Minimal Active Space: NOSCF and NOSI in Multistate Density Functional Theory

In this Perspective, we introduce a minimal active space (MAS) for the lowest N eigenstates of a molecular system in the framework of a multistate density functional theory (MSDFT), consisting of no more than N2 nonorthgonal Slater determinants. In comparison with some methods in wave function theory in which one seeks to expand the ever increasing size of an active space to approximate the wave functions, it is possible to have an upper bound in MSDFT because the auxiliary states in a MAS are used to represent the exact N-dimensional matrix density D(r). In analogy to Kohn-Sham DFT, we partition the total Hamiltonian matrix functional H[D] into an orbital-dependent part, including multistate kinetic energy Tms and Coulomb exchange energy EHx plus an external potential energy R dr v(r)D(r), and a correlation matrix density functional Ec[D]. The latter accounts for the part of correlation energy not explicitly included in the minimal active space. However, a major difference from Kohn-Sham DFT is that state interactions are necessary to represent the N-matrix density D(r) in MSDFT, rather than a non-interacting reference state for the scalar ground-state density ρo(r). Two computational approaches are highlighted. We first derive a set of non-orthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of H[D]. We introduce the multistate correlation potential, as the functional derivative of Ec[D], which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a non-orthogonal state interaction (NOSI) procedure, in which the determinant functions are optimized separately. Both computational methods are useful for determining the exact eigenstate energies and for constructing variational diabatic states, provided that the universal correlation matrix functional is known. It is hoped that this discussion would stimulate developments of approximate multistate density functionals both for the ground and excited states.