In this presentation I have tried to combine scientific (like in hardcopy
journals) and informational (communication) potential of **Web** technology.
By clicking the word "Fullerenes" in the first sentence you can obtain
interesting information about fullerenes. You always can return to the page
where you were before by using "back" button on Netscape or MOSAIC browsers.

David Danovich (e-mail: dodik@yfaat.ch.huji.ac.il) 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 C_{60}
as a prototype, have attracted great interest in various areas of research
[1].
Understanding the electronic structure of C_{60}
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
[2]. 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
[3], 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
[4]. 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
[4].
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
[7], AM1
[8] and^{
}PM3 [9]
(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 C_{60}
and from 5.1 to 8.7 eV for C_{70}
[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
[19] method
for neutral and charged C_{60} and C_{70}.
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 I_{h}
and D_{5h} isomers of C_{60}
and C_{70} 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
[4]. 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.

where

=

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 [20], 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 C_{60} and C_{70}
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 C_{60} 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
[21]. The
calculated IPs of C_{60} 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
[21], are summarized in Table 1.

Experimental and calculatedi (by different semiempirical methods) ionization
energies of C_{60} (I_{h})^{(a)}

MO | AM1 | OVGF(AM1)^{(b)} |
MNDO | OVGF(MNDO) | PM3 | OVGF(PM3) | SAM1 | INDO/S^{(c)} |
INDO^{(d)} |
INDO(GF)^{(d)} |
Exp^{(e)} |
---|---|---|---|---|---|---|---|---|---|---|---|

h_{u} |
-9.64 | 8.78(1.18) | -9.13 | 8.27(0.67) | -9.48 | 8.72(1.12) | -8.47(0.87) | -6.57(-1.03) | -9.99 | 8.85(1.25) | 7.6 |

h_{g} |
-10.54 | 9.54(0.59) | -9.90 | 8.91(0.04) | -10.36 | 9.48(0.53) | -9.60(0.87) | -7.89(-1.06) | -10.80 | 9.71(0.76) | 8.95 |

g_{g} |
-10.91 | 9.73(0.78) | -10.28 | 9.11(0.16) | -10.74 | 9.68(0.73) | -9.95(1.00) | -8.05(-0.90) | -11.12 | 9.82(0.87) | 8.95 |

g_{u} |
-12.73 | 11.21 | -11.85 | 10.37 | -12.49 | 11.14 | -10.90 | -10.38 | -12.86 | -11.17 | 10.82(f) |

t_{2u} |
-12.92 | 11.45 | -11.97 | 10.56 | -12.70 | 11.39 | -11.17 | -10.97 | -13.10 | -11.53 | (f) |

h_{u} |
-13.17 | 11.56 | -12.87 | 11.27 | -13.54 | 12.15 | -12.27 | -10.84 | -13.38 | -11.66 | (f) |

h_{g} |
-13.35 | 11.73 | -12.99 | 11.39 | -13.69 | 12.29 | -12.37 | -11.05 | -13.44 | -11.76 | 11.59(f) |

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 h_{g} and g_{g}
MOs is 0.35 eV and between g_{u} and h_{g}
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 C_{60} 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 C_{70} is much more
complicated than for C_{60}. The ionization
peacks in the outer valence region below 12 eV are grouped into three roughly
separate bands of overlapping ionizations
[22]. 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 C_{60}
by SCF(MNDO), SCF(AM1), SCF(PM3), SCF(SAM1), **OVGF**(MNDO), **OVGF**(AM1)
and **OVGF**(PM3), as well as previous computational
[17] and
experimental
[22] data are summarized in Table 2.

Experimental and calculated (by different semiempirical methods) ionization
energies of C_{70} (D_{5h})^{(a)}

MO | AM1 | OVGF(AM1)^{(b)} |
MNDO | OVGF(MNDO) | PM3 | OVGF(PM3) | SAM1 | DZP^{(c)} |
Exp^{(d)} |
---|---|---|---|---|---|---|---|---|---|

a_{2}" |
-9.14 | 8.39(0.92) | -8.68 | 7.90(0.43) | -9.01 | 8.34(0.87) | -7.93(0.46) | -7.59(0.12) | 7.47 |

e_{1}" |
-9.35 | 8.52 | -8.86 | 8.01(0.54) | -9.21 | 8.48 | -8.11(0.64) | -7.59(0.12) | 7.47 |

a_{2}' |
-9.65 | 8.79 | -9.10 | 8.24(0.56) | -9.48 | 8.72 | -8.48(0.80) | -7.94(0.26) | 7.68 |

e_{2}' |
-9.95 | 8.98 | -9.40 | 8.43(0.47) | -9.79 | 8.95 | -8.78(0.82) | -8.46(0.50) | 7.96 |

