  We introduce an efficient method to construct optimal and system adaptive
basis sets for use in electronic structure and quantum Monte Carlo
calculations. The method is based on an embedding scheme in which a reference
atom is singled out from its environment, while the entire system (atom and
environment) is described by a Slater determinant or its antisymmetrized
geminal power (AGP) extension. The embedding procedure described here allows
for the systematic and consistent contraction of the primitive basis set into
geminal embedded orbitals (GEOs), with a dramatic reduction of the number of
variational parameters necessary to represent the many-body wave function, for
a chosen target accuracy. Within the variational Monte Carlo method, the Slater
or AGP part is determined by a variational minimization of the energy of the
whole system in presence of a flexible and accurate Jastrow factor,
representing most of the dynamical electronic correlation. The resulting GEO
basis set opens the way for a fully controlled optimization of many-body wave
functions in electronic structure calculation of bulk materials, namely,
containing a large number of electrons and atoms. We present applications on
the water molecule, the volume collapse transition in cerium, and the
high-pressure liquid hydrogen.
