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David Danovich (e-mail: firstname.lastname@example.org) and Sason Shaik
Department of Organic Chemistry, The Hebrew University, Jerusalem 91904, Israel
Presented at the 210th ACS National Meeting in Chicago, August 21, 1995
Fullerenes, with C60 as a prototype, have attracted great interest in various areas of research . Understanding the electronic structure of C60 and related fullerenes is central to the emerging chemical and physical picture of these fascinating molecules. The ionization of a molecule provides the most direct experimental probe of the electronic structure. Photoelectron spectroscopy (PES) has developed during the last few decades into an extremely useful experimental technique for studying the electronic structure of atoms and molecules, and in particular for analyzing the bonding characteristics of orbitals and their mutual interactions . In order to interpret a PE spectrum and to fully exploit the information contained in it, a theoretical determination of the ionization potentials (IPs) is required.
Usually Koopmans' theorem , equating the IP with the negative value of the energy of the molecular orbital from which the electron is removed, is used in the interpretation of IPs. A large number of theoretical methods and techniques, semiempirical as well as ab initio, are available for calculating the energies of molecular orbitals. However, in many cases these theoretical methods fail to provide the quantitative IPs correctly, or even to reproduce the correct orbital ordering. Cederbaum and his coworkers have shown that the combination of ab initio techniques with Green's function method can improve significantly the ability of theoretical calculations to accurately predict IPs, for a variety of systems . In this method many-body perturbation theory is used to derive the equations that calculate IPs including corrections for electron correlation and for orbital relaxation effects . Hartree-Fock solutions obtained from ab initio calculations serve as the zeroth order approximation in the perturbation series.
Unfortunately, the application of ab initio-Green's function method is still prohibitively expensive for the large size molecules. To overcome this difficulty, one of us has suggested recently [5,6] to couple the Outer-Valence Green's Function (OVGF) approach with semiempirical methods such as MNDO , AM1  and PM3  (denoted in this presentation as OVGF(corresponding semiempirical method)). The OVGF(semiempirical) method was recently successfully applied to calculate the IPs of a variety of nitrogen-containing heterocycles, such as substituted pyridines (30 compounds), pyrimidines, pyridazines and azoles [5,6]. More recently we applied the method to calculate the IPs of a series of substituted triazines, tetrazines [10a], cyclic and linear polysilanes [10b] as well as for calculating the IPs of radicals and the electron affinities of various neutral compounds [10c]. In general it was found that inclusion of the Green's function method in the calculation improves substantially the quantitative agreement between theoretical and experimental data [5,6,10]. A computer program combining the OVGF formalism with the NDDO based semiempirical methods is available from QCPE [11a], and it was also incorporated in the new MOPAC-93 [11b] and AMPAC 4.5 [11c] packages.
The current theoretical understanding of the spectroscopy of fullerenes relies on a series of semiempirical [12-15] and ab initio calculations [16-18]. Estimates of the first ionization energy based on electronic structure calculations range from 4.5 to 9.1 eV for C60 and from 5.1 to 8.7 eV for C70 [12-18]. Thus, there are considerable differences among the calculations in the prediction of the first IPs in this class of molecules. Existent semiempirical methods are unable in generally to reproduce quantitatively the PES of fullerenes. From this point of view it is interesting to evaluate the potential of the OVGF(semiempirical) approach to reproduce correctly the PES of fullerenes.
This talk reports OVGF calculations coupled with the semiempirical MNDO, AM1 and PM3 methods as well as with the new SAM1  method for neutral and charged C60 and C70. We compare our computational results with the available experimental IP values, as well as with previous semiempirical and ab initio calculations. The focus here is on the highly symmetrical and most stable Ih and D5h isomers of C60 and C70 correspondingly.
The OVGF method is described in detail in ref. 4. The application of this method for the case of semiempirical wave functions was discussed in refs. 5 and 6. Here we briefly present some of the main points about the computational method and the interested reader is referred to ref. 4 for more details.
The OVGF technique was used with the self-energy part extended to include third-order perturbation corrections . The higher order contributions were estimated by the renormalization procedure. The actual expression used to calculate the self-energy part, , chosen in the diagonal form, is given in equation 1, where and are the second- and third-order corrections, and A is the screening factor accounting for all the contributions of higher orders.
The particular expression which was used for the second-order corrections is given in equation 2.
In equation 2, i,j - denote occupied orbitals, a,b- denote virtual orbitals, p - denotes orbitals with unspecified occupancy, - denotes the orbital energy and w - is the corrected IP. The equations were solved by an iterative procedure which is given in equation 3.
