Effective models offer useful descriptions of complex materials, but selecting the right model and parameters to accurately describe a particular material can be challenging. Our group's recently developed method of density matrix downfolding yields effective models with quantitative accuracy from a set of quantum Monte Carlo calculations. Several times more data can be extracted per calculation by computing derivatives of the energy and density matrix with respect to parameters of each trial wave function and incorporating these derivatives into the fitting procedure. We demonstrate the use of parameter derivatives on silicon and determine energies for several single-particle states at the $\Gamma$ and X points.

While QMC accurately captures correlation energy in materials, full band structure calculations remain prohibitively expensive. Density matrix downfolding can be used to produce band structures from QMC by fitting and solving a tight binding model. We demonstrate this approach by generating a band structure for silicon.