Density Functional Approximations Research Articles

Density functional theory beyond the Born–Oppenheimer approximation: exact mapping onto an electronically non-interacting Kohn–Sham molecule

This work presents an alternative, general, and in-principle exact extension of electronic Kohn–Sham density functional theory (KS-DFT) to the fully quantum-mechanical molecular problem. Unlike in existing multi-component or exact-factorization-based DFTs of electrons and nuclei, both nuclear and electronic densities are mapped onto a fictitious electronically non-interacting molecule (referred to as KS molecule), where the electrons still interact with the nuclei. Moreover, in the present molecular KS-DFT, no assumption is made about the mathematical form (exactly factorized or not) of the molecular wavefunction. By expanding the KS molecular wavefunction à la Born–Huang, we obtain a self-consistent set of ‘KS beyond Born–Oppenheimer’ electronic equations coupled to nuclear equations that describe nuclei interacting among themselves and with non-interacting electrons. An exact adiabatic connection formula is derived for the Hartree-exchange-correlation energy of the electrons within the molecule and, on that basis, a practical adiabatic density-functional approximation is proposed and discussed.

Characterization of Deformational Isomerization Potential and Interconversion Dynamics with Ultrafast X-ray Solution Scattering.

Dimeric complexes composed of d8 square planar metal centers and rigid bridging ligands provide model systems to understand the interplay between attractive dispersion forces and steric strain in order to assist the development of reliable methods to model metal dimer complexes more broadly. [Ir2 (dimen)4]2+ (dimen = para-diisocyanomenthane) presents a unique case study for such phenomena, as distortions of the optimal structure of a ligand with limited conformational flexibility counteract the attractive dispersive forces from the metal and ligand to yield a complex with two ground state deformational isomers. Here, we use ultrafast X-ray solution scattering (XSS) and optical transient absorption spectroscopy (OTAS) to reveal the nature of the equilibrium distribution and the exchange rate between the deformational isomers. The two ground state isomers have spectrally distinct electronic excitations that enable the selective excitation of one isomer or the other using a femtosecond duration pulse of visible light. We then track the dynamics of the nonequilibrium depletion of the electronic ground state population─often termed the ground state hole─with ultrafast XSS and OTAS, revealing a restoration of the ground state equilibrium in 2.3 ps. This combined experimental and theoretical study provides a critical test of various density functional approximations in the description of bridged d8-d8 metal complexes. The results show that density functional theory calculations can reproduce the primary experimental observations if dispersion interactions are added, and a hybrid functional, which includes exact exchange, is used.

Current density functional framework for spin-orbit coupling: Extension to periodic systems.

Spin-orbit coupling induces a current density in the ground state, which consequently requires a generalization for meta-generalized gradient approximations. That is, the exchange-correlation energy has to be constructed as an explicit functional of the current density, and a generalized kinetic energy density has to be formed to satisfy theoretical constraints. Herein, we generalize our previously presented formalism of spin-orbit current density functional theory [Holzer et al., J. Chem. Phys. 157, 204102 (2022)] to non-magnetic and magnetic periodic systems of arbitrary dimension. In addition to the ground-state exchange-correlation potential, analytical derivatives such as geometry gradients and stress tensors are implemented. The importance of the current density is assessed for band gaps, lattice constants, magnetic transitions, and Rashba splittings. In the latter, the impact of the current density may be larger than the deviation between different density functional approximations.

Thermodynamics of the gas-phase dimerization of formic acid: Fully anharmonic finite temperature calculations at the CCSD(T) and many DFT levels.

A proof-of-concept study is undertaken to demonstrate the utility of the machine learning combined with the thermodynamic perturbation theory (MLPT) to test the accuracy of electronic structure methods in finite-temperature thermodynamic calculations. As a test example, formic acid dimer is chosen, which is one of the systems included in the popular benchmark set S22 [Jurečka et al., Phys. Chem. Chem. Phys. 8, 1985-1993 (2006)]. Starting from the explicit molecular dynamics and thermodynamic integration performed at the PBE + D2 level, the MLPT is used to obtain fully anharmonic dimerization free and internal energies at the reference quality CCSD(T) level and 19 different density functional approximations, including GGA, meta-GGA, non-local, and hybrid functionals with and without dispersion corrections. Our finite-temperature results are shown to be both qualitatively and quantitatively different from those obtained using the conventional benchmarking strategy based on fixed structures. The hybrid functional HSE06 is identified as the best performing approximate method tested, with the errors in free and internal energies of dimerization being 36 and 41meV, respectively.

