  We derive a semiclassical expression for an energy smoothed autocorrelation
function defined on a group of eigenstates of the Schr\"odinger equation. The
system we considered is an energy-conserved Hamiltonian system possessing
time-invariant symmetry. The energy smoothed autocorrelation function is
expressed as a sum of three terms. The first one is analogous to Berry's
conjecture, which is a Bessel function of the zeroth order. The second and the
third terms are trace formulae made from special trajectories. The second term
is found to be direction dependent in the case of spacing averaging, which
agrees qualitatively with previous numerical observations in high-lying
eigenstates of a chaotic billiard.
