Constrained Density Functional Theory and Its Application in Long-Range Electron Transfer

Highly Accurate and Robust Constraint-Based Orbital-Optimized Core Excitations

We adapt our recently developed constraint-based orbital-optimized excited-state method (COOX) for the computation of core excitations. COOX is a constrained density functional theory (cDFT) approach based on excitation amplitudes from linear-response time-dependent DFT (LR-TDDFT), and has been shown to provide accurate excitation energies and excited-state properties for valence excitations within a spin-restricted formalism. To extend COOX to core-excited states, we introduce a spin-unrestricted variant which allows us to obtain orbital-optimized core excitations with a single constraint. Using a triplet purification scheme in combination with the constrained unrestricted Hartree-Fock formalism, scalar-relativistic zero-order regular approximation corrections, and a semiempirical treatment of spin-orbit coupling, COOX is shown to produce highly accurate results for K- and L-edge excitations of second- and third-period atoms with subelectronvolt errors despite being based on LR-TDDFT, for which core excitations pose a well-known challenge. L- and M-edge excitations of heavier atoms up to uranium are also computationally feasible and numerically stable, but may require more advanced treatment of relativistic effects. Furthermore, COOX is shown to perform on par with or better than the popular ΔSCF approach while exhibiting more robust convergence, highlighting it as a promising tool for inexpensive and accurate simulations of X-ray absorption spectra.

A Constrained CASSCF(2,2) Approach to Study Electron Transfer between a Molecule and Metal Cluster

We have implemented a constrained CASSCF(2,2) calculation so as to study thermal electron transfer between a chlorine ion and a cluster of lithium atoms of variable size (from 1 to 17). Our calculations illustrate how the geometry of the ground state-charge transfer state crossing point (as well as the strength of a diabatic coupling) can depend sensitively on the number of metal ions (i.e., the size of the cluster) and the relative positioning of the donor and acceptor. Thus, this set of calculations is an initial step toward understanding the transition from homogeneous to heterogeneous electron transfer. In the future, these constrained calculations should allow us to model still far larger systems, ideally opening up a pathway to study meaningful electrochemical phenomena.

An Efficient Algorithm for Constrained CASSCF(1,2) and CASSCF(3,2) Simulations as Relevant to Electron and Hole Transfer Problems

We propose an efficient algorithm for the recently published electron/hole-transfer Dynamical-weighted State-averaged Constrained CASSCF (eDSC/hDSC) method studying charge transfer states and D1-D0 crossings for systems with odd numbers of electrons. By separating the constrained minimization problem into an unconstrained self-consistent-field (SCF) problem and a constrained nonself-consistent-field (nSCF) problem, as well as accelerating the direct inversion in the iterative subspace (DIIS) technique to solve the SCF problem, the overall computational cost is reduced by a factor of 8-20 compared with directly using sequential quadratic programming (SQP). This approach should be applicable for other constrained minimization problems, and in the immediate future, once gradients are available, the present eDSC/hDSC algorithm should allow for speedy nonadiabatic dynamics simulations.

ADCHα-I population analysis and constrained dipole moment density functional theory in force fields for molecular simulations

A new population analysis, ADCHα-I, based on the interpolation between the Hirshfeld (H) and the iterative Hirshfeld (H-I) methods through a parameter α and on the atomic dipole moment corrected Hirshfeld (ADCH) methodology is proposed, in combination with the constrained dipole moment density functional theory (CD-DFT) previously developed, to determine the charge distributions of force fields. Following this approach, the electronic density of the isolated molecule is determined for the value of the dipole moment that reproduces the experimental dielectric constant, in order to incorporate through this property the effects of the surrounding molecules in the liquid, and to carry on this information to the molecular simulation, the new population analysis is built to obtain the set of charges that reproduces this dipole moment. By selecting α = 1/2, one is led to charges that are larger than the ones obtained through H and ADCH and smaller than those of H-I and that incorporate, at the local level, information about the response of isolated atoms to donate or to accept charge, which is not considered in ADCH. The results obtained for several liquid properties indicate that the combination of CD-DFT with this population analysis leads to a good description of the charge distributions in force fields used in molecular simulations.

A Constraint-Based Orbital-Optimized Excited State Method (COOX)

In this work, we present a novel method to directly calculate targeted electronic excited states within a self-consistent field calculation based on constrained density functional theory (cDFT). The constraint is constructed from the static occupied-occupied and virtual-virtual parts of the excited state difference density from (simplified) linear-response time-dependent density functional theory calculations (LR-TDDFT). Our new method shows a stable convergence behavior, provides an accurate excited state density adhering to the Aufbau principle, and can be solved within a restricted SCF for singlet excitations to avoid spin contamination. This also allows the straightforward application of post-SCF electron-correlation methods like MP2 or direct RPA methods. We present the details of our constraint-based orbital-optimized excited state method (COOX) and compare it to similar schemes. The accuracy of excitation energies will be analyzed for a benchmark of systems, while the quality of the resulting excited state densities is investigated by evaluating excited state nuclear forces and excited state structure optimizations. We also investigate the performance of the proposed COOX method for long-range charge transfer excitations and conical intersections with the ground-state.

