Leveling the Mountain Range of Excited-State Benchmarking through Multistate Density Functional Theory.

The performance of multistate density functional theory (MSDFT) with nonorthogonal state interaction (NOSI) is assessed for 100 vertical excitation energies against the theoretical best estimates extracted to the full configuration interaction accuracy on the database developed by Loos et al. in 2018 (Loos2018). Two optimization techniques, namely, block-localized excitation and target state optimization, are examined along with two ways of estimating the transition density functional (TDF) for the correlation energy of the Hamiltonian matrix density functional. The results from the two optimization methods are similar. It was found that MSDFT-NOSI using the spin-multiplet degeneracy constraint for the TDF of spin-coupling interaction, along with the M06-2X functional, yields a root-mean-square error (RMSE) of 0.22 eV, which performs noticeably better than time-dependent density functional theory (DFT) at an RMSE of 0.43 eV using the same functional and basis set on the Loos2018 database. In comparison with wave function theory, NOSI has smaller errors than CIS(D∞), LR-CC2, and ADC(3) all of which have an RMSE of 0.28 eV, but somewhat greater than STEOM-CCSD (RMSE of 0.14 eV) and LR-CCSD (RMSE of 0.11 eV) wave function methods. In comparison with Kohn-Sham (KS) DFT calculations, the multistate DFT approach has little double counting of correlation. Importantly, there is no noticeable difference in the performance of MSDFT-NOSI on the valence, Rydberg, singlet, triplet, and double-excitation states. Although the use of another hybrid functional PBE0 leads to a greater RMSE of 0.36 eV, the deviation is systematic with a linear regression slope of 0.994 against the results with M06-2X. The present benchmark reveals that density functional approximations developed for KS-DFT for the ground state with a noninteracting reference may be adopted in MSDFT calculations in which the state interaction is key.

Ultrafast nonequilibrium dynamics of short-range protein electron transfer in flavodoxin.

Short-range protein electron transfer (ET) is ubiquitous in biology and is often observed in photosynthesis, photoreceptors and photoenzymes. These ET processes occur on an ultrafast timescale from femtoseconds to picoseconds at a short donor-acceptor distance within 10 Å, and thus couple with local environmental fluctuations. Here, we use oxidized Anabaena flavodoxin as a model system and have systematically studied the photoinduced redox cycle of the wild type and seven mutant proteins by femtosecond spectroscopy. We observed a series of ultrafast dynamics from the initial charge separation in 100-200 fs, subsequent charge recombination in 1-2 ps and final vibrational cooling process of the products in 3-6 ps. We further characterized the active-site solvation and observed the relaxations in 1-200 ps, indicating a nonergodic ET dynamics. With our new ET model, we uncovered a minor outer (solvent) reorganization energy and a large inner (donor and acceptor) reorganization energy, suggesting a frozen active site in the initial ultrafast ET while the back ET couples with the environment relaxations. The vibronically coupled back ET dynamics was first reported in D. vulgaris flavodoxin and here is observed in Anabaena flavodoxin again, completely due to the faster ET dynamics than the cooling relaxations. We also compared the two flavodoxin structures, revealing a stronger coupling with the donor tyrosine in Anabaena. All ultrafast ET dynamics are from the large donor-acceptor couplings and the minor activation barriers due to the reaction free energies being close to the inner reorganization energies. These observations should be general to many redox reactions in flavoproteins.

Exact eigenenergies of a model of vibronically coupled electron transfer reactions.

We present the first exact solution to the time-independent Schrödinger equation of a model Hamiltonian consisting of a vibrational mode coupled to three electronic states. This Hamiltonian serves as a generic model for photo-induced electronic transfer reactions. The solution is non-perturbative and can be applied to ET reactions with weak and strong electronic and vibrational coupling strengths. This work suggests a new direction towards understanding the vibronic effects in ET dynamics beyond the non-adiabatic limit and Condon approximation.

Nonequilibrium dynamics of photoinduced forward and backward electron transfer reactions.

