We report a rigorous formulation of multi-state density functional theory (MSDFT) that treats ground and excited states on an equal footing. We prove the existence of a Hamiltonian matrix functional H[D] of the multi-state matrix density D(r) in the N-dimensional Hilbert subspace V^N. Here, the matrix density D(r) is defined as the projection of the electron density operator within the space V^N, being an N-dimensional matrix with state densities in the diagonal and transition densities in the off-diagonal. Then, we establish the variational principle of MSDFT, which states that variational minimization of the subspace energy, equal to the trace of H[D], with respect to the matrix density D(r) yields both the exact eigen-energies and densities of the N lowest eigenstates. To construct the Hamiltonian matrix functional H[D], we introduce two alternative representations of the fundamental variable of MSDFT, which employs a set of auxiliary wavefunctions to represent the matrix density. Based on each representation of D(r), H[D] is constructed by the Hamiltonian matrix of the auxiliary wavefunctions accompanied with a correlation matrix functional E_c[D], which includes correlation effects outside the space of auxiliary wavefunctions. With the expression of H[D], the variational principle of MSDFT gives rise to the self-consistent-field equations (MS-SCF). We also explore the possibilities of using functional approximations of Kohn-Sham DFT in practical applications of MSDFT. The method successfully reproduces the point of avoided crossing with the correct energy splitting between the ground and first-excited states of Lithium Fluoride (LiF), as well as the conical intersection between the ground and first-excited states during the homolytic dissociation of NH₃into NH₂ and H. This work provides a rigorous foundation for developments of approximated functionals in MSDFT and excited-state simulations in density functional theory.

Theoretical Structure of Multi-State Density Functional Theory

Yangyi Lu

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