Density Functional Research Articles

Neural density functionals: Local learning and pair-correlation matching.

Recently, Dijkman etal. [arXiv:2403.15007] proposed training classical neural density functionals via bulk pair-correlation matching. We show their method to be an efficient regularizer for neural functionals based on local learning of inhomogeneous one-body direct correlations [Sammüller etal., Proc. Natl. Acad. Sci. USA 120, e2312484120 (2023)0027-842410.1073/pnas.2312484120]. While Dijkman etal. demonstrated pair-correlation matching of a global neural free-energy functional, we argue in favor of local one-body learning for flexible neural modeling of the full Mermin-Evans density-functional map. Using spatial localization gives access to accurate neural free-energy functionals, including convolutional neural networks, that transcend the training box.

Deciphering TADF Mechanisms in Metal-Free Organic Emitters BrA-HBI: Utilizing Optimally Tuned Range-Separated Functionals for Insight

Abstract Metal-free organic emitters, characterized by their thermally activated delayed fluorescence (TADF) properties, offer considerable promise for the creation of highly efficient organic light-emitting diodes (OLEDs). Recently, Shao et al. presented a novel excited state intramolecular proton transfer (ESIPT) system BrA-HBI, demonstrating an emission quantum yield of up to 50% [Adv. Funct. Mater. 32, 2201256 (2022)]. However, many open issues cannot be answered solely by experimental means only and require detailed theoretical investigations. For instance, what causes the activation of TADF from the Keto* tautomer and leads to fluorescence quenching in the Enol* form? Herein, we provide a theoretical investigation on the TADF mechanism of the BrA-HBI molecule by optimally tuned range-separated functionals. Our findings reveal that ESIPT occurs in the BrA-HBI molecule. Moreover, we have disclosed the reason for the fluorescence quenching of the Enol* form and determined that the T 2 state plays a dominant role in the TADF phenomenon. In addition, double hybrid density functionals method was utilized to verify the reliability of optimally tuned range separation functionals on the calculation of the TADF mechanism in BrA-HBI. These findings not only provide a theoretical reference for development of highly efficient organic light-emitting diodes, but also demonstrate the effectiveness of the optimally tuned range-separated functionals in predicting the luminescence properties of TADF molecules.

Range-separated hybrid van der Waals density functionals to describe Cu[formula omitted]O[formula omitted]-complexes

Dimeric copper centers are commonly used to activate molecular oxygen in enzymatic and heterogeneous catalysis where Cu2 is embedded in porous structures. Here we evaluate the ability of recently developed range-separated hybrid van der Waals density functionals to describe activated O2 in a crystal with the Cu2O2 unit and in a ▪ complex embedded in a chabzite zeolite. The electronic and geometrical structure of the Cu2O2 unit depends sensitively on the exchange–correlation functional and a sufficiently large amount of short-ranged Fock-exchange is needed to describe the experimentally observed Cu2O2 structures.

Application of the Adiabatic Connection Random Phase Approximation to Electron-Nucleus Hyperfine Coupling Constants.

The electron-nucleus hyperfine coupling constant is a challenging property for density functional methods. For accurate results, hybrid functionals with a large amount of exact exchange are often needed and there is no clear "one-for-all" functional which describes the hyperfine coupling interaction for a large set of nuclei. To alleviate this unfavorable situation, we apply the adiabatic connection random phase approximation (RPA) in its post-Kohn-Sham fashion to this property as a first test. For simplicity, only the Fermi-contact and spin-dipole terms are calculated within the nonrelativistic and the scalar-relativistic exact two-component framework. This requires to solve a single coupled-perturbed Kohn-Sham equation to evaluate the relaxed density matrix, which comes with a modest increase in computational demands. RPA performs remarkably well and substantially improves upon its Kohn-Sham (KS) starting point while also reducing the dependence on the KS reference. For main-group systems, RPA outperforms global, range-separated, and local hybrid functionals─at similar computational costs. For transition-metal compounds and lanthanide complexes, a similar performance as for hybrid functionals is observed. In contrast, related post-Hartree-Fock methods such as Møller-Plesset perturbation theory or CC2 perform worse than semilocal density functionals.

Efficient real space formalism for hybrid density functionals.

