Short-range screened density matrix functional for proper descriptions of thermochemistry, thermochemical kinetics, nonbonded interactions, and singlet diradicals.

Common one-electron reduced density matrix (1-RDM) functionals that depend on Coulomb and exchange-only integrals tend to underestimate dynamic correlation, preventing reduced density matrix functional theory (RDMFT) from achieving comparable accuracy to density functional theory in main-group thermochemistry and thermochemical kinetics. The recently developed ωP22 functional introduces a semi-local density functional to screen the erroneous short-range portion of 1-RDM functionals without double-counting correlation, potentially providing a better treatment of dynamic correlation around equilibrium geometries. Herein, we systematically evaluate the performance of this functional model, which consists of two parameters, on main-group thermochemistry, thermochemical kinetics, nonbonded interactions, and more. Tests on atomization energies, vibrational frequencies, and reaction barriers reveal that the ωP22 functional model can reliably predict properties at equilibrium and slightly away from equilibrium geometries. In particular, it outperforms commonly used density functionals in the prediction of reaction barriers, nonbonded interactions, and singlet diradicals, thus enhancing the predictive power of RDMFT for routine calculations of thermochemistry and thermochemical kinetics around equilibrium geometries. Further development is needed in the future to refine short- and long-range approximations in the functional model in order to achieve an excellent description of properties both near and far from equilibrium geometries.

Targeted Spin-State Regulation to Boost Oxygen Reduction Reaction.

Catalytic reactions are known to be significantly affected by spin states and their variations during reaction processes, yet the mechanisms behind them remain not fully understood, thus preventing the rational optimization of catalysis. Here, we explore the relationship between the spin states of active sites and their catalytic performance, taking the oxygen reduction reaction as an example. We demonstrate that the catalytic performance is spin-state-dependent and can be improved by adjusting spin states during the catalytic process. To this end, we further investigate the possibility of altering the spin states of transition metals through the application of external fields, such as adsorbed species. By studying the influence of the strength of adsorbed ligands on spin states and its impact on catalytic performance, our results show that optimal catalytic performance is achieved when the strength of the external field is neither too strong nor too weak, forming a volcano-like relationship between the catalytic performance and the external field strength. Our findings can have far-reaching implications for the rational design of high-performance catalysis.

The Role of Self-Catalysis Induced by Co Doping in Nonaqueous Li-O2 Batteries.

This work systematically studies the product self-catalysis of in situ electrochemical cobalt doping of Li2O2 and reveals its potential mechanism for improving the performance of lithium-oxygen (Li-O2) batteries. Theoretical calculations demonstrate that the discharge products contain substituted and interstitial Co impurities, which serve as active sites to promote the formation of Li3O4 crystallization, thus switching the nucleation mechanism from the main discharge product Li2O2 to Li3O4. This Co-doping behavior leads to the thermodynamically favorable and dynamically stable formation of Li3O4 crystals during the discharge process. Through systematic investigation of the structural, energetic, electronic, diffusive, and catalytic properties of the Co-doped Li2O2 and Li3O4 compounds, we found that Li3O4 has better charge/mass transport and a lower overpotential for the Li3O4 formation/decomposition reaction. Consequently, this work elucidates that Co doping provides a simple and effective approach for increasing the proportion of Li3O4, which can significantly improve the Li-O2 battery performance.

Explicit-by-Implicit Treatment of Natural Orbital Occupations Using First- and Second-Order Optimization Algorithms: A Comparative Study.

