Summary: We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity \(f(\rho) = \rho^\sigma\), where \(\rho =|\psi|^2\) is the density with \(\psi\) the wave function and \(\sigma > 0\) is the exponent of the semi-smooth nonlinearity. Under the assumption of \(H^2\)-solution of the NLSE, we prove error bounds at \(O(\tau^{\frac{1}{2}+\sigma} + h^{1+2\sigma})\) and \(O(\tau + h^2)\) in \(L^2\)-norm for \(0<\sigma \leq \frac{1}{2}\) and \(\sigma \geq \frac{1}{2}\), respectively, and an error bound at \(O(\tau^\frac{1}{2} + h)\) in \(H^1\)-norm for \(\sigma \geq \frac{1}{2}\), where \(h\) and \(\tau\) are the mesh size and time step size, respectively. In addition, when \(\frac{1}{2}<\sigma <1\) and under the assumption of \(H^3\)-solution of the NLSE, we show an error bound at \(O(\tau^{\sigma} + h^{2\sigma})\) in \(H^1\)-norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional \(L^2\)-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of \(0 < \sigma \leq \frac{1}{2}\), and to establish an \(l^\infty\)-conditional \(H^1\)-stability to obtain the \(l^\infty\)-bound of the numerical solution by using the mathematical induction and the error estimates for the case of \(\sigma \geq \frac{1}{2}\); and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.