We present an efficient real space formalism for hybrid exchange-correlation functionals in generalized Kohn-Sham density functional theory (DFT). In particular, we develop an efficient representation for any function of the real space finite-difference Laplacian matrix by leveraging its Kronecker product structure, thereby enabling the time to solution of associated linear systems to be highly competitive with the fast Fourier transform scheme while not imposing any restrictions on the boundary conditions. We implement this formalism for both the unscreened and range-separated variants of hybrid functionals. We verify its accuracy and efficiency through comparisons with established planewave codes for isolated as well as bulk systems. In particular, we demonstrate up to an order-of-magnitude speedup in time to solution for the real space method. We also apply the framework to study the structure of liquid water using abinitio molecular dynamics, where we find good agreement with the literature. Overall, the current formalism provides an avenue for efficient real-space DFT calculations with hybrid density functionals.

Theoretical modeling of electronic absorption spectra of ionized species of β-diketones.

Direct DFT evaluation of negatively charged or highly π-delocalized systems is intrinsically problematic. Single ionized (anionic) forms of β-diketones combine both these issues. In this article, we have summarized and analyzed the experimental and theoretical spectral dataset for anionic forms of different substituted perfluorinated β-diketones that were obtained by our team during the past few years. Using previously collected experimental data, spectra of anion of diketones containing phenol, 2-furoyl, 2-thiophen, 2-selenophen, 2-tellorphe, 2-pyridin, 2-naphthyl, and N-methyl-pyrrole substituted rings were considered and chosen for the simulation. The X3LYP density functional demonstrates good results in both ultraviolet and visible parts of the diketones spectra. Minnesota family show predictable ability within the reasonable errors. The linear correlation between the Hartree-Fock exchange and the errors of estimation has been observed. Density functionals with a low contribution of HF weight (0-30%) provide better prediction accuracy. Quantum chemistry calculations were performed under eighteen density functionals (Slater, Becke, OPTX, LYP, PW91C, BPE0, B3LYP, CAM-B3LYP, X3LYP, TPSS, revTPSS, TPSSh, M06-L, M06, M06-2X, M06-HF, M11-L, MN15-L), paired with the SMD solvation model implemented in GAMESS US program package. Def2-SVP basis set functions were applied to light atoms, and CRENBL effective core potential was used to Tellurium.

Training Machine-Learned Density Functionals on Band Gaps.

The systematic underestimation of band gaps is one of the most fundamental challenges in semilocal density functional theory (DFT). In addition to hindering the application of DFT to predicting electronic properties, the band gap problem is intimately related to self-interaction and delocalization errors, which make the study of charge transfer mechanisms with DFT difficult. To expand the range of available tools for addressing the band gap problem, we design an approach for machine learning density functionals based on Gaussian processes to explicitly fit single-particle energy levels. We also introduce nonlocal features of the density matrix that are expressive enough to fit these single-particle levels. Combining these developments, we train a machine-learned functional for the exact exchange energy that predicts molecular energy gaps and reaction energies of a wide range of molecules in excellent agreement with reference hybrid DFT calculations. In addition, while being trained solely on molecular data, our model predicts reasonable formation energies of polarons in solids, showcasing its transferability and robustness. We discuss how this approach can be generalized to full exchange-correlation functionals, thus paving the way to the design of state-of-the-art functionals for the prediction of electronic properties of molecules and materials.

Absorption Intensities of Organic Molecules from Electronic Structure Calculations versus Experiments: the Effect of Solvation, Method, Basis Set, and Transition Moment Gauge.

Recently, we derived experimental oscillator strengths (OSs) from well-defined UV-visible absorption spectral peaks of 100 molecules in solution. Here, we focus on a subset of transitions with the highest reliability to further benchmark the OSs from several wave function methods and density functionals. We consider multiple basis sets, transition moment gauges (length, velocity, and mixed), and solvent corrections. Most transitions in the comparison set come from conjugated molecules and have π → π* character. We use an automated algorithm to assign computed transitions to experimental bands. OSs computed using the Tamm-Dancoff approximation (TDA), CIS, or EOM-CCSD exhibited a strong gauge dependence, which is diminished in linear response theories (TD-DFT, TD-HF, and to a smaller degree LR-CCSD). OSs calculated from TD-DFT with PCM solvent models are systematically larger than apparent OSs derived from experimental spectra. For example, fcomp from hybrid functionals and PCM have mean absolute errors that are ∼10% of n·fexp, where n is a solvent refractive index factor that arises from the energy flux of the radiation field in a dielectric (solvent). Theoretical cavity field corrections considering spherical cavities do not improve the agreement between computed and experimental data. Corrections that account for the molecular shape and the direction of transition dipole moments, or that explicitly account for the effect of solvent molecules on the local field, should be more appropriate.

