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The exchange coupling between the valence electrons of the fullerene cage and the electrons of the N atoms in N@C60^{-1,3}

MCSCF calculations are performed in order to determine the exchange coupling between the 2p electrons of the N atom and the LUMOs of the fullerene cage in the case of mono- and tri-anions of N@C60. The exchange coupling resulted by our calculations is large compared to the hyperfine interaction. The strong coupling can explain the missing EPR signal of the nitrogen in paramagnetic anions.


I. INTRODUCTION
Since the discovery of the first endohedral fullerene a lot of interests have been attracted by this area of the nanotechnology. Many metal atom can be encapsulated by using discharge techniques or ion implantation. In all the cases the metal atom interacts strongly with the fullerene and acts as an electron donor occupying an 'off-centered' position inside the cage.
In contrast the nitrogen in N@C 60 is situated at the center of the molecule and retains its S=3/2 spin quartet atomic state 1 . This amazing property of the encapsulated N atom triggered several research on its possible application in quantum computing and spin labeling.
Several publications 2,3,4,5 studied the promise and limitations of using endohedral fullerenes as quantum information carriers. Mehring et al. 6 recently pointed out experimentally the entanglement of the nuclear spin and the electronic spin of the encaged N atom.
The changes of the characteristic EPR signal of the quartet electronic spin of the N atom makes it an ideal probe for monitoring chemical reactions of C60 7 . During the last decade a great deal of excitement has been brought by the discovery of the superconductivity of the alkali-doped fullerenes. In this type of fullerene compounds the valence electrons of the ionized alkali atoms partially occupy the bands formed by the LUMOs of the C 60 molecules.
The applicability of the quartet atomic state of the N atom as a spin label depends on the strength of the interaction between the 2p electrons of the N atom and the valence electron of the fullerene cage. An interaction which is small compared to the hyper-fine interaction, results in a line width effect of the EPR signal and the N@C 60 is a good candidate for a spin labeling agent. In the case of strong coupling the EPR signal of the system is completely changed and the lines corresponding to the valence electrons of the N atom are hard to identify in the signal of the paramagnetic system.
The interaction between the 2p electrons of the N atom and the valence electrons of the C 60 can be described by a Heisenberg like effective Hamiltonian H int = JS N S C 60 where S N and S C 60 denote the spin operator for the valence electrons of the N atom and the the C 60 , respectively, and J is the exchange coupling characterizing the strength of the interaction. The aim of the present paper is to determine theoretically the exchange coupling of the effective Hamiltonian. The exchange coupling has importance not only for EPR measurements but it plays essential role in the description of the transport through magnetic molecules 8 which is particularly interesting from the point of view of spintronics.

III. RESULTS AND DISCUSSIONS
It has been shown experimentally that in the highly reduced states of the N@C 60 the excess electrons occupy the LUMOs of the fullerene and the N atom inside the cage remains in spin quartet state 13 . In order to check the consistency of our calculations to the experimental findings we performed a set of ROHF calculations on the mono-and tri-anions populating at first the 2p orbitals of the nitrogen and then populating the LUMOs of the C 60 . The results are summarized in Table I. The valence electrons of the nitrogen referred as N2p in Table I occupy the 7t 1u orbitals of the endohedral complex between the 6h u HOMO and 8t 1u LUMO of the C 60 in agreement with the result of ref 15 . Rather different value for the one-electron energy of the N2p orbitals is reported by Greer 16 . This discrepancy is originated from the different treatment of the open-shell problem as it is discussed in ref. 17 . In the case of the mono-anion the energy of the two triplet states were compared while in the case of the triply ionized molecule the energy of the singlet state with fully occupied valence orbitals of N was compared to the high spin state of the N@C −3 60 . For both ions the system with intact N atom were energetically more favorable in agreement with the EPR measurements 13 .
The interaction between the electrons of the nitrogen atom and the valence electrons on the C 60 anion is described by a Heisenberg-like effective Hamiltonian: where J is the coupling constant, S N and S c 60 are the spin of the nitrogen atom and the C 60 anion, respectively. The square of the total spin operator S 2 = (S N + S c 60 ) 2 commutes with the Hamiltonian of the full system H = H N + H C 60 + H int and ,consequently, its eigenvalue is a good quantum number. Expressing the interaction in terms of the spin of the subsystems and the spin of the whole molecule: the energy can be simply given as: where E 0 denotes the energy of the separated systems and the subscript S indicates the explicit dependence of the energy on the multiplicity. Since our interaction Hamiltonian can describe only such processes in which S N and S c 60 are unchanged the excitations altering the spin of the subsystems, namely the hole and the particle are on different species, has to be excluded from the configuration space. In the case of N@C −1 60 S N = 3/2 and S C 60 = 1/2 spanning an 8 dimensional direct product space. The total spin can have the values of S = 1 or S = 2 with the corresponding energies Comparing the energy of the triplet and quintet state one can easily extract the exchange coupling as: The results of the MCSCF calculations using 631g and DH basis are summarized in Table   II. Although the application of the double zeta basis resulted in considerably deeper total energy the deviation of the exchange couplings is small.
In    60 corresponding to different multiplicity and the parabola fitted to the data points. The energy E 0 independent of spin is subtracted.