To address the convergence issues in the natural occupation optimization of reduced density matrix functional theory (RDMFT), we recently proposed the explicit-by-implicit (EBI) idea to handle the ensemble N-representability constraint (Yao et al. J. Phys. Chem. Lett. 2021, 12, 6788). This work continues to focus on these issues that can affect the reliability of the electronic structure description in RDMFT; further explores the combination of EBI, as well as the (augmented) Lagrangian methods (both LM and ALM), with both first- and second-order numerical optimization algorithms; and carefully evaluates their performances in natural occupation optimizations of various systems, including strongly correlated systems and large molecules. By comparing both converged energies and elapsed times, it can be seen that the LM and ALM have serious convergence issues for systems of different sizes. In contrast, the optimizations of EBI can converge to better energies with fewer iterations. However, due to the local convergence nature of the Newton's Method (NM) algorithm, EBI@NM still suffers from the local minimum issue for both strongly correlated systems and large molecules. Overall, the combination of EBI with the simple first-order algorithm of gradient descent (GD), namely EBI@GD, consistently provides the lowest converged energies for different types of systems, with the lowest computational scaling. These tests demonstrate the advantages of EBI in the calculations of transition states, strongly correlated systems, and large molecules. Meanwhile, the insights gained from this work are helpful to further develop more efficient algorithms for RDMFT.

Describing Chemical Reactivity with Frontier Molecular Orbitalets.

Locality in physical space is critical in understanding chemical reactivity in the analysis of various phenomena and processes in chemistry, biology, and materials science, as exemplified in the concepts of reactive functional groups and active sites. Frontier molecular orbitals (FMOs) pinpoint the locality of chemical bonds that are chemically reactive because of the associated orbital energies and thus have achieved great success in describing chemical reactivity, mainly for small systems. For large systems, however, the delocalization nature of canonical molecular orbitals makes it difficult for FMOs to highlight the locality of the chemical reactivity. To obtain localized molecular orbitals that also reflect the frontier nature of the chemical processes, we develop the concept of frontier molecular orbitalets (FMOLs) for describing the reactivity of large systems. The concept of orbitalets was developed recently in the localized orbital scaling correction method, which aims for eliminating the delocalization error in common density functional approximations. Orbitalets are localized in both physical and energy spaces and thus contain both orbital locality and energy information. The FMOLs are thus the orbitalets with energies highest among occupied orbitalets and lowest among unoccupied ones. The applications of FMOLs to hexadeca-1,3,5,7,9,11,13,15-octaene in its equilibrium geometry, inter- and intra-molecular charge-transfer systems, and two transition states of a bifurcating reaction demonstrate that FMOLs can connect quantum mechanical treatments of chemical systems and chemical reactivities by locating the reactive region of large chemical systems. Therefore, FMOLs extend the role of FMOs for small systems and describe the chemical reactivity of large systems with energy and locality insight, with potentially broad applications.

Computational Insight into Metallated Graphynes as Single Atom Electrocatalysts for Nitrogen Fixation.

The electrochemical nitrogen reduction reaction (NRR) is expected to achieve sustainable ammonia synthesis via direct nitrogen fixation; however, the high-quality catalysts that play a crucial role in the NRR are still lacking. The emerging transition metal-1,3,5-triethynylbenzene (TM-TEB) frameworks offer attractive possibilities in the electrochemical catalysis due to the featured atomic and electronic structures. This work presents a comprehensive first-principles study of the TM-TEB systems for TMs from the first three d-block series and systematically explores their potential applications as NRR electrocatalysts. By designing a hierarchical screening strategy, the TM-TEB systems are evaluated based on the NRR catalytic activity as well as the competition from the hydrogen evolution reaction. In addition, in order to have a deeper understanding of the catalytic activities of the TM-TEB systems, diverse possible NRR paths on the TM-TEB surfaces are completely analyzed as well. Our analysis reveals that the TM-TEB systems with TM = V, Mo, Tc, W, and Os are electrocatalysts with a high NRR catalytic activity, while among them, only Mo- and V-TEB show promising NRR selectively. This work demonstrates the great potential of the TM-TEB systems as electrocatalysts in the NRR process, which improves the understanding of the TM-TEB systems and can motivate further exploration of their application in catalysis.

Revealing intrinsic spin coupling in transition metal-doped graphene.

