Correlation Energy Density Research Articles

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.

Self-consistent implementation of locally scaled self-interaction-correction method.

Recently proposed local self-interaction correction (LSIC) method [Zope et al., J. Chem. Phys. 151, 214108 (2019)] is a 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. The LSIC method is exact for the one-electron densities, also recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to the well-known Perdew-Zunger SIC (PZSIC) method as a special case. This article presents the self-consistent implementation of the LSIC method using the ratio of Weizsäcker and Kohn-Sham kinetic energy densities as an iso-orbital indicator. The atomic forces as well as the forces on the Fermi-Löwdin orbitals are also implemented for the LSIC energy functional. Results show that LSIC with the simplest local spin density functional predicts atomization energies of the AE6 dataset better than some of the most widely used generalized-gradient-approximation (GGA) functional [e.g., Perdew-Burke-Ernzerhof (PBE)] and barrier heights of the BH6 database better than some of the most widely used hybrid functionals (e.g., PBE0 and B3LYP). The LSIC method [a mean absolute error (MAE) of 0.008 Å] predicts bond lengths of a small set of molecules better than the PZSIC-LSDA (MAE 0.042 Å) and LSDA (0.011 Å). This work shows that accurate results can be obtained from the simplest density functional by removing the self-interaction-errors using an appropriately designed SIC method.

Fast Estimation of Møller-Plesset Correlation Energies Based on Atomic Contributions.

Dynamic correlation plays an important role in the accurate calculation of chemical compounds such as the description of equilibrium structures in chemical systems. A model for the fast estimation of dynamic correlation energy is introduced in this work. This model is based on the idea of decomposition of the contribution of dynamic correlation energy calculated by nth order Møller-Plesset perturbation (MPn) theory with respect to atomic regions. Multiple levels of theory, including MP2, MP2.5, and MP4, are used as the reference, and the corresponding correlation energy densities are calculated. The proposed model is concise, fast, and promising for practical use, such as the prediction of reaction energies. It can also work as a baseline model or pretrained model for follow-up studies of machine learning.

Machine-learned electron correlation model based on frozen core approximation.

The machine-learned electron correlation (ML-EC) model is a regression model in the form of a density functional that reproduces the correlation energy density based on wavefunction theory. In a previous study [T. Nudejima et al., J. Chem. Phys. 151, 024104 (2019)], the ML-EC model was constructed using the correlation energy density from all-electron calculations with basis sets including core polarization functions. In this study, we applied the frozen core approximation (FCA) to the correlation energy density to reduce the computational cost of the response variable used in machine learning. The coupled cluster singles, doubles, and perturbative triples [CCSD(T)] correlation energy density obtained from a grid-based energy density analysis was analyzed within FCA and correlation-consistent basis sets without core polarization functions. The complete basis set (CBS) limit of the correlation energy density was obtained using the extrapolation and composite schemes. The CCSD(T)/CBS correlation energy densities based on these schemes showed reasonable behavior, indicating its appropriateness as a response variable. As expected, the computational time was significantly reduced, especially for systems containing elements with a large number of inner-shell electrons. Based on the density-to-density relationship, a large number of data (5 662 500 points), which were accumulated from 30 molecules, were sufficient to construct the ML-EC model. The valence-electron correlation energies and reaction energies calculated using the constructed model were in good agreement with the reference values, the latter of which were superior in accuracy to density functional calculations using 71 exchange-correlation functionals. The numerical results indicate that the FCA is useful for constructing a versatile model.

Erratum: Simple self-interaction correction to random-phase-approximation-like correlation energies [Phys. Rev. A 100, 022515 (2019)

