versión On-line ISSN 0717-9707
J. Chil. Chem. Soc. v.48 n.3 Concepción sep. 2003
J. Chil. Chem. Soc., 48, N 3 (2003) ISSN 0717-9324
A COMPARISON OF SEMIEMPIRICAL AND AB INITIO METHODS FOR
CALCULATING THE ELECTRONIC STRUCTURE OF C60 AND C70 FULLERENES.
Juan S. Gómez-Jeria*, Nelson Gonzalez-Tejeda and Francisco Soto-Morales
Universidad de Chile, Facultad de Ciencias
Casilla 653 Santiago CHILE.
The aim of this work is to find a method suitable at least for obtaining the isolated molecule band structure approximated by a Gaussian broadening of the discrete eigenvalues, to apply it for a first scan of bigger and more complex structures. We compared the results of several semiempirical and ab initio quantum-chemical methods to calculate the band structure of isolated C60 and C70 fullerenes. Theoretical results were compared with experimental photoemission and inverse photoemission spectra.
The results show that Extended Hückel Theory is the best method of all analyzed here. It compares well with experimental results related to valence and conduction bands for fullerene but underestimates the valence-conduction band gap by about 50%. Therefore, it is suitable for a first screening of fullerene-like molecules. Finally, EHT is employed to predict the electronic structure of several hypothetical molecules (P60O60, N70, C30N30 and N60 ).
Carbon is one of the most fascinating chemical elements. It is the basis of all life and of all organic chemistry. Twenty years ago only two carbon allotropes were known: diamond and graphite. Today we are faced with a large family of carbon allotropes, such as amorphous carbon, graphite, diamond, carbyne, and the newly discovered fullerenes and carbon nanotubes. These discoveries have opened up a new research area in chemistry, physics and materials science. Examples of applications are: quantum wires, nanoprobes, sensors, gas storage nanodevices, electron field emitters for ultra-thin TV screens, and parts of nanomachines, among others.
The first buckminsterfullerene C60 was discovered in 1985 . On the other hand, it now has become clear that fullerenes have existed for much longer than mankind. They have been found in interstellar dust and meteor rocks and seem to be present everywhere in the universe. In 1990 Krätschmer and Huffmann found a method that allows the production of C60 and other fullerenes in bulk quantities . In 1991, the alkali metal-intercalated material K3 C60 was found to become superconducting at 18K, a record for organic superconductors . This research was recognized when The Royal Swedish Academy of Sciences awarded the Nobel Prize in chemistry to R.F. Curl, H.W. Kroto and R.E. Smalley for the discovery of fullerenes.
The discovery of carbon nanotubes stretches back only a decade. In 1991, Japanese electron microscopist Sumio Iijima, of the NEC Corporation, experimented with the technique that had enabled the C60 researchers to make their new form of carbon. By passing electrical sparks between two closely spaced graphite rods, Iijima vaporized them and allowed the carbon to condense in a sooty mass. But when he looked at the soot through the microscope, he found something altogether unexpected. Amongst the debris, where others had found C60, were tiny tubes of pure carbon, just a few nanometers across. These nanotubes were hollow but many-layered with tubes inside tubes, like nested Russian dolls, and their ends sealed with conical caps . A year later Thomas Ebbesen and P. M. Ajayan at NEC found a way to produce nanotubes in higher yields and make them available for studies by different techniques. Subsequently they found a way to purify them. [5,6]. It is clear now that the discovery of giant fullerenes (in the form of nanotubes and polyhedra), together with nanocones and several combinations of all these compounds, has greatly expanded the realm of this new form of carbon. Based on the remarkable mechanical properties of carbon nanotubes [7,8], there is some reason to believe that a focused effort to develop fullerene nanotechnology could yield materials with remarkable properties. The nanotubes are particularly interesting because their ideal geometrical form, and nearly faultless structures provides them with unique mechanical, electrical and thermal properties. These tubes are expected, on the basis of theoretical calculations, to have strongly anisotropic electrical and thermal conductivities. The mechanical properties are also interesting