---
abstract: |
    The electronic and vibrational contributions to the static dipole polarizability of C$_{70}$ fullerene are determined using the [*finite-field* ]{} method within the density functional formalism. Large polarized Gaussian basis sets augmented with diffuse functions are used and the exchange-correlation effects are described within the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA). The calculated polarizability ($
    \alpha$) of C$_{70}$ is 103 , in excellent agreement with the experimental value of 102 and is completely determined by the electronic part, vibrational contribution being negligible. The ratio $\alpha(C_{70})/\alpha( C_{60})$ is 1.26. The comparison of polarizability calculated with only local terms (LDA) in the PBE functional to that obtained with PBE-GGA shows that LDA is sufficient to determine the static dipole polarizability of C$_{70}$.
address: 'Department of Physics, University of Texas at El Paso, El Paso 79968'
author:
- 'Rajendra R. Zope'
bibliography:
- 'C70\_ref.bib'
- 'list1.bib'
- 'dunlap.bib'
- 'list2.bib'
- 'delta\_scf.bib'
- 'ADFT.bib'
- 'diamondoid.bib'
title: ' Static dipole polarizability of C$_{70}$ fullerene '
---

Introduction
============

C$_{70}$ is perhaps the most studied carbon fullerene after the C$_{60}$ fullerene. Several studies have addressed electric response properties of C$_{70}$ and its more famous cousin C$_{60}$ [@C70:378; @C70:377; @C70:375; @C70:382; @C70:379; @C70:385; @C70:386; @C70:387; @C70:393; @C70:390; @C70:391; @C70:381; @C70:389; @C70:383; @C70:376; @C70:374; @C70:388]. Static dipole polarizability of C$_{70}$, which is the subject of present work, was recently measured in gas phase by Compagnon and coworkers[@C70:375]. Using the molecular beam deflection technique, they reported the mean static dipole polarizability to be 102 Å$^3$ with an error bar of $\pm 14$ Å$^3$. Theoretically, polarizability of C$_{70}$ has been subject of several investigations. Only two of these studies are, however, at the [*ab initio*]{} level of theory. The first study is by Jonsson and coworkers[@C70:379], who using the self-consistent-field (SCF), and the multi-configuration self-consistent field (MCSCF) theories in combination with the 6-31++G basis set reported polarizability of C$_{70}$ to be 89.8 Å$^3$. The second study is due to van Fassen [@C70:374] who used time dependent density functional theory (TDDFT) and three different basis sets to calculate of [@C70:374]. Using the largest basis (triple zeta with additional field induced polarization function TZVP+) they found to be 104.8 Å$^3$. These authors also computed using the current-dependent Vignale-Kohn functional and concluded that polarizabilities calculated using the current-dependent functional gives good agreement with experimental values for and C$_{70}$ . The geometric structure of C$_{70}$ was not optimized in both these works.

The present article complements these earlier studies and reports the static dipole polarizability of C$_{70}$ calculated within density functional formalism, using large polarized Gaussian basis sets augmented by diffuse functions. The calculations are performed within the generalized gradient approximation using Perdew-Burke-Ernzerhof parametrization[@PBE]. Unlike in previous works, we first determine the equilibrium structure of the C$_{70}$ fullerene at the same level of theory. We then compute the static dipole polarizability. We investigate the vibrational contribution to the polarizability as well as the electronic contribution. The former, which has not yet been computed, is computationally significantly more expensive than the latter as it requires multiple optimization of C$_{70}$ geometry in the presence of electric field or requires the calculation of full vibrational spectrum.

Computational details and Results
=================================

The geometry of C$_{70}$ was fully optimized using the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm[@lbfgs] using the NRLMOL suit of codes developed by Pederson and coworkers[@NRLMOL1; @NRLMOL2; @NRLMOL_Basis]. The code employs an efficient scheme for numerical quadrature for the exchange-correlation integrals[@NRLMOL_MESH]. The $D_{5h}$ symmetry[@C70:380] of C$_{70}$ fullerene was exploited to reduce the computational expenses during structure optimization. The molecular orbitals in NRLMOL are expressed as a linear combination of 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 basis functions used for the structure optimization is 2450 and that used for polarizability calculations is 2870. The exponents in the basis set are optimized iteratively by performing a self-consistent calculation on isolated atoms[@NRLMOL_Basis]. More details about basis set and its construction can be found in Ref. [@NRLMOL_Basis; @PBAS05; @PB_POL]. The fully optimized structure of is shown in Fig. 1. There are five inequivalent atoms in C$_{70}$. The optimized positions 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$_{70}$ fullerene geometry can be generated using the $xyz$ coordinates of inequivalent atoms and symmetry operations of point group $D_{5h}$, where the five-fold highest symmetry axis is the $z-$ axis. In Table I, the calculated bond distances are compared with some experimental and theoretical values reported in literature. Agreement with experimental bond distances is quite good. The equatorial bond distance of 1.45 Å is smaller than the 1.538 Å measured in the gaseous electron diffraction experiment. However, the smaller values comparable to the prediction of the present calculation, have been reported in other experiments. The distance between the polar pentagons is 7.869 Å in good agreement with experimental value of 7.906 Å from the most recent experimental measurement[@C70:380].