Quantum chemical package Jaguar: A survey of recent developments and unique features

This paper is dedicated to the quantum chemical package Jaguar, which is commercial software developed and distributed by Schrödinger, Inc. We discuss Jaguar's scientific features that are relevant to chemical research as well as describe those aspects of the program that are pertinent to the user interface, the organization of the computer code, and its maintenance and testing. Among the scientific topics that feature prominently in this paper are the quantum chemical methods grounded in the pseudospectral approach. A number of multistep workflows dependent on Jaguar are covered: prediction of protonation equilibria in aqueous solutions (particularly calculations of tautomeric stability and pKa), reactivity predictions based on automated transition state search, assembly of Boltzmann-averaged spectra such as vibrational and electronic circular dichroism, as well as nuclear magnetic resonance. Discussed also are quantum chemical calculations that are oriented toward materials science applications, in particular, prediction of properties of optoelectronic materials and organic semiconductors, and molecular catalyst design. The topic of treatment of conformations inevitably comes up in real world research projects and is considered as part of all the workflows mentioned above. In addition, we examine the role of machine learning methods in quantum chemical calculations performed by Jaguar, from auxiliary functions that return the approximate calculation runtime in a user interface, to prediction of actual molecular properties. The current work is second in a series of reviews of Jaguar, the first having been published more than ten years ago. Thus, this paper serves as a rare milestone on the path that is being traversed by Jaguar's development in more than thirty years of its existence.

The energy landscape of magnetic materials

Magnetic materials can display many solutions to the electronic-structure problem, corresponding to different local or global minima of the energy functional. In Hartree-Fock or density-functional theory different single-determinant solutions lead to different magnetizations, ionic oxidation states, hybridizations, and inter-site magnetic couplings. The vast majority of these states can be fingerprinted through their projection on the atomic orbitals of the magnetic ions. We have devised an approach that provides an effective control over these occupation matrices, allowing us to systematically explore the landscape of the potential energy surface. We showcase the emergence of a complex zoology of self-consistent states; even more so when semi-local density-functional theory is augmented - and typically made more accurate - by Hubbard corrections. Such extensive explorations allow to robustly identify the ground state of magnetic systems, and to assess the accuracy (or not) of current functionals and approximations.

The use of constrained methods to analyze the molecular reactivity and to define a new type of pseudo atoms

CONTEXT

Constrained methods in electronic structure methodologies add terms to the variational equations and generate solutions that represent distorted electronic distributions. In some cases, the new solutions can be used to study the chemical reactivity of parts of the molecule. Additionally, this contribution presents the use of population constraints to define pseudo atoms in a molecule. The effects of the pseudo atom on the molecular properties are analyzed. The pseudo atoms are used to simulate the inductive effect of the substituent in a group of carbonyl molecules and their effect on the stability of the complexes between these organic species and one molecule of water. A discussion on the assumptions involved in the present definition of pseudo atoms is also included.

METHOD

The constrained RHF computations are done in a modified Hartree-Fock code for Gaussian basis sets. The selected basis set is STO-6 G.

Near-infrared absorption and radiative cooling of naphthalene dimers (C10H8)2

The radiative cooling of naphthalene dimer cations, (C10H8)2+ was studied experimentally through action spectroscopy using two different electrostatic ion-beam storage rings, DESIREE in Stockholm and Mini-Ring in Lyon. The spectral characteristics of the charge resonance (CR) band were observed to vary significantly with a storage time of up to 30 seconds in DESIREE. In particular, the position of the CR band shifts to the blue, with specific times (inverse of rates) of 0.64 s and 8.0 s in the 0-5 s and 5-30 s storage time ranges, respectively. These long-time scales indicate that the internal energy distribution of the stored ions evolves by vibrational radiative cooling, which is consistent with the absence of fast radiative cooling via recurrent fluorescence for (C10H8)2+. Density functional based tight binding calculations with local excitations and configuration interactions (DFTB-EXCI) were used to simulate the absorption spectrum for ion temperatures between 10 and 500 K. The evolution of the bandwidth and position with temperature is in qualitative agreement with the experimental findings. Furthermore, these calculations yielded linear temperature dependencies for both the shift and the broadening. Combining the relationship between the CR band position and the ion temperature with the results of the statistical model, we demonstrate that the observed blue shift can be used to determine the radiative cooling rate of (C10H8)2+.

Critical assessment of the x-ray restrained wave function approach: Advantages, drawbacks, and perspectives for density functional theory and periodic ab initio calculations

The x-ray restrained wave function (XRW) method is a quantum crystallographic technique to extract wave functions compatible with experimental x-ray diffraction data. The approach looks for wave functions that minimize the energies of the investigated systems and also reproduce sets of x-ray structure factors. Given the strict relationship between x-ray structure factors and electron distributions, the strategy practically allows determining wave functions that correspond to given (usually experimental) electron densities. In this work, the capabilities of the XRW approach were further tested. The aim was to evaluate whether the XRW technique could serve as a tool for suggesting new exchange-correlation functionals for density functional theory or refining existing ones. Additionally, the ability of the method to address the influences of the crystalline environment was also assessed. The outcomes of XRW computations were thus compared to those of traditional gas-phase, embedding quantum mechanics/molecular mechanics, and fully periodic calculations. The results revealed that, irrespective of the initial conditions, the XRW computations practically yield a consensus electron density, in contrast to the currently employed density functional approximations (DFAs), which tend to give a too large range of electron distributions. This is encouraging in view of exploiting the XRW technique to develop improved functionals. Conversely, the calculations also emphasized that the XRW method is limited in its ability to effectively address the influences of the crystalline environment. This underscores the need for a periodic XRW technique, which would allow further untangling the shortcomings of DFAs from those inherent to the XRW approach.

A fast and smooth one-electron approach for investigating charge transfer states and D1-D0 crossings for systems with odd numbers of electrons

We propose an efficient version of ensemble Hartree-Fock/density functional theory to calculate a set of two charge-transfer states for systems with odd-numbers of electrons. The approach does require definitions of donor/acceptor fragments; however, the approach is not very sensitive to such definitions-even in the limit of very strong electronic coupling. The key ansatz is that, by mandating that the vector space spanned by the active orbitals projects equally onto the donor and acceptor fragments, such a constraint eliminates all intra-molecular local excitations and makes it far easier to generate potential energy surfaces that are smooth over a wide region of configuration space. The method is fast, working with only two electron configurations, and should be useful for ab initio non-adiabatic dynamics in the near future.

