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.

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.

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.

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.

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.

Describing strong correlation with fractional-spin correction in density functional theory.

An effective fractional-spin correction is developed to describe static/strong correlation in density functional theory. Combined with the fractional-charge correction from recently developed localized orbital scaling correction (LOSC), a functional, the fractional-spin LOSC (FSLOSC), is proposed. FSLOSC, a correction to commonly used functional approximations, introduces the explicit derivative discontinuity and largely restores the flat-plane behavior of electronic energy at fractional charges and fractional spins. In addition to improving results from conventional functionals for the prediction of ionization potentials, electron affinities, quasiparticle spectra, and reaction barrier heights, FSLOSC properly describes the dissociation of ionic species, single bonds, and multiple bonds without breaking space or spin symmetry and corrects the spurious fractional-charge dissociation of heteroatom molecules of conventional functionals. Thus, FSLOSC demonstrates success in reducing delocalization error and including strong correlation, within low-cost density functional approximation.

Doubly hybrid density functionals that correctly describe both density and energy for atoms.

Recently, it was argued [Medvedev MG, et al. (2017) Science 355:49-52] that the development of density functional approximations (DFAs) is "straying from the path toward the exact functional." The exact functional should yield both exact energy and density for a system of interest; nevertheless, they found that many heavily fitted functionals for molecular energies actually lead to poor electron densities of atoms. They also observed a trend that, for the nonempirical and few-parameter functionals, densities can be improved as one climbs up the first four rungs of the Jacob's ladder of DFAs. The XYG3 type of doubly hybrid functionals (xDHs) represents a less-empirical and fewer-parameter functional on the top fifth rung, in which both the Hartree-Fock-like exchange and the second-order perturbative (MP2-like) correlation are hybridized with the low rung functionals. Here, we show that xDHs can well describe both density and energy for the same atomic set of Medvedev et al., showing that the latter trend can well be extended to the top fifth rung.

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.

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.

Plasmon-Enhanced Catalysis: Distinguishing Thermal and Nonthermal Effects.

In plasmon-enhanced heterogeneous catalysis, illumination accelerates reaction rates by generating hot carriers and hot surfaces in the constituent nanostructured metals. In order to understand how photogenerated carriers enhance the nonthermal reaction rate, the effects of photothermal heating and thermal gradients in the catalyst bed must be confidently and quantitatively characterized. This is a challenging task considering the conflating effects of light absorption, heat transport, and reaction energetics. Here, we introduce a methodology to distinguish the thermal and nonthermal contributions from plasmon-enhanced catalysts, demonstrated by illuminated rhodium nanoparticles on oxide supports to catalyze the CO2 methanation reaction. By simultaneously measuring the total reaction rate and the temperature gradient of the catalyst bed, the effective thermal reaction rate may be extracted. The residual nonthermal rate of the plasmon-enhanced reaction is found to grow with a superlinear dependence on illumination intensity, and its apparent quantum efficiency reaches ∼46% on a Rh/TiO2 catalyst at a surface temperature of 350 °C. Heat and light are shown to work synergistically in these reactions: the higher the temperature, the higher the overall nonthermal efficiency in plasmon-enhanced catalysis.

Development of New Density Functional Approximations.

Kohn-Sham density functional theory has become the leading electronic structure method for atoms, molecules, and extended systems. It is in principle exact, but any practical application must rely on density functional approximations (DFAs) for the exchange-correlation energy. Here we emphasize four aspects of the subject: (a) philosophies and strategies for developing DFAs; (b) classification of DFAs;

Accurate potential energy surfaces for hydrogen abstraction reactions: A benchmark study on the XYG3 doubly hybrid density functional.

The potential energy surface (PES) for the H + CH4 system has been constructed with the recently developed XYG3 doubly hybrid functional, while those with the standard B3LYP hybrid functional, and the Møller-Plesset perturbation theory up to the second order (MP2) are also presented for comparison. Quantum dynamics studies demonstrated that satisfactory results on the reaction probabilities and the rate coefficients can be obtained on top of the XYG3-PES, as compared to the results based on the highly accurate, yet expensive, CCSD(T)-PES (Li et al., J. Chem. Phys. 2015, 142, 204302). Further investigation suggested that the XYG3 functional is useful in providing accurate rate coefficients for some larger systems involving H atom abstractions. © 2017 Wiley Periodicals, Inc.

