Mark R. Pederson,1 Tunna Baruah,2 B.J. Powell3 and W.A. Anderson4

1Center for Computational Materials Science, Code 6392, Naval Research Laboratory, Washington DC 20375-5345

2Department of Physics, University of Texas at El Paso, El Paso, TX 79968 3Department of Physics, University of Queensland, Brisbane AU

4Center for Computational Science, Code 5594, Naval Research Laboratory, Washington DC

Summary: The development of environmentally friendly, inexpensive, lightweight solar cells would significantly enhance both sea- and land- based DoD operations and directly enhance global security through a reduction of competition for carbon-based fuels. For example, capturing one out of every 10000 photons from the sun would provide all the power the world currently uses. Approximately 75% of the solar radiation striking the upper atmosphere (1368 W/m2 = 8555 eV/sec/nm2 ) reaches the surface of the earth and most of this is in the form of photons with more than 1 eV of energy since water vapor and other atmospheric constituents effectively absorb energy below this threshold. In order to produce a one micron photovoltaic molecular film (composed of approximately 1000 layers of a molecule) with 10-20 percent efficiency, each molecule would have to create 1-3 eV/sec. Since the available solar radiation is in the 1-4 eV range, this implies ~1-3 electron-hole pairs per second which biological systems currently achieve. As such, various organic molecules, some of which are biologically inspired, have been proposed as alternative building blocks for solar energy materials.

In the first phase of our challenge project, we have performed calculations on a molecular triad composed of a fullerene, a porphyrin and a carotenoid polyene. By calculating electronic structures, approximate excited states and respective dipole transition rates, we have simulated charge transfer dynamics in a collection of molecules exposed to an appropriate bath of solar photons. The resulting time constants associated with capture of solar radiation in the form of a charge-separated state have been determined. In the first phase of this challenge project, only electronic excitations were considered in the kinetic Monte Carlo modeling. Furthermore, the low symmetry in the molecular triad made it relatively easy to identify charge transfer excitations. In this phase of the project we report four new computationally intensive investigations that are aimed at validating and extending the formalism. First, to justify the use of an approximate excited state formalism, we have performed analogous calculations of excited states on a much larger selection of molecules and atoms and compared these results to experiment. Second we have performed calculations entirely analogous to our earlier molecular-triad results on highly idealized and high-symmetry fullerene tubules. Third we have used and extended a recently developed method[15] to calculate all electron-hole-phonon interactions in these carbon nanotubes as well as the light-harvesting molecular triad. Fourth, in analogy to symmetry-breaking methods used for density-functional treatments of ferro- and ferri- magnetic ordering in molecular magnets, we have developed and tested a massively parallel computational method, employing coarse and fine-grain strategies, for identifying self trapped ferroelectric excited states in highly symmetric carbon nanotubes. These techniques allow us to determine charge-transfer excitations and to determine whether excited-states are to be viewed within a Franck-Condon or Marcus-Hush picture.

Massively Parallel Simulation of Ferroelectric States and Light Induced Charge Transfer in Molecules

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I. Introduction Biological light-harvesting systems demonstrate amazing efficiency which is unmatched by the man-made solar cells. Therefore deployment of organic light-harvesting components in a solar cell device is a logical step to achieve highly efficient solar power converter. However, mimicking the photosynthesis process with high efficiency in the laboratory has proved to be a difficult task. The key to achieving such a device is in understanding the basic electronic processes in the system. One way to do so is to develop computational tools to simulate the basic processes and thereby understand the role of various components such as solvent molecules, polarization of the medium, and coupling between vibrational, electronic and polarization processes.

