  We present a spectral scheme for atomic structure calculations in
pseudopotential Kohn-Sham density functional theory. In particular, after
applying an exponential transformation of the radial coordinates, we employ
global polynomial interpolation on a Chebyshev grid, with derivative operators
approximated using the Chebyshev differentiation matrix, and integrations using
Clenshaw-Curtis quadrature. We demonstrate the accuracy and efficiency of the
scheme through spin-polarized and unpolarized calculations for representative
atoms, while considering local, semilocal, and hybrid exchange-correlation
functionals. In particular, we find that $\mathcal{O}$(200) grid points are
sufficient to achieve an accuracy of 1 microhartree in the eigenvalues for
optimized norm conserving Vanderbilt pseudopotentials spanning the periodic
table from atomic number $Z = 1$ to $83$.
