  The accurate computation of forces and other energy derivatives has been a
long-standing challenge for quantum Monte Carlo methods. A number of technical
obstacles contribute to this challenge. We discuss how these obstacles can be
removed with the auxiliary-field quantum Monte Carlo (AFQMC) approach. AFQMC is
a general, high-accuracy, many-body total-energy method for molecules and
solids. The implementation of back-propagation for pure estimators allows
direct calculation of gradients of the energy via the Hellmann-Feynman theorem.
A planewave basis with norm-conserving pseudopotentials is used for the study
of periodic bulk materials. Completeness of the planewave basis minimizes the
effect of so-called Pulay terms. The ionic pseudopotentials, which can be
incorporated in AFQMC in exactly the same manner as in standard
independent-electron methods, regulate the force and stress estimators and
eliminate any potential divergence of the Monte Carlo variances. The resulting
approach allows applications of full geometry optimizations in bulk materials.
It also paves the way for many-body computations of the phonon spectrum in
solids.