Many-Body Expansion-Based Quantum Mechanical Force Field for Cyclotrimethylene Trinitramine under High Pressure.

RDX undergoes pressures of approximately 30-50 GPa during detonation, leading to significant changes in intermolecular interactions. Accurately describing these interactions is crucial for understanding the energy transfer in the detonation process. To address this, this work introduces a many-body expansion-based quantum mechanical force field (MB-QMFF) to accurately describe RDX's intermolecular interactions under high pressures. Using MB-QMFF, we evaluated various density functionals and found that the M062X functional with GD3 dispersion correction provided the highest accuracy. Regarding intermolecular forces, two-body interactions were the most significant, with three-body interactions being negligible. Additionally, we investigated intermolecular energy variations at different densities (or pressures). The results clearly demonstrate an accurate description of intermolecular interactions by the MB-QMFF scheme. Therefore, we believe that the MB-QMFF scheme can serve as a foundation for the development of RDX-specific force fields and pave the way for future studies on the detonation process of RDX.

Exchange-correlation kernel for perturbation dependent auxiliary functions in auxiliary density perturbation theory

ContextAnalytic exchange-correlation kernel formulations are of the outermost importance for density functional theory (DFT) perturbation calculations. In this paper, the working equation for the exchange-correlation kernel of the generalized gradient approximation (GGA) for perturbation dependent auxiliary functions is derived and discussed in the framework of auxiliary density functional theory (ADFT). The presented new formulation is extended to the unrestricted approach, too. A comprehensive discussion of the implementation of the GGA ADFT kernel, using either the native exchange-correlation functional implementations in deMon2k or the ones from the LibXC library, is given. Calculations with analytic exchange-correlation kernels are compared to their finite difference counterparts. The obtained results are in quantitative agreement. Nevertheless, analytic GGA ADFT kernel implementations show substantial improvement in the computational performance. Similar results are reported for analytic second derivatives of effective core potential (ECP) and model core potential (MCP) matrix elements when compared to their finite difference counterparts in molecular frequency analyses.MethodAll calculations are performed in the framework of ADFT as implemented in deMon2k. In the ADFT analytic frequency calculations, auxiliary density perturbation theory was used. The underlying two-center exchange-correlation kernel matrix elements are calculated by numerical integration either with analytic or finite difference kernel expressions. Validation calculations are performed with the VWN and PBE functionals employing DFT-optimized DZVP basis sets in conjunction with automatically generated GEN-A2 auxiliary density function sets. In the (Pt3Cu)n cluster benchmark calculations, the RPBE functional was used. For Pt atoms, the quasi-relativistic LANL2DZ effective core potential with the corresponding valence basis set was employed, whereas for Cu atoms, the all-electron DFT-optimized TZVP basis was applied. The auxiliary density was expanded by the automatically generated GEN-A2* auxiliary function set. We run all benchmark calculations in parallel on 24 cores.

Ab-initio investigation of stability and electronic properties of new yttrium-based solid solution of double transition metal YMXTx (M= Ti and zr; X= C and N; Tx= H, O, and F) MXenes

In this study, an extensive examination was carried out to explore the stability, and physical characteristics of a novel solid solution comprising two transition metals, namely YMXTx (M = Ti and Zr; X = C and N; Tx = H, O, and F) MXenes. The Quantum-Espresso computational package which is based on density functional theory (DFT) and pseudopotential techniques has been employed for performing calculations. The results obtained from the investigation of the structural, dynamic, and mechanical stability revealed that the YTiX and YZrX MXenes exhibited remarkable stability. These findings strongly suggest the feasibility and potential success of experimental synthesis of these materials. Furthermore, the result of cohesive energy indicated that Nitride-based MXenes (Y(Ti/Zr)N) are more stable than Carbide-based MXenes (Y(Ti/Zr)C). By performing comprehensive calculations of the total density of states and band structures, using various approximations such as LDA, GGA, and GGA + HSE06, the investigation revealed a metallic behavior in YMX MXenes. Additionally, the analysis of the TDOS revealed the presence of a pseudogap close to the Fermi level. This observation provides further evidence for the structural stability of the investigated mentioned MXenes. After undergoing saturation with different functional groups, it was observed that the YTiNH2 and YTiNF2 MXenes displayed semiconducting behavior, whereas YZrNO2 is classified as a non-conductive material, and the remaining monolayers exhibit metallic properties. Analyses of charge density and electron localization functions confirmed the metallic behavior of the studied MXenes. Modifying the electronic band gap of 2D MXenes, while also preserving structural stability, could lead to enhancing the application of these materials in optoelectronic devices.

