  In this paper, we analyse a new exponential-type integrator for the nonlinear
cubic Schr\"odinger equation on the $d$ dimensional torus $\mathbb T^d$. The
scheme has recently also been derived in a wider context of decorated trees in
[Y. Bruned and K. Schratz, arXiv:2005.01649]. It is explicit and efficient to
implement. Here, we present an alternative derivation, and we give a rigorous
error analysis. In particular, we prove second-order convergence in
$H^\gamma(\mathbb T^d)$ for initial data in $H^{\gamma+2}(\mathbb T^d)$ for any
$\gamma > d/2$. This improves the previous work in [Kn\"oller, A. Ostermann,
and K. Schratz, SIAM J. Numer. Anal. 57 (2019), 1967-1986].
  The design of the scheme is based on a new method to approximate the
nonlinear frequency interaction. This allows us to deal with the complex
resonance structure in arbitrary dimensions. Numerical experiments that are in
line with the theoretical result complement this work.
