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## Journal of the Chilean Chemical Society

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*On-line version* ISSN 0717-9707

### J. Chil. Chem. Soc. vol.51 no.4 Concepción Dec. 2006

#### http://dx.doi.org/10.4067/S0717-97072006000400016

J. Chil. Chi. Soc., 51, N°.4 (2006), p.1061-1064
Universidad de Chile, Facultad de Ciencias, Departamento de Química. Casilla 653 Santiago, Chile; and Programa de Doctorado en Fisicoquímica Molecular, Facultad de Ecología y Recursos Naturales, Universidad Andrés Bello, Santiago, Chile. facien03@uchile.cl
Two medium-sized molecular systems X60 (X=C,N) were analyzed using Extended Hückel, Density Functional and Keywords: Extended Hückel Method, N_{60}, C_{60}, Total Density of States, Density Functional Method.
In 1963, Hoffmann developed a semiempirical quantum mechanical method known as the Extended Hückel Theory (EHT) [1]. EHT is the simplest and most primitive of all all-valence-electron methodologies. EHT models all the valence orbitals based on the orbital overlaps and experimental electron affinities and ionization potentials. The diagonal elements of the Hamiltonian are taken as the negative of the normal first ionization energy of the atom corrected by spectroscopic terms to deal with the situation where the normal ionization is not removing the electron from the orbital in question. The off-diagonal matrix elements of the Hamiltonian are calculated according to the modified Wolfsberg-Helmholz formula [2]. The use of experimental ionization energies for the atoms in the molecule implies that the correlation energy is taken care of. It is known that, in general, EHT performs rather poorly at predicting energy differences between isomers or even correct molecular geometries [3], but there are some exceptions [4]. Charge differences, particularly between atoms of very different electronegativity, can be grossly exaggerated. However it does give some useful results in some cases. The strength of EHT is that it gives a good qualitative picture of the molecular orbitals (MOs). It is now known that for the occupied MOs the corresponding eigenvalues agree reasonably well with experimentally determined ionization energies from photoelectron spectroscopy (PES). In 1988 a study found that EHT is also useful to determine unoccupied levels [5]. It is concluded that EHT can be used to study both occupied and unoccupied orbitals of a molecule, since it is directly useful for the calculation of excitation energies [5]. The reasons for the good performance of EHT have elicited some interesting theoretical analyses. It was suggested that the EHT may be regarded as a method of simulating Hartree-Fock (HF) calculations by guessing the elements of the HF Hamiltonian matrix through the use of the Wolfsberg-Helmholz approximation [6]. More recently, it was shown that, within the Hartree-Fock-Rüdenberg picture (HFR), EHT is compatible with the nonempirical Hartree-Fock-Roothaan method [7-8]. HFR thus explains why EHT turned out to be qualitatively successful [see Ref. 9 for example]. With the above mentioned considerations, EHT is still undoubtedly a useful tool in areas where SCF calculations will not be feasible for some time to come. We must keep in mind that, as the EHT formalism does not take into account explicitly the electron-electron interaction, we should expect that the molecular orbital energies will be shifted downwards. Also a decrease in the energy difference between any pair of MOs is to be expected. In a previous paper we have introduced the concept of “minimal length” (or minimal size) to set the boundary between a big molecule and a material properly speaking [10]. At least for the case of armchair and zigzag nanotubes this boundary implies that for a material we have to obtain the wave function of a system composed of more than one hundred atoms. Computing capabilities preclude for the time being the use of Hartree-Fock or Density Functional calculations. Regarding molecular systems composed of hundreds or thousands of atoms in which a large or very large number of electrons are delocalized the following may be securely stated. Such systems have an extremely large number of molecular orbitals. The result, as the number of levels tends to infinity, is that MOs become very similar in energy over a certain range, forming an almost continuous band. In this case, and if we define zero energy as the midpoint between the Highest Occupied MO (HOMO) and the Lowest Unoccupied MO (LUMO) energies, the eigenvalues about zero will be almost the same in EHT, HF or DFT calculations. In the case of medium size aromatic molecules, like the first members of the fullerene family, no work has been done to discuss band formation from the MO eigenvalues. Thereby, in this paper we analyze the performance and the limits of EHT to obtain molecular band structures for fullerene and fullerene-like molecules. Special emphasis is placed on the comparison of the EHT and DFT energy distributions of the eigenvalues. Also we examine whether or not at this molecular size level the MO eigenvalues have begun to form an almost continuous band. This last aspect is important because it provides data that could be applied to molecules whose study by
