The third-order polarizability gamma of the C-60 molecule has been calculated for the third harmonic generation at several incident frequencies using two different schemes of the sum-over-state (SOS) method, and the contribution of the low-lying singlet excited states as the second intermediate states has been analyzed. The group-theoretical analysis of the SOS expression clarifies that the (1)A(g) and H-1(g) states are the only states that contribute to the gamma as the second intermediate states. For the numerical analysis, the electronic states previously obtained in the semiempirical CNDO/S approximation with the singly and doubly excited configuration interaction method are used. It is found that the inclusion of the doubly excited configurations is essential in evaluating the gamma. Those excited (1)A(g) and H-1(g) states which make significant positive contributions to the gamma are practically the doubly excited states. In order to secure error-free SOS calculations, equivalent but independent calculations have been also carried out with the frequency-dependent moment method. (C) 2000 American Institute of Physics. [S0021-9606(00)30515-3].