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 multistate density functional theory (MSDFT), consisting of no more than N 2 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 Ndimensional matrix density function D(r). Here, we partition the total Hamiltonian matrix functional [ ] D into an orbital-dependent part, including multistate kinetic energy T ms and Coulomb-exchange energy E Hx plus an external potential energy ∫ dr v(r)D(r), and a correlation matrix density functional [ ] D c . The latter accounts for the part of correlation energy not explicitly included in the minimal active space. 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 noninteracting reference state for the scalar ground-state density ρ o (r). Two computational approaches are highlighted. We first derive a set of nonorthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of [ ] D . We introduce the multistate correlation potential, as the functional derivative of [ ] D c, which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a nonorthogonal 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

“…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 ), double-struckE normalV = double-struckE [ ρ V ] , whereas an arbitrary weighting function is further introduced . Although the optimization target, the multistate energy E MS [ D ], is the same, Theophilou’s subspace theory and MSDFT are fundamentally different; ρ V ( r ) is the trace of D ( r ) (eq ), 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?…”

“…The variational principle in terms of the matrix functional of eq distinguishes that from the Theophilou subspace theory and the state-weighted ensemble in that all eigenstates and density vectors in the subspace double-struckV are simultaneously determined exactly by diagonalization of scriptH [ D ] (provided that the universal matrix functional scriptF [ D ] is known). Importantly, the eigenvalues and densities (eqs and ) 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 ). , …”

Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States

“…TD-DFT can behave poorly in describing double and higher-order excitations, − and long-range charge-transfer (LRCT) and Rydberg states. − Furthermore, TD-DFT is not well-suited for treating core excitations, having errors greater than 10 eV due to the lack of orbital relaxation. , Of course, some of the issues may be ameliorated to some extent with long-range corrected exchange-correlation (XC) functionals. − Nevertheless, there is continuing interest in developing reliable and efficient optimization techniques in DFT. − In this work, we present a target state optimization (TSO) method for treating excited states that cannot be adequately modeled by TD-DFT. The method can also be used to generate diabatic states in combined quantum mechanical and molecular mechanical (QM/MM) simulations of electron transfer and excited energy transfer reactions as well as basis configurations in multistate density functional theory (MSDFT). − …”

“…It is rooted in the BLW and GBLO procedures, but the present TSO method is significantly more flexible both in configuration definition and user-friendliness. Importantly, because of orbital-subspace partition, a form of constraint in orbital space, the SCF optimization is robust without the possibility of variational collapse, making it essentially a black-box model for defining excited basis configurations in multistate nonorthogonal state interaction (NOSI) calculations. , In the following, we first introduce the concept and theoretical background of the TSO method. This is followed by presenting the details of SCF optimization of the target state orbitals.…”

Nonorthogonal Active Space Decomposition of Wave Functions with Multiple Correlation Mechanisms

“…It is important to note that the full tensor product in eq may produce larger determinant expansions than necessary, and these additional determinants may in fact lead to greater errors. As demonstrated in the context of the multistate density functional theory (DFT), a minimal active space with half projection of open-shell determinants is all that should be required. − However, here, we opt for complete active space (CAS) expansions to ensure the connection with the active space decomposition philosophy embodied by eq and to enable direct comparison to CAS wave functions.…”