  We study the Hessian of the fundamental solution to the parabolic problem for
weighted Schr\"odinger operators of the form $\frac 12 \Delta+\nabla h-V$
proving a second order Feynman-Kac formula and obtaining Hessian estimates. For
manifolds with a pole, we use the Jacobian determinant of the exponential map
to offset the volume growth of the Riemannian measure and use the
semi-classical bridge as a delta measure at $y_0$ to obtain exact Gaussian
estimates.
  These estimates are in terms of bounds on $Ric-2 Hess (h)$, on the curvature
operator, and on the cyclic sum of the gradient of the Ricci tensor.
