  A practical method for finding free energy barriers for transitions in
high-dimensional classical and quantum systems is presented and used to
calculate the dissociative sticking probability of H2 on a metal surface within
transition state theory (TST). The reversible work involved in shifting the
system confined to a hyperplane from the reactant region towards products is
evaluated directly. Quantum mechanical degrees of freedom are included by using
Feynman Path Integrals with the hyperplane constraint applied to the centroid
of the cyclic paths. An optimal dividing surface for the rate estimated by TST
is identified naturally in the course of the reversible work evaluation. The
free energy barrier is determined relative to the reactant state directly, so
an estimate of the transition rate can be obtained without requiring a solvable
reference model for the transition state. The method has been applied to
calculations of the sticking probability of a thermalized hydrogen gas on a
Cu(110) surface. The two hydrogen atoms were included quantum mechanically, and
over two hundred atoms in the Cu crystal where included classically. The
activation energy for adsorption and desorption was determined and found to be
significantly lowered by tunneling at low temperature. The calculated values
agree quite well with experimental estimates. Dynamical corrections to the
classical TST rate estimate were evaluated and found to be small.