Self-Consistent Convolutional Density Functional Approximations: Application to Adsorption at Metal Surfaces.

The exchange-correlation (XC) functional in density functional theory is used to approximate multi-electron interactions. A plethora of different functionals are available, but nearly all are based on the hierarchy of inputs commonly referred to as "Jacob's ladder." This paper introduces an approach to construct XC functionals with inputs from convolutions of arbitrary kernels with the electron density, providing a route to move beyond Jacob's ladder. We derive the variational derivative of these functionals, showing consistency with the generalized gradient approximation (GGA), and provide equations for variational derivatives based on multipole features from convolutional kernels. A proof-of-concept functional, PBEq, which generalizes the PBE framework with being a spatially-resolved function of the monopole of the electron density, is presented and implemented. It allows a single functional to use different GGAs at different spatial points in a system, while obeying PBE constraints. Analysis of the results underlines the importance of error cancellation and the XC potential in data-driven functional design. After testing on small molecules, bulk metals, and surface catalysts, the results indicate that this approach is a promising route to simultaneously optimize multiple properties of interest.

Machine learning for accuracy in density functional approximations.

Machine learning techniques have found their way into computational chemistry as indispensable tools to accelerate atomistic simulations and materials design. In addition, machine learning approaches hold the potential to boost the predictive power of computationally efficient electronic structure methods, such as density functional theory, to chemical accuracy and to correct for fundamental errors in density functional approaches. Here, recent progress in applying machine learning to improve the accuracy of density functional and related approximations is reviewed. Promises and challenges in devising machine learning models transferable between different chemistries and materials classes are discussed with the help of examples applying promising models to systems far outside their training sets.

Symmetry breaking and self-interaction correction in the chromium atom and dimer.

Density functional approximations to the exchange-correlation energy can often identify strongly correlated systems and estimate their energetics through energy-minimizing symmetry-breaking. In particular, the binding energy curve of the strongly correlated chromium dimer is described qualitatively by the local spin density approximation (LSDA) and almost quantitatively by the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA), where the symmetry breaking is antiferromagnetic for both. Here, we show that a full Perdew-Zunger self-interaction-correction (SIC) to LSDA seems to go too far by creating an unphysical symmetry-broken state, with effectively zero magnetic moment but non-zero spin density on each atom, which lies ∼4eV below the antiferromagnetic solution. A similar symmetry-breaking, observed in the atom, better corresponds to the 3d↑↑4s↑3d↓↓4s↓ configuration than to the standard 3d↑↑↑↑↑4s↑. For this new solution, the total energy of the dimer at its observed bond length is higher than that of the separated atoms. These results can be regarded as qualitative evidence that the SIC needs to be scaled down in many-electron regions.

Spurious Oscillations Caused by Density Functional Approximations: Who is to Blame? Exchange or Correlation?

We analyze the varying susceptibilities of different density functional approximations (DFAs) to present spurious oscillations on the profiles of several vibrational properties. Among other problems, these spurious oscillations cause significant errors in harmonic and anharmonic IR and Raman frequencies and intensities. This work hinges on a judicious strategy to dissect the exchange and correlation components of DFAs and pinpoint the origins of these oscillations. We identify spurious oscillations in derivatives of all energy components with respect to nuclear displacements, including those energy terms that do not involve numerical integrations. These indirect spurious oscillations are attributed to suboptimal electron densities resulting from a self-consistent field procedure using a DFA that exhibits direct spurious oscillations. Direct oscillations stem from inaccurate numerical integration of the exchange and correlation energy density functionals. A thorough analysis of direct spurious oscillations reveals that only a handful of exchange and correlation components are insensitive to spurious oscillations, giving rise to three families of functionals, BH&H, LSDA, and BLYP. Among the functionals in these families, we encounter four widespread DFAs: BLYP, B3LYP, LC-BLYP, and CAM-B3LYP. Certain DFAs like PBE appear less sensitive to spurious oscillations due to compensatory cancellations between their energy components. Additionally, we found non-negligible but small oscillations in PBE and TPSS, which could be safely employed provided a sufficiently large integration grid is used in the calculations. These findings hint at the key components of current approximations to be improved and emphasize the necessity to develop accurate DFAs suitable for studying molecular spectroscopies.