Depending upon the nature of the electronic and vibrational processes the charge transfer (CT) process may be described in terms of extended Marcus-Hush-Mulliken[1,2,3] or Franck-Condon formalisms. Requisite to either theory is very specific information about both ground and excited energies and potential energy surfaces and, in the latter, electronic transition rates. In this article, we discuss a methodology based on the density functional theory for determining this information. However, this methodology can be easily adapted for other quantum chemical methods. Density functional theory (DFT) is the choice here due to its inherent simplicity and computational efficiency. As we will show below, we apply these methods to a large molecular system comprising 80-700 atoms and therefore multi-determinantal quantum chemical calculations can not be used. One molecule which shows light induced charge transfer contains three different components – carotene, porphyrin, and C60 [4,5,6,7,8]. The porphyrin behaves as a chromophore absorbing energy and then the excited electron is transferred to the C60. The beta-carotene donates one electron to the C60 leading to a charge separated state with a huge dipole moment of 153 Debye. Our earlier calculations have shown that in this molecular system the effect of polarization plays a very important role in the charge transfer process. The effect of the phonons have not yet been investigated so far in this molecule. In the next section we describe in brief the earlier results to set the background and in the subsequent sections we describe our recently developed methods for calculation of excited-state energies, electron-phonon couplings and apply these methods to a carbon nanotube (CNT) model system to test the validity of the methods. The chosen model system is small enough for easy reproducibility.

II. Background

The geometries of the optimized structures of the molecular triad are shown in Fig. 1. The linear structure (Fig. 1(b)) has a large dipole moment of similar magnitude as determined by experiment. The electronic structure shows that the molecular orbitals are localized on the parent components and therefore charge transfer from one component to another results in a large change in dipole moments. Our calculations on a gas-phase molecule shows [9,10] the dipole moment for a charge-separated state can be as large as 180 Debye. It was found that the electronic transitions leading to a large dipole state is a rare event but the transition is also subject to the effects of the polarization of the environment. A detailed account of our calculations can be found in Ref.[9].

Table 1 . The excited state energies for closed shell atoms. The GGA and HF energies are calculated with a delta-SCF

Figure 2 Two optimized structures of the light harvesting molecular triad.

Figure 1 Model for one step of charge transfer process from the ground state of donor-acceptor (DA) to cationic donor and anionic acceptor (A). Q is the nuclear coordinate.

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method.

Our calculations are based on density functional

theory (DFT). While DFT has the advantage that it is computationally inexpensive compared to other first principles quantum chemical methods, a major problem of applying DFT for charge transfer systems is in the calculation of the excited states. DFT is based on the principle that the ground state electron density is unique for each system and all the system properties can be extracted from the ground-state density. While the ground state energy is obtained variationally using an approximation to the exchange-correlation functional, there is no practical method for obtaining the excited state energies. Time-dependent DFT has been used to calculate excited state energies for a number of systems, however it was found to be inadequate for description of charge separated states [11]. So we have developed a scheme for calculation of the energies of singly excited particle-hole states in which the particle-hole interactions are accounted for explicitly. A detailed account of the method is given below in section IV.

The vibrations of a molecule play an important role in the charge transfer process. However, for the purpose of simulations for the CT process, calculation of the vibrational modes of a molecule of the size of the triad is an expensive step, even only for the ground state. To make systematic progress in this area, a method for determining curvatures of excited-state potential energy

curves from calculations at the ground-state geometry are most desirable.

The charge-transfer process can be described by a donor-acceptor model as shown in Fig. 2. We have recently developed a method for the calculation of electron-phonon couplings which can be readily generalized to incorporate particle-hole excitations. The computational efforts in this method is of the order 1 and therefore it will be relatively inexpensive to use this method to calculate the electron-hole-phonon couplings for the molecular triad. This method is described in section V. The transitions between the ground and the charge separated state can be described in two different regimes. In the generalized Marcus regime [1,2,3], the regime where the system polarizations are important and the Frank-Condon regime where the ground and excited states are nearly at the same point in the 3N dimensional space. We apply the above mentioned methods to a small system so that the results can be easily checked. This work is described in section VI. In the future, these methods and analysis techniques will be applied to the molecular triad. III. Software and Hardware All the calculations described below were performed using the Naval Research Laboratory Molecular Orbital Library package known as NRLMOL [12,13,14]. A few salient points about this software are the use of atom-centered Gaussian basis functions, a mesh variationally optimized for integrations up to a given accuracy, analytical calculation of the Coulomb potential and massive parallelization of the code. The code has been developed by Pederson and his group over last two decades. Several HPCMO high performance computing platforms were used for these calculations. This includes the ARL Intel Xeon (jvn), the NRL Cray XD1 (kamala), and the MHPCC IBM SP (tempest). IV. Method for particle-hole states As mentioned earlier, the Kohn-Sham DFT can not be used for description of excited states, except for the cases where the excited states are the lowest ones of different symmetry. For the triad molecule, we have chosen a few states in the transition region near the Fermi energy. We have considered only single excitations so far. As pointed out in Ref. [9,10], the DFT eigenvalue difference puts the energy of a carotene-C60 excited state as 0.18 eV which underestimates the actual value by a factor of 10. This underestimation is due to the fact that as an electron is moved to another component in the triad, there should be an attractive