Framework for Laplacian-Level Noninteracting Free-Energy Density Functionals.

A framework for orbital-free Laplacian-level meta-generalized-gradient approximation (meta-GGA) for the noninteracting free-energy-density functionals based upon analysis of the fourth-order gradient expansion is developed. The framework presented here provides a new tool for developing advanced orbital-free thermal functionals at the meta-GGA level of theory. A nonempirical meta-GGA functional, which in the slowly varying density limit correctly reduces to the finite-temperature fourth-order gradient expansion for the noninteracting free energy, is constructed. Application to warm dense helium demonstrates that the developed meta-GGA functional drastically increases the accuracy of orbital-free-density functional theory simulations at temperatures below 40 eV as compared to the lower Thomas-Fermi and GGA rungs.

Revisiting Artifacts of Kohn-Sham Density Functionals for Biosimulation.

We revisit the problem of unphysical charge density delocalization/fractionalization induced by the self-interaction error of common approximate Kohn-Sham (KS) density functional theory functionals on simulation of small to medium-sized proteins in a vacuum. Aside from producing unphysical electron densities and total energies, the vanishing of the HOMO-LUMO gap associated with the unphysical charge delocalization leads to an unphysical low-energy spectrum and catastrophic failure of most popular solvers for the KS self-consistent field (SCF) problem. We apply a robust quasi-Newton SCF solver [ Phys. Chem. Chem. Phys. 2024, 26, 6557] to obtain solutions for some of these difficult cases. The anatomy of the charge delocalization is revealed by the natural deformation orbitals obtained from the density matrix difference between the Hartree-Fock and KS solutions; the charge delocalization not only can occur between charged fragments (such as in zwitterionic polypeptides) but also involves neutral fragments. The vanishing-gap phenomenon and troublesome SCF convergence are both attributed to the unphysical KS Fock operator eigenspectra of molecular fragments (e.g., amino acids or their side chains). Analysis of amino acid pairs suggests that the unphysical charge delocalization can be partially ameliorated by the use of some range-separated hybrid functionals but not by semilocal or standard hybrid functionals. Last, we demonstrate that solutions without the unphysical charge delocalization can be located even for semilocal KS functionals highly prone to such defects, but such solutions have non-Aufbau character and are unstable with respect to mixing of the non-overlapping "frontier" orbitals. Caution should be exercised when unexpectedly small (or vanishing) HOMO-LUMO gaps and atypical SCF convergence patterns (e.g., oscillatory) are observed in KS DFT simulations in any context (bio or otherwise).

Density function theory investigation of bimetallic phthalocyanine as a potential sensor and scavenger for nitrogen-containing toxic gases

Urbanization and industrialization processes inevitably release a large amount of nitrogen-containing toxic gases (NTGs) such as NO, NO2, NH3 and HCN, posing a serious threat to the ecological environment. Therefore, it is necessary to develop new gas sensors to promptly detect the emission and leakage of these toxic gases. Herein, four bimetallic phthalocyanines (BMPc = FePdPc, FePtPc, MnPdPc and MnPtPc) were designed by introducing pairs of precious metal (Pd/Pt) and inexpensive metal (Fe/Mn) atoms into the phthalocyanine (Pc) monolayer. Subsequently, the adsorption behavior and sensing properties of these BMPc towards NTGs were investigated using density functional theory (DFT) method. The studied results show that the four BMPc exhibit excellent stability and display chemical adsorption for these target gases due to the synergistic effect of bimetallic active centers, with adsorption energy greater than −0.70 eV. The adsorption strength follows the order NO > NO2 > NH3 > HCN, which is further confirmed by electronic properties such as charge density difference and density of states. Additionally, the analyses of work function, bandgap, and sensing response demonstrate the superior sensitivity of BMPc towards NO, NO2, NH3, and HCN. Meanwhile, the MnPtPc monolayer presents suitable adsorption strength for HCN at 298 K, resulting in a short recovery time of only 0.68 s. When the temperature is increased to 348 K, FePdPc and MnPdPc exhibit high reversibility for HCN, while MnPtPc displays favorable performance for NH3. Therefore, MnPtPc is identified as a recyclable HCN gas-sensing material at room temperature, while FePdPc, FePtPc, and MnPdPc are regarded as reusable gas-sensing materials for HCN at high temperature, with their sensing properties remaining unaffected in a humid environment. By contrast, the prolonged recovery times of NOx (x=1, 2) adsorption systems make these four BMPc more suitable as efficient scavengers for NOx. This study provides a new strategy for designing and developing phthalocyanine-based gas sensors and scavengers for the NTGs and other harmful substances.

