Bond energy decomposition analysis for subsystem density functional theory

Accurately describing intermolecular interactions within the framework of Kohn-Sham density functional theory (KS-DFT) has resulted in numerous benchmark databases over the past two decades. By far, the largest efforts have been spent on closed-shell, neutral dimers for which today, the interaction energies and geometries can be accurately reproduced by various combinations of dispersion-corrected density functional approximations (DFAs). In sharp contrast, charged, open-shell dimers remain a challenge as illustrated by the analysis of the OREL26rad benchmark set consisting of pi-dimer radical cations. Aside from the methodological aspect, achieving a proper description of radical cationic complexes is appealing due to their role as models for charge carriers in organic semiconductors. In the interest of providing an assessment of more realistic dimer systems, we construct a dataset of large radical cationic dimers (CryOrel) and jointly train the 19 parameters of a dispersion corrected, range-separated hybrid density functional (wB97X-dDsC), with the objective of providing the maximum balance between the treatment of long-range London dispersion and reduction of the delocalization error. These conditions are essential to obtain accurate energy profiles and binding energies of charged, open-shell dimers. Comparisons with the performance of the parent wB97X functional series and state-of-the-art wavefunction based methods are provided.

Kohn-Sham (KS) density functional theory (DFT) is a very efficient method for calculating various properties of solids as, for instance, the total energy, the electron density, or the electronic band structure. The KS-DFT method leads to rather fast calculations; however, the accuracy depends crucially on the chosen approximation for the exchange and correlation (xc) functional Exc and/or potential vxc. Here, an overview of xc methods to calculate the electronic band structure is given, with the focus on the so-called semilocal methods that are the fastest in KS-DFT and allow to treat systems containing up to thousands of atoms. Among them, there is the modified Becke-Johnson potential that is widely used to calculate the fundamental bandgap of semiconductors and insulators. The accuracy for other properties like magnetic moment or electron density, that are also determined directly by vxc, is also discussed.

Frozen density embedding (FDE) has become a popular subsystem density functional theory (DFT) method for systems with weakly overlapping charge densities. The failure of this method for strongly interacting and covalent systems is due to the approximate kinetic energy density functional (KEDF), although the need for approximate KEDFs may be eliminated if each subsystem's Kohn-Sham (KS) orbitals are orthogonal to the other, termed external orthogonality (EO). We present an implementation of EO into the FDE framework within the Amsterdam density functional program package, using the level-shift projection operator method. We generalize this method to remove the need for orbital localization schemes and to include multiple subsystems, and we show that the exact KS-DFT energies and densities may be reproduced through iterative freeze-and-thaw cycles for a number of systems, including a charge delocalized benzene molecule starting from atomic subsystems. Finally, we examine the possibility of a truncated basis for systems with and without charge delocalization, and found that subsystems require a basis that allows them to correctly describe the supermolecular delocalized orbitals.

