Complete Shroedinger equation for any real system can not be solved exactly. In such sense, there is no true ab-initio method for electronic structure calculation. The concept of ab-initio methods usually denotes a set of quantum chemical methods, which use both wavefunctions composition and Hamiltonian parameters as varied parameters to achieve convergence in calculation procedure. In crystal-field methods the basis (mathematically complete infinite set of spherical functions) is fixed and limited number of parameters in Hamiltonian is varied to get agreement with experiment.
From first principals. Ideally this means just using a small number of physical constants (Planck's constant, the speed of light, the charge on the electron, etc) to solve the Schrödinger equation. In practice, a number of approximations are also employed.