Graphene materials offer attractive possibilities in spintronics due to their unique atomic and electronic structures, which is in contrast to their limited applications in the design of sophisticated spintronic devices. This should be attributed to the lack of knowledge about the intrinsic characteristics of graphene materials, especially the diverse correlations between sites within the materials and their roles in spin-signal generation and propagation. This work comprehensively studies the spin couplings between transition metal atoms doped on graphene and reveals their potential application in spintronic device design through the realization of various logic gates. In addition, the effects of the distance between doped metal atoms and the number of carbon layers on the logic gate implementation further verify that the spin-coupling effect can exhibit a certain distance dependence and space propagation. The achievements in this work uncover the potential value of graphene materials and are expected to open up new avenues for exploring their application in the design of sophisticated spintronic devices.

Combining localized orbital scaling correction and Bethe-Salpeter equation for accurate excitation energies.

We applied localized orbital scaling correction (LOSC) in Bethe-Salpeter equation (BSE) to predict accurate excitation energies for molecules. LOSC systematically eliminates the delocalization error in the density functional approximation and is capable of approximating quasiparticle (QP) energies with accuracy similar to or better than GW Green's function approach and with much less computational cost. The QP energies from LOSC, instead of commonly used G0W0 and evGW, are directly used in BSE. We show that the BSE/LOSC approach greatly outperforms the commonly used BSE/G0W0 approach for predicting excitations with different characters. For the calculations of Truhlar-Gagliardi test set containing valence, charge transfer, and Rydberg excitations, BSE/LOSC with the Tamm-Dancoff approximation provides a comparable accuracy to time-dependent density functional theory (TDDFT) and BSE/evGW. For the calculations of Stein CT test set and Rydberg excitations of atoms, BSE/LOSC considerably outperforms both BSE/G0W0 and TDDFT approaches with a reduced starting point dependence. BSE/LOSC is, thus, a promising and efficient approach to calculate excitation energies for molecular systems.

Functional-Based Description of Electronic Dynamic and Strong Correlation: Old Issues and New Insights.

Approximate functionals in Kohn-Sham density functional theory (KS-DFT) and reduced density matrix functional theory (RDMFT) have advantages in dealing with dynamic correlation and strong correlation, respectively; their combination can benefit from complementarity while suffering from the problem of correlation double-counting. Herein, a short-range corrected reduced density matrix (1-RDM) functional is developed to take advantage of the functionals in KS-DFT and RDMFT without double-counting. The resulting functional, denoted as ωP22, outperforms other 1-RDM functionals for the tests of thermochemistry, nonbonded interactions, and bond dissociation energy. In particular, ωP22 shows much less systematic error for systems involving fractional spins, and it can properly predict the energies at both equilibrium and dissociated distances for different single and multiple bonds, which cannot be achieved by commonly used KS-DFT and RDMFT functionals. Therefore, ωP22 is demonstrated effective in balance handling dynamic and strong correlation, and the advances in this work would create new possibilities for the development and application of approximate functionals.

LibSC: Library for Scaling Correction Methods in Density Functional Theory.

In recent years, a series of scaling correction (SC) methods have been developed in the Yang laboratory to reduce and eliminate the delocalization error, which is an intrinsic and systematic error existing in conventional density functional approximations (DFAs) within density functional theory (DFT). On the basis of extensive numerical results, the SC methods have been demonstrated to be capable of reducing the delocalization error effectively and producing accurate descriptions for many critical and challenging problems, including the fundamental gap, photoemission spectroscopy, charge transfer excitations, and polarizability. In the development of SC methods, the SC methods were mainly implemented in the QM4D package that was developed in the Yang laboratory for research development. The heavy dependency on the QM4D package hinders the SC methods from access by researchers for broad applications. In this work, we developed a reliable and efficient implementation, LibSC, for the global scaling correction (GSC) method and the localized orbital scaling correction (LOSC) method. LibSC will serve as a lightweight and open-source library that can be easily accessed by the quantum chemistry community. The implementation of LibSC is carefully modularized to provide the essential functionalities for conducting calculations of the SC methods. In addition, LibSC provides simple and consistent interfaces to support multiple popular programing languages, including C, C++, and Python. In addition to the development of the library, we also integrated LibSC with two popular and open-source quantum chemistry packages, the Psi4 package and the PySCF package, which provides immediate access for general users to perform calculations with SC methods.

