  The Van Vleck formula is a semiclassical approximation to the integral kernel
of the propagator associated to a time-dependent Schr\"odinger equation. Under
suitable hypotheses, we present a rigorous treatment of this approximation
which is valid on "Ehrenfest time scales", i.e. $\hbar$-dependent time
intervals which most commonly take the form $|t| \leq c|\log\hbar|$. Our
derivation is based on an approximation to the integral kernel often called the
"Herman-Kluk approximation", which realizes the kernel as an integral
superposition of Gaussians parameterized by points in phase space. As was shown
by Robert, this yields effective approximations over Ehrenfest time intervals.
In order to derive the Van Vleck approximation from the Herman-Kluk
approximation, we are led to develop stationary phase asymptotics where the
phase functions depend on the frequency parameter in a nontrivial way, a result
which may be of independent interest.
