  A general and rigorous methodology to compute the quantum equilibrium isotope
effect is described. Unlike standard approaches, ours does not assume
separability of rotational and vibrational motions and does not make the
harmonic approximation for vibrations or rigid rotor approximation for the
rotations. In particular, zero point energy and anharmonicity effects are
described correctly quantum mechanically. The approach is based on the
thermodynamic integration with respect to the mass of isotopes and on the
Feynman path integral representation of the partition function. An efficient
estimator for the derivative of free energy is used whose statistical error is
independent of the number of imaginary time slices in the path integral,
speeding up calculations by a factor of 60 at 500 K. We describe the
implementation of the methodology in the molecular dynamics package Amber 10.
The method is tested on three [1,5] sigmatropic hydrogen shift reactions.
Because of the computational expense, we use ab initio potentials to evaluate
the equilibrium isotope effects within the harmonic approximation, and then the
path integral method together with semiempirical potentials to evaluate the
anharmonicity corrections. Our calculations show that the anharmonicity effects
amount up to 30% of the symmetry reduced reaction free energy. The numerical
results are compared with recent experiments of Doering and coworkers,
confirming the accuracy of the most recent measurement on
2,4,6,7,9-pentamethyl-5-(5,5-$^2$H$_2$)methylene-11,11a-dihydro-12H-naphthacene
as well as concerns about compromised accuracy, due to side reactions, of
another measurement on
2-methyl-10-(10,10-$^2$H$_2$)methylenebicyclo[4.4.0]dec-1-ene.