Self-consistent field for fragmented quantum mechanical model of large molecular systems.

Fragment-based linear scaling quantum chemistry methods are a promising tool for the accurate simulation of chemical and biomolecular systems. Because of the coupled inter-fragment electrostatic interactions, a dual-layer iterative scheme is often employed to compute the fragment electronic structure and the total energy. In the dual-layer scheme, the self-consistent field (SCF) of the electronic structure of a fragment must be solved first, then followed by the updating of the inter-fragment electrostatic interactions. The two steps are sequentially carried out and repeated; as such a significant total number of fragment SCF iterations is required to converge the total energy and becomes the computational bottleneck in many fragment quantum chemistry methods. To reduce the number of fragment SCF iterations and speed up the convergence of the total energy, we develop here a new SCF scheme in which the inter-fragment interactions can be updated concurrently without converging the fragment electronic structure. By constructing the global, block-wise Fock matrix and density matrix, we prove that the commutation between the two global matrices guarantees the commutation of the corresponding matrices in each fragment. Therefore, many highly efficient numerical techniques such as the direct inversion of the iterative subspace method can be employed to converge simultaneously the electronic structure of all fragments, reducing significantly the computational cost. Numerical examples for water clusters of different sizes suggest that the method shall be very useful in improving the scalability of fragment quantum chemistry methods.

Error accumulations in adhesive energies of dihydrogen molecular chains: performances of the XYG3 type of doubly hybrid density functionals.

We have systematically analyzed the error accumulations in the adhesive energies for a series of hydrogen molecular chains calculated by various kinds of density functional theory (DFT) methods. In particular, we have focused on some representative doubly hybrid (DH) functionals of either the B2PLYP type (B2PLYP, B2PLYP-D, and B2GP-PLYP) or the XYG3 type (XYG3, XYGJ-OS, and xDH-PBE0). The hydrogen molecular chain models have recently been proposed by Zheng et al. (J. Chem. Phys. 2012, 137, 214106) to identify the delocalization errors (DEs) in thermodynamic properties. From the perspective of DEs, it is shown here that the XYG3 type of DH functionals yield good performance on the calculated adhesive energies due to the minimizing effects of DEs, highlighting the underlying physics for the successes or failures of the approximate functionals. Examination was also extended to HF-DFT, where DFT energies are evaluated with the Hartree-Fock (HF) densities.

H + H2 quantum dynamics using potential energy surfaces based on the XYG3 type of doubly hybrid density functionals: validation of the density functionals.

The potential energy surfaces for the H + H2 exchange reaction are determined, using the standard Becke-3_parameter-Lee-Yang-Parr (B3LYP) hybrid exchange-correlation functional and the recently developed XYG3 type (xDH) doubly hybrid functionals. Quantum dynamical results calculated on these density functional theory surfaces revealed that the accuracy for the surfaces from the xDH functionals is quite satisfactory as compared to the benchmark Boothroyd-Keogh-Martin-Peterson-2 surface, whereas B3LYP is wholly inadequate in describing this simplest reaction despite the fact that it has been widely used to study reactions in complex systems. It is shown that further improvements can be achieved if the xDH functional parameters are fine-tuned to be reaction specific.

Perturbative treatment of anharmonic vibrational effects on bond distances: an extended Langevin dynamics method.

In this work, we first review the perturbative treatment of an oscillator with cubic anharmonicity. It is shown that there is a quantum-classical correspondence in terms of mean displacement, mean-squared displacement, and the corresponding variance in the first-order perturbation theory, provided that the amplitude of the classical oscillator is fixed at the zeroth-order energy of quantum mechanics EQM (0). This correspondence condition is realized by proposing the extended Langevin dynamics (XLD), where the key is to construct a proper driving force. It is assumed that the driving force adopts a simple harmonic form with its amplitude chosen according to EQM (0), while the driving frequency chosen as the harmonic frequency. The latter can be improved by using the natural frequency of the system in response to the potential if its anharmonicity is strong. By comparing to the accurate numeric results from discrete variable representation calculations for a set of diatomic species, it is shown that the present method is able to capture the large part of anharmonicity, being competitive with the wave function-based vibrational second-order perturbation theory, for the whole frequency range from ∼4400 cm(-1) (H2 ) to ∼160 cm(-1) (Na2 ). XLD shows a substantial improvement over the classical molecular dynamics which ceases to work for hard mode when zero-point energy effects are significant.