The SCF energies and the corresponding integrals, which were calculated by a particular semiempirical method (MNDO, AM1 or PM3), were taken as the zeroth order approximation for the Green's function expansion and a maximum of 80 occupied MOs and 80 virtuals MOs were included in the active space for the OVGF calculations.
The SCF orbital energies of the open-shell fullerenes were calculated by the half-electron method , using doublet correction for ionization potentials. The expressions used for and A are given in ref. 4b. In this study only second-order correction was used.
The geometries of the parent C60 and C70 were fully optimized with the specified semiempirical methods using the MOPAC or AMPAC-4.5 programs. The main criteria which will be used to compare the experimental and the computational data and to assess the performance of the various computational methods; are the deviation , between the experimental and theoretical ionization potentials for a particular orbital, or the mean deviation  for all the orbitals of a particular fullerene.
The PES of C60 in the outer valence region below 12 eV shows three well separated bands at 7.61, 8.95 eV and mixed bands in the range of 10.82-11.59 eV . The calculated IPs of C60 by SCF(MNDO), SCF(AM1), SCF(PM3), SCF(SAM1), OVGF(MNDO), OVGF(AM1) and OVGF(PM3), as well as previous data; computational [13,14] and experimental , are summarized in Table 1.
Experimental and calculatedi (by different semiempirical methods) ionization energies of C60 (Ih)(a)
a)All values in eV. In parentheses are presented the deviations between theoretical and experimental values. b) In all OVGF calculations the active space is a window with 80 occupied MOs and 80 virtual MOs. c) From ref. 13. d) From ref. 14. e) From ref. 21. f) Four mixed bands are in the range of 10.82 - 11.59 eV.
At the SCF level, with MNDO, AM1, PM3 and INDO methods, the calculated ionization energies exhibit substantial deviations from the experimental data. For some of the particular orbitals this deviation is as high as 2 eV. The SCF(SAM1) gives better estimates of IPs in comparison with the other SCF calculations; having a mean deviation  for the three first occupied MO of ca. 0.84 eV. But even for this novel method the mean deviation is substantial and the method is unable to reproduce correctly the features of the experimental PES; differences between two experimentally degenerated hg and gg MOs is 0.35 eV and between gu and hg MOs is 1.47 in comparison with 0.77 eV in experimental PES.
The addition of Green's function improves significantly the agreement between theoretical and experimental results (by more than 1 eV for every orbital) for all methods. But, even after OVGF correction the deviation from experiment is quite large for OVGF(AM1) and OVGF(PM3), especially for first three occupied MOs. For C60 the best agreement between theory and experiment is achieved with OVGF(MNDO) method; the mean deviation  for first three HOMOs decreases to 0.29 eV. It is important to note that OVGF(MNDO) is able to reproduce quite well the features of the experimental PES.
The valence HeI PES of C70 is much more complicated than for C60. The ionization peacks in the outer valence region below 12 eV are grouped into three roughly separate bands of overlapping ionizations . The first band runs from about 7.4 to 8.8 eV, the second one from 8.9 to 10.3 eV and third continues from 10.3 to 12 eV. The calculated IPs for C60 by SCF(MNDO), SCF(AM1), SCF(PM3), SCF(SAM1), OVGF(MNDO), OVGF(AM1) and OVGF(PM3), as well as previous computational  and experimental  data are summarized in Table 2.
Experimental and calculated (by different semiempirical methods) ionization energies of C70 (D5h)(a)
a) All values in eV. Values in parentheses are the deviations between theoretical and experimental values. b) For all OVGF calculations the active space is a window with 80 occupied MOs and 80 virtual MOs .c) Ab initio results using a double zeta plus polarization basis set from ref. 17 d) From ref. 22.
At the SCF level, with MNDO, AM1, PM3 and INDO, the calculated ionization energies exhibit substantial deviations from the experimental data. The SCF(SAM1) method is again quite good for the first ionization potentials, but disagreement between theoretical and experimental data increases for lower lying MOs. The mean deviation  for the six HOMOs is 0.69 eV in the case of SCF(SAM1) calculation and it increases to 0.80 eV for eleven occupied MOs. Very good agreement was obtained also for ab initio calculations with DZP basis set . The mean deviation  for the six HOMOs is only 0.31 eV, but calculated IP of the MO of a1' symmetry by this method is very bad. Including the result for this orbital the mean deviation for seven occupied MOs increases to 0.51 eV. It is important to note that the experimental data here are generally estimated due to strong overlap of the ionization peacks  and this estimation was carried out on the base of above ab initio calculations.