The rise and fall of stretched bond errors: Extending the analysis of Perdew-Zunger self-interaction corrections of reaction barrier heights beyond the LSDA.

Incorporating self-interaction corrections (SIC) significantly improves chemical reaction barrier height predictions made using density functional theory methods. We present a detailed orbital-by-orbital analysis of these corrections for three semi-local density functional approximations (DFAs) situated on the three lowest rungs of Jacob's ladder of approximations. The analysis is based on Fermi-Löwdin Orbital Self-Interaction Correction (FLOSIC) calculations performed at several steps along the reaction pathway from the reactants (R) to the transition state (TS) to the products (P) for four representative reactions selected from the BH76 benchmark set. For all three functionals, the major contribution to self-interaction corrections of the barrier heights can be traced to stretched bond orbitals that develop near the TS configuration. The magnitude of the ratio of the self-exchange-correlation energy to the self-Hartree energy (XC/H) for a given orbital is introduced as an indicator of one-electron self-interaction error. XC/H = 1.0 implies that an orbital's self-exchange-correlation energy exactly cancels its self-Hartree energy and that the orbital, therefore, makes no contribution to the SIC in the FLOSIC scheme. For the practical DFAs studied here, XC/H spans a range of values. The largest values are obtained for stretched or strongly lobed orbitals. We show that significant differences in XC/H for corresponding orbitals in the R, TS, and P configurations can be used to identify the major contributors to the SIC of barrier heights and reaction energies. Based on such comparisons, we suggest that barrier height predictions made using the strongly constrained and appropriately normed meta-generalized gradient approximation may have attained the best accuracy possible for a semi-local functional using the Perdew-Zunger SIC approach.

Performance of Density Functionals for Excited-State Properties of Isolated Chromophores and Exciplexes: Emission Spectra, Solvatochromic Shifts, and Charge-Transfer Character.

This study assesses the performance of various meta-generalized gradient approximation (meta-GGA), global hybrid, and range-separated hybrid (RSH) density functionals in capturing the excited-state properties of organic chromophores and their excited-state complexes (exciplexes). Motivated by their uses in solar energy harvesting and photoredox CO2 reduction, we use oligo-(p-phenylenes) and their excited-state complexes with triethylamine as model systems. We focus on the fluorescence properties of these systems, specifically emission energies. We also consider solvatochromic shifts and wave function characteristics. The latter is described by using reduced quantities such as natural transition orbitals (NTOs) and exciton descriptors. The functionals are benchmarked against the experimental fluorescence spectra and the equation-of-motion coupled-cluster method with single and double excitations. Both in isolated chromophores and in exciplexes, meta-GGA functionals drastically underestimate the emission energies and exhibit significant exciton delocalization and anticorrelation between electron and hole motion. The performance of global hybrid functionals is strongly dependent on the percentage of exact exchange. Our study identifies RSH GGAs as the best-performing functionals, with ωPBE demonstrating the best agreement with experimental results. RSH meta-GGAs often overestimate emission energies in exciplexes and yield larger hole NTOs. Their performance can be improved by optimally tuning the range-separation parameter.

Beyond Electrons: Correlation and Self-Energy in Multicomponent Density Functional Theory.