The random phase approximation (RPA) is exact for the exchange energy of a many-electron ground state, but RPA makes the correlation energy too negative by about 0.5 eV/electron. That large short-range error, which tends to cancel out of iso-electronic energy differences, is largely corrected by an exchange-correlation kernel, or (as in RPA+) by an additive local or semilocal correction. RPA+ is by construction exact for the homogeneous electron gas, and it is also accurate for the jellium surface. RPA+ often gives realistic total energies for atoms or solids in which spin-polarization corrections are absent or small. RPA and RPA+ also yield realistic singlet binding energy curves for H2 and N2, and thus RPA+ yields correct total energies even for spin-unpolarized atoms with fractional spins and strong correlation, as in stretched H2 or N2. However, RPA and RPA+ can be very wrong for spin-polarized one-electron systems (especially for stretched H2+), and also for the spin-polarization energies of atoms. The spin-polarization energy is often a small part of the total energy of an atom, but important for ionization energies, electron affinities, and the atomization energies of molecules. Here we propose a computationally efficient generalized RPA+ (gRPA+) that changes RPA+ only for spin-polarized systems by making gRPA+ exact for all one-electron densities, in the same simple semilocal way that the correlation energy densities of many meta-generalized gradent approximations are made self-correlation free. By construction, gRPA+ does not degrade the exact RPA+ description of jellium. gRPA+ is found to greatly improve upon RPA and RPA+ for the ionization energies and electron affinities of light atoms. Many versions of RPA with an approximate exchange-correlation kernel fail to be exact for all one-electron densities, and they can also be self-interaction corrected in this way.

Long-range behavior of a nonlocal correlation-energy density functional based on the random-phase approximation

We propose a density-based nonlocal correlation energy functional, corresponding to a revised form of the seamless van der Waals (vdW) density functional developed by Dobson and coauthors in the late nineties. The functional, termed as rDW99 in this work, has the same energy expression as the random phase approximation (RPA), but its key ingredient---the noninteracting density response function---is expressed entirely in terms of the electron density and density gradient. This is enabled by generalizing the Lindhard function from the homogeneous electron gas (HEG) to inhomogeneous electron systems. Compared to the original DW99 functional of Dobson and coauthors, we have introduced two revisions in rDW99: (1) A gap correction is incorporated into the generalized Lindhard function, making the functional more suitable for describing inhomogeneous, and in particular insulating, systems; (2) an approximate, yet fairly accurate, analytical form of the Lindhard function in real space is developed which facilitates the numerical evaluation of the rDW99 functional. In contrast with previously developed vdW density functionals, the nonlocal polarization effect is naturally captured in the rDW99 functional. The functional form, as it stands right now, is computationally still rather involving in a straightforward implementation. However, we anticipate that numerical techniques can be developed to significantly reduce the computational cost. As a first step, we assess the quality of rDW99 for describing the vdW interactions in the nonoverlapping asymptotic regime by computing the ${C}_{6}$ coefficients for a set of 43 atoms and molecules. With only one adjustable parameter, the mean absolute percentage error (MAPE) of the ${C}_{6}$ coefficients is estimated to be $8.7%$. In this work, ${C}_{6}$ coefficients corresponding to the standard RPA method are also computed for the same test set, and the MAPE for the RPA ${C}_{6}$ coefficients is estimated to be $13.2%$.

Simple self-interaction correction to random-phase-approximation-like correlation energies

The random-phase approximation (RPA) is exact for the exchange energy of a many-electron ground state, but RPA makes the correlation energy too negative by about 0.5 eV/electron. That large short-range error, which tends to cancel out of isoelectronic energy differences, is largely corrected by an exchange-correlation kernel, or (as in RPA+) by an additive local or semilocal correction. RPA+ is by construction exact for the homogeneous electron gas, and it is also accurate for the jellium surface. RPA+ often gives realistic total energies for atoms or solids in which spin-polarization corrections are absent or small. RPA and RPA+ also yield realistic singlet binding energy curves for ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$, and thus RPA+ yields correct total energies even for spin-unpolarized atoms with fractional spins and strong correlation, as in stretched ${\mathrm{H}}_{2}$ or ${\mathrm{N}}_{2}$. However, RPA and RPA+ can be very wrong for spin-polarized one-electron systems (especially for stretched ${{\mathrm{H}}_{2}}^{+}$), and also for the spin-polarization energies of atoms. The spin-polarization energy is often a small part of the total energy of an atom, but important for ionization energies, electron affinities, and the atomization energies of molecules. Here we propose a computationally efficient generalized RPA+ (gRPA+) that changes RPA+ only for spin-polarized systems by making gRPA+ exact for all one-electron densities, in the same simple semilocal way that the correlation energy densities of many metageneralized gradient approximations are made self-correlation free. By construction, gRPA+ does not degrade the exact RPA+ description of jellium. gRPA+ is found to greatly improve upon RPA and RPA+ for the ionization energies and electron affinities of light atoms. Many versions of RPA with an approximate exchange-correlation kernel fail to be exact for all one-electron densities, and they can also be self-interaction corrected in this way.

