  We present a construction of the Anisotropic Gaussian Semi-Classical
Schr\"{o}dinger Propagator, emblematic of a class of Fourier Integral Operators
of quadratic phase kernels related to the Schr\"{o}dinger equation. We deduce a
set of algebraic relations of the variational matrices, solutions of the
variational system pertaining to single Gaussian wave packet semi-classical
time evolution, representing the symplectic and other invariances of the
dynamics, which are subsequently used to derive the Van Vleck formula from the
semi-classical propagator, as an argument for the practical importance of the
later relations in the relevant wave packet calculus.