Localized orbital scaling correction for periodic systems.

Density functional theory offers accurate structure prediction at acceptable computational cost, but commonly used approximations suffer from delocalization error; this results in inaccurate predictions of quantities such as energy band gaps of finite and bulk systems, energy level alignments, and electron distributions at interfaces. The localized orbital scaling correction (LOSC) was developed to correct delocalization error by using orbitals localized in space and energy. These localized orbitals span both the occupied and unoccupied spaces and can have fractional occupations in order to correct both the total energy and the one-electron energy eigenvalues. We extend the LOSC method to periodic systems, in which the localized orbitals employed are dually localized Wannier functions. In light of the effect of the bulk environment on the electrostatic interaction between localized orbitals, we modify the LOSC energy correction to include a screened Coulomb kernel. For a test set of semiconductors and large-gap insulators, we show that the screened LOSC method consistently improves the band gap compared to the parent density functional approximation.

Perturbation theory made efficient and effective for predictions of ionization potential and electron affinity.

Ionization potential and electron affinity are essential molecular properties. The most straightforward method is to calculate them by taking the total energy differences of the initial and final states according to the definition. However, it often suffers from a serious convergence problem due to the requirement of the self-consistent field (SCF) calculations for the ionic states with non-Aufbau choices of occupations. In the present work, we have constructed a theoretical framework in view of perturbation theory to bypass the SCF calculations of the ionic states. To address the imbalance issue that arises from the precisely treated neutral ground state followed by the truncated perturbative treatment of the ionic states, an accurate yet effective method has been developed here, which adds back some terms from the higher order perturbations into the lower order to cancel out the most computationally cost terms in the truncated expansion, thus reaching a better convergence with less computation. The validity of the present methodology has been tested out by applying it to the Hartree-Fock (HF) method in combination with the correlation effect described at the second-order Møller-Plesset level in a frozen-orbital approximation. All the derivations in this work are given in a general framework, which are applicable not only to HF but also to a wide range of density functional theory methods from semi-local functionals to hybrid and doubly hybrid functionals.

Handling Ensemble N-Representability Constraint in Explicit-by-Implicit Manner.

The convergence issues caused by the improper treatment of the ensemble N-representability constraint have severely affected the applicability and reproducibility of reduced density matrix functional theory (RDMFT). Unlike the commonly used Lagrange methods explicitly bringing the constraint into the objective functions, we present a different idea to handle the constraint in an implicit manner, which is achieved by introducing implicit functions to exactly consider the nonlinear unclear connection embedded in the explicit constraint. This explicit-by-implicit idea, denoted as EBI, thus transforms the constrained optimization problem into an unconstrained minimization problem. The tests on different systems, initial guesses, and functionals demonstrate the superiority of EBI in the treatment of the ensemble N-representability constraint. Therefore, EBI solves the convergence issues of the Lagrange methods, which is essential for further development and application of RDMFT. Besides, the idea of EBI is helpful for the treatment of different constrained problems in modern physics and chemistry.

The Role of Range-Separated Correlation in Long-Range Corrected Hybrid Functionals.

The extensive application of long-range corrected hybrid functionals highlights the importance of further improving their accuracy. Unlike common long-range corrected hybrid functionals mainly focusing on the exchange part, range-separated correlation and its role in long-range corrected hybrid functionals are the main concerns of this work. To this end, we present theory on the derivation of the range-separated correlation, whose reliability and validity are proved by the agreement with the full CI on the test of the short-range correlation energy. The tests on various properties indicate that the long-range part of the LYP functional cannot effectively capture the long-range correlation effect required in LC-BLYP, whose absence instead results in a better XC functional. This new functional significantly improves LC-BLYP on all the tests in this work, with an accuracy on par with or even greater than the widely recognized CAM-B3LYP method for some applications, while maintaining the important -1/r asymptotic behavior of the XC potential. The advances and insights gained in this work are useful for the application and development of long-range corrected hybrid functionals, while emphasizing the significance of developing effective and low-cost long-range correlation functionals.