Three molecular systems were selected to achieve our objective: C The calculations were performed as follows. For the effects of comparison, the geometry of the molecules was fully optimized with Molecular Mechanics (MM), with the AM1 semiempirical method and with an The Hyperchem package was employed for MM, AM1 and EHT calculations [11]. The RHF 6-31G** and B3LYP/6-311G** (hereafter DFT) calculations were performed with the Gaussian package [12]. We also used DFT C The valence (VB) and conduction (CB) bands were obtained separately through a convolution of the occupied and empty MO energies with a Gaussian function [14]. A value of 0.1 eV was used for the broadening parameter and the scanning distance [13]. For the sake of comparison, in the case of the VB the HOMO energy was placed at E=0.0 eV for both EHT and DFT results. For the CB the LUMO energy was placed at E=0.0 eV.
The first point to note is that we could not obtain a stable icosahedral structure for B The second point to stress is that Molecular Mechanics was not able to produce a stable icosahedral structure for N DFT, AM1 and This first important result is that the order of the degeneracies of the MOs is not the same for B3LYP/6-311G**//6-31G** and BP/DZP//BP/DZP calculations. Tables 1 and 2 show, respectively, the degeneracies of the first 10 occupied and empty MOs. We can see that the HOMO and HOMO-1 multiplicities of C
Given that BP/DZP//BP/DZP results agree with other theoretical studies we suggest that the source of error in the B3LYP/6-311G**//6-31G** results lies in employing different levels of calculation for single point and geometry optimization calculations. Therefore in the following we shall compare only EHT and BP/DZP//BP/DZP results. Notice that in the case of N From earlier results we know that the DOS spectrum of C In the case of the valence region of C In the case of the first valence band of N Therefore, the first general conclusion of this work is that EHT is only reliable for the first valence and conduction bands of medium sized molecules. In this kind of molecules the eigenvalues are still well separated in energy. As there is still no "compaction" of the occupied MO energies, the main source of error of EHT calculations comes directly from the non-inclusion of the electron-electron interactions. This source of error should disappear only in the case when the occupied MOs become very close in energy (i.e., in bigger molecules). Note that in C Is there any mathematical relationship between the X60 (X=C,N) set of EHT eigenvalues and the DFT ones? If we can find one, we may "correct" the EHT eigenvalues to get a better DOS spectrum. But we may stress that this kind of relationship will not correct the disagreements in the relative ordering of the MO degeneracies. To explore this idea, and keeping in mind that we are interested in the DOS around the Fermi Level, we performed the following linear fits: a) Between the first sixty occupied DFT and EHT eigenvalues, for C b) Between the first thirty empty DFT and EHT eigenvalues for C
We may see immediately that the standard deviation is too high in all equations. We conclude therefore that it is not possible to build a "corrected" set of EHT eigenvalues. The main conclusion of this work is that the use of the EHT eigenvalues to get the total DOS curve for medium sized molecules is not reliable despite the interesting results regarding the first valence and conduction bands. All the error sources reported here should disappear when the occupied and empty molecular orbitals become very similar in energy over a certain range, forming an almost continuous band, as happens in bigger molecular systems.
The author is grateful to Dr. Boris Weiss-López (Faculty of Sciences, University of Chile) for providing access to the Gaussian Package and computing time of the Computer Cluster of his Laboratory. Prof. Dr. Ramiro Arratia-Pérez (Universidad Andrés Bello) is thanked for providing access to the ADF Package. Prof. Dr. Bruce Cassels (Faculty of Sciences, University of Chile) is thanked for helpful comments.
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