Atomic configuration

ΔGGA (eV)

ΔHF (eV)

This theory (eV)

Expt. (eV)

He (1s1)(2s1)

19.49 18.71 20.75 19.82

He (1s1)(2p1)

20.66 19.87 22.41 20.96

Ne [He]2s22p53s

16.84 14.95 18.35 16.84

Ne [He]2s22p53p

18.80 21.10 18.72

Ne [He]2s22p54s

19.78 19.33 19.66

Mg [Ne]3s3p

2.71 2.68 2.72

Mg [Ne]3s4s

4.93 5.87 5.10

Ar [Ne]3s23p54s

11.09 10.89 11.68 11.82

Ca [Ar]4s3d

2.12 1.91 2.51

Ca [Ar]4s4p

1.84 1.81 1.88

Zn [Ar]3d104s4p

4.15 4.64 4.07

Zn [Ar]3d104s5s

6.48 7.13 6.91

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interaction between the particle and the hole which is completely missed in the DFT eigenvalues. To alleviate this problem we use a perturbative approach in which it is assumed that the difference between the excited state potential and the ground state potential is small. The energy of the excited state is calculated as in the following. Ignoring the relaxation of the orbitals, the total energy difference (∆E) between the ground and the excited state can be expressed as :

phhp

hphphhpp

r

rFFE

!!!!

!!!!!!!!

|1|

|1|||||

12

12

+

""=#

where the subscripts p, h, and g to single-particle orbital φ stand for the particle, hole and ground, respectively. F is the Fock operator [16]. We calculate this energy by taking a perturbative approach. We have carried out some extensive tests on different systems ranging from inert atoms, other closed-shell atoms and small molecules. In Table I and II the results are summarized. We compare our results with a frozen occupation self-consistent field (the so-called ΔSCF [17]) calculation. The results for small molecules are compared in Table

II. The results show that the approximations work well for larger systems. The energies for the gas-phase triad molecule is also calculated. While direct comparison with experiment is not possible since the experiments

are carried out in solutions with polar solvent, our result is in accord with what is expected from known electron affinities and ionization energies of the molecular subunits. To make direct contact with experiment, a

correction due to the environment polarization is necessary. An approximate polarization correction based on the Drude model brings the states close to experiment. However, the corrections depend on the ratio of molecular polarizability to volume and thus subject to molecular concentration. V. Electron-phonon coupling

The electronic energy can formally be expanded in a

Taylor’s series as function of excess charge (n) and vibrational displacement (Q). We have earlier shown that the electron-phonon coupling appears as the change in Hellman-Feynman force as a function of extra charge [15]. That implies that the electron-phonon coupling can now be calculated by performing one SCF calculation on a charged molecule. While the method represents a major computational enhancement, validation through a series of frozen phonon calculations requires the full resources of the HPCMO DoD computational platforms. This formulation can easily be generalized to a charge transfer system where n is now the amount of the charge being transferred rather than removed or added. For more details we refer the interested reader to Ref. [15]. To second-order, the spring-constant matrix is identical for both the ground and the CT state. As such this formulation leads to an easy estimation of the excited state energies as a function of Q, where Q is an arbitrary displacement from the equilibrium point. For charge transfer systems, the transition coordinate is nearly equal to the displacement between the ground and excited-state equilibrium geometries. The charge transfer process can be described by a donor-acceptor model as shown in Fig. 2. The donor for the porphyrin-C60 charge transfer is the porphyrin and C60 is the acceptor. The photo absorption process leads to a porphyrin-C60 acceptor state which is in Frank-Condon regime. The charge transfer from the porphyrin to C60 however can have a large polarization-induced reorganization energy leading to a state at a large Q (Marcus regime). This reorganization energy will be strongly dependent on the environmental polarization. It is worth mentioning here that experimental observation of the CT state is strongly dependent on the solvent. Solvent with a large permanent dipole moment is observed to assist the charge transfer process. As shown in Fig. 2, the vibrational transitions also play an important role here. It is important to understand at which regime the charge-transfer in the triad takes place. Since the calculation of the normal modes for the triad is time-consuming, we have tested the above mentioned theories on a much smaller system – a finite carbon nanotube. These calculations are now described.