Mechanistic investigation of methanol-to-olefins conversion catalyzed by H-ZSM-5 zeolite: a DFT study.

The mechanisms for the formation of the first C - C bond and lower olefins on methanol to olefins (MTO) conversion on H-ZSM-5 had been focused in dispute. In this paper, density functional theory has been used to study the reaction mechanisms of methanol to olefins on ZSM-5. The configurations of reactants, intermediates, products and transition state of the numerous reactions involved in such a process have been optimized, as well as the elementary reactions related to these configurations were determined by the calculation of corresponding activation energy barriers and reaction heats. Here, two different kinds of the mechanisms were proposed for the formation of dimethylether (DME), one involving an associative interaction of two methanol molecules with the zeolite Brønsted acid sites and the other occurring via a surface methoxy species and a methanol molecule. A critical intermediate of the methoxy methyl cation was theoretically verified by the reaction of the methoxy species and dimethylether. Besides, it was found that the first intermediates containing a C - C bond were 1,2-dimethoxyethane and 2-methoxy-ethanolare, in which the former was formed from methoxy species with dimethyl ether and the latter was formed from methanol by onium ions((CH3)2O+CH2CH2OCH3), respectively. For the whole reaction mechanism, the results in this paper indicated that the ethene formation is more favorable than propylene formation due to the low activation energy barrier for ethene formation (123.49 vs. 162.09kJ.mol-1). From these calculations, it would be concluded that ethene is the first alkene product that induces the occurrence of the hydrocarbon pool mechanism. All the periodic density function theory (DFT) calculations were performed by the Vienna Ab Initio Simulation package (VASP). The interaction between nucleus and valence electron was described using the pseudopotentials found in the projector augmented wave (PAW) method. PBE-D3 was used in the whole DFT calculations and CI-NEB was used to locate transition state.

One dimensional lattice fluid mixture with nearest neighbour interactions

We present an exact derivation of the free energy functional of a fluid mixture of hard rods with arbitrary sizes on a one-dimensional lattice. Our approach is based on the Wertheim cluster theory which consists of mapping a system with finite range interactions to the system with pure hard-core interaction but with modified activities. We show that the free energy functional has the same form as the fundamental measure functional. The interactions part of the free energy has two contributions, one from the one-particle cavity restricted to the hard rod or hard-sphere diameter and a second from the two-particle cavity which includes the finite range of the interaction. In the limit of a one-component system, our results reduce to the one derived using the Markov chain approach. For vanishing interactions, the density functionals coincide exactly with the previously derived for the mixture of hard rods with pure hard-core interaction.

Small Representative Databases for Testing and Validating Density Functionals and Other Electronic Structure Methods.

Broad and diverse sets of accurate data provide useful metrics for assessing the performance of new theoretical methods. However, assessing methods against large databases can be an arduous task. Here, we present 17 representative energetic databases, defined as small databases whose errors and error spreads are representative of larger databases and which therefore can serve as efficient benchmarks for developing and testing electronic structure methods and density functionals. In 15 cases, the representative databases have 6 entries while being representative of larger databases with 14-107 entries, and in the other two cases, they have 14 entries while being representative of larger databases with 418-455 entries. The mean unsigned error (MUE) of 100 electronic structure methods on a given representative database is typically within about 8% of the MUE on its parent database, and the root-mean-square error (RMSE) is typically within about 11% of the RMSE on the parent database. Thus, the representative databases are quite successful in indicating accuracy while maintaining good diversity. The databases include both main-group and transition-metal compounds and reactions, and they include bond energies, reaction energies, barrier heights, noncovalent interactions, ionization potentials, and absolute energies.