Coumarin derivatives (CoDe) exhibit significant biological activities, and the positions of different substituents as well as the introduction of methyl groups may influence their biological activity and efficacy. By investigating the relationship between substituent positions, methylation, and hydrogen bond properties, the specific mechanisms and rules governing hydrogen bonds in intermolecular interactions can be further elucidated. This provides a theoretical basis for the synthesis of novel coumarin derivatives with tailored hydrogen bond characteristics and biological activities, as well as a research foundation for the development of new drug molecules. Therefore, the hydrogen bonding interactions between 3-hydroxycoumarin (3-HyCo), 4-hydroxycoumarin (4-HyCo), 7-hydroxycoumarin (7-HyCo), and 7-hydroxy-4-methylcoumarin (4-Me-7-HyCo) with ethanol and water were studied using density functional theory (DFT) and time-dependent density functional theory (TD-DFT). Firstly, the molecular structure of the coumarin (Co) monomer and the charge properties and molecular electrostatic potential extreme value of its four derivatives were analyzed. The stable complex structures of the four derivatives with ethanol and water were obtained through optimization, and the presence of hydrogen bonds in these complexes was confirmed. Secondly, the hydrogen bond lengths, frontier molecular orbitals (FMOs), interaction energies, atoms in molecules (AIM) topological parameters, and hydrogen bond vibration spectra in both the ground and excited states were examined. It was found that among the derivatives with different hydroxyl positions, the hydrogen bond structure of the 4-hydroxycoumarin complex was the most stable, and the introduction of a methyl group further enhanced the stability of the hydrogen bond structure. Finally, the hydrogen bond vibration spectra were found to undergo a red shift in the excited state, rendering the hydrogen bond structure more stable than that in the ground state. All computational studies in this paper were performed using the Gaussian 16 software package (Revision B.01), while GaussView 6.0.16 was employed for the analysis and visualization of compounds. The DFT method was adopted to optimize all compounds, utilizing the B3LYP functional combined with the 6-311++G(d,p) basis set. Multiwfn 3.8 (dev) and VMD 1.9.3 software were used to analyze and create the molecular electrostatic potential extreme value diagrams and frontier molecular orbital (FMO) images. Basis set superposition error (BSSE) correction was introduced during the calculation of interaction energies. Topological analysis was conducted using Multiwfn 3.8 (dev) software, and the excited-state vibration spectra were obtained based on the TD-DFT method.

Ionic liquids (ILs) hold great promise in many fields including energy storage and generation, mechanical, pharmaceutical, synthetic and separation applications to name just a few. For any given application, the desired physical properties of the ideal IL may differ substantially from others and no widely applicable patterns or trends to facilitate intuitive design. The origins of physical properties lie in the characteristics of the intermolecular energetics, which consist of a complex interplay between electrostatic and dispersion forces. This thesis investigates and develops computational methodologies for calculating a reliable description of the intermolecular interactions for this challenging class of solvents and electrolytes of the future. The electrostatic approximation used in classical molecular dynamics (MD) where atomic partial charges are assigned was investigated in terms of methods based on density matrix partitioning, and the restrained electrostatic potentials (RESP) approaches. It was found that the “geodesic” atomic partial charge scheme, part of the RESP family, produced the most accurate charges. This was measured in terms of (a) charge convergence with increasing basis set size; (b) charge invariance with changes to the coordinate system; (c) insensitivity to minor structural changes on the resulting charges; (d) adequate capture of charge transfer effects; and (e) the preservation of symmetric of charges in symmetric molecules. Although charges can vary dramatically depending on the scheme used, the careful use of atomic partial charge schemes may still produce reliable forcefields, or at least serve as a rapid diagnostic tool to quantify electrostatic interactions and charge transfer. In moving towards unbiased a priori descriptions of IL intermolecular interactions, second-order Moller-Plesset perturbation theory (MP2) was used with the linear-scaling fragment molecular orbital (FMO) framework to assess the extent to which dispersion forces play a role in the intermolecular energetics. ILs of increasing size were examined such that the many-body effects may be captured. The dispersion energy contribution formed up to 20% of the total interaction energy. Furthermore, the interaction energy as produced by FMO was within 1 kJ mol−1 of the full-wavefunction MP2 interaction energy when three-body effects were included. As the dispersion interaction is purely a quantum mechanical phenomenon, correlated quantum mechanical methods, such as MP2 or coupled-cluster approaches, are required to provide an unbiased account of these effects. Ab initio methods such as MP2 and CCSD(T) scale formally as N⁵ an N⁷, respectively, with chemical system size. While the FMO approach provides a marked improvement in efficiency, the counterpoise (CP) approach to correcting the basis set superposition error (BSSE) is not amenable to fragmented approaches and requires each ion in the cluster to be calculated in the basis set of the entire system. In order to remove this bottleneck, the spin-component scaled second-order Moller Plesset perturbation theory (SCS-MP2) methodology was refined by fitting 174 non-CP corrected interaction energies at the MP2/cc-pVTZ level of theory to CP corrected CCSD(T)/CBS benchmark energies. This has resulted in an implicit BSSE correction that may be used within the highly efficient FMO framework, and is shown to yield results on par with or exceeding the accuracy of MP2/cc-pVQZ for clusters of two and four ion pairs (IPs). This new approach as been termed SCS-IL-MP2. An alternative dispersion corrected density functional theory (DFT) approach, DFT- D3, was assessed and refined in view of producing accurate interaction energies at the same CCSD(T)/CBS quality. The same test set of 174 ILs was used to fit the SCS-IL-MP2 approach was used to refit the DFT-D3 approach for both the Hartree-Fock (HF) wavefunction and the PBE and BLYP density functionals (DFs). In most cases, the selection of the DF and associated DFT-D version 3 (DFT-D3) parameters differed negligibly with all reaching within 1 to 2 kJ mol−1 per IP. HF-D3 parameters, on the other hand, showed a substantial improvement, particularly when used with the Becke-Johnson (BJ) damping function. Refitted HF-D3 and the BJ damping function was able to consistently provide interaction energy errors below 5 kJ mol−1 per IP. It would be worthwhile further investigating the application of the refined HF-D3 in its ability to produce reliable energies and geometries over a more diverse set of ILs. Both the SCS-IL-MP2 and new DFT-D3 approaches may be applied to the highly scal- able FMO framework. These form the core elements of the quantum chemistry toolbox for the study and understanding of the physicochemical properties of ILs that have so far been only superficially characterised by electronic structure theory. From this starting point, a rigorous and unbiased a priori understanding of the intermolecular interactions and resulting physicochemical properties may be predicted by means of efficient ab initio molecular dynamics (AIMD) techniques.