since the bulk module as well as Young's and bending moduli are expected to be very high simply due to the cylindrical geometry and the rigidity of the hexagonal lattice. Axially, the tubes are expected to have strengths and stiffnesses greater than those of diamond. A recent experimental determination of the stiffness of high quality tubes observed to vibrate in a TEM (Transmission Electron Microscopy), generated values of Young's modulus of several terapascals . Strengths are predicted to be at least 20 GPa. . However, the tubes are flexible laterally, an effect related to reversible buckling of the atomic layers . Ideal tubes would also have an isotropic coefficient of thermal expansion due to the geometrical constraints on the structure. Also, a given nanotube shell is expected to be either conducting or semiconducting with a small band-gap depending critically on the exact arrangement of the carbon layer. Measurements using scanning tunneling electron microscopes seem to confirm this , whilst conductivity measurements have also been performed on individual tubes  and on crystalline ropes of single-walled tubes. Finally, carbon nanotubes are diamagnetic with the axial direction having the more negative susceptibility, probably as a result of currents circulating around the circumference .
One on the properties used to characterize these systems is their band structure. As is well known, two of the bands, the valence band (VB) and the conduction band (CB) are useful to classify a molecule as an insulator, a semiconductor, or a metal. An associated concept is that of density of states (DOS), which describes the energy levels per unit energy increment.
In this paper we present the results of the comparison, for the first time, of several semiempirical and ab initio quantum-chemical methods used to approximate, starting from the discrete eigenvalues, the band structure of the isolated C60 and C70 fullerenes (Figure 1). The aim of this work is to find a method suitable at least for obtaining the band structure to apply it for a first scan of bigger and more complex structures (i.e., from 500 atoms up). Also it is important to investigate if the theoretical results coming from isolated systems compare well or not with experimetal results from solids. This is important because the accelerated downscaling of electronic devices has reached the single molecule domain. As a consequence, the investigation of the mechanism by which a single molecule carries an electric current becomes crucial in view of the possible exploitation of molecular electronic circuits. Based on our results we present an analysis of the band structure of three new molecules, not yet synthesized. We must caution the reader about the following fact. The classification of insulators, semiconductors and metals is applied to solid state materials. In the following we shall employ these terms by analogy to refer to the possibility of electron movement inside the isolated systems.
2. Methods and calculations.
|Figure 1. C60 and C70.|
The geometry of the C60 and C70 fullerenes was fully optimized by means of the AM1 methodology. To compute the density of states, a calculation was done first to get the wave functions and the associated eigenvalues. The wavefunctions for both molecules were obtained with a set of semiempirical (EHT, INDO, ZINDO/1, ZINDO/S) and ab initio (STO-3G, 4-31G** and 6-31G** basis sets) methods. Semiempirical calculations were done with the Hyperchem package  and the ab initio ones with the Gaussian 98 program ..
For each method and molecule the band structure was approximated by a Gaussian broadening of the discrete eigenvalues through the following expression :
where D(E) is the density of states, E1 is the energy of the i-th molecular orbital and s the Gaussian half-width (0.5 eV here). The theoretical reason to proceed in this way is the following. Let us consider a non-periodic molecule. The number of discrete molecular orbitals grows as the number of atoms in the molecule increases. When the number becomes very large, the energy states are so close together that they «blend» and become a continuum of states. This continuum is employed to build bands with the above method. The density of states in these bands is found to be non-uniform across the band. The reason for this is that the discrete levels are packed more closely together at some energies than others as a result of overlap of the molecular orbitals.