Post-Kohn--Sham methods are used to evaluate the ground-state correlation energy and the orbital self-energy of systems consisting of multiple flavors of different fermions. Starting from multicomponent density functional theory, suitable ways to arrive at the corresponding multicomponent random-phase approximation and the multicomponent Green's function GW approximation, including relativistic effects, are outlined. Given the importance of both of this methods in the development of modern Kohn--Sham density functional approximations, this work will provide a foundation to design advanced multicomponent density functional approximations. Additionally, the GW quasiparticle energies are needed to study light-matter interactions with the Bethe--Salpeter equation.

Benchmark Study on the Calculation of 207Pb NMR Chemical Shifts.

A benchmark set for the computation of 207Pb nuclear magnetic resonance (NMR) chemical shifts is presented. The PbS50 set includes conformer ensembles of 50 lead-containing molecular compounds and their experimentally measured 207Pb NMR chemical shifts. Various bonding motifs at the Pb center with up to seven bonding partners are included. Six different solvents were used in the measurements. The respective shifts lie in the range between +10745 and -5030 ppm. Several calculation settings are assessed by evaluating computed 207Pb NMR shifts for the use with different density functional approximations (DFAs), relativistic approaches, treatment of the conformational space, and levels for geometry optimization. Relativistic effects were included explicitly with the zeroth order regular approximation (ZORA), for which only the spin-orbit variant was able to yield reliable results. In total, seven GGAs and three hybrid DFAs were tested. Hybrid DFAs significantly outperform GGAs. The most accurate DFAs are mPW1PW with a mean absolute deviation (MAD) of 429 ppm and PBE0 with an MAD of 446 ppm. Conformational influences are small as most compounds are rigid, but more flexible structures still benefit from Boltzmann averaging. Including explicit relativistic treatments such as SO-ZORA in the geometry optimization does not show any significant improvement over the use of effective core potentials (ECPs).

Computational study of oxygen evolution reaction on flat and stepped surfaces of strontium titanate

Oxygen evolution reaction (OER) is reconciled with the bottleneck in hydrogen production via photo- or electrocatalytic water splitting. One of the remedies to overcome this shortcoming is to develop catalytically efficient anode materials. Strontium titanate (STO) is a potential candidate for the OER by referring to recent experimental reports on enhanced photocatalytic activity of faceted nanoparticles. In this paper, we perform electronic structure calculations in the density functional theory approximation to study the OER on flat and stepped surfaces, such as encountered in nano-sized STO. We demonstrate that stepped surfaces reveal higher OER activity than flat surfaces, which is consistent with experimental data. We also observe partial breaking of OER scaling relation on stepped surfaces, albeit this does not necessarily cause higher catalytic activity. Finally, we propose recommendations for thorough implementation of zero-point energy calculations as the calculation method may impact the free-energy landscape of the OER.

A numerical Poisson solver with improved radial solutions for a self-consistent locally scaled self-interaction correction method

The universal applicability of density functional approximations is limited by self-interaction error made by these functionals. Recently, a novel one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities was proposed. The locally scaled SIC (LSIC) method is exact for the one-electron densities, and unlike the well-known Perdew–Zunger SIC (PZSIC) method recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to PZSIC method as a special case when isoorbital indicator is set to the unity. Here, we present a numerical scheme that we have adopted to evaluate the Coulomb potential of the electron density scaled by the iso-orbital indicator required for the self-consistent LSIC calculations. After analyzing the behavior of the finite difference method (FDM) and the green function solution to the radial part of the Poisson equation, we adopt a hybrid approach that uses the FDM for the Coulomb potential due to the monopole and the GF for all higher-order terms. The performance of the resultant hybrid method is assessed using a variety of systems. The results show improved accuracy than earlier numerical schemes. We also find that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies.