Simulating electron transfer at reactive solid-liquid interfaces under constant electrochemical potentials of the constituents (electrons, ions, solvent etc.) is crucial to understanding the formation, functional and failure of electrochemical devices. Albeit being sufficiently accurate in decreasing breaking and formation of chemical bond at solid surfaces, existing methods based on Kohn-Sham density functional theory (DFT) are unsatisfactory in system consistency, namely, simulating the open solid-liquid interface under grand-canonical conditions, as well as in scaling up the simulation due to its high computational cost. Herein, to improve the system consistency and computational efficiency, we develop density-potential functional theoretic (DPFT) schemes out of Kohn-Sham DFT, drawing upon ideas of orbital-free DFT, frozen density embedding theory, and tight-binding DFT. The DPFT transforms an all-atom, Kohn-Sham DFT description of the nonreactive electrolyte solution into a coarse-grained, field-based description, while retaining a Kohn-Sham DFT description for the reactive subsystem. In the absence of surface reactions, the solid electrode can be described using orbital-free DFT, reducing the computational cost further. On the conceptual level, the physical meaning of potential in DPFT is examined. Recent applications of DPFT to simulate EDLs with mesoscopic roughness and EDLs at supported nanoparticles will be introduced. Related papers Zhang, Y., Binninger, T., Huang, J., & Eikerling, M. H. (2025). Theory of Electro-Ionic Perturbations at Supported Electrocatalyst Nanoparticles. Physical Review Letters, 134(6), 066201.Zhang, M., Chen, Y., Eikerling, M., & Huang, J. (2025). Structured solvent on a split electron tail: A semiclassical theory of electrified metal-solution interfaces. Physical Review Applied, 23(2), 024009.Huang, J., Domínguez-Flores, F., & Melander, M. (2024). Variants of surface charges and capacitances in electrocatalysis: Insights from density-potential functional theory embedded with an implicit chemisorption model. PRX Energy, 3(4), 043008. Acknowledgements Our research is supported financially by the Initiative and Networking Fund of the Helmholtz Association (no. VH-NG-1709), and European Research Council (ERC) Starting Grant (MESO-CAT, Grant agreement No. 101163405).