Machine-learned electron correlation model based on correlation energy density at complete basis set limit.

We propose a machine-learned correlation model that is built using the regression between density variables such as electron density and correlation energy density. The correlation energy density of coupled cluster singles, doubles, and perturbative triples [CCSD(T)] is derived based on grid-based energy density analysis. The complete basis set (CBS) limit is estimated using the composite method, which has been reported to calculate the total correlation energy. The numerical examination revealed that the correlation energy density of the CCSD(T)/CBS level is appropriate for the response variable of machine learning. In addition to the density variables used in the exchange-correlation functionals of the density functional theory, the Hartree-Fock (HF) exchange energy density and electron density based on the fractional occupation number of molecular orbitals were employed as explanatory variables. Numerical assessments confirmed the accuracy and efficiency of the present correlation model. Consequently, the present protocol, namely, learning the CCSD(T)/CBS correlation energy density using density variables obtained by the HF calculation with a small basis set, yields an efficient correlation model.

Towards density functional approximations from coupled cluster correlation energy densities.

(Semi)local density functional approximations (DFAs) are the workhorse electronic structure methods in condensed matter theory and surface science. The correlation energy density ϵc(r) (a spatial function that yields the correlation energy Ec upon integration) is central to defining such DFAs. Unlike Ec, ϵc(r) is not uniquely defined, however. Indeed, there are infinitely many functions that integrate to the correct Ec for a given electron density ρ. The challenge for constructing useful DFAs is thus to find a suitable connection between ϵc(r) and ρ. Herein, we present a new such approach by deriving ϵc(r) directly from the coupled-cluster (CC) energy expression. The corresponding energy densities are analyzed for prototypical two-electron systems. As a proof-of-principle, we construct a semilocal functional to approximate the numerical CC correlation energy densities. Importantly, the energy densities are not simply used as reference data but guide the choice of the functional form, leading to a remarkably simple and accurate correlation functional for the helium isoelectronic series. While the resulting functional is not transferable to many-electron systems (due to a lack of same-spin correlation), these results underscore the potential of the presented approach.

Correlation energy functionals from adiabatic connection formalism

We propose a modified interaction strength interpolation method that is more robust in the quasi-two-dimensional density regime and more accurate than the original method [M. Seidl et al., Phys. Rev. Lett. 84, 5070 (2000)] for the correlation energy and adiabatic integrand of small atoms. This method also allows us to construct correlation energy densities as simple local interpolations along the adiabatic connection. As an example, we develop a generalized gradient approximation correlation functional that performs in line with, and often better than, popular semilocal correlation functionals.

Coulomb Energy Density Functionals for Nuclear Systems: Recent Studies of Coulomb Exchange and Correlation Functionals

The Coulomb exchange and correlation energy density functionals for electron systems are applied to nuclearsystems. It is found that the exchange functionals in the generalized gradient approximation provide agreements with the exact-Fock energy with one adjustable parameter within a few dozen keV accuracy, whereas the correlation functionals are not directly applicable to nuclear systems due to the existence of the nuclear force.

Embedded Cluster Density Approximation for Exchange-Correlation Energy: A Natural Extension of the Local Density Approximation.