Making quantum chemistry compressive and expressive: Toward practical ab‐initio simulation

AbstractAb‐initio quantum chemistry simulations are essential for understanding electronic structure of molecules and materials in almost all areas of chemistry. A broad variety of electronic structure theories and implementations has been developed in the past decades to hopefully solve the many‐body Schrödinger equation in an approximate manner on modern computers. In this review, we present recent progress in advancing low‐rank electronic structure methodologies that rely on the wavefunction sparsity and compressibility to select the important subset of electronic configurations for both weakly and strongly correlated molecules. Representative chemistry applications that require the many‐body treatment beyond traditional density functional approximations are discussed. The low‐rank electronic structure theories have further prompted us to highlight compressive and expressive principles that are useful to catalyze idea of quantum learning models. The intersection of the low‐rank correlated feature design and the modern deep neural network learning provides new feasibilities to predict chemically accurate correlation energies of unknown molecules that are not represented in the training dataset. The results by others and us are discussed to reveal that the electronic feature sets from an extremely low‐rank correlation representation, which is very poor for explicit energy computation, are however sufficiently expressive for capturing and transferring electron correlation patterns across distinct molecular compositions, bond types and geometries.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry Software > Simulation Methods

The role of the coupling matrix elements in time-dependent density functional theory on the simulation of core-level spectra of transition metal complexes

Time-dependent density functional theory (TD-DFT) stands out as an efficient tool for computing core-level spectra in large molecules, particularly transition metal complexes. However, despite their relatively moderate computational demands, TD-DFT methods can still pose challenges for typical computations involving transition metal complexes with over a thousand basis functions. In this study, we investigate the role of the Coulomb, Hartree–Fock exchange, and exchange-correlation kernel contributions to the TD-DFT coupling matrix elements when simulating core-level spectra in transition metal complexes. Our observations reveal that the exchange-correlation kernel contribution, responsible for more than 50% of the computational time in a hybrid TD-DFT calculation, surprisingly has no discernible impact on the qualitative aspects of the calculated spectra. While the Coulomb term plays a crucial role in describing L2,3 -edge spectra, its significance becomes negligible when considering K, L 1, and M4,5 edges. In contrast, the scaled Hartree–Fock exchange is demonstrated to be the most influential term, underscoring the necessity for hybrid density functional approximations in accurately simulating core-level spectra. These trends hold irrespective of the chosen basis set and exchange-correlation functional, providing valuable insights for the development of approximate methods for incorporating two-electron interactions within the realm of core-level spectroscopies.

Correlation consistent basis sets designed for density functional theory: Third-row atoms (Ga-Br).

The correlation consistent basis sets (cc-pVnZ with n = D, T, Q, 5) for the Ga-Br elements have been redesigned, tuning the sets for use for density functional approximations. Steps to redesign these basis sets for an improved correlation energy recovery and efficiency include truncation of higher angular momentum functions, recontraction of basis set coefficients, and reoptimization of basis set exponents. These redesigned basis sets are compared with conventional cc-pVnZ basis sets and other basis sets, which are, in principle, designed to achieve systematic improvement with respect to increasing basis set size. The convergence of atomic energies, bond lengths, bond dissociation energies, and enthalpies of formation to the Kohn-Sham limit is improved relative to other basis sets where convergence to the Kohn-Sham limit is typically not observed.

A density-fitting implementation of the density-based basis-set correction method.

This work reports an efficient density-fitting implementation of the density-based basis-set correction (DBBSC) method in the MOLPRO software. This method consists in correcting the energy calculated by a wave-function method with a given basis set by an adapted basis-set correction density functional incorporating the short-range electron correlation effects missing in the basis set, resulting in an accelerated convergence to the complete-basis-set limit. Different basis-set correction density-functional approximations are explored and the complementary-auxiliary-basis-set single-excitation correction is added. The method is tested on a benchmark set of reaction energies at the second-order Møller-Plesset (MP2) level and a comparison with the explicitly correlated MP2-F12 method is provided. The results show that the DBBSC method greatly accelerates the basis convergence of MP2 reaction energies, without reaching the accuracy of the MP2-F12 method but with a lower computational cost.

Employing neural density functionals to generate potential energy surfaces.