The biological functions of photoenzymes are often triggered by photoinduced electron transfer (ET) reactions. An ultrafast backward ET (BET) reaction follows the initial photoinduced forward ET (FET), which dissipates the energy of absorbed photons and terminates the biological function in vain. Based upon our previous works, we reasoned that the dynamics of the BET is coupled with that of the FET and other local motions. In this work, the dynamics of the FET and BET is modeled as the master equation of the reduced density operator of a three-state system coupled with a classical harmonic reservoir. The coupling of the FET and BET is reflected in the time-evolution of the charge-transfer state's population, which is generated by a source, the reaction flux for the FET, and annihilated by a sink, the reaction flux for the BET. Surprisingly, numerical simulations show that when the BET is in the Marcus normal region, the BET can be accelerated by nonequilibrium local motions and becomes faster than what is predicted by the Marcus theory. The experimental confirmation of this novel dynamics would provide qualitative evidence for nonequilibrium effects on ultrafast ET dynamics. Additionally, the effects of quantum vibrational modes on the dynamics are discussed. This work can help understand the dynamical interactions between the chain of ultrafast reactions and the complex local environmental motions, revealing the physical nature underlying biological functions.

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

A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree-Fock theory and in Kohn-Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO-DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Multistate Energy Decomposition Analysis of Molecular Excited States.

A multistate energy decomposition analysis (MS-EDA) method is described to dissect the energy components in molecular complexes in excited states. In MS-EDA, the total binding energy of an excimer or an exciplex is partitioned into a ground-state term, called local interaction energy, and excited-state contributions that include exciton excitation energy, superexchange stabilization, and orbital and configuration-state delocalization. An important feature of MS-EDA is that key intermediate states associated with different energy terms can be variationally optimized, providing quantitative insights into widely used physical concepts such as exciton delocalization and superexchange charge-transfer effects in excited states. By introducing structure-weighted adiabatic excitation energy as the minimum photoexcitation energy needed to produce an excited-state complex, the binding energy of an exciplex and excimer can be defined. On the basis of the nature of intermolecular forces through MS-EDA analysis, it was found that molecular complexes in the excited states can be classified into three main categories, including (1) encounter excited-state complex, (2) charge-transfer exciplex, and (3) intimate excimer or exciplex. The illustrative examples in this Perspective highlight the interplay of local excitation polarization, exciton resonance, and superexchange effects in molecular excited states. It is hoped that MS-EDA can be a useful tool for understanding photochemical and photobiological processes.

Fundamental Variable and Density Representation in Multistate DFT for 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. Phys. Chem. Lett. 2022, 13, 7762) recently proved that the Hamiltonian projected on to this subspace is a matrix functional H[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 H[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).

Multistate Density Functional Theory of Excited States.

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 H[D] 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 N2 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.

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 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 function D(r). Here, 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 ∫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. 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 ρo(r). Two computational approaches are highlighted. We first derive a set of nonorthogonal 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 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.

Ultrafast Dynamics of Nonequilibrium Short-Range Electron Transfer in Semiquinone Flavodoxin.

Short-range protein electron transfer (ET) is crucially important in light-induced biological processes such as in photoenzymes and photoreceptors and often occurs on time scales similar to those of environment fluctuations, leading to a coupled dynamic process. Herein, we use semiquinone Anabaena flavodoxin to characterize the ultrafast photoinduced redox cycle of the wild type and seven mutants by ultrafast spectroscopy. We have found that the forward and backward ET dynamics show stretched behaviors in a few picoseconds (1-5 ps), indicating a coupling with the local protein fluctuations. By comparison with the results from semiquinone D. vulgaris flavodoxin, we find that the electronic coupling is crucial to the ET rates. With our new nonergodic model, we obtain smaller values of the outer reorganization energy (λoγ) of environment fluctuations and the reaction free energy force (ΔGγ), a signature of nonequilibrium ET dynamics.