Conspectus Ab initio modeling of matter has become a pillar of chemical research: with ever-increasing computational power, simulations can be used to accurately predict, for example, chemical reaction rates, electronic and mechanical properties of materials, and dynamical properties of liquids. Many competing quantum mechanical methods have been developed over the years that vary in computational cost, accuracy, and scalability: density functional theory (DFT), the workhorse of solid-state electronic structure calculations, features a good compromise between accuracy and speed. However, approximate exchange-correlation functionals limit DFT's ability to treat certain phenomena or states of matter, such as charge-transfer processes or strongly correlated materials. Furthermore, conventional DFT is purely a ground-state theory: electronic excitations are beyond its scope. Excitations in molecules are routinely calculated using time-dependent DFT linear response; however applications to condensed matter are still limited. By contrast, many-electron wavefunction methods aim for a very accurate treatment of electronic exchange and correlation. Unfortunately, the associated computational cost renders treatment of more than a handful of heavy atoms challenging. On the other side of the accuracy spectrum, parametrized approaches like tight-binding can treat millions of atoms. In view of the different (dis-)advantages of each method, the simulation of complex systems seems to force a compromise: one is limited to the most accurate method that can still handle the problem size. For many interesting problems, however, compromise proves insufficient. A possible solution is to break up the system into manageable subsystems that may be treated by different computational methods. The interaction between subsystems may be handled by an embedding formalism. In this Account, we review embedded correlated wavefunction (CW) approaches and some applications. We first discuss our density functional embedding theory, which is formally exact. We show how to determine the embedding potential, which replaces the interaction between subsystems, at the DFT level. CW calculations are performed using a fixed embedding potential, that is, a non-self-consistent embedding scheme. We demonstrate this embedding theory for two challenging electron transfer phenomena: (1) initial oxidation of an aluminum surface and (2) hot-electron-mediated dissociation of hydrogen molecules on a gold surface. In both cases, the interaction between gas molecules and metal surfaces were treated by sophisticated CW techniques, with the remainder of the extended metal surface being treated by DFT. Our embedding approach overcomes the limitations of conventional Kohn-Sham DFT in describing charge transfer, multiconfigurational character, and excited states. From these embedding simulations, we gained important insights into fundamental processes that are crucial aspects of fuel cell catalysis (i.e., O2 reduction at metal surfaces) and plasmon-mediated photocatalysis by metal nanoparticles. Moreover, our findings agree very well with experimental observations, while offering new views into the chemistry. We finally discuss our recently formulated potential-functional embedding theory that provides a seamless, first-principles way to include back-action onto the environment from the embedded region.

We explore a new formalism to study the nonlinear electronic density response based on Kohn–Sham density functional theory (KS-DFT) at partially and strongly quantum degenerate regimes. It is demonstrated that the KS-DFT calculations are able to accurately reproduce the available path integral Monte Carlo simulation results at temperatures relevant for warm dense matter research. The existing analytical results for the quadratic and cubic response functions are rigorously tested. It is demonstrated that the analytical results for the quadratic response function closely agree with the KS-DFT data. Furthermore, the performed analysis reveals that currently available analytical formulas for the cubic response function are not able to describe simulation results, neither qualitatively nor quantitatively, at small wavenumbers q < 2qF, with qF being the Fermi wavenumber. The results show that KS-DFT can be used to describe warm dense matter that is strongly perturbed by an external field with remarkable accuracy. Furthermore, it is demonstrated that KS-DFT constitutes a valuable tool to guide the development of the nonlinear response theory of correlated quantum electrons from ambient to extreme conditions. This opens up new avenues to study nonlinear effects in a gamut of different contexts at conditions that cannot be accessed with previously used path integral Monte Carlo methods.

