2 109.2 1.22 Ref. [4]Valence effective Hamiltonian 154.0 214.3 1.39 Ref. [44]Monopole-dipole 60.8 73.8 1.21 Ref. [14]MNDO/PM3 63.9 79.0 1.24 Ref. [9]HF 6-31+G 75.1 89.8 1.20 Ref. [32]TDDFT/SAOP 83 101 1.22 Ref. [16]TDCDFT/VK 76 91 1.51 Ref. [16]PBE/NRLMOL 82.9 102.8 1.24 This work [3] Compagnon I, Antoine R, Broyer M, Dugourd P, Lerme J and Rayane D 2001
Physical Review a art. no.–025201[4] Guha S, Menendez J, Page J B and Adams G B 1996 Physical Review B Journal of Chemical Physics
Optics Communications
Chemical Physics Letters
Jetp Letters Journal of Physical Chemistry Physical Review B Physical Review B Surface Review and Letters tatic dipole polarizability of C fullerene [13] Ramaniah L M, Nair S V and Rustagi K C 1993 Optics Communications Journal of Physical Chemistry Journal of Chemical Physics
Chemical Physics Letters
International Journal of Modern Physics B Physical Review Letters Math. Programming Physical Review B Physical Review B Physical Review A Phys. Rev. B Journal of theAmerican Chemical Society
Journal of Chemical Theory and Computation Lecture Series in Computer and Computational Sciences The Journal of Chemical Physics S34–S39[30] Blundell S A, Guet C and Zope R R 2000
Phys. Rev. Lett. Phys. Rev. A The Journal of Chemical Physics
The Journal of Chemical Physics arXiv:cond-mat/0701466 [35] Mckenzie D R, Davis C A, Cockayne D J H, Muller D A and Vassallo A M 1992
Nature
Chemical Physics Letters
Journal De Physique i Chemical Physics Letters
Chem. Phys. Lett. Applied Physics Letters tatic dipole polarizability of C fullerene [41] Sohmen E, Fink J and Krtschmer W 1992 Zeitschrift fr Physik B Condensed Matter Chemical Physics Letters
The Journal of Chemical Physics Phys. Rev. B46