The theoretical results were then compared with the experimental ones. Experimental techniques which produce information on the DOS are X-ray and ultraviolet photoelectron spectroscopy (PES), inverse photoemission spectroscopy, and energy loss near edge structure. PES gives detailed information on the variation of the density of states at the valence band. It is possible to measure the conduction band structure using the technique of inverse photoemission, which yields the energy and momentum of a photon emitted when an electron makes the transition from the conduction band to the valence band. For our study we have selected the experimental results appearing in references 17 to 19.
3. Results and discussion.
Before presenting our results a word must be said about the experimental results we used for comparison. The line shapes of the resulting spectra are a function of the incident particle energy (photon or electron). Low energy particles will show more detailes of the band structure at the valence (photons) or conduction (electrons) band level. As our main interest resides in these two bands, we have selected the experimental spectra obtained at low energies. It is then clear that we cannot expect, with one experimental result, to make a full comparison with the whole theoretical band results.
The Highest Occupied Molecular Orbital C60 (HOMO) of C60 is a five-fold degenerate p orbital with hu symmetry. The LUMO is triply degenerate with t1u symmetry . The VB-CB gap for C60 has been experimentally determined as 2.15 eV , 1.85 eV  or 2.3-2.7 eV [23-26]. Local density approximation (LDA) calculations predict a band gap of about 1.5 eV [27-29]. Table I shows some features of C60 band structure and the Fermi level (see Footnote 1).
|a. Ref. 21 b. Calculated from ref. 17 c. Calculated from Ref. 19.|
Figure 2 shows the C60 band structure calculated with ab initio methods. We may see that the band gap for all methods is about 5.5 eV.
| ||Figure 2. Ab initio density of states for C60.|
We may conclude that these methods are not suitable for a first screening of fullerene and fullerene-like molecules. Figure 3 shows the semiempirical results. At a first sight only the EHT (1.25 eV) and ZINDO/S (2.0 eV) methods give a reasonable band gap. Now let us consider the conduction bandwidth (see Footnote 2).
| ||Figure 3. Semiempirical density of states for C60.|
Pseudopotential local-density calculations  show that the CB width is about 0.47 eV, but because of the approximations the true band gap should be larger. LDA calculations give a value of ~1eV . Our results with the EHT method give a CB width of about 1.9 eV. ZINDO/S gives 5.5 eV for this same property. Therefore, the conclusion is that the EHT method is more suitable for a preliminary study of this system.
Figure 4 displays the experimental spectrum of C60 together with the EHT band structure. The experimental spectrum was scanned and amplified without any deformation in such a way that both curves are represented on the same scale. The procedure was as follows. The experimental spectra were first scanned. The theoretical EHT curve was obtained with the Origin v. 6.0 program . Both curves where merged and scaled without any distortion with the Adobe Photoshop v.5.5 program .
| ||Figure 4. Experimental and EHT density of states for C60. The experimental spectrum was taken from Refs. 17 and 19 with permission.|
We have stressed that the experimental setting does not cover the whole range of energies covered by the theoretical calculations. Therefore, we cannot expect a perfect fit of those two curves but we may comment on the following. At 65 eV  and within ~7 eV of the HOMO, experimental results show three bands: the intensity increases from the VB to the third band. This feature is well represented but EHT cannot totally resolve bands two and three. The relative locations of the three bands are well represented by EHT results. Theoretical EHT results show a wide band between 10 and 15 eV. This should correspond to a band with s - ps character. High-energy experimental results show a wide band between 10 and 20 eV encompassing s - ps and probably s-derived bands .
Regarding the empty states, we may see that the first two curves of the experimental spectrum (at 15.25 eV, Ref. 19) may correspond to the two overlapping curves of the EHT band structure. Note that both experimental and theoretical bands have the same height. The third experimental band agrees with the theoretical result. Moreover, this band is higher than the first two in both curves.