The elements of the static dipole polarizability tensor are $$\alpha_{ij} = -\frac{\partial^2 E }{{\partial F_i } {\partial F_j}}.
             \label{eq:alpha_ij}$$ Here, E is the total molecular energy and F$_i$ is the $i^{th}$ component of the electric field. Eq. (\[eq:alpha\_ij\]) is the coefficient of the second term in Taylor’s expansion of total energy E in the presence of field: $$E(F) = E_0 + \sum_i \Bigl ( \frac {\partial E}{\partial F_i} \Bigr ) F_i + 
          \frac{1}{2} \sum_{i,j}\Bigl ( \frac {\partial^2 E}{\partial F_i \partial F_j}\Bigr )  F_i  F_j
          + \cdots
           \label{eq:Taylor}$$ Alternatively, the polarizability tensor elements could also be obtained from the induced dipole moments. A number of methods have been formulated to obtain and several review articles describing the details and applications of these methods exist[@Bonin; @Karna]. In this work we compute numerically using finite-difference formula. This requires calculation of total energy of the molecule for various field values. The electronic contribution to the polarizability is then obtained using a suitable approximation to compute the second derivative in Eq. \[eq:alpha\_ij\]. Alternatively, least-square fitting technique can also be used to extract the polarizability[@cohen:s34; @Na_Pol; @C70:390]. This way of calculating is known as the [*finite-field*]{} (FF) method. The implementation of FF method is fairly straight-forward. The Hamiltonian of the system is augmented by the term $-\vec{F}\cdot\vec{r}$ that represent interaction of an electron with applied electric field $\vec{F}$. The self-consistent solution is performed to obtain total energy for a given value of electric field. The self-consistent process takes into account the field-induced polarization or the so-called [*screening*]{} effects. The drawback is that addition of $-\vec{F}\cdot\vec{r}$ to the Hamiltonian, in general, breaks the full point-group symmetry. Thus computational advantage of the full point group symmetry is lost. In some cases, it may be still possible to use lower symmetry for applied fields along a specific direction. The present calculations of of did not use any symmetry. The electric field step was chosen by ensuring that the calculated polarizability is accurate and uncontaminated by higher polarizability. The obtained from induced dipole moments agrees with that obtained from energy within 1%. The basis set used during structure optimization was augmented by a $d-$ type diffuse function with 0.0772097 exponent. This methodology has been found to provide a good description of of C$_{60}$ and several molecules and clusters[@C70:390; @C70:391; @Jackson; @PBAS05; @PB_POL]. For more details of methodology and application we refer reader to a recent review by Pederson and Baruah[@PB_POL]. The computational scheme used in this work has also been applied to fullerene. In fact, one of the earliest calculation of polarizability of by Pederson and Quong used the same methodology[@C70:390; @C70:391]. As mentioned earlier, the nuclear positions were assumed to be frozen during the calculation of . The relaxation of nuclear positions in the presence of applied electric field also contribute to the polarizability. This contribution is often called vibrational polarizability and usually is the second largest contribution of the polarizability. In some cases, particularly for the system with ionic or hydrogen bonding it could be even larger than . Here, we compute within double-harmonic approximation. A full account of the formulation of calculation of used in this work can be found in Ref. [@PBAS05].

