Multistate Density Functional Theory: Theory, Methods, and Applications

A quantum theory of density functionals and its applications is presented. By introducing a matrix density $D\left(r\right)$ of rank $N$ as the fundamental variable, a one-to-one correspondence has been established between $D\left(r\right)$ and the Hamiltonian matrix representing $N$ electronic states-that is, a matrix density functional $\mathscr{H}\left[D\right]$ . Moreover, no more than $N^2$ Slater determinants are sufficient to represent $D\left(r\right)$ exactly, giving rise to the concept of minimal active space (MAS). The use of a MAS naturally leads to the definition of correlation matrix functional ${ℰ}^c\left[D\right]$ , the multi-state extension of the exchange-correlation functional in Kohn-Sham DFT. Variational minimization of the multistate energy, which is defined as the trace of the Hamiltonian matrix functional, yields the exact energies and densities of the lowest $N$ eigenstates. A nonorthogonal state interaction (NOSI) algorithm has been developed to optimize the orbitals associated with $D\left(r\right)$ and to approximate the correlation matrix functional. The MSDFT-NOSI method is demonstrated across a range of applications, particularly in cases where KS-DFT and linear-response time-dependent DFT fail, with its accuracy validated through comparison with high-level multiconfigurational wave function theory.