We report a rigorous formulation of density functional theory for excited states, providing a theoretical foundation for a multistate density functional theory. We prove the existence of a Hamiltonian matrix functional of the multistate matrix density D(r) in the subspace spanned by the lowest N eigenstates. Here, D(r) is an N-dimensional matrix of state densities and transition densities. Then, a variational principle of the multistate subspace energy is established, whose minimization yields both the energies and densities of the individual N eigenstates. Furthermore, we prove that the N-dimensional matrix density D(r) can be sufficiently represented by N 2 nonorthogonal Slater determinants, based on which an interacting active space is introduced for practical calculations. This work establishes that the ground and excited states can be treated on an equal footing in density functional theory.

Short-range electron transfer (ET) in proteins is an ultrafast process on the similar time scales as local protein-solvent fluctuation, and thus the two dynamics are coupled. Here we use semiquinone flavodoxin and systematically characterized the photoinduced redox cycle with 11 mutations of different aromatic electron donors (tryptophan and tyrosine) and local residues to change redox properties. We observed the forward and backward ET dynamics in a few picoseconds, strongly following a stretched behavior resulting from a coupling between local environment relaxations and these ET processes. We further observed the hot vibrational-state formation through charge recombination and the subsequent cooling dynamics also in a few picoseconds. Combined with the ET studies in oxidized flavodoxin, these results coherently reveal the evolution of the ET dynamics from single to stretched exponential behaviors and thus elucidate critical time scales for the coupling. The observed hot vibration-state formation is robust and should be considered in all photoinduced back ET processes in flavoproteins.

A variety of electron transfer (ET) reactions in biological systems occurs at short distances and is ultrafast. Many of them show behaviors that deviate from the predictions of the classic Marcus theory. Here, we show that these ultrafast ET dynamics highly depend on the coupling between environmental fluctuations and ET reactions. We introduce a dynamic factor, γ (0 ≤ γ ≤ 1), to describe such coupling, with 0 referring to the system without coupling to a “frozen” environment, and 1 referring to the system’s complete coupling with the environment. Significantly, this system’s coupling with the environment modifies the reaction free energy, ΔGγ, and the reorganization energy, λγ, both of which become smaller. This new model explains the recent ultrafast dynamics in flavodoxin and elucidates the fundamental mechanism of nonequilibrium ET dynamics, which is critical to uncovering the molecular nature of many biological functions.

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.

Complementary to the theorems of Hohenberg and Kohn for the ground state, Theophilou’s subspace theory establishes a one-to-one relationship between the total eigenstate energy and density ρV(r) of the subspace spanned by the lowest N eigenstates. However, the individual eigenstate energies are not directly available from such a subspace density functional theory. Lu and Gao (J. Chem. Phys. Lett. 2022, 13, 7762) recently proved that the Hamiltonian projected on to this subspace is a matrix functional scriptH [ D ] of the multistate matrix density D (r) and that variational optimization of the trace of the Hamiltonian matrix functional yields exactly the individual eigenstates and densities. This study shows that the matrix density D (r) is the necessary fundamental variable in order to determine the exact energies and densities of the individual eigenstates. Furthermore, two ways of representing the matrix density are introduced, making use of nonorthogonal and orthogonal orbitals. In both representations, a multistate active space of auxiliary states can be constructed to exactly represent D (r) with which an explicit formulation of the Hamiltonian matrix functional scriptH [ D ] is presented. Importantly, the use of a common set of orthonormal orbitals makes it possible to carry out multistate self-consistent-field optimization of the auxiliary states with singly and doubly excited configurations (MS-SDSCF).