The mean or average polarizability , is one-third of the trace of the polarizability matrix: $$\alpha_m = \frac{1}{3} (\alpha_{xx} + \alpha_{yy} + \alpha_{zz}).$$ The calculated values of the polarizability components are $\alpha_{xx} = \alpha_{yy} =99$ and $\alpha_{zz} = 111$, where $z-$ axis is along the five-fold axis. The mean polarizability is 103 Å$^3$ . We have also calculated unscreened polarizability using the sum-over-states expression with excitation energies approximated by eigenvalue differences. As in case of C$_{60}$ fullerene[@C70:390; @C70:391], the unscreened polarizability turns out to be 330 , roughly three times larger than the screened polarizability. In Table II calculated is compared with earlier published theoretical and experimental values. Recent measurements of gas-phase polarizability of indicate to be 102 with a rather larger error bar of about 14 . The predicted (PBE/NRLMOL) value of 103 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 agree well with time dependent density functional theory calculation with statistically averaged orbital potential (TDDFT/SAOP) polarizability[@C70:374]. But, it is larger than the predictions by Hartree-Fock (HF) theory (HF/6-31+G)[@jonsson:572] and time dependent current density functional theory (TDCDFT/VK)[@C70:374]models. The differences in the structure of C$_{70}$ 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 molecular structure[@maroulis:583]. The other possible cause of differences is in the modeling of many body effects. In our recent study that compared static dipole polarizabilities of 142 small molecules in the HF and PBE models, the HF polarizabilities were found to be smaller than their PBE counterparts[@SOS]. The larger value of in the PBE/NRLMOL than in the HF model is consistent with this observation. The obtained within the time-dependent current density functional model (TDCFT/VK) falls intermediate between the HF/6-31+G and PBE/NRLMOL predictions. We repeated calculations with only local terms in the PBE functional, to estimate the effect of gradient correction to the exchange-correlation functional. This approximation gives to be 100 , indicating that local approximation is sufficient to determine the dipole polarizability of C$_{70}$. The vibrational contribution to the polarizability tensor within the double harmonic approximation[@PBAS05] is given as $\alpha_{vib,i,j} = \sum_{\mu} Z_{i,\mu} \omega^{-2}_{\mu} 
                    Z^{T}_{j,\mu}. $ Here, $\omega_{\mu}$ is the frequency of the $\mu$th vibrational mode, $Z_{i,\mu}$ is the effective charge tensor (See Ref. [@PBAS05] for details). The vibrational contribution along the five-fold axis is $\alpha_{zz}^{vib}$ is 0.43 while that along the transverse axis is 0.74 . Thus, in comparison with electronic polarizability, vibrational contribution to the dipole polarizability of is negligible and the static dipole polarizability of is completely determined by the electronic polarizability. This result in consistent with the finding of Pederson and coworkers[@PBAS05] for the and is relevant for the dielectric response of carbon nanotubes.

To conclude, static dipole polarizability of is calculated within density functional formalism using large polarized Gaussian basis sets. The calculated values of electronic ($\alpha_{el} = 102.8$ ) and vibrational ($\alpha_{vib} = 4.3$ ) polarizabilities indicate that the vibrational contribution to the total polarizability of is very small.

This work is supported in part by the National Science Foundation through CREST grant, 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 ONR 05PR07548-00 grant.

  Bond distance     GED     SED     ND     X-ray   SCF/DZP   BP86/TZP   PBE/NRLMOL
  --------------- ------- ------- ------- ------- --------- ---------- ------------
  C1-C2            1.461   1.464   1.460   1.458    1.451     1.454       1.439
  C1-C6            1.388   1.37    1.382   1.380    1.375     1.401       1.389
  C6-C7            1.453   1.47    1.449   1.459    1.446     1.450       1.436
  C10-C12          1.386   1.37    1.396   1.370    1.361     1.395       1.383
  C7-C8            1.468   1.46    1.464   1.460    1.457     1.449       1.433
  C8-C9            1.425   1.47    1.420   1.430    1.415     1.441       1.426
  C9-C31           1.405   1.39    1.415   1.407    1.407     1.424       1.410
  C31-C32          1.538   1.41    1.477   1.476    1.475     1.471       1.452

  Method                               C$_{60}$        C$_{70}$        C$_{70}$/C$_{60}$              Reference          
  ------------------------------- -- ------------ -- ------------- -- ------------------- -- --------------------------- --
  Gas phase                           76.5$\pm$8      102.$\pm$14            1.33                  Ref. [@C70:375]       
  Ellipsometry                           79.0            97.0                1.23                  Ref. [@Ren:124]       
  EELS                                   83.0            103.5               1.25                  Ref. [@Sohmen]        
  [**Theory**]{}                                                                                                         
  Coupled Hartree-Fock/STO-3G            45.6            57.0                1.25                  Ref. [@C70:395]       
  Pople-Parr-Pariser model               49.4            63.8                1.29              Ref. [@willaime:6369]\]   
  Tight binding                          77.0            91.6                1.19                  Ref. [@C70:378]       
  Bond polarizability model              89.2            109.2               1.22                  Ref. [@C70:382]       
  Valence effective Hamiltonian         154.0            214.3               1.39             Ref. [@PhysRevB.46.16135]  
  Monopole-dipole                        60.8            73.8                1.21                  Ref. [@C70:383]       
  MNDO/PM3                               63.9            79.0                1.24                  Ref. [@C70:393]       
  HF 6-31+G                              75.1            89.8                1.20                Ref. [@jonsson:572]     
  TDDFT/SAOP                              83              101                1.22                  Ref. [@C70:374]       
  TDCDFT/VK                               76              91                 1.51                  Ref. [@C70:374]       
  PBE/NRLMOL                             82.9            102.8               1.24                     This work          

references {#references .unnumbered}
==========