Machine learning (ML) of kinetic energy functionals (KEFs), in particular kinetic energy density (KED) functionals, is a promising way to construct KEFs for orbital-free density functional theory (DFT). Neural networks and kernel methods including Gaussian process regression (GPR) have been used to learn Kohn-Sham (KS) KED from density-based descriptors derived from KS DFT calculations. The descriptors are typically expressed as functions of different powers and derivatives of the electron density. This can generate large and extremely unevenly distributed datasets, which complicates effective application of ML techniques. Very uneven data distributions require many training datapoints, can cause overfitting, and can ultimately lower the quality of an ML KED model. We show that one can produce more accurate ML models from fewer data by working with smoothed density-dependent variables and KED. Smoothing palliates the issue of very uneven data distributions and associated difficulties of sampling while retaining enough spatial structure necessary for working within the paradigm of KEDF. We use GPR as a function of smoothed terms of the fourth order gradient expansion and KS effective potential and obtain accurate and stable (with respect to different random choices of training points) kinetic energy models for Al, Mg, and Si simultaneously from as few as 2000 samples (about 0.3% of the total KS DFT data). In particular, accuracies on the order of 1% in a measure of the quality of energy-volume dependence B'=EV0-ΔV-2EV0+E(V0+ΔV)ΔV/V02 (where V0 is the equilibrium volume and ΔV is a deviation from it) are obtained simultaneously for all three materials.

Research Article| January 01, 2010 Density Functional Theory of Electronic Structure: A Short Course for Mineralogists and Geophysicists John P. Perdew; John P. Perdew Department of Physics and Engineering Physics Tulane University New Orleans, Louisiana 70118, U.S.A., perdew@tulane.edu Search for other works by this author on: GSW Google Scholar Adrienn Ruzsinszky Adrienn Ruzsinszky Department of Physics and Engineering Physics Tulane University New Orleans, Louisiana 70118, U.S.A., perdew@tulane.edu Search for other works by this author on: GSW Google Scholar Reviews in Mineralogy and Geochemistry (2010) 71 (1): 1–18. https://doi.org/10.2138/rmg.2010.71.1 Article history first online: 09 Mar 2017 Cite View This Citation Add to Citation Manager Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Search Site Citation John P. Perdew, Adrienn Ruzsinszky; Density Functional Theory of Electronic Structure: A Short Course for Mineralogists and Geophysicists. Reviews in Mineralogy and Geochemistry 2010;; 71 (1): 1–18. doi: https://doi.org/10.2138/rmg.2010.71.1 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentBy SocietyReviews in Mineralogy and Geochemistry Search Advanced Search Mineralogists and geophysicists need to understand and predict the properties of solids and liquids at normal and especially at high pressures and temperatures. For example, they need to know the equilibrium structure, equation of state, phase transitions, and vibrational properties of solids, and the interatomic or intermolecular interaction needed for a molecular dynamics study of liquids (Stixrude et al. 1994; Soederlind and Ross 2000; Karki et al. 2001; Alfè et al. 2002; Steinle-Neumann et al. 2004; Sha and Cohen 2006; Carrier et al. 2007). This information, in sufficient detail, is not always... You do not have access to this content, please speak to your institutional administrator if you feel you should have access.

Density functional theory has been employed to investigate the interactions between copper(I) bis(trifluoromethylsulfonyl)imide (Cu[Tf 2 N]) and C 3 H 8 / C 3 H 6 .The calculations were based on the model of Cu[Tf 2 N], and performed at X3LYP/6-31++G** level.The calculated interaction energies indicated that the adsorption ability of Cu[Tf 2 N] for C 3 H 8 /C 3 H 6 followed the order of C 3 H 8 < C 3 H 6 .The noncovalent interactions in real space, based on the electron density and its derivatives, reveal the nature of interactions between Cu[Tf 2 N] and C 3 H 8 /C 3 H 6 .The natural bond orbital (NBO) analysis of the second-order perturbation energy has been performed to investigate the interaction mechanism of C 3 H 8 /C 3 H 6 on ionic liquid Cu[Tf 2 N].

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.