The tale of HORTON: Lessons learned in a decade of scientific software development

HORTON is a free and open-source electronic-structure package written primarily in Python 3 with some underlying C++ components. While HORTON's development has been mainly directed by the research interests of its leading contributing groups, it is designed to be easily modified, extended, and used by other developers of quantum chemistry methods or post-processing techniques. Most importantly, HORTON adheres to modern principles of software development, including modularity, readability, flexibility, comprehensive documentation, automatic testing, version control, and quality-assurance protocols. This article explains how the principles and structure of HORTON have evolved since we started developing it more than a decade ago. We review the features and functionality of the latest HORTON release (version 2.3) and discuss how HORTON is evolving to support electronic structure theory research for the next decade.

Exploring the Reactivity of Donor-Acceptor Systems through a Combined Conceptual and Constrained DFT Approach

In the context of the conceptual density functional theory (cDFT) and based on the computational efficiency of the constrained DFT (CDFT), we demonstrate that chemical reactivity can be governed by the difference between the local interacting chemical potentials of the reactants (referred as Edual), in agreement with Sanderson's equalization principle. In a proof-of-concept study, we investigated illustrative examples involving typical non-covalent donor-acceptor systems and reactive systems are provided. For the selected systems, our approach reveals significant mimicking between Edual and the DFT-computed intermolecular interaction energy profiles. We further evaluate the influence of the Coulomb and exchange-correlation contributions in Edual. These latter results suggest that numerous potential energy surfaces of clusters can be explored using a Sanderson-like model only based on classical interactions between molecular orbitals domains. To conclude, this study achieved a deeper understanding of the principles of cDFT and assessed, in a wider context, its efficiency in predicting the chemical reactivity.

Probing Nonadiabaticity of Proton-Coupled Electron Transfer in Ribonucleotide Reductase

The enzyme ribonucleotide reductase, which is essential for DNA synthesis, initiates the conversion of ribonucleotides to deoxyribonucleotides via radical transfer over a 32 Å pathway composed of proton-coupled electron transfer (PCET) reactions. Previously, the first three PCET reactions in the α subunit were investigated with hybrid quantum mechanical/molecular mechanical (QM/MM) free energy simulations. Herein, the fourth PCET reaction in this subunit between C439 and guanosine diphosphate (GDP) is simulated and found to be slightly exoergic with a relatively high free energy barrier. To further elucidate the mechanisms of all four PCET reactions, we analyzed the vibronic and electron-proton nonadiabaticities. This analysis suggests that interfacial PCET between Y356 and Y731 is vibronically and electronically nonadiabatic, whereas PCET between Y731 and Y730 and between C439 and GDP is fully adiabatic and PCET between Y730 and C439 is in the intermediate regime. These insights provide guidance for selecting suitable rate constant expressions for these PCET reactions.

Tryptophan to Tryptophan Hole Hopping in an Azurin Construct

Electron transfer (ET) between neutral and cationic tryptophan residues in the azurin construct [ReI(H126)(CO)3(dmp)](W124)(W122)CuI (dmp = 4,7-Me2-1,10-phenanthroline) was investigated by Born-Oppenheimer quantum-mechanics/molecular mechanics/molecular dynamics (QM/MM/MD) simulations. We focused on W124•+ ← W122 ET, which is the middle step of the photochemical hole-hopping process *ReII(CO)3(dmp•-) ← W124 ← W122 ← CuI, where sequential hopping amounts to nearly 10,000-fold acceleration over single-step tunneling (ACS Cent. Sci. 2019, 5, 192-200). In accordance with experiments, UKS-DFT QM/MM/MD simulations identified forward and reverse steps of W124•+ ↔ W122 ET equilibrium, as well as back ET ReI(CO)3(dmp•-) → W124•+ that restores *ReII(CO)3(dmp•-). Strong electronic coupling between the two indoles (≥40 meV in the crossing region) makes the productive W124•+ ← W122 ET adiabatic. Energies of the two redox states are driven to degeneracy by fluctuations of the electrostatic potential at the two indoles, mainly caused by water solvation, with contributions from the protein dynamics in the W122 vicinity. ET probability depends on the orientation of Re(CO)3(dmp) relative to W124 and its rotation diminishes the hopping yield. Comparison with hole hopping in natural systems reveals structural and dynamics factors that are important for designing efficient hole-hopping processes.

Simulating Electron Transfer Reactions in Solution: Radical-Polar Crossover

Single-electron transfer (SET) promotes a wide variety of interesting chemical transformations, but modeling of SET requires a careful treatment of electronic and solvent effects to give meaningful insight. Therefore, a combined constrained density functional theory and molecular mechanics (CDFT/MM) tool is introduced specifically for SET-initiated reactions. Mechanisms for two radical-polar crossover reactions involving the organic electron donors tetrakis(dimethylamino)ethylene (TDAE) and tetrathiafulvalene (TTF) were studied with the new tool. An unexpected tertiary radical intermediate within the TDAE system was identified, relationships between kinetics and substitution in the TTF system were explained, and the impact of the solvent environments on the TDAE and TTF reactions were examined. The results highlight the need for including solvent dynamics when quantifying SET kinetics and thermodynamics, as a free energy difference of >20 kcal/mol was observed. Overall, the new method informs mechanistic analysis of SET-initiated reactions and therefore is envisioned to be useful for studying reactions in the condensed phase.

