  We introduce hybrid gausslet/Gaussian basis sets, where a standard Gaussian
basis is added to a gausslet basis in order to increase accuracy near the
nuclei while keeping the spacing of the grid of gausslets relatively large. The
Gaussians are orthogonalized to the gausslets, which are already orthonormal,
and approximations are introduced to maintain the diagonal property of the two
electron part of the Hamiltonian, so that it continues to scale as the second
power of the number of basis functions, rather than the fourth. We introduce
several corrections to the Hamiltonian designed to enforce certain exact
properties, such as the values of certain two-electron integrals. We also
introduce a simple universal energy correction which compensates for the
incompleteness of the basis stemming from the electron-electron cusps, based on
the measured double occupancy of each basis function. We perform a number of
Hartree Fock and full configuration interaction (full-CI) test calculations on
two electron systems, and Hartree Fock on a ten-atom hydrogen chain, to
benchmark these techniques. The inclusion of the cusp correction allows us to
obtain complete basis set full-CI results, for the two electron cases, at the
level of several microHartrees, and we see similar apparent accuracy for
Hartree Fock on the ten-atom hydrogen chain.
