  Relativistic atomic structure calculations are carried out in
alkaline-earth-metal ions using a semiempirical-core-potential approach. The
systems are partitioned into frozen-core electrons and an active valence
electron. The core orbitals are defined by a Dirac-Hartree-Fock calculation
using the grasp2k package. The valence electron is described by a Dirac-like
Hamiltonian involving a core-polarization potential to simulate the
core-valence electron correlation. The associated equation is solved with the
Lagrange-mesh method, which is an approximate variational approach having the
form of a mesh calculation because of the use of a Gauss quadrature to
calculate matrix elements. Properties involving the low-lying metastable
$^2D_{3/2,5/2}$ states of Ca$^{+}$, Sr$^{+}$, and Ba$^{+}$ are studied, such as
polarizabilities, one- and two-photon decay rates, and lifetimes. Good
agreement is found with other theory and observation, which is promising for
further applications in alkali-like systems.
