  In this paper, we study a second-order accurate and linear numerical scheme
for the nonlocal Cahn-Hilliard equation. The scheme is established by combining
a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation
for the temporal discretization, and by applying the Fourier spectral
collocation to the spatial discretization. In addition, two stabilization terms
in different forms are added for the sake of the numerical stability. We
conduct a complete convergence analysis by using the higher-order consistency
estimate for the numerical scheme, combined with the rough error estimate and
the refined estimate. By regarding the numerical solution as a small
perturbation of the exact solution, we are able to justify the discrete
$\ell^\infty$ bound of the numerical solution, as a result of the rough error
estimate. Subsequently, the refined error estimate is derived to obtain the
optimal rate of convergence, following the established $\ell^\infty$ bound of
the numerical solution. Moreover, the energy stability is also rigorously
proved with respect to a modified energy. The proposed scheme can be viewed as
the generalization of the second-order scheme presented in an earlier work, and
the energy stability estimate has greatly improved the corresponding result
therein.
