  We implement and benchmark the frozen core approximation, a technique
commonly adopted in electronic structure theory to reduce the computational
cost by means of mathematically fixing the chemically inactive core electron
states. The accuracy and efficiency of this approach are well controlled by a
single parameter, the number of frozen orbitals. Explicit corrections for the
frozen core orbitals and the unfrozen valence orbitals are introduced,
safeguarding against seemingly minor numerical deviations from the assumed
orthonormality conditions of the basis functions. A speedup of over two-fold
can be achieved for the diagonalization step in all-electron density-functional
theory simulations containing heavy elements, without any accuracy degradation
in terms of the electron density, total energy, and atomic forces. This is
demonstrated in a benchmark study covering 103 materials across the periodic
table, and a large-scale simulation of CsPbBr3 with 2,560 atoms. Our study
provides a rigorous benchmark of the precision of the frozen core approximation
(sub-meV per atom for frozen core orbitals below -200 eV) for a wide range of
test cases and for chemical elements ranging from Li to Po. The algorithms
discussed here are implemented in the open-source Electronic Structure
Infrastructure software package.