We developed a local correlation method in the framework of Kohn-Sham density functional theory (KS-DFT). The method is termed ''embedded cluster density approximation'' (ECDA) and is a logical extension of the local density approximation. In ECDA, an embedded cluster is defined for each atom based on the finite-temperature density functional embedding theory. The clusters' exchange-correlation (XC) energy densities are calculated using high-level XC functionals. The system's XC energy is then constructed by patching these locally computed high-level XC energy densities over the system in an atom-by-atom manner. We derive the relationship between the embedding potential and the system's KS potential. We show how to efficiently compute the system's XC potential, which is the functional derivative of the patched XC energy with respect to the system's electron density. The accuracy of ECDA is examined by patching the exact exchange (EXX) and the random phase approximation (RPA) correlation energy densities in a one-dimensional hydrogen chain, as well as by patching EXX energy densities in several molecules. The agreement between ECDA and KS-DFT on total energies, dipole moments, XC potentials, and KS eigenvalues is good in general as the clusters are made larger. Based on these encouraging results, we expect ECDA to be a simple yet effective method to scale up high-level KS-DFT simulations in large systems.

Information theoretic approach provides a reliable description for kinetic component of correlation energy density functional

AbstractIn recent years, much efforts have been invested to apply information theory for different physical and chemical problems. In this regard, one can make use of one or many of the information theoretic quantities together within an approach so‐called information functional theory to describe the energetic components and electronic properties of various systems. In the present contribution, several information theoretic quantities such as Fisher information, Shannon entropy, Onicescu information energy, and Ghosh–Berkowitz–Parr entropy based on the two representations of electron density and shape function are utilized for the prediction of kinetic component of the correlation energy density functional as an important quantity in density functional theory. Taking the atoms and isoelectronic series as benchmark sets we find that with more or less different accountabilities of the considered quantities they can be introduced as reliable measures for the kinetic energy functional. Concerning different natures of the information theoretic quantities with variety of scaling properties and physiochemical propensities, it is shown that instead of describing all data using one of such quantities individually, considering all of them concurrently provides a better view on the prediction of the kinetic component of correlation energy density functional. Hopefully, the information theoretic approach can provide an alternative pathway toward the theoretical prediction and rationalization of important quantities in density functional calculations from the perspective of information theory.

Application of a Coulomb energy density functional for atomic nuclei: Case studies of local density approximation and generalized gradient approximation

We test the Coulomb exchange and correlation energy density functionals of electron systems for atomic nuclei in the local density approximation (LDA) and the generalized gradient approximation (GGA). For the exchange Coulomb energies, it is found that the deviation between the LDA and GGA ranges from around $ 11 \, \% $ in $ {}^{4} \mathrm{He} $ to around $ 2.2 \, \% $ in $ {}^{208} \mathrm{Pb} $, by taking the Perdew-Burke-Ernzerhof (PBE) functional as an example of the GGA\@. For the correlation Coulomb energies, it is shown that those functionals of electron systems are not suitable for atomic nuclei.

Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional.

We analyzed various possibilities to improve upon the SCAN meta-generalized gradient approximation density functional obeying all known properties of the exact functional that can be satisfied at this level of approximation. We examined the necessity of locally satisfying a strongly tightened lower bound for the exchange energy density in single-orbital regions, the nature of the error cancellation between the exchange and correlation parts in two-electron regions, and the effect of the fourth-order term in the gradient expansion of the correlation energy density. We have concluded that the functional can be modified to separately reproduce the exchange and correlation energies of the helium atom by locally releasing the strongly tightened lower bound for the exchange energy density in single-orbital regions, but this leads to an unbalanced improvement in the single-orbital electron densities. Therefore, we decided to keep the FX ≤ 1.174 exact condition for any single-orbital density, where FX is the exchange enhancement factor. However, we observed a general improvement in the single-orbital electron densities by revising the correlation functional form to follow the second-order gradient expansion in a wider range. Our new revSCAN functional provides more-accurate atomization energies for the systems with multireference character, compared to the SCAN functional. The nonlocal VV10 dispersion-corrected revSCAN functional yields more-accurate noncovalent interaction energies than the VV10-corrected SCAN functional. Furthermore, its global hybrid version with 25% of exact exchange, called revSCAN0, generally performs better than the similar SCAN0 for reaction barrier heights. Here, we also analyzed the possibility of the construction of a local hybrid from the SCAN exchange and a specific locally bounded nonconventional exact exchange energy density. We predict compatibility problems since this nonconventional exact exchange energy density does not really obey the strongly tightened lower bound for the exchange energy density in single-orbital regions.