State ΔSCF (eV) (GGA)

Present (eV)

Expt. (eV)

N2 S (H-L) 8.58 7.50 8.04

T (H-L) 7.53 7.99 9.31

CO S (H-L) 6.84 6.80 8.51

T (H-L) 5.96 6.03 6.32

H2O S (H-L) 7.30 7.28 7.40

T(H-L) 7.07 7.09 7.20

C2H4 S(H-L) 4.46 4.47 4.36 S(H-L+1) 7.54 7.51 7.11

T(H-L+1) 7.40 7.40 6.98

Table 2. The excited state energies for test molecules. S and T denote singlet and triplet state and H and L are abreviations for HOMO and LUMO.

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VI. Applications to Carbon Nanotubes

Carbon nanotubes [18,19] (CNT) have received significant attention lately for a variety of DoD relevant applications. Applications related to chemical, biological and radiological sensing will require an understanding of how the current-voltage (IV) characteristics and/or charge-separated excited states depend upon these environmental changes. Light, either in the form of solar-, laser-, or other radiation will create excited states. The light-induced excited states which cause conduction or long-lived metastable charge separated states can form the basis of new technologies. In addition to the impact on emerging technologies, calculations on photo-induced charge transport in carbon nanotubes provides a good means for testing the order-1 method for electron-phonon coupling and the excited state method. We have performed calculations on a large class of fullerene nanotubes ranging in sizes from ~80 to ~700 atoms.

These are pure carbon nanotubes (e.g. devoid of

hydrogen terminators). All of the fullerene tubules considered have relaxed to a geometry with short pi-bonded C-C bonds near the CNT tips. The emergence of these short bonds is responsible for the opening of the energy gaps between the highest occupied and lowest unoccupied molecular orbitals (HOMO/LUMO gaps) and the presence of some high-energy vibrational modes as well. Pictured in Fig. 3 is the vibrational density of states for one of these tubules. Near the energy of 1867 cm-1, there is a strong infra-red (IR) active vibrational mode due to the carbons at the CNT tip beating against one another in an asymmetric fashion. Since the C-C triple bonds on one end stretch while the those on the other contract, there is relatively strong charge transfer induced across the tip which is responsible for the IR activity. In the Table (below), the HOMO/LUMO gap as a function of CNT length and diameter is presented. These calculations correspond to fully relaxed CNTs. Table 3. Diameters and HOMO-LUMO gaps of

the carbon nanotubes that are studied here. CNT Diameter Number Energy

Length (nm)

(nm) of Atoms Gap (eV)

1.46 (1) 0.82 120 0.54 1.57 (2) 0.4 84 0.68 3.54 (3) 0.4 180 0.17 6.97 (4) 0.4 348 0.13 2.54 (5) 2.0 704 0.40 In terms of applications which employ carbon nanotubes, or other high-symmetry polymers, for photovoltaics, it is first important to determine whether the molecule in question is capable of accommodating a long-lived charge-separated excited state. This question can be answered using NRLMOL, the DoD HPC computing platforms and the recently developed O(N) electron-phonon method.

As an initial test of the order-N method discussed above we have calculated the vibrational spectrum and electronic excited states of CNT-2 as a function of electric field. There are a total of 18 low-energy particle-hole excitations that could participate in light-induced charge transfer. There are many possible particle-hole excitations that lead to a CNT that has no dipole moment. To find the states which may have long-

lived dipole moments we have modified a method used in our DFT-based investigations of molecular magnets. For the case of ferro- and ferri- magnetically ordered states in molecules, it is necessary to apply some symmetry breaking potential or localized magnetic field that allows the system to order ferro-magnetically. Once the ordering is achieved, the field is removed and the symmetry broken state either heals itself (i.e. returns to a

Figure 3 Vibrational density of states for the CNT investigated here. The spring-constant matrix, which determines the obsevable IR and Raman spectra, may also be used in conjunction with electron-hole-phonon interactions to determine transition pathways.