Constrained DFT-based magnetic machine-learning potentials for magnetic alloys: a case study of Fe-Al

We propose a machine-learning interatomic potential for multi-component magnetic materials. In this potential we consider magnetic moments as degrees of freedom (features) along with atomic positions, atomic types, and lattice vectors. We create a training set with constrained DFT (cDFT) that allows us to calculate energies of configurations with non-equilibrium (excited) magnetic moments and, thus, it is possible to construct the training set in a wide configuration space with great variety of non-equilibrium atomic positions, magnetic moments, and lattice vectors. Such a training set makes possible to fit reliable potentials that will allow us to predict properties of configurations in the excited states (including the ones with non-equilibrium magnetic moments). We verify the trained potentials on the system of bcc Fe-Al with different concentrations of Al and Fe and different ways Al and Fe atoms occupy the supercell sites. Here, we show that the formation energies, the equilibrium lattice parameters, and the total magnetic moments of the unit cell for different Fe-Al structures calculated with machine-learning potentials are in good correspondence with the ones obtained with DFT. We also demonstrate that the theoretical calculations conducted in this study qualitatively reproduce the experimentally-observed anomalous volume-composition dependence in the Fe-Al system.

Hybrid Quantum Mechanical/Molecular Mechanical Methods For Studying Energy Transduction in Biomolecular Machines

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods have become indispensable tools for the study of biomolecules. In this article, we briefly review the basic methodological details of QM/MM approaches and discuss their applications to various energy transduction problems in biomolecular machines, such as long-range proton transports, fast electron transfers, and mechanochemical coupling. We highlight the particular importance for these applications of balancing computational efficiency and accuracy. Using several recent examples, we illustrate the value and limitations of QM/MM methodologies for both ground and excited states, as well as strategies for calibrating them in specific applications. We conclude with brief comments on several areas that can benefit from further efforts to make QM/MM analyses more quantitative and applicable to increasingly complex biological problems.

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.

Fe-Fe Double-Atom Catalysts for Murine Coronavirus Disinfection: Nonradical Activation of Peroxides and Mechanisms of Virus Inactivation

Peroxides find broad applications for disinfecting environmental pathogens particularly in the COVID-19 pandemic; however, the extensive use of chemical disinfectants can threaten human health and ecosystems. To achieve robust and sustainable disinfection with minimal adverse impacts, we developed Fe single-atom and Fe-Fe double-atom catalysts for activating peroxymonosulfate (PMS). The Fe-Fe double-atom catalyst supported on sulfur-doped graphitic carbon nitride outperformed other catalysts for oxidation, and it activated PMS likely through a nonradical route of catalyst-mediated electron transfer. This Fe-Fe double-atom catalyst enhanced PMS disinfection kinetics for inactivating murine coronaviruses (i.e., murine hepatitis virus strain A59 (MHV-A59)) by 2.17-4.60 times when compared to PMS treatment alone in diverse environmental media including simulated saliva and freshwater. The molecular-level mechanism of MHV-A59 inactivation was also elucidated. Fe-Fe double-atom catalysis promoted the damage of not only viral proteins and genomes but also internalization, a key step of virus lifecycle in host cells, for enhancing the potency of PMS disinfection. For the first time, our study advances double-atom catalysis for environmental pathogen control and provides fundamental insights of murine coronavirus disinfection. Our work paves a new avenue of leveraging advanced materials for improving disinfection, sanitation, and hygiene practices and protecting public health.

An Efficient Multilayer Approach to Model DNA-Based Nanobiosensors

In this work, we present a full computational protocol to successfully obtain the one-electron reduction potential of nanobiosensors based on a self-assembled monolayer of DNA nucleobases linked to a gold substrate. The model is able to account for conformational sampling and environmental effects at a quantum mechanical (QM) level efficiently, by combining molecular mechanics (MM) molecular dynamics and multilayer QM/MM/continuum calculations within the framework of Marcus theory. The theoretical model shows that a guanine-based biosensor is more prone to be oxidized than the isolated nucleobase in water due to the electrostatic interactions between the assembled guanine molecules. In addition, the redox properties of the biosensor can be tuned by modifying the nature of the linker that anchor the nucleobases to the metal support.

Transformation of Graphitic Carbon Nitride by Reactive Chlorine Species: "Weak" Oxidants Are the Main Players

Graphitic carbon nitride (g-C₃N₄) nanomaterials hold great promise in diverse applications; however, their stability in engineering systems and transformation in nature are largely underexplored. We evaluated the stability, aging, and environmental impact of g-C₃N₄ nanosheets under the attack of free chlorine and reactive chlorine species (RCS), a widely used oxidant/disinfectant and a class of ubiquitous radical species, respectively. g-C₃N₄ nanosheets were slowly oxidized by free chlorine even at a high concentration of 200-1200 mg L^-1, but they decomposed rapidly when ClO· and/or Cl₂^•- were the key oxidants. Though Cl₂^•- and ClO· are considered weaker oxidants in previous studies due to their lower reduction potentials and slower reaction kinetics than ·OH and Cl·, our study highlighted that their electrophilic attack efficacy on g-C₃N₄ nanosheets was on par with ·OH and much higher than Cl·. A trace level of covalently bonded Cl (0.28-0.55 at%) was introduced to g-C₃N₄ nanosheets after free chlorine and RCS oxidation. Our study elucidates the environmental fate and transformation of g-C₃N₄ nanosheets, particularly under the oxidation of chlorine-containing species, and it also provides guidelines for designing reactive, robust, and safe nanomaterials for engineering applications.

