Extension of Kohn-Sham theory to excited states by means of an off-diagonal density array

Previous work extending the Kohn-Sham theory to excited states was based on replacing the study of the ground-state energy as a functional of the ground-state density by a study of an ensemble average of the Hamiltonian as a functional of the corresponding average density. We suggest and develop an alternative to this description of excited states that utilizes the matrix of the density operator taken between any two estates of the included space. Such an approach provides more detailed information about the states included, for example transition probabilities between discrete states of local one-body operators. The new theory is also based on a variational principle for the trace of the Hamiltonian over the space, of states that we wish to describe, viewed, however, as a functional of the associated array of matrix elements of the density. This finds expression in a matrix version of the Kohn-Sham theory. To illustrate the formalism, we study a suitably defined weak-coupling limit, which is our equivalent of the linear response approximation. On this basis, we derive an eigenvalue equation that has the same form as an equation derived directly from the time-dependent Kohn-Sham equation and applied recently with considerable success to molecular excitations. We provide an independent proof, within the defined approximations, that the eigenvalues can be interpreted as true excitation energies.