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high symmetry state) or relaxes self-consistently to an optimal magnetically ordered state. Here we adopt the same strategy but use a large applied electric field to prepare symmetry breaking states in the CNT. We tested several electric fields to obtain an electron density and Kohn-Sham wavefunctions that break the intrinsic inversion symmetry of the CNT. Then, we create particle-hole excited states from the occupied and unoccupied molecular orbitals and perform SCF calculations on these structures as a function of applied field. These calculations were performed in parallel on several DoD platforms (ARL JVN, NRL CRAY XD1) using coarse-grained parallelism (over electric fields and particle-hole pairs) and fine-grained (MPI honey-bee algorithms) for each configuration. Examination of the energy, dipole moments and polarizabilities as a function of electric field identified possible candidates for charge-transfer excitations.

Table 4. The dipole moments and polarizabilities of the ground and excited states of the CNT.

The energies, dipole moment and polarizabilities of

these states, at zero field, are shown in the Table (4). These calculations identified states 6, 7 and 13 as ones with interesting field-induced signatures. State 7 and 13 have large permanent dipoles and represent the states that would be most likely to participate in light-induced charge separation. The appearance of a small polarizability associated with state 6 indicates that a higher-energy excitation crosses this state when the CNT is exposed to a large electric field. These states will also

vary strongly when placed in a polarizable medium, counter ions or electric fields.

To further refine the description of the charge-transfer excited states, we have used the spring-constant matrix (used for determining the vibrational modes in Fig 2), the particle-hole-vibron interactions and the calculated excited states to determine 3N-dimensional displacements between ground and excited-geometries. In Fig 5, we show the calculations for one of the interesting excited states. For both the ground and excited states we show the parabolic curves as a function of transition coordinate (black dashed lines). The classical crossing of these parabolas is generally referred to as Qt. In eighteen cases studied we find that the crossing appears after crossing the equilibrium geometry of the excited state. In other words, the excitations of relevance to this CNT should be described in a Franck-Condon rather than Marcus-Hush picture. This conclusion could change for CNT’s embedded in a dielectric matrix.

Under these conditions the charge transfer excited states would be further stabilized by the long-range interactions between the dipole and the medium. Also pictured in Fig. 4 are the energies (red and blue respectively) of the ground and excited-state manifolds calculated self consistently along the transition coordinate. This shows that the two energy surfaces repel each other and do not lead to a crossing that is attainable at low temperatures. Prior to developing time-dependent models of charge transfer it is necessary to determine which excitations are to be treated with a Franck-Condon method and

State Energy (eV)

Dipole (Debye)

Polarizability (Ang3)

1 (grnd) 0 0 363 2 0.70 0.05 451 3 0.52 0 387 4 0.98 0.03 463 5 0.94 0 384 6 0.71 0.1 112 7 1.10 -0.98 396 8 1.15 -0.04 553 9 1.02 0 388 10 0.88 0 383 11 0.74 0 392 12 1.95 0.03 436 13 2.07 1.35 501 14 1.86 0 380 15 1.96 0 384 16 2.32 0 402 17 2.41 0.02 394 18 2.43 0 406

Figure 4 Energy of ground and one charge-transfer excited state as a function of transition displacement. Black dashes show the curves predicted from order-1 algorithm described within. Blue and red curves show frozen-phonon-based validation of the method.

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which are to be treated within a Marcus/Hush approach. Here we illustrate how the parallel order-1 determination of electron-phonon interactions can be used to efficiently determine the character of intersecting excited and ground state energy surfaces.

In Fig 5, we show the charge density contours of the charge transfer excitation that has been predicted in this work. Based on the results of these calculations we expect that there will be two low lying dipole carrying excited states in this CNT.