Constrained Density Functional Theory: A Potential-Based Self-Consistency Approach

Chemical reactions, charge transfer reactions, and magnetic materials are notoriously difficult to describe within Kohn-Sham density functional theory, which is strictly a ground-state technique. However, over the last few decades, an approximate method known as constrained density functional theory (cDFT) has been developed to model low-lying excitations linked to charge transfer or spin fluctuations. Nevertheless, despite becoming very popular due to its versatility, low computational cost, and availability in numerous software applications, none of the previous cDFT implementations is strictly similar to the corresponding ground-state self-consistent density functional theory: the target value of constraints (e.g., local magnetization) is not treated equivalently with atomic positions or lattice parameters. In the present work, by considering a potential-based formulation of the self-consistency problem, the cDFT is recast in the same framework as Kohn-Sham DFT: a new functional of the potential that includes the constraints is proposed, where the constraints, the atomic positions, or the lattice parameters are treated all alike, while all other ingredients of the usual potential-based DFT algorithms are unchanged, thanks to the formulation of the adequate residual. Tests of this approach for the case of spin constraints (collinear and noncollinear) and charge constraints are performed. Expressions for the derivatives with respect to constraints (e.g., the spin torque) for the atomic forces and the stress tensor in cDFT are provided. The latter allows one to study striction effects as a function of the angle between spins. We apply this formalism to body-centered cubic iron and first reproduce the well-known magnetization amplitude as a function of the angle between local magnetizations. We also study stress as a function of such an angle. Then, the local collinear magnetization and the local atomic charge are varied together. Since the atomic spin magnetizations, local atomic charges, atomic positions, and lattice parameters are treated on an equal footing, this formalism is an ideal starting point for the generation of model Hamiltonians and machine-learning potentials, computation of second or third derivatives of the energy as delivered from density-functional perturbation theory, or for second-principles approaches.

Diabatic Decomposition Perspective on the Role of Charge Transfer and Local Excitations in Thermally Activated Delayed Fluorescence

Thermally activated delayed fluorescence (TADF) is a phenomenon that relies on the upconversion of triplet excitons to singlet excitons by means of reverse intersystem crossing (rISC). It has been shown both experimentally and theoretically that the TADF mechanism depends on the interplay between charge transfer and local excitations. However, the difference between the diabatic and adiabatic character of the involved excited states is rarely discussed in the literature. Here we develop a diabatization procedure to implement a four-state model Hamiltonian to a set of TADF molecules. We provide physical interpretations of the Hamiltonian elements and show their dependence on the electronic state of the equilibrium geometry. We also demonstrate how vibrations affect the TADF efficiency by modifying the diabatic decomposition of the molecule. Finally, we provide a simple model that connects the diabatic Hamiltonian to the electronic properties relevant to TADF and show how this relationship translates into different optimization strategies for rISC, fluorescence, and overall TADF performance.

On the termination of the X-ray constrained wavefunction procedure: reformulation of the method for an unequivocal determination of λ

The X-ray constrained/restrained wavefunction (XCW/XRW) approach of quantum crystallography is revisited by introducing the stationary condition of the Jayatilaka functional with respect to the Lagrange multiplier λ. The theoretical derivation has unequivocally shown that the right value of λ is a maximum stationary point of the functional to optimize, thus enabling the solution of the longstanding problem of establishing the point at which to halt the XCW/XRW procedure. Based on the new finding, a reformulation of the X-ray constrained wavefunction algorithm is proposed and its implementation is envisaged. In addition to relying on more solid mathematical grounds, the new variant of the method will be intrinsically more physically meaningful, allowing a straightforward evaluation of the highest level of confidence with which the experimental X-ray diffraction data can be possibly reproduced.

Modeling and Characterization of Exciplexes in Photoredox CO2 Reduction: Insights from Quantum Chemistry and Fluorescence Spectroscopy

Interactions between excited-state arenes and amines can lead to the formation of structures with a distinct emission behavior. These excited-state complexes or exciplexes can reduce the ability of the arene to participate in other reactions, such as CO₂ reduction, or increase the likelihood of degradation via Birch reduction. Exciplex geometries are necessary to understand photophysical behavior and probe degradation pathways but are challenging to calculate. We establish a detailed computational protocol for calculation, verification, and characterization of exciplexes. Using fluorescence spectroscopy, we first demonstrate the formation of exciplexes between excited-state oligo-(p-phenylene) (OPP), shown to successfully carry out CO₂ reduction, and triethylamine. Time-dependent density functional theory is employed to optimize the geometries of these exciplexes, which are validated by comparing both emission energies and their solvatochromism with the experiment. Excited-state energy decomposition analysis confirms the predominant role played by charge transfer interactions in the red shift of emissions relative to the isolated excited-state OPP*. We find that although the exciplex emission frequency depends strongly on solvent dielectric, the extent of charge separation in an exciplex does not. Our results also suggest that the formation of solvent-separated ionic radical states upon complete electron transfer competes with exciplex formation in higher-dielectric solvents, thereby leading to reduced exciplex emission intensities in fluorescence experiments.

Charge-Transfer Excitation within a Hybrid-(G)KS Framework through Cartesian Grid DFT

Organic molecules that exhibit charge-transfer (CT) excited states are known to play an important role in processes linked to electron transfer properties and molecular conductance. In this article, we present a simple technique based on "Becke's excitation theorem" that offers an accurate picture of these electronic states. It expresses the correlated energy splitting between triplet and its corresponding singlet states by a two-electron integral, which is numerically evaluated by our recently developed strategy on Cartesian grid. We first examine the consistency of our adopted numerical strategy to evaluate the integral with the original prescribed technique. Then we assess the method on weakly bound CT complexes with three different functionals (BLYP, B3LYP, and LC-BLYP). The accuracy on asymptotic limit of CT excitation is also explored. Finally in order to illustrate the strength and feasibility, it is further extended to a few "challenging" molecules. The method, when employed with hybrid B3LYP functional, turns out to be quite accurate to describe CT excitation energy.

