  The numerical solution of a linear Schr\"odinger equation in the
semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft
of modern computational methods are precise and affordable, while conserving
energy and resolving high oscillations very well. This, however, is far from
the case with regard to its solution in $\mathbb{R}^d$, a setting more suitable
for many applications. In this paper we extend the theory of splitting methods
to this end. The main idea is to derive the solution using a spectral method
from a combination of solutions of the free Schr\"odinger equation and of
linear scalar ordinary differential equations, in a symmetric Zassenhaus
splitting method. This necessitates detailed analysis of certain orthonormal
spectral bases on the real line and their evolution under the free
Schr\"odinger operator.