OVGF technique applied to AM1 and PM3 calculations improves considerably results of corresponding SCF calculations, but disagreement between theoretical and experimental values is still quite significant. The best results over the all semiempirical methods was obtained with OVGF(MNDO) method in the case of C60 as well as C70. The mean deviation for six HOMOs is only 0.41 eV with a good qualitative agreement with the experimental PES. For the eleven occupied MOs the mean deviation  decreases considerably to 0.26 eV.
The dependence of the calculated IPs on the number of the active orbitals taken into OVGF calculations are presented in Table 3 for OVGF(MNDO) method, which can be considered as the best method for calculating IPs of fullerenes.
C60 and C70 OVGF(MNDO) calculations with a variable window size.
|Window size||Window size|
a) a / b, a - the number of occupied MOs which were taken for OVGF calculations, b - the number of virtual MOs which were taken for OVGF calculations.
Inspection of Table 3 indicates that increasing the number of the active orbitals improves the results of OVGF calculations. This conclusion is true especially for low lying occupied orbitals. Increasing the size of the active orbital window from 40 HOMO - 40 LUMO to 80 HOMO - 80 LUMO decreases the theoretical experimental disagreement by 0.18 eV for the HOMO whereas for HOMO-6 this reduction is 0.88 eV. Unfortunately increasing the size of the active orbital window requires using the swap space on our workstation and slows down the calculations considerably.
The data presented in Tables 4 and 5 demonstrate that the effect of electron correlation introduced into the calculations by the OVGF technique is quite large for both fullerenes as well as for their anion radicals.
First IP of C60(Ih) and C60-(Ih) calculated with SCF, SCF and OVGF techniques(a)
a) All values in eV. b) The calculations are performed on the geometry of neutral molecule. c) Difference between Koopmans' and SCF values. d) For all OVGF calculations the active space is a window with 80 occupied MOs and 80 virtual MOs . e) Difference between Koopmans' and OVGF values. f) From ref. 23.
First IP of C70(D5h) and C70-(D5h) calculated with SCF, SCF and OVGF techniques(a)
a) All values in eV. b) The calculations are performed on the geometry of neutral molecule. c) Difference between Koopmans' and SCF values. d) For all OVGF calculations the active space is a window with 80 occupied MOs and 80 virtual MOs. e) Difference between Koopmans' and OVGF values. f) From ref. 23.
In comparison with neutral molecules, electron correlation effect is much smaller in the corresponding anion radical for the first IP. For both neutral molecules and their anion radicals the orbital relaxation effect, which can be estimated by SCF procedure, is very small and almost the same for neutral and charged molecules. According to our own semiempirical calculations and to Green's function calculation coupled with INDO  the Koopmans' defect exceedi by 1 eV for higher lying orbitals and increases to 1.5 eV for lower lying MOs.
Inspection of Table 4 shows that first C60- IP which can be described also as the vertical electron affinity (EA) of the neutral molecule can not be reproduced well by Koopmans' theorem with all the semiempirical methods, eventhough the disagreement is now considerably smaller than for the neutral fullerenes. Good results were obtained for OVGF(AM1), OVGF(PM3) and SCF(MNDO), with deviations of less than 0.1 eV. The deviations with SCF(SAM1), SCF(SAM1) and OVGF(MNDO) methods are considerably (ca. 0.4 eV) higher. On the other hand, SCF(SAM1) and OVGF(MNDO) give excellent agreement within 0.01 - 0.03 eV for EA of C70- (see Table 5).
We have studied the ability of semiempirical calculations coupled with OVGF techniques to predict correctly the PE spectra of the valence electrons of fullerenes. The SCF(SAM1) method is found to produce reliable results only for first higher lying occupied MOs and fails for lower lying orbitals. It also gives much higher results for EA of C60. The best and the most consistent results for neutral and charged fullerenes are obtained with the OVGF(MNDO) method which produces much impoved results over the SCF(MNDO) and SCF(SAM1) calculations. OVGF(MNDO) and SCF(SAM1) methods calculates the first IP to be lower by up to ca. 0.70 ev for C60 and 0.45 for C70.
This implies that even at this level of theory the magnitude of the destabilizing interactions between (C-C) bonds are underestimated. We conclude that the OVGF technique coupled with semiempirical methods (especially with MNDO) as the zeroth approximation is a very reliable and economic computational method for predicting the IPs and EAs of fullerenes. We are currently applying these methods to study the electronic properties of other fullerenes. The experience with this technique projects its potential as a routine method for studing the ionization patterns of large molecules.
This research is supported by the Volkswagen Stiftung. administered by the Israel Academy of Science and Humanities.
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