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We present a many-body Bethe-Salpeter equation eigenstates basedsum-over-states method to calculate the linear and nonlinear optical propertiesof solids. Excitonic and local field effects are included in the calculations.As applications, we calculate the one-photon absorption, third harmonicgeneration, degenerate four-wave mixing spectra of solid C60 fullerene. Theoverall agreement between the theoretical and experimental results is very goodfor all three calculated spectra. By comparisons with the independent particleapproximation based sum-over-states method, we show that excitonic effects mixthe independent particle transition peaks to new excitonic ones. The positionand intensity of spectral peaks are modified significantly. By tracing thesum-over-states progress, we determine the type of nonlinear polarizationresonances for the characteristic peaks of third harmonic generation process,which may clear up a discrepancy in two experimental results.

Excitonic effects on the third-order nonlinear optical properties of solids: Theory and application