The VB-CB gap for C70 has been experimentally determined as 1.65 eV . An LDA calculation reports a gap of 0.7 eV . We must note that this calculation underestimates the real gap because LDA does not correctly take into account electron correlations. Table 2 shows the VB-CB gap and the VB and CB widths for C70. Figures 5 and 6 show, respectively, the ab initio and semiempirical band structure of C70. The same reasoning used for C60 leads to the conclusion that the EHT method is the best (and faster) method for a first approach to the study of band structure in these compounds. Figure 7 shows the C70 experimental spectra  together with the EHT band structure (For occupied states the photon energy is 65 eV. For empty states the electron energy is 27.25 eV). Experimental results, at all energies, show that the third occupied band is higher than the first two. This fact is well represented in the EHT curve as well as the relative band locations. Nevertheless, EHT calculations cannot separate totally the BV and the next one because their overlap. In the case of this system, experimental results show a considerable variation of intensities with the probe energy at both sides of the Fermi level . For empty states the energy probe at 27.25 eV (Fig. 7) shows that the second band is higher than the CB. Nevertheless, at lower energies (19.25 eV) both bands have the same intensity . EHT results show a wide band located in the same energy interval.
a. Ref. 30. b. Calculated from Ref. 18.
Figure 5. Ab initio density of states for C70.
Figure 6. Semiempirical density of states for C70.
| ||Figure 7. Experimental and EHT density of states for C70. The experimental spectrum was taken from Ref. 18 with permission.|
The conclusion then is that Extended Hückel Theory is a method that can be employed to screen the band structure of fullerene-like molecules before using more elaborate theoretical models (see Footnote 3 for details about he EHT version used here).
We must keep in mind that EHT underestimates the BV-CB gap by about 50%. This is probably due to the fact that EHT neglects electron-electron interactions. We may add that other preliminary results, not reported here, of the application of the EHT method to nanotubes and graphene surfaces (300 to 500 atoms) are giving very promising results.
4. Application of the EHT methodology for analyzing new molecular systems.
Quantum chemistry has reached definitively the status of a predictive branch of science. Eiji Osawa, President of the NanoCarbon Research Institute, Ltd., is known as the first scientist to conceive C60 as a molecule with special aromatic stability in 1970. For the prediction of C60, he was awarded the Chu-nichi Culture Prize in 2001. Also quantum chemical calculations predicted the existence of C60 more than a decade before its discovery . Successful predictions were also made by Ramiro Arratia-Pérez et al. of the magnetic anisotropies of a silver trimer cluster  and the luminescence of several rhenium clusters . In the field of quantum pharmacology we may cite our successful quantitative prediction of the biological activity of a molecule which experimentalists had neglected believing it to be uninteresting . The study of molecules not yet synthesized is therefore a valid attitude provided we remain in the realm of chemical composition rules.
The discovery of fullerenes has raised the possibility that other elements form stable cage structures. Recently a C60 fullerene-like molecule was proposed but composed only of nitrogen atoms: N60 . No electronic structure was calculated. N60 is a high-energy system with possible applications as an explosive and/or a rocket propellant. No synthesis is yet known but a mechanism involving bringing together six N10 molecules was suggested .
Considering C60 and C70 geometry and bonding, phosphorus seems to be a good candidate to form a cage-like structure. Replacing the sixty carbon atoms of C60 by phosphorous atoms produces P60. We found that no quantum chemical method was able to stabilize P60 into a C60 cage-like structure. We suggest then that this structure is not stable under normal circumstances. The next step was to consider that phosphorus also has a pentavalent bonding capacity. Therefore we replaced each carbon atom by a P=O unit producing P60O60 (see Fig. 8) which we have named Urchinen. For this system, the CNDO/2 method was employed for full geometry optimization due to the well known shortcoming of AM1 to treat P=O bonds. A stable structure was obtained. For the case of phosphorus only a P+21 structure was recently synthesized .
| ||Figure 8. P60O60. Urchinen. The spherical core is composed by P atoms. The external ones are O atoms.|
Two more molecules were generated in this way: N70 which is the natural extension of , and a hybrid cage-like structure with formula C30N30 obtained by replacing one of the hemispheres of C60 by nitrogen atoms. We generated several more similar structures but, as we are interested only in showing the goodness of the EHT method to deal with band structures, we present here the most interesting ones. The geometry of N70 and C30N30 was fully optimized with the AM1 methodology. All systems are stable. We also included N60 for the sake of completeness. These molecules are interesting candidates for nanoelectronic devices when coupled with other molecular systems.