General synthesis of 2D rare-earth oxide single crystals with tailorable facets.

Two-dimensional (2D) rare-earth oxides (REOs) are a large family of materials with various intriguing applications and precise facet control is essential for investigating new properties in the 2D limit. However, a bottleneck remains with regard to obtaining their 2D single crystals with specific facets because of the intrinsic non-layered structure and disparate thermodynamic stability of different facets. Herein, for the first time, we achieve the synthesis of a wide variety of high-quality 2D REO single crystals with tailorable facets via designing a hard-soft-acid-base couple for controlling the 2D nucleation of the predetermined facets and adjusting the growth mode and direction of crystals. Also, the facet-related magnetic properties of 2D REO single crystals were revealed. Our approach provides a foundation for further exploring other facet-dependent properties and various applications of 2D REO, as well as inspiration for the precise growth of other non-layered 2D materials.

How rare and common risk variation jointly affect liability for autism spectrum disorder.

BACKGROUND: Genetic studies have implicated rare and common variations in liability for autism spectrum disorder (ASD). Of the discovered risk variants, those rare in the population invariably have large impact on liability, while common variants have small effects. Yet, collectively, common risk variants account for the majority of population-level variability. How these rare and common risk variants jointly affect liability for individuals requires further study. METHODS: To explore how common and rare variants jointly affect liability, we assessed two cohorts of ASD families characterized for rare and common genetic variations (Simons Simplex Collection and Population-Based Autism Genetics and Environment Study). We analyzed data from 3011 affected subjects, as well as two cohorts of unaffected individuals characterized for common genetic variation: 3011 subjects matched for ancestry to ASD subjects and 11,950 subjects for estimating allele frequencies. We used genetic scores, which assessed the relative burden of common genetic variation affecting risk of ASD (henceforth "burden"), and determined how this burden was distributed among three subpopulations: ASD subjects who carry a potentially damaging variant implicated in risk of ASD ("PDV carriers"); ASD subjects who do not ("non-carriers"); and unaffected subjects who are assumed to be non-carriers. RESULTS: Burden harbored by ASD subjects is stochastically greater than that harbored by control subjects. For PDV carriers, their average burden is intermediate between non-carrier ASD and control subjects. Both carrier and non-carrier ASD subjects have greater burden, on average, than control subjects. The effects of common and rare variants likely combine additively to determine individual-level liability. LIMITATIONS: Only 305 ASD subjects were known PDV carriers. This relatively small subpopulation limits this study to characterizing general patterns of burden, as opposed to effects of specific PDVs or genes. Also, a small fraction of subjects that are categorized as non-carriers could be PDV carriers. CONCLUSIONS: Liability arising from common and rare risk variations likely combines additively to determine risk of any individual diagnosed with ASD. On average, ASD subjects carry a substantial burden of common risk variation, even if they also carry a rare PDV affecting risk.

Effects of nonequilibrium fluctuations on ultrafast short-range electron transfer dynamics.

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.

Understanding Short-Range Electron-Transfer Dynamics in Proteins.

Short-range electron-transfer (ET) reactions in biological systems are usually ultrafast, having transfer rates comparable to or even faster than corresponding environmental fluctuations, and often display nonexponential behaviors. To understand these nonequilibrium ET dynamics, we carried out detailed theoretical analyses based on the Sumi-Marcus model. It is shown that the ET dynamics is largely determined by the relative time scales of the ET reaction and its surrounding motions. Significantly, different environmental fluctuations can produce a variety of apparent ET dynamics even with the same driving force, Δ Go, and reorganization energy, λ. We applied our analyses to an ultrafast ET process in DNA repair by (6-4) photolyase and directly obtained the inner and outer reorganization energies (λi and λo) as well as the free energy Δ Go of various mutants, providing mechanical insight into ultrafast short-range ET reactions in proteins.

Short-Range Electron Transfer in Reduced Flavodoxin: Ultrafast Nonequilibrium Dynamics Coupled with Protein Fluctuations.

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.