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

Abstract

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.

Fundamental Variable and Density Representation in Multistate DFT for Excited States

“…Importantly, the eigenvalues and densities (eqs 9 and 10) are not dependent on the number of states N in the subspace because they are obtained based on the variation principle (theorem 2 of ref 14). 14,16 Note that, in general, an N-matrix functional D [ ] is defined as an N × N matrix whose element G AB is a functional of the full matrix variable, here, the N-matrix density function D(r). This implicit matrix functional dependence is denoted as D AB [ ] rather than a term-by-term functional of the element…”

“…For comparison, Theophilou's theory of subspace DFT states that the minimal expectation energy of the subspace is a functional of the subspace density ρ V (r), V V = [ ], 5 whereas an arbitrary weighting function is further introduced. 6 Although the optimization target, the multistate energy E MS [D], is the same, Theophilou's subspace theory and MSDFT are fundamentally different; 16 ρ V (r) is the trace of D(r) (eq 5), but the latter cannot be obtained from ρ V (r) alone. Then, what is the difference between D(r) and ρ V (r) as the fundamental variables in the excited-state DFT?…”