Energetic Information from Information-Theoretic Approach in Density Functional Theory as Quantitative Measures of Physicochemical Properties.

The Hohenberg-Kohn theorem of density functional theory (DFT) stipulates that energy is a universal functional of electron density in the ground state, so energy can be thought of having encoded essential information for the density. Based on this, we recently proposed to quantify energetic information within the framework of information-theoretic approach (ITA) of DFT (J. Chem. Phys. 2022, 157, 101103). In this study, we systematically apply energetic information to a variety of chemical phenomena to validate the use of energetic information as quantitative measures of physicochemical properties. To that end, we employed six ITA quantities such as Shannon entropy and Fisher information for five energetic densities, yielding twenty-six viable energetic information quantities. Then, they are applied to correlate with physicochemical properties of molecular systems, including chemical bonding, conformational stability, intermolecular interactions, acidity, aromaticity, cooperativity, electrophilicity, nucleophilicity, and reactivity. Our results show that different quantities of energetic information often behave differently for different properties but a few of them, such as Shannon entropy of the total kinetic energy density and information gain of the Pauli energy density, stand out and strongly correlate with several properties across different categories of molecular systems. These results suggest that they can be employed as quantitative measures of physicochemical properties. This work not only enriches the body of our knowledge about the relationship between energy and information, but also provides scores of newly introduced explicit density functionals to quantify physicochemical properties, which can serve as robust features for building machine learning models in future studies.

Benchmarking Charge-Transfer Excited States in TADF Emitters: ΔDFT Outperforms TD-DFT for Emission Energies.

Charge-transfer (CT) excited states are crucial to organic light-emitting diodes (OLEDs), particularly to those based on thermally activated delayed fluorescence (TADF). However, accurately modeling CT states remains challenging, even with modern implementations of (time-dependent) density functional theory [(TD-)DFT], especially in a dielectric environment. To identify shortcomings and improve the methodology, we previously established the STGABS27 benchmark set with highly accurate experimental references for the adiabatic energy gap between the lowest singlet and triplet excited states (ΔEST). Here, we diversify this set to the STGABS27-EMS benchmark by including experimental emission energies (Eem) and use this new set to (re)-evaluate various DFT-based approaches. Surprisingly, these tests demonstrate that a state-specific (un)restricted open-shell Kohn-Sham (U/ROKS) DFT coupled with a polarizable continuum model for perturbative state-specific nonequilibrium solvation (ptSS-PCM) provides exceptional accuracy for predicting Eem over a wide range of density functionals. In contrast, the main workhorse of the field, Tamm-Dancoff-approximated TD-DFT (TDA-DFT) paired with the same ptSS-PCM, is distinctly less accurate and strongly functional-dependent. More importantly, while TDA-DFT requires the choice of two very different density functionals for good performance on either ΔEST or Eem, the time-independent U/ROKS/PCM approaches deliver excellent accuracy for both quantities with a wide variety of functionals.

Chemically Accurate Singlet-Triplet Gaps of Arylcarbenes from Local Hybrid Density Functionals.

Singlet-triplet (ST) gaps are key descriptors of carbenes, because their properties and reactivity are strongly spin-dependent. However, the theoretical prediction of ST gaps is challenging and generally thought to require elaborate correlated wave function methods or double-hybrid density functionals. By evaluating two recent test sets of arylcarbenes (AC12 and AC18), we show that local hybrid functionals based on the "common t" local mixing function (LMF) model achieve mean absolute errors below 1 kcal/mol at a computational cost only slightly higher than that of global hybrid functionals. An analysis of correlation contributions to the ST gaps suggests that the accuracy of the common t-LMF model is mainly due to an improved description of nondynamical correlation which, unlike exchange, is not additive in each spin-channel. Although spin-nonadditivity can be achieved using the local spin polarization alone, using the "common", i.e., spin-unresolved, iso-orbital indicator t for constructing the LMF is found to be critical for consistent accuracy in ST gaps of arylcarbenes. The results support the view of LHs as vehicles to improve the description of nondynamical correlation rather than sophisticated exchange mixing approaches.