Introductory lecture: when the density of the noninteracting reference system is not the density of the physical system in density functional theory.

A major challenge in density functional theory (DFT) is the development of density functional approximations (DFAs) to overcome errors in existing DFAs, leading to more complex functionals. For such functionals, we consider roles of the noninteracting reference systems. The electron density of the Kohn-Sham (KS) reference with a local potential has been traditionally defined as being equal to the electron density of the physical system. This key idea has been applied in two ways: the inverse calculation of such a local KS potential for the reference from a given density and the direct calculation of density and energy based on given DFAs. By construction, the inverse calculation can yield a KS reference with the density equal to the input density of the physical system. In application of DFT, however, it is the direct calculation of density and energy from a DFA that plays a central role. For direct calculations, we find that the self-consistent density of the KS reference defined by the optimized effective potential (OEP), is not the density of the physical system, when the DFA is dependent on the external potential. This inequality holds also for the density of generalized KS (GKS) or generalized OEP reference, which allows a nonlocal potential, when the DFA is dependent on the external potential. Instead, the density of the physical system, consistent with a given DFA, is given by the linear response of the total energy with respect to the variation of the external potential. This is a paradigm shift in DFT on the use of noninteracting references: the noninteracting KS or GKS references represent the explicit computational variables for energy minimization, but not the density of the physical system for external potential-dependent DFAs. We develop the expressions for the electron density so defined through the linear response for general DFAs, demonstrate the results for orbital functionals and for many-body perturbation theory within the second-order and the random-phase approximation, and explore the connections to developments in DFT.

Revisiting the Hole Size in Double Helical DNA with Localized Orbital Scaling Corrections.

The extent of electronic wave function delocalization for the charge carrier (electron or hole) in double helical DNA plays an important role in determining the DNA charge transfer mechanism and kinetics. The size of the charge carrier's wave function delocalization is regulated by the solvation induced localization and the quantum delocalization among the π stacked base pairs at any instant of time. Using a newly developed localized orbital scaling correction (LOSC) density functional theory method, we accurately characterized the quantum delocalization of the hole wave function in double helical B-DNA. This approach can be used to diagnose the extent of delocalization in fluctuating DNA structures. Our studies indicate that the hole state tends to delocalize among 4 guanine-cytosine (GC) base pairs and among 3 adenine-thymine (AT) base pairs when these adjacent bases fluctuate into degeneracy. The relatively small delocalization in AT base pairs is caused by the weaker π-π interaction. This extent of delocalization has significant implications for assessing the role of coherent, incoherent, or flickering coherent carrier transport in DNA.

Preserving Symmetry and Degeneracy in the Localized Orbital Scaling Correction Approach.

Symmetry is a fundamental concept that plays a critical role in many chemical and physical phenomena and processes, which highlights the importance of theoretical methods to correctly handle symmetry. The recently developed localized orbital scaling correction (LOSC1) shows great improvement on the description of band gaps, photoemission spectra, and dissociation limits of cationic species. However, issues remain with LOSC1 in dealing with the symmetry and degeneracy of electronic states, which are also relevant to other methods using localization. In this work, we utilize a new method that deals with the physical-space and the energy-space localization on an equal footing. The resulting localized orbitals, i.e., orbitalets, are able to maintain more symmetry and the desired state degeneracy, which is important in calculating the electronic structure of both molecules and periodic bulk systems. Furthermore, the curvature matrix is redefined to improve potential energy curves for systems with stretched bonds, while retaining the correct dissociation limits. This new approach, termed LOSC2, includes only two fitting parameters. It maintains accuracy similar to that of LOSC1 over many properties, while overcoming LOSC1's deficiencies in symmetry and degeneracy. Our tests have shown that LOSC2 orbitalets possess the full- or subgroup of molecular symmetry if allowed, which preserves the state degeneracy. Tests on differently sized planar annulenes, odd-numbered allenes, and triphenylene again verify that LOSC2 is able to maintain the state degeneracy, while LOSC1 cannot. All the tests demonstrate the advantage of LOSC2 in the calculation of molecular systems and its potential for application to periodic bulk systems.

Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional Theory.

Vertical ionization potential (IP) and electron affinity (EA) are fundamental molecular properties, while the Δ method and the direct method are the widely used approaches to compute these properties. The Δ method is calculated by taking the total energy difference of the initial and final states, whose reliability is seriously affected by the issue associated with the imbalanced treatment of these two states. The direct method based on the derivatives involving only one single-state calculation can yield a quasi-particle spectrum whose accuracy, on the other hand, is mostly affected by the levels of approximate molecular structure theories. Because of the aforementioned issues, EA prediction can be particularly problematic. Here we present, for the first time, an analytic theory on the derivation and realization of generalized Kohn-Sham (KS) eigenvalues of doubly hybrid (DH) functionals that depend on both occupied and unoccupied orbitals. The method based on the KS eigenvalues of neutral systems, termed the NKS method, is found to suffer little from the imbalance issue, while it is only the NKS method that can offer accurate EA prediction from a good functional approximation, such as the XYG3 type of DH functionals. Being less sensitive to the size of basis sets, the NKS method is of great significance for its application to large systems. The insights gained in this work are useful for the calculation of properties associated with small energy differences while emphasizing the importance of the development of generalized functionals that rely on both occupied and unoccupied orbitals.

Renormalized Singles Green's Function for Quasi-Particle Calculations beyond the G0 W0 Approximation.

Quasi-particle energies and band gaps in particular are critical for investigating novel materials. Commonly used density functional approximations (DFAs) systematically underestimate band gaps, and GW approximation is the established method of choice for good accuracy and reliability. However, G0 W0 has some undesired dependence on the DFA, while self-consistent GW (sc GW) is expensive and not consistent in accuracy improvement. Here a simple and efficient GRS W0 approach has been developed: a subspace diagonalization of the Hartree-Fock (HF) Hamiltonian with the DFA density matrix provides the new reference Green's function GRS that incorporates the effect of all single excitation contributions to the self-energy, thereby essentially eliminating the starting-point dependence. Calculations for molecules and large band gap solids demonstrate the significant improvement over G0 W0 and greatly reduced dependence on the initial DFA. GRS W0 approach also improve results for other bulks over G0 W0, but to a lesser extent, which could be due to the limitations in current implementation for bulks. The results demonstrate that to achieve good accuracy, it is not necessary to use hybrid DFA, which is expensive for bulks. This work should be greatly significant in making GW a more robust approach.

Approximating Quasiparticle and Excitation Energies from Ground State Generalized Kohn-Sham Calculations.

Quasiparticle energies and fundamental band gaps in particular are critical properties of molecules and materials. It was rigorously established that the generalized Kohn-Sham HOMO and LUMO orbital energies are the chemical potentials of electron removal and addition and thus good approximations to band edges and fundamental gaps from a density functional approximation (DFA) with minimal delocalization error. For other quasiparticle energies, their connection to the generalized Kohn-Sham orbital energies has not been established but remains highly interesting. We provide the comparison of experimental quasiparticle energies for many finite systems with calculations from the GW Green function and localized orbitals scaling correction (LOSC), a recently developed correction to semilocal DFAs, which has minimal delocalization error. Extensive results with over 40 systems clearly show that LOSC orbital energies achieve slightly better accuracy than the GW calculations with little dependence on the semilocal DFA, supporting the use of LOSC DFA orbital energies to predict quasiparticle energies. This also leads to the calculations of excitation energies of the N-electron systems from the ground state DFA calculations of the ( N - 1)-electron systems. Results show good performance with accuracy similar to TDDFT and the delta SCF approach for valence excitations with commonly used DFAs with or without LOSC. For Rydberg states, good accuracy was obtained only with the use of LOSC DFA. This work highlights the pathway to quasiparticle and excitation energies from ground density functional calculations.