  We develop a formalism to directly evaluate the matrix of force constants
within a Quantum Monte Carlo calculation. We utilize the matrix of force
constants to accurately relax the positions of atoms in molecules and determine
their vibrational modes, using a combination of Variational and Diffusion Monte
Carlo. The computed bond lengths differ by less than 0.007{\AA} from the
experimental results for all four tested molecules. For hydrogen and hydrogen
chloride, we obtain fundamental vibrational frequencies within 0.1% of
experimental results and ~10 times more accurate than leading computational
methods. For carbon dioxide and methane, the vibrational frequency obtained is
on average within 1.1% of the experimental result, which is at least 3 times
closer than results using Restricted Hartree-Fock and Density Functional Theory
with a Perdew-Burke-Ernzerhof (PBE) functional and comparable or better than
Density Functional Theory with a semi-empirical functional.
