  We present a new kind of basis function for discretizing the Schr\"odinger
equation in electronic structure calculations, called a gausslet, which has
wavelet-like features but is composed of a sum of Gaussians. Gausslets are
placed on a grid and combine advantages of both grid and basis set approaches.
They are orthogonal, infinitely smooth, symmetric, polynomially complete, and
with a high degree of locality. Because they are formed from Gaussians, they
are easily combined with traditional atom-centered Gaussian bases. We also
introduce diagonal approximations which dramatically reduce the computational
scaling of two-electron Coulomb terms in the Hamiltonian.