A Simple Local Correlation Energy Functional for Spherically Confined Atoms from ab Initio Correlation Energy Density.

We propose a simple method of calculating the electron correlation energy density ec (r) and the correlation potential Vc (r) from second-order Møller-Plesset amplitudes and its generalization for the case of a configuration interaction wavefunction, based on Nesbet's theorem. The correlation energy density obtained by this method for free and spherically confined Be and He atoms was employed to fit a local analytical density functional based on Wigner's functional. The functional is capable of producing a strong increase in the correlation energy with decreasing confined radius for the Be atom.

The coupling constant averaged exchange–correlation energy density

ABSTRACTThe exchange–correlation energy, central to density-functional theory, may be represented in terms of the coupling constant averaged (CCA) exchange–correlation energy density. We present an approach to calculate the CCA energy density using accurate ab initio methods and its application to simple atomic systems. This function provides a link between intrinsically non-local, many-body electronic structure methods and simple local and semi-local density-functional approximations (DFAs). The CCA energy density is resolved into separate exchange and correlation terms and the features of each compared with those of quantities commonly used to construct DFAs. In particular, the more complex structure of the correlation energy density is found to exhibit features that align well with those present in the Laplacian of the density, suggesting its role as a key variable to be used in the construction of improved semi-local correlation functionals. The accurate results presented in this work are also compared with those provided by the Laplacian-dependent Becke–Roussel model for the exchange energy.

Global hybrids from the semiclassical atom theory satisfying the local density linear response.

We propose global hybrid approximations of the exchange-correlation (XC) energy functional which reproduce well the modified fourth-order gradient expansion of the exchange energy in the semiclassical limit of many-electron neutral atoms and recover the full local density approximation (LDA) linear response. These XC functionals represent the hybrid versions of the APBE functional [Phys. Rev. Lett. 2011, 106, 186406] yet employing an additional correlation functional which uses the localization concept of the correlation energy density to improve the compatibility with the Hartree-Fock exchange as well as the coupling-constant-resolved XC potential energy. Broad energetic and structural testing, including thermochemistry and geometry, transition metal complexes, noncovalent interactions, gold clusters and small gold-molecule interfaces, as well as an analysis of the hybrid parameters, show that our construction is quite robust. In particular, our testing shows that the resulting hybrid, including 20% of Hartree-Fock exchange and named hAPBE, performs remarkably well for a broad palette of systems and properties, being generally better than popular hybrids (PBE0 and B3LYP). Semiempirical dispersion corrections are also provided.

Polar and nonpolar structures of BiCrO3 from first-principles calculations

By first-principles calculations, we find that BiCrO3 can stabilize in three structures when relaxing from different experimental crystal lattice. The experimental undetermined structure is identified to be noncentrosymmetric R3c which permits the spontaneous ferroelectric polarization. The atomic positions within the unit cell were calculated using the GGA, GGA+U, LDA and LDA+U exchange–correlation energy density functionals, then the lattice constants along with the atomic positions were relaxed to find the ground state structure. All the calculations indicate that the Pnma structure is lower in energy than R3c and C2/c.

Generalized Gradient Approximation Correlation Energy Functionals Based on the Uniform Electron Gas with Gap Model

We studied uniform electron gas with a gap model in the context of density functional theory. On the basis of this analysis, we constructed two local gap models that are used in generalized gradient approximation (GGA) correlation functionals that satisfy numerous exact constraints for correlation energy. The first one, named GAPc, fulfills the full second-order correlation gradient expansion at any density regime and is very accurate for jellium surfaces, comparable to state-of-the-art GGAs for atomic systems and molecular systems, and is well compatible with known semilocal exchanges. The second functional, named GAPloc, satisfies the same exact conditions, except that the second-order gradient expansion is sacrificed for a better behavior under the Thomas-Fermi scaling and a more realistic correlation energy density of the helium atom. The GAPloc functional displays a high accuracy for atomic correlation energies, still preserving a reasonable behavior for jellium surfaces. Moreover, it shows a higher compatibility with the Hartree-Fock exchange than other semilocal correlation functionals. This feature is explained in terms of the real-space analysis of the GAPloc correlation energy.