Path Integrals for Nonadiabatic Dynamics: Multistate Ring Polymer Molecular Dynamics

This review focuses on a recent class of path-integral-based methods that simulate nonadiabatic dynamics in the condensed phase using only classical molecular dynamics trajectories in an extended phase space. Specifically, a semiclassical mapping protocol is used to derive an exact, continuous, Cartesian variable path-integral representation for the canonical partition function of a system in which multiple electronic states are coupled to nuclear degrees of freedom. Building on this exact statistical foundation, multistate ring polymer molecular dynamics methods are developed for the approximate calculation of real-time thermal correlation functions. The remarkable promise of these multistate ring polymer methods, their successful applications, and their limitations are discussed in detail.Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Resolving Chemical Dynamics in Biological Energy Conversion: Long-Range Proton-Coupled Electron Transfer in Respiratory Complex I

Biological energy conversion is catalyzed by membrane-bound proteins that transduce chemical or light energy into energy forms that power endergonic processes in the cell. At a molecular level, these catalytic processes involve elementary electron-, proton-, charge-, and energy-transfer reactions that take place in the intricate molecular machineries of cell respiration and photosynthesis. Recent developments in structural biology, particularly cryo-electron microscopy (cryoEM), have resolved the molecular architecture of several energy transducing proteins, but detailed mechanistic principles of their charge transfer reactions still remain poorly understood and a major challenge for modern biochemical research. To this end, multiscale molecular simulations provide a powerful approach to probe mechanistic principles on a broad range of time scales (femtoseconds to milliseconds) and spatial resolutions (10¹–10⁶ atoms), although technical challenges also require balancing between the computational accuracy, cost, and approximations introduced within the model. Here we discuss how the combination of atomistic (aMD) and hybrid quantum/classical molecular dynamics (QM/MM MD) simulations with free energy (FE) sampling methods can be used to probe mechanistic principles of enzymes responsible for biological energy conversion. We present mechanistic explorations of long-range proton-coupled electron transfer (PCET) dynamics in the highly intricate respiratory chain enzyme Complex I, which functions as a redox-driven proton pump in bacterial and mitochondrial respiratory chains by catalyzing a 300 Å fully reversible PCET process. This process is initiated by a hydride (H^–) transfer between NADH and FMN, followed by long-range (>100 Å) electron transfer along a wire of 8 FeS centers leading to a quinone biding site. The reduction of the quinone to quinol initiates dissociation of the latter to a second membrane-bound binding site, and triggers proton pumping across the membrane domain of complex I, in subunits up to 200 Å away from the active site. Our simulations across different size and time scales suggest that transient charge transfer reactions lead to changes in the internal hydration state of key regions, local electric fields, and the conformation of conserved ion pairs, which in turn modulate the dynamics of functional steps along the reaction cycle. Similar functional principles, which operate on much shorter length scales, are also found in some unrelated proteins, suggesting that enzymes may employ conserved principles in the catalysis of biological energy transduction processes.

Perspective and challenges in electrochemical approaches for reactive CO2 separations

The desire toward decarbonization and renewable energy has sparked research interests in reactive CO2 separations, such as direct air capture that utilize electricity as opposed to conventional thermal and pressure swing processes, which are energy-intensive, cost-prohibitive, and fossil-fuel dependent. Although the electrochemical approaches in CO2 capture that support negative emissions technologies are promising in terms of modularity, smaller footprint, mild reaction conditions, and possibility to integrate into conversion processes, their practice depends on the wider availability of renewable electricity. This perspective discusses key advances made in electrolytes and electrodes with redox-active moieties that reversibly capture CO2 or facilitate its transport from a CO2-rich side to a CO2-lean side within the last decade. In support of the discovery of new heterogeneous electrode materials and electrolytes with redox carriers, the role of computational chemistry is also discussed.

Flexible boundary layer using exchange for embedding theories. II. QM/MM dynamics of the hydrated electron

The FlexiBLE embedding method introduced in Paper I [Z. Shen and W. J. Glover, J. Chem. Phys. 155, 224112 (2021)] is applied to explore the structure and dynamics of the aqueous solvated electron at an all-electron density functional theory Quantum Mechanics/Molecular Mechanics level. Compared to a one-electron mixed quantum/classical description, we find the dynamics of the many-electron model of the hydrated electron exhibits enhanced coupling to water OH stretch modes. Natural bond orbital analysis reveals this coupling is due to significant population of water OH σ^* orbitals, reaching 20%. Based on this, we develop a minimal frontier orbital picture of the hydrated electron involving a cavity orbital and important coupling to 4-5 coordinating OH σ^* orbitals. Implications for the interpretation of the spectroscopy of this interesting species are discussed.

Extended Mulliken-Hush Method with Applications to the Theoretical Study of Electron Transfer

A novel adiabatic-to-diabatic (ATD) transformation strategy, namely, the extended Mulliken-Hush (XMH) method, is proposed to evaluate diabatic properties including electronic couplings, potential energy surfaces, and their crossings. The XMH method is developed by adopting our recently proposed ATD transformation formula of a general vectorial physical observable, in which a useful ATD transformation is further determined by using an auxiliary dipole between localized frontier orbitals as a simple approximation of the diabatic transition dipole. The XMH method is simple and practical that provides a flexible way to construct diabatic states. To some extent, it can be regarded as an extension of the generalized Mulliken-Hush (GMH) method since the latter takes a stronger approximation, in which the diabatic transition dipole is assumed to be vanishing. Test calculations on the HeH₂⁺ system show that the electronic couplings predicted by the XMH method are closer to the ones calculated by the valence bond block-diagonalization approach than the GMH ones since the XMH method takes into account both the magnitude and direction of the diabatic transition dipole, which is consistent with the properties of this molecule. In the study of electron transfer in the two kinds of donor-bridge-acceptor systems, the XMH method maintains the simplicity of the GMH method and gives reasonable results even when the latter fails, wherein the diabatic transition dipole is nearly perpendicular to the difference of the initial and final adiabatic dipoles. More importantly, the XMH method can be easily combined with high-level electronic structure methods, in which the properties of the ground and excited states may be more accurately calculated, and hence, one may expect that further development of the XMH method would result in a general computational model for studying electron transfer reactions.