With the union of machine learning (ML) and quantum chemistry, amid the debate between machine-learned functionals and human-designed functionals in density functional theory (DFT), this paper aims to demonstrate the generation of potential energy surfaces using computations with machine-learned density functional approximation (ML-DFA). A recent research trend is the application of ML in quantum sciences in the design of density functionals such as DeepMind's Deep Learning model (DeepMind21, DM21). Though science reported the state-of-the-art performance of DM21, the opportunity to utilize DeepMind's pretrained DM21 neural networks in computations in quantum chemistry has not yet been tapped. So far in the literature, the Deep Learning density functionals (DM21) have not been applied to generate potential energy surfaces. While the superior accuracy of DM21 has been reported, there is still a scarcity of publications that apply DM21 in calculations in the field. In this context, for the first time in literature, neural density functionals inferring 2D potential energy surfaces (ML-DFA-PES) based on machine-learned DFA-based computational method is contributed in this paper. This paper reports the ML-DFA-generated PES for C4H8, H2O, H2, and H2+ by employing a pretrained DM21m TensorFlow model with cc-pVDZ basis set. In addition, we also analyze the long-range behavior of DM21 based PES to investigate the ability to describe a system at long ranges. Furthermore, we compare PES diagrams from DM21 with popular DFT functionals (b3lyp/ PW6B95) and CCSD(T). In this method, 2D potential energy surfaces are obtained using a method that relies upon the neural network's ability to accurately learn the mapping between 3D electron density and exchange-correlation potential. By inserting Deep Learning inference in DFT with a pretrained neural network, self-consistent field (SCF) energy at different geometries along the coordinates of interest is computed, and then, potential energy surfaces are plotted. In this method, first, the electron density is computed mathematically, and this computed 3D electron density is used as a ML feature vector to predict the exchange correlation potential as a ML inference computed by a forward pass of pre-trained DM21 TensorFlow computational graph, followed by the computation of self-consistent field energy at multiple geometries, and then, SCF energies at different bond lengths/angles are plotted as 2D PES. We implement this in a python source code using frameworks such as PySCF and DM21. This paper contributes this implementation in open source. The source code and DM21-DFA-based PES are contributed at https://sites.google.com/view/MLfunctionals-DeepMind-PES .

Extending the Scope of Conceptual Density Functional Theory with Second Order Analytical Methodologies.

In the context of the growing impact of conceptual density functional theory (DFT) as one of the most successful chemical reactivity theories, response functions up to second order have now been widely applied; in recent years, among others, particular attention has been focused on the linear response function and also extensions to higher order have been put forward. As the larger part of these studies have been carried using a finite difference approach to compute these concepts, we now embarked on (an extension of) an analytical approach to conceptual DFT. With the ultimate aim of providing a complete set of analytically computable second order properties, including the softness and hardness kernels, the hardness as the simplest second order response function is scrutinized again with numerical results highlighting the difference in nature between the analytical hardness (referred to as hardness condition) and the Parr-Pearson absolute chemical hardness. The hardness condition is investigated for its capability to gauge the (de)localization error of density functional approximations (DFAs). The analytical Fukui function, besides overcoming the difficulties in the finite difference approach in treating negatively charged systems, also showcases the errors of deviating from the straight-line behavior using fractional occupation number calculations. Subsequently, the softness kernel and its atom-condensed inverse, the hardness matrix, are accessed through the Berkowitz-Parr relation. Revisiting the softness kernel confirms and extends previous discussions on how Kohn's Nearsightedness of Electronic Matter principle can be retrieved and identified as the physicist's version of the chemist's "transferability of functional groups" concept. The accurate, analytical hardness matrix evaluation on the other hand provides further support for the basics of Nalewajski's charge sensitivity analysis. Based on Parr and Liu's functional expansion of the energy functional, a new energy decomposition is introduced with an order of magnitude analysis of the different terms for a series of simple molecules both at their equilibrium geometry and upon variation in bond length and dihedral angle. Finally, for the first time, the perturbation expansion of the energy functional is studied numerically up to second order now that all response functions and integration techniques are at hand. The perturbation expansion energies are in excellent agreement with those obtained directly from DFA calculations giving confidence in the convergence of the perturbation series and its use in judging the importance of the different terms in reactivity investigations.