Fractional charge behavior and band gap predictions with the XYG3 type of doubly hybrid density functionals.

In this work, we examine the fractional charge behaviors of doubly hybrid (DH) functionals. By plotting the ground-state energies E and energy derivatives for atoms and molecules with fractional electron numbers N, we directly quantify the delocalization errors of some representative DH functionals such as B2PLYP, XYG3, and XYGJ-OS. Numerical assessments on ionization potentials (IPs), electron affinities (EAs), and fundamental gaps, from either integer number calculations or energy derivative calculations, are provided. It is shown that the XYG3 type of DH functionals gives good agreement between their energy derivatives and the experimental IPs, EAs, and gaps, as expected from their nearly straight line fractional charge behaviors.

Construction of a parameter-free doubly hybrid density functional from adiabatic connection.

In this work, the adiabatic connection (AC) formalism, coordinate scaling relations, and the second order Görling-Levy perturbation theory (GL2) are first reviewed. Emphasis is laid upon the construction of the AC integrand (Wxc(λ)), with suitable input data {Wxc(0), Wxc'(0), Wxc(1) and the proper asymptotic behavior on λ(-1/2) as λ → ∞. This leads to a non-empirical DH functional, namely, PBE-ACDH. The PBE-ACDH functional is unique in that it explicitly considers contributions from density scaling and singles, and it utilizes density and orbital information from the PBE functional, which has a local multiplicative potential, being most compatible with the GL2 theory. Systematical tests on heats of formation, bond dissociation enthalpies, reaction barrier heights, and nonbonded interactions, using some well-established benchmarking sets, suggest that PBE-ACDH is a significant improvement over its parent functional PBE, as well as PBE0, Becke's Half-and-Half (PBE-HH), and GL2. The physical insight gained in the present work should prove useful for the further development of new functionals.

Analytic derivatives for the XYG3 type of doubly hybrid density functionals: Theory, implementation, and assessment.

We present a theoretical development of the equations required to perform an analytic geometry optimization of a molecular system using the XYG3 type of doubly hybrid (xDH) functionals. In contrast to the well-established B2PLYP type of DH functionals, the energy expressions in the xDH functionals are constructed by using density and orbital information from another standard Kohn-Sham (KS) functional (e.g., B3LYP) for doing the self-consistent field calculations. Thus, the xDH functionals are nonvariational in both the hybrid density functional part and the second-order perturbation part, each of which requires formally to solve a coupled-perturbed KS equation. An implementation is reported here which combines the two parts by defining a total Lagrangian such that only a single set of the Z-vector equations need to be solved. The computational cost with our implementation is of the same order as those for the conventional Møller-Plesset theory to the second order (MP2) and B2PLYP. Systematic test calculations are provided for covalently bonded molecules as well as compounds involving the intramolecular nonbonded interactions for the main group elements. Satisfactory performance of the xDH functionals demonstrates that the extra computer time on top of the conventional KS procedure is well-invested, in particular, when the standard KS functionals and MP2 as well, are problematic.

A comparison of geometric parameters from PBE-based doubly hybrid density functionals PBE0-DH, PBE0-2, and xDH-PBE0.

We present a systematic investigation on the optimized geometric parameters for covalently bonded molecules, nonbonded intermolecular complexes, and transition state structures from three PBE (Perdew-Burke-Ernzerhof)-based doubly hybrid (DH) density functionals, namely PBE0-DH, PBE0-2, and xDH-PBE0. While the former two are the B2PLYP-type of DH functionals with no fit parameters, the latter is the XYG3-type of DH functional (xDH for short) with three fit parameters, whose energy expression is constructed by using density and orbital information from another standard (general) Kohn-Sham functional (i.e., PBE0) for doing the self-consistent field calculations. Generally good performances have been obtained with all three DH functionals, in particular, with xDH-PBE0.