Pseudospectral implementations of long-range corrected density functional theory

We have implemented pseudospectral density-functional theory (DFT) with long-range corrected DFT functionals (PS-LRC) in quantum mechanics package Jaguar, and applied it in the calculations of geometry optimizations, dimmer interaction energies, polarizabilities and first-order hyperpolarizabilities, harmonic vibrational frequencies, S₁ and T₁ excitation energies, singlet-triplet gaps, charge transfer numbers, oscillator strengths, reaction barrier heights, electron-transfer couplings, and charge-transfer excitation energies. From our accuracy benchmark analysis, PS grids, PS dealiasing functions, PS atomic corrections, PS multigrid strategy, PS length scales, and PS cutoff scheme perform well in PS DFT with LRC density functionals with very small and ignorable deviations when compared to the conventional spectral (CS) method. The timing benchmark study of S₁ excitation energy calculations of fullerenes (C_n , n up to 540) demonstrates that PS-LRC achieves 1.4-8.4-fold speedups in SCF, 22-32-fold speedups in Tamm-Dancoff approximation, and 6-15-fold speedups in total wall clock time with an average error 0.004 eV of excitation energies compared to the CS method.

The redox potential of a heme cofactor in Nitrosomonas europaea cytochrome c peroxidase: a polarizable QM/MM study

Redox reactions are crucial to biological processes that protect organisms against oxidative stress. Metalloenzymes, such as peroxidases which reduce excess reactive oxygen species into water, play a key role in detoxification mechanisms. Here we present the results of a polarizable QM/MM study of the reduction potential of the electron transfer heme in the cytochrome c peroxidase of Nitrosomonas europaea. We have found that environment polarization does not substantially affect the computed value of the redox potential. Particular attention has been given to analyzing the role of electrostatic interactions within the protein environment and the solvent on tuning the redox potential of the heme co-factor. We have found that the electrostatic interactions predominantly explain the fluctuations of the vertical ionization/attachment energies of the heme for the sampled configurations, and that the long range electrostatic interactions (up to 40 Å) contribute substantially to the absolute values of the vertical energy gaps.

Evaluation of optical band gaps and dopant state energies in transition metal oxides using oxidation-state constrained density functional theory

We report the use of oxidation-state constrained density functional theory (OS-CDFT) to calculate the optical band gaps of transition metal oxides and dopant state energies of different doped anatase. OS-CDFT was used to control electron transfer from the valence band maximum of the transition metal system to the conduction band minimum or to the dopant state in order to calculate the band gap or the dopant state energies respectively. The calculation of the dopant state energies also allows identification of the transition responsible for the reduced band gap of the doped system in ambiguous cases. We applied this approach to the band gap calculation in TiO₂anatase and rutile, vanadium pentoxide (V₂O₅), chromium(III) oxide (Cr₂O₃), manganese(IV) oxide (MnO₂), ferric oxide (Fe₂O₃), ferrous oxide (FeO) and cobalt(II) oxide (CoO). The dopant state energies calculations were carried out in the V-, Cr-, Mn-, and Fe-doped anatase.

How the Donor/Acceptor Spin States Affect the Electronic Couplings in Molecular Charge-Transfer Processes?

The electronic coupling matrix element H_AB is an essential ingredient of most electron-transfer theories. H_AB depends on the overlap between donor and acceptor wave functions and is affected by the involved states’ spin. We classify the spin-state effects into three categories: orbital occupation, spin-dependent electron density, and density delocalization. The orbital occupancy reflects the diverse chemical nature and reactivity of the spin states of interest. The effect of spin-dependent density is related to a more compact electron density cloud at lower spin states due to decreased exchange interactions between electrons. Density delocalization is strongly connected with the covalency concept that increases the spatial extent of the diabatic state’s electron density in specific directions. We illustrate these effects with high-level ab initio calculations on model direct donor–acceptor systems relevant to metal oxide materials and biological electron transfer. Obtained results can be used to benchmark existing methods for H_AB calculations in complicated cases such as spin-crossover materials or antiferromagnetically coupled systems.

Revealing Charge-Transfer Dynamics at Electrified Sulfur Cathodes Using Constrained Density Functional Theory

To understand and control the behavior of electrochemical systems, including batteries and electrocatalysts, we seek molecular-level details of the charge transfer mechanisms at electrified interfaces. Recognizing some key limitations of standard equilibrium electronic structure methods to model materials and their interfaces, we propose applying charge constraints to effectively separate electronic and nuclear degrees of freedom, which are especially beneficial to the study of conversion electrodes, where electronic charge carriers are converted to much slower polarons within a material that is nonmetallic. We demonstrate the need for such an approach within the context of sulfur cathodes and the arrival of Li ions during discharge of a Li-S cell. The requirement that electronic degrees of freedom are arrested is justified by comparison with real-time evolution of the electronic structure. Long-lived metastable configurations provide plenty of time for nuclear dynamics and relaxation in response to the electrification of the interface, a process that would be completely missed without applying charge constraints. This approach will be vital to the study of dynamics at electrified interfaces which may be created deliberately, adding charge to the electrode, or spontaneously, due to finite temperature dynamics in the electrolyte.

