  We introduce nested gausslet (NG) bases, an improvement on previous gausslet
bases which can treat systems containing atoms with much larger atomic number.
We also introduce pure Gaussian distorted gausslet bases, which allow the
Hamiltonian integrals to be performed analytically, as well as hybrid bases in
which the gausslets are combined with standard Gaussian-type bases. All these
bases feature the diagonal approximation for the electron-electron
interactions, so that the Hamiltonian is completely defined by two $N_b\times
N_b$ matrices, where $N_b \approx 10^4$ is small enough to permit fast
calculations at the Hartree-Fock level. In constructing these bases we have
gained new mathematical insight into the construction of one-dimensional
diagonal bases. In particular we have proved an important theorem relating four
key basis set properties: completeness, orthogonality, zero-moment conditions,
and diagonalization of the coordinate operator matrix. We test our basis sets
on small systems with a focus on high accuracy, obtaining, for example, an
accuracy of $2\times10^{-5}$ Ha for the total Hartree-Fock energy of the neon
atom in the complete basis set limit.