e_{2}" |
-10.09 | 9.02 | -9.55 | 8.49(0.37) | -9.94 | 9.01 | -8.90(0.78) | -8.59(0.47) | 8.12 |

e_{1}' |
-10.17 | 9.17 | -9.58 | 8.59(0.16) | -10.01 | 9.13 | -9.05(0.62) | -8.84(0.41) | 8.43 |

e_{1}' |
-10.87 | 9.70 | -10.19 | 9.03(0.01) | -10.67 | 9.63 | -9.91(0.87) | 9.04 | |

e_{2}" |
-11.16 | 9.91 | -10.45 | 9.23(0.05) | -10.98 | 9.88 | -10.28(1.0) | 9.28 | |

e_{1}" |
-11.54 | 10.21 | -10.81 | 9.51(0.09) | -11.35 | 10.18 | -10.62(1.02) | 9.60 | |

e_{2}' |
-11.75 | 10.34 | -10.99 | 9.62(0.02) | -11.55 | 10.31 | -10.73(1.13) | 9.60 | |

a_{1}' |
-11.85 | 10.47 | -11.08 | 9.72(0.12) | -11.69 | 10.45 | -10.54(0.70) | 11.56(1.72) | 9.84 |

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
[17]. 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
[22] 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 C_{60} as well as C_{70}.
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.

C_{60} and C_{70}
**OVGF**(MNDO) calculations with a variable window size.

C_{60} |
C_{70} |
||||||
---|---|---|---|---|---|---|---|

Window size | Window size | ||||||

MO | 40/40(a) | 60/60 | 80/80 | MO | 40/40 | 60/60 | 80/80 |

h_{u} |
8.45 | 8.34 | 8.27 | a_{2}" |
8.06 | 7.96 | 7.90 |

h_{g} |
9.12 | 8.99 | 8.91 | e_{1}" |
8.19 | 8.08 | 8.01 |

g_{g} |
9.35 | 9.20 | 9.11 | a_{2}' |
8.42 | 8.31 | 8.24 |

g_{u} |
10.71 | 10.49 | 10.37 | e_{2}' |
8.65 | 8.51 | 8.43 |

t_{2u} |
10.96 | 10.66 | 10.56 | e_{2}" |
8.73 | 8.57 | 8.49 |

h_{u} |
12.15 | 11.59 | 11.27 | e_{1}' |
8.82 | 8.68 | 8.59 |

h_{g} |
12.23 | 11.74 | 11.39 | e_{1}' |
9.28 | 9.13 | 9.03 |

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 C_{60}(I_{h})
and C_{60}^{-}(I_{h})
calculated with SCF, SCF and OVGF
techniques^{(a)}

AM1 | PM3 | MNDO | SAM1 | |||||
---|---|---|---|---|---|---|---|---|

Charge | 0 | -1 | 0 | -1 | 0 | -1 | 0 | -1 |

SCF | 9.64 | 3.19 | 9.48 | 3.10 | 9.13 | 2.80 | 8.47 | 2.20 |

SCF^{(b)} |
9.46 | 3.07 | 9.32 | 2.99 | 8.97 | 2.68 | 8.27 | 2.07 |

_{SCF}^{(c)} |
0.18 | 0.12 | 0.16 | 0.11 | 0.16 | 0.12 | 0.20 | 0.13 |

IP_{ovgf}^{(d)} |
8.78 | 2.63 | 8.72 | 2.61 | 8.27 | 2.24 | ||

_{ovgf}^{(d,e)} |
0.86 | 0.56 | 0.76 | 0.49 | 0.86 | 0.56 | ||

Exp^{(f)} |
7.60 | 2.67 |

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 C_{70}(D_{5h})
and C_{70}^{-}(D_{5h})
calculated with SCF, SCF and OVGF
techniques^{(a)}

AM1 | PM3 | MNDO | SAM1 | |||||
---|---|---|---|---|---|---|---|---|

Charge | 0 | -1 | 0 | -1 | 0 | -1 | 0 | -1 |

SCF | 9.14 | 3.81 | 9.01 | 3.30 | 8.68 | 3.18 | 7.93 | 2.69 |

SCF^{(b)} |
8.94 | 3.52 | 8.89 | 3.20 | 8.53 | 3.01 | 7.75 | 2.42 |

_{SCF}^{(c)} |
0.20 | 0.29 | 0.12 | 0.10 | 0.15 | 0.17 | 0.18 | 0.27 |

IP_{ovgf}^{(d)} |
8.39 | 3.32 | 8.34 | 2.90 | 7.90 | 2.71 | ||

_{ovgf}^{(d,e)} |
0.75 | 0.49 | 0.67 | 0.40 | 0.78 | 0.47 | ||

Exp^{(f)} |
7.47 | 2.68 |

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 [14] 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 C_{60}^{-}
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 C_{70}^{-}
(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 C_{60}. 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 C_{60}
and 0.45 for C_{70}.

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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