Figure 9 shows the DOS for C30N30 and Fig. 10 the DOS for the rest of the systems. For the sake of comparison, the energy of the highest molecular orbital (HOMO) of all the molecules has been placed at zero.
Figure 9. Density of states for C30N30.
Figure 10. Density of states for N60, N70 and P60O60.
In the case of C30N30 we may observe a narrow band centered at about 3.5 eV which contains a large number of electronic p states. Also there is a small band about E = 0 that is not completly filled with electrons indicating that this system must present semiconducting or quasi-metallic properties. It is suggested that the place where electrons are almost free to move is the C-N junction which is the natural place to bind this system to another one. More theoretical work is needed to know the exact nature (closed shell vs open shell) of this interesting system. Finally, a word about a possible mechanism for the synthesis of C30N30: it would be possible to design, within a limaçon  or nautilus  model for fullerene growth, a way to add N atoms after a C30 curved sheet is formed.
N60 shows an overlap between the VB and the CB. Therefore this molecule should be, at least, a semiconductor. N70 presents a VB-CB gap of about 1.85 eV and could possibly have less semiconducting properties than N60. The VB is about 10 eV and the CB 12 eV wide, both composed of several overlapping bands. Moreover, as in the case of N60, this molecule also could be a candidate for explosives or rocket propellants. In both molecules the p bonding is reflected in the large number of states lying in the 0 to 5 eV interval.
P60O60 is a quite interesting system because it has no gap between the valence and conduction bands. This indicates that it could have metallic properties. Nevertheless, its implementation into a electronic device seems difficult because of its reactivity. The only way to connect it to a molecular device is by removing an oxygen atom (to bind Urchinen to another molecule through a C=P bond) or by keeping it singly bonded to its P atom and to another molecule.
Footnote 1. In solid state physics the chemical potential, m, is often called the Fermi level. The value of m at zero temperature is written as m(t=0) =eF We call eF the Fermi energy. In a system of many independent orbitals and at t=0, all orbitals below the Fermi energy are occupied and all above it unoccupied. Here we have calculated the Fermi level with the working definition m=(eHOMO + eLUMO) /2.
Footnote 2. In our case the bandwidths were calculated as follows. From the numerical set obtained from the Gaussian broadening we inspect visually the energies at which the DOS of a given band tends to zero at both sides. Their difference is the bandwidth. In the case of overlapping bands we used the energy of the middle point of the overlap.
This is an approximation that could be ameliorated in the future.
Footnote 3. In the EHT Hamiltonian the diagonal elements are the experimental valence state ionization energies (VSIE) with opposite sign. The off-diagonal elements are the interaction energies between two atomic orbitals. Interaction energies are the average of the binding energies (ie, the VSIE) multiplied by the overlap integral over the two atomic orbitals. Hyperchem scales this result by the Hückel constant, k, which has a default value of 1.75.
The conclusions of this work are:
a. Extended Hückel Theory compares well with experimental results related to Valence and Conduction bands for fullerenes. Therefore, it is suitable for a first screening of fullerene-like molecules.
b. Nevertheless, EHT underestimates the valence-conduction band gap by about 50%.
c. The prediction of the band structure of new fullerene-like structures using EHT is a valid approach.
We thank the Department of Chemistry, Faculty of Sciences, University of Chile, for financial help. The first part of this work corresponds to the Research Rotation of Nelson Gonzalez-Tejeda. We also thank Prof. Dr. John A. Weaver for his written permission to reproduce some figures of this work.
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