where r α α = r − − τ τ α α is the position vector relative to the position of atom nucleus α α . An LAPW basis function is thus a plane wave in the IR and a linear combination of the radial functions u l , α α ( r α α , E l , α α ) and u ˙ ˙ l , α α ( r α α , E l , α α ) multiplied by spherical harmonics Y l , m in each MT sphere. The radial function u l , α α ( r α α , E l , α α ) is hereby the solution of the Kohn-Sham Hamiltonian for the spherically averaged potential with regular behavior at the nucleus for the given energy parameter E l , α α . Together with its energy derivative u ˙ ˙ l , α α ( r α α , E l , α α ) these augmentations of the plane wave in each MT sphere enable a representation of the Kohn-Sham orbitals at arbitrary eigenenergies linearized around the energy parameters. The coefficients a l , m k , G , α α and b l , m k , G , α α are automatically determined by enforcing the basis function to be continuously differentiable for the respective ( l , m ) channel. The set of LAPW basis functions is defined by specifying a cutoff parameter K max = | k + G | max . In each MT sphere, the expansion into spherical harmonics is limited to a maximum number of angular momenta l max , α α ≈ ≈ K max R MT α α , where R MT α α is the muffin-tin radius of atom α α . The choice of this cutoff is connected to the decay of expansion coefficients for growing l in the Rayleigh expansion of plane waves into spherical harmonics.