  The quantum instanton approximation is a type of quantum transition state
theory that calculates the chemical reaction rate using the reactive flux
correlation function and its low order derivatives at time zero. Here we
present several path-integral estimators for the latter quantities, which
characterize the initial decay profile of the flux correlation function. As
with the internal energy or heat capacity calculation, different estimators
yield different variances (and therefore different convergence properties) in a
Monte Carlo calculation. Here we obtain a virial-type estimator by using a
coordinate scaling procedure rather than integration by parts, which allows
more computational benefits. We also consider two different methods for
treating the flux operator, i.e., local-path and global-path approaches, in
which the latter achieves a smaller variance at the cost of using second-order
potential derivatives. Numerical tests are performed for a one-dimensional
Eckart barrier and a model proton transfer reaction in a polar solvent, which
illustrates the reduced variance of the virial estimator over the corresponding
thermodynamic estimator.