A non-perturbative pairwise-additive analysis of charge transfer contributions to intermolecular interaction energies

A non-perturbative scheme for complete decomposition of energy and charge associated with charge transfer interaction into pairwise additive components.

Block-Localized Excitation for Excimer Complex and Diabatic Coupling

We describe a block-localized excitation (BLE) method to carry out constrained optimization of block-localized orbitals for constructing valence bond-like, diabatic excited configurations using multistate density functional theory (MSDFT). The method is an extension of the previous block-localized wave function method through a fragment-based ΔSCF approach to optimize excited determinants within a molecular complex. In BLE, both the number of electrons and the electronic spin of different fragments in a whole system can be constrained, whereas electrostatic, exchange, and polarization interactions among different blocks can be fully taken into account of. To avoid optimization collapse to unwanted states, a ΔSCF projection scheme and a maximum overlap of wave function approach have been presented. The method is illustrated by the excimer complex of two naphthalene molecules. With a minimum of eight spin-adapted configurational state functions, it was found that the inversion of L_a^- and L_b^- states near the optimal structure of the excimer complex is correctly produced, which is in quantitative agreement with DMRG-CASPT2 calculations and experiments. Trends in the computed transfer integrals associated with excited-state energy transfer both in the singlet and triplet states are discussed. The results suggest that MSDFT may be used as an efficient approach to treat intermolecular interactions in excited states with a minimal active space (MAS) for interpretation of the results and for dynamic simulations, although the selection of a small active space is often system dependent.

Excited state diabatization on the cheap using DFT: Photoinduced electron and hole transfer

Excited state electron and hole transfer underpin fundamental steps in processes such as exciton dissociation at photovoltaic heterojunctions, photoinduced charge transfer at electrodes, and electron transfer in photosynthetic reaction centers. Diabatic states corresponding to charge or excitation localized species, such as locally excited and charge transfer states, provide a physically intuitive framework to simulate and understand these processes. However, obtaining accurate diabatic states and their couplings from adiabatic electronic states generally leads to inaccurate results when combined with low-tier electronic structure methods, such as time-dependent density functional theory, and exorbitant computational cost when combined with high-level wavefunction-based methods. Here, we introduce a density functional theory (DFT)-based diabatization scheme that directly constructs the diabatic states using absolutely localized molecular orbitals (ALMOs), which we denote as Δ-ALMO(MSDFT2). We demonstrate that our method, which combines ALMO calculations with the ΔSCF technique to construct electronically excited diabatic states and obtains their couplings with charge-transfer states using our MSDFT2 scheme, gives accurate results for excited state electron and hole transfer in both charged and uncharged systems that underlie DNA repair, charge separation in donor-acceptor dyads, chromophore-to-solvent electron transfer, and singlet fission. This framework for the accurate and efficient construction of excited state diabats and evaluation of their couplings directly from DFT thus offers a route to simulate and elucidate photoinduced electron and hole transfer in large disordered systems, such as those encountered in the condensed phase.

Computational Analysis of Electron Transfer Kinetics for CO2 Reduction with Organic Photoredox Catalysts

We present a fundamental description of the electron transfer (ET) step from substituted oligo(p-phenylene) (OPP) radical anions to CO₂, with the larger goal of assessing the viability of underexplored, organic photoredox routes for utilization of anthropogenic CO₂. This work varies the electrophilicity of para-substituents to OPP and probes the dependence of rate coefficients and interfragment interactions on the substituent Hammett parameter, σ_p, using constrained density functional theory (CDFT) and energy decomposition analysis (EDA). Large electronic couplings across substituents indicates an adiabatic electron transfer process for reactants at contact. As one might intuitively expect, free energy changes dominate trends in ET rate coefficients in most cases, and rates increase with substituent electron-donating ability. However, we observe an unexpected dip in rate coefficients for the most electron-donating groups, due to the combined impact of flattening free energies and a steep increase in reorganization energies. Our analysis shows that, with decreasing σ_p, flattening OPP LUMO levels lower the marginal increase in free energy. EDA reveals trends in electrostatics and charge transfer interactions between the catalyst and substrate fragments that influence free energy changes across substituents. Reorganization energies do not exhibit a direct dependence on σ_p and are largely similar across systems, with the exception of substituents containing lone pairs of electrons that exhibit significant deformation upon electron transfer. Our study therefore suggests that while a wide range of ET rates are observed, there is an upper limit to rate enhancements achievable by only tuning the substituent electrophilicity.

Neutral excitation density-functional theory: an efficient and variational first-principles method for simulating neutral excitations in molecules

We introduce neutral excitation density-functional theory (XDFT), a computationally light, generally applicable, first-principles technique for calculating neutral electronic excitations. The concept is to generalise constrained density functional theory to free it from any assumptions about the spatial confinement of electrons and holes, but to maintain all the advantages of a variational method. The task of calculating the lowest excited state of a given symmetry is thereby simplified to one of performing a simple, low-cost sequence of coupled DFT calculations. We demonstrate the efficacy of the method by calculating the lowest single-particle singlet and triplet excitation energies in the well-known Thiel molecular test set, with results which are in good agreement with linear-response time-dependent density functional theory (LR-TDDFT). Furthermore, we show that XDFT can successfully capture two-electron excitations, in principle, offering a flexible approach to target specific effects beyond state-of-the-art adiabatic-kernel LR-TDDFT. Overall the method makes optical gaps and electron-hole binding energies readily accessible at a computational cost and scaling comparable to that of standard density functional theory. Owing to its multiple qualities beneficial to high-throughput studies where the optical gap is of particular interest; namely broad applicability, low computational demand, and ease of implementation and automation, XDFT presents as a viable candidate for research within materials discovery and informatics frameworks.