The energy profile of the lowest energy path connecting the slipped parallel (C2h) and T-shaped (C2v) dimers of carbon dioxide was investigated using a reasonably large 6-311+G(2df) basis set with the MP2 level electron correlation correction and with the counterpoise correction. There was no inversion barrier and the T-shaped dimer was a transition state connecting the two slipped parallel dimers. The calculated interaction energies of the two dimers (−1.290 and −1.072 kcal/mol) were larger than those reported previously. The calculated energy difference of the two dimers (0.218 kcal/mol) was also larger than the previous values (less than 0.1 kcal/mol). The effects of basis set, electron correlation, and basis set superposition error were studied in detail. The second-order Mo/ller–Plesset (MP2) interaction energies of the two dimers were close to the coupled cluster calculations using single and double substitutions including noniteratively triple excitations [CCSD(T)] ones. The choice of the basis set greatly affected the calculated interaction energies. Small basis sets underestimated the attractive interaction energies. Diffuse polarization functions were essential to evaluate the attractive interaction. The calculated MP2 interaction energies of the two dimers with the counterpoise correction increased by the augmentation of the diffuse polarization functions to the 6-311G* basis set. These values were close to the estimated MP2 interaction energies at the complete basis set (−1.362 and −1.140 kcal/mol). The 6-311+G(2df) basis set was reasonably large to evaluate the attractive interactions. Although the basis set augmented with diffuse polarization functions had large basis set superposition error (BSSE), the calculated interaction energies with the counterpoise correction were close to those calculated with the nearly BSSE free cc-pV5Z and aug-cc-pVQZ basis sets and to the expected values at the complete basis set.

The equilibrium structures, vibrational properties, and interaction energies for four bimolecular complexes of HCN and HNC (HCN···HCN, HNC···HNC, HCN···HNC, and HNC···HCN) were studied with ab initio methods. Different electron correlation levels and basis sets up to CCSD(T)/6-311++G(2d,2p) were used in the geometry optimizations and interaction energy calculations, and the vibrational frequencies were calculated at the MP2 and MP3 levels of theory. To study the nature of the intermolecular interactions, an energy decomposition analysis was carried out. For the HNC···HCN complex, which possesses a hydrogen bridge between two carbons, the counterpoise-corrected interaction energy and its decomposition has been calculated as a function of the intermolecular distance. The effect of subunit deformation on the interaction energy has also been considered. The results indicate that the HNC dimer has considerably stronger intermolecular interaction than the HCN dimer, and the HNC···HCN complex is bonded as strongly as the HCN dimer. The comparison of the various properties of the complexes and the energy decomposition analysis in particular indicate that carbon can act as both a hydrogen acceptor and donor, and the resulting hydrogen bonds in these complexes are relatively strong.

Weak intermolecular interactions in phenylenediamine dimer (pdd) clusters are studied by dispersion-corrected density functional theory (DFT) calculations. Along with the optimization of geometric structures and the calculation of interaction energies, we employ molecular electrostatic potential (MEP) mapping, natural bond orbital (NBO) analysis and quantum theory of atoms in molecule (AIM) to analyze the origin and relative energetic contributions of the weak interactions in these pdd systems. It is revealed that the most stable o-phenylenediamine dimer (opdd) cluster is dominated by N-HN hydrogen bonds, the p-phenylenediamine dimer (ppdd) cluster is largely stabilized by N-Hπ and ππ stacking interactions, while the m-phenylenediamine dimer (mpdd) cluster is mainly held by a combination of n → π*, C-Hπ and C-HN interactions. Energy decomposition analysis (EDA) of the total interaction energies of these clusters further demonstrates that the weak intermolecular interactions are associated with electrostatic and dispersion contributions. Structural spectroscopic analysis is also addressed depicting the coexistence of multiple intermolecular interactions which give rise to the spectral variation in wavenumbers of the infrared and Raman activities. Insights into the weak interactions of pdds help us to understand the molecular mechanisms involved in biochemistry and self-assembly materials.