Multi-State Density Functional Theory for Ground and Excited States

We report a rigorous formulation of multi-state density functional theory (MSDFT) that extends the Kohn-Sham (KS) energy functional for the ground state to a Hamiltonian matrix functional H[D] of the density matrix D in the space spanned by the lowest N adiabatic states. We establish a variational principle of MSDFT, which guarantees that the variational optimization results in a Hamiltonian matrix, whose eigenvalues are the lowest N eigen-energies of the system. We present an explicit expression of H[D] and introduce the correlation matrix functional. Akin to KS-DFT for the ground state, a universal multi-state correlation potential is derived for a two-state system as an illustrative example. This work shows that MSDFT is an exact density functional theory that treats the ground and excited states on an equal footing and provides a framework for practical applications and future developments of approximate functionals for excited states.