VII Summary

We have used a suite of massively parallel computational techniques to determine charge-transfer states, their energies and transition rates in organic molecules that may form the basis for future DoD photovoltaic applications. In the first phase of our challenge proposal, we have performed calculations on a molecular triad composed of a fullerene, a porphyrin and a carotenoid polyene. By calculating electronic structures, approximate excited states and respective dipole transition rates, we have simulated charge transfer dynamics in a collection of the triad molecules exposed to an appropriate bath of solar photons. The resulting time constants associated with capture of solar radiation

in the form of a charge-separated state have been determined. In the second phase of the proposal, we have performed three new types of calculations. First, to justify the use of an approximate excited state formalism, we have performed analogous calculations of excited states on a much larger selection of molecules and compared these results to experiment. The results show that our method is appropriate for large systems as the molecular triad. Second, we have recently developed a method to calculate the HOMO and LUMO electron-phonon couplings which scales as order 1. This method is then generalized to calculate the electron-phonon interactions for charge-transfer excitations. Third, we have tested the developed methods by performing similar calculations on highly idealized and high-symmetry fullerene tubules. Fourth, in analogy to symmetry-breaking methods used for density-functional treatments of ferro- and ferri- magnetic ordering in molecular magnets, we have developed and tested a massively parallel computational method, employing coarse and fine-grain strategies, for identifying self trapped ferroelectric excited states in highly symmetric carbon nanotubes. We find that our method to predict the charge-transfer state from a few calculations on the ground state geometry yields results comparable to a frozen-phonon calculation. A combination of coarse-grained and fine-grained parallelization has been used to predict the complicated charge-separated excitations in these systems. The results are promising and the methods will be applied to large molecular calculations. In future, methods will be developed and tested to further reduce the computational efforts. Another area of concentration will be an accurate description of solvent polarization.

ACKNOWLEDGMENTS

We thank Dr. J. Osburn and P. Shingler for crucial and immeasurable support in a variety of capacities related to DoD HPCMO resources. We thank Dr. C. Ashman and N. O. Jones for help with technical details related to running NRLMOL on the HPCMO platforms. Authors acknowledge ONR (Grant No. N000140211046) and DoD CHSSI Program. TB acknowledges NSF Grants No. HRD-0317607 and NIRT-0304122. The graphics were created through MOLEKEL [18] package, gnuplot and xmgrace.

REFERENCES [1] R. A. Marcus, J. Chem. Phys. 24, 966 (1956).

Figure 5. Charge transfer plots for two ferroelectric charge transfer states predicted using the DOD HPCMO computer resources and new algorithms developed at NRL.

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[2] R. A. Marcus, Annu. Rev. Phys. Chem. 15, 155 (1964).

[3] N. S. Hush, Ann. N. Y. Acad. Sci. 1006, 1 (2003). [4] P.A. Liddell, D. Kuciauskas, J. P. Sumida, B. Nash,

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[7] D. Carbonera, M. D. Valentin, C. Corvaja, G. Agostini, G. Giacometti, P. A. Liddell, D. Kuciauskas, A. L. Moore, T. A. Moore, D. Gust, J. Am. Chem. Soc. 120, 4398 (1998).

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[9] T. Baruah, M. R. Pederson, W. A. Anderson, Proc. HPCMP Users Group Conference , p. 11(2005).

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[12] M. R. Pederson and K. A. Jackson, Phys. Rev. B. 41, 7453 (1990); ibid 43, 7312 (1991); K. A. Jackson and M. R. Pederson, ibid 42, 3276 (1990).

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[14] M. R. Pederson and K. A. Jackson, Phys. Rev. B, 41, 7453 (1990).

[15] B. J. Powell, M. R. Pederson, and T. Baruah, submitted to Phys. Rev. Lett.

[16] Any book on Quantum Mechanics. See, for example, I. N. Levine, Quantum Chemistry, 4th ed.; Prentice Hall (2000) .

[17] R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).

[18] S. Iijima, Nature 354, 56 (1991). [19]M. R. Pederson and J. Q. Broughton, Phys. Rev. Lett. 69, 2889 (1992). [20] MOLEKEL 4.3, P. Flükiger, H.P. Lüthi, S. Portmann, J. Weber, Swiss Center for Scientific Computing, Manno (Switzerland), 2000-2002. Stefan Portmann & Hans Peter Lüthi. MOLEKEL: An Interactive Molecular Graphics Tool. CHIMIA (2000) 54 766-770.