Excited Cr impurity states in Al2O3 from constraint density functional theory

The excited states, T-4(2g) and E-2(g), of a Cr impurity in Al2O3 were treated by constraint density functional theory by imposing a density matrix constraint (constraint field) to control the electron occupation numbers of the d orbitals. The calculated excitation energies, directly calculated from the self- consistent total energies of the (4)A(2g) ground states and the various excited states, correctly reproduce the experimental ordering. In addition, we find that there is no stationary solution for the excited T-4(2g) state corresponding to the crystal-field transition state in the usual Kohn-Sham equation, i.e., with no constraint field. By contrast, the excited E-2(g) state of the spin-flip transition state is a (meta-) stable stationary solution, and may be responsible for the long radiative decay lifetime observed in experiments on ruby.