Interaction-strength Interpolation Research Articles

Adiabatic connection interaction strength interpolation method made accurate for the uniform electron gas.

The adiabatic connection interaction strength interpolation (ISI)-like method provides a high-level expression for the correlation energy, being, in principle, exact not only in the weak-interaction limit, where it recovers the second-order Görling-Levy perturbation term, but also in the strong-interaction limit that is described by the strictly correlated electron approach. In this work, we construct a genISI functional made accurate for the uniform electron gas, a solid-state physics paradigm that is a very difficult test for ISI-like correlation functionals. We assess the genISI functional for various jellium spheres with the number of electrons Z ≤ 912 and for the non-relativistic noble atoms with Z ≤ 290. For the jellium clusters, the genISI is remarkably accurate, while for the noble atoms, it shows a good performance, similar to other ISI-like methods. Then, the genISI functional can open the path using the ISI-like method in solid-state calculations.

Modified Interaction-Strength Interpolation Method as an Important Step toward Self-Consistent Calculations

The modified point charge plus continuum (mPC) model [ConstantinL. A.; Phys. Rev. B2019, 99, 085117] solves the important failures of the original counterpart, namely, the divergences when the reduced gradient of the density is large, such as in the tail of the density and in quasi-dimensional density regimes. The mPC allows us to define a modified interaction-strength interpolation (mISI) method inheriting these good features, which are important steps toward the full self-consistent treatment. Here, we provide an assessment of mISI for molecular systems (i.e., considering thermochemistry properties, correlation energies, vertical ionization potentials, and several noncovalent interactions), harmonium atoms, and functional derivatives in the strong-interaction limit. For all our tests, mISI provides a systematic improvement over the original ISI method. Semilocal approximations of the second-order Görling–Levy (GL2) perturbation theory are also considered in the mISI method, showing considerable worsening of the results. Possible further development of mISI is briefly discussed.

Erratum: "Assessment of interaction-strength interpolation formulas for gold and silver clusters" [J. Chem. Phys. 148, 134106 (2018)

With this Erratum, we provide corrections to certain mistakes appearing in the original publication.1 Namely, in Eq. (12), the -dependent Hamiltonian for the Hartree-Fock adiabatic connection lacks the external potential term. The correct equation is [Formula Presented]. The typos listed in the following all appear in the appendix. The parameter c in Eq. (A6) is wrong by a factor of two; the correct expression reads as follows: [Formula Presented]. In Eq. (A13), we have incorrectly used the parameter “c”; “c” has to be replaced by as defined in Eq. (A12). Also, the expression lacks the exchange energy. Thus, the proper expression reads as follows: [Formula Presented]. Finally, in Eqs. (A14) and (A15), the fractional powers appearing in the first terms are powers of the density and not of the position vector. Hence, [Formula Presented]. These corrections do not alter any of the results or conclusions shown in the paper.

Assessment of interaction-strength interpolation formulas for gold and silver clusters.

The performance of functionals based on the idea of interpolating between the weak- and the strong-interaction limits the global adiabatic-connection integrand is carefully studied for the challenging case of noble-metal clusters. Different interpolation formulas are considered and various features of this approach are analyzed. It is found that these functionals, when used as a correlation correction to Hartree-Fock, are quite robust for the description of atomization energies, while performing less well for ionization potentials. Future directions that can be envisaged from this study and a previous one on main group chemistry are discussed.

Interaction-Strength Interpolation Method for Main-Group Chemistry: Benchmarking, Limitations, and Perspectives

We have tested the original interaction-strength-interpolation (ISI) exchange-correlation functional for main group chemistry. The ISI functional is based on an interpolation between the weak and strong coupling limits and includes exact-exchange as well as the Görling-Levy second-order energy. We have analyzed in detail the basis-set dependence of the ISI functional, its dependence on the ground-state orbitals, and the influence of the size-consistency problem. We show and explain some of the expected limitations of the ISI functional (i.e., for atomization energies) but also unexpected results, such as the good performance for the interaction energy of dispersion-bonded complexes when the ISI correlation is used as a correction to Hartree-Fock.

Accurate calculation and modeling of the adiabatic connection in density functional theory

Using a recently implemented technique for the calculation of the adiabatic connection (AC) of density functional theory (DFT) based on Lieb maximization with respect to the external potential, the AC is studied for atoms and molecules containing up to ten electrons: the helium isoelectronic series, the hydrogen molecule, the beryllium isoelectronic series, the neon atom, and the water molecule. The calculation of AC curves by Lieb maximization at various levels of electronic-structure theory is discussed. For each system, the AC curve is calculated using Hartree-Fock (HF) theory, second-order Moller-Plesset (MP2) theory, coupled-cluster singles-and-doubles (CCSD) theory, and coupled-cluster singles-doubles-perturbative-triples [CCSD(T)] theory, expanding the molecular orbitals and the effective external potential in large Gaussian basis sets. The HF AC curve includes a small correlation-energy contribution in the context of DFT, arising from orbital relaxation as the electron-electron interaction is switched on under the constraint that the wave function is always a single determinant. The MP2 and CCSD AC curves recover the bulk of the dynamical correlation energy and their shapes can be understood in terms of a simple energy model constructed from a consideration of the doubles-energy expression at different interaction strengths. Differentiation of this energy expression with respect to the interaction strength leads to a simple two-parameter doubles model (AC-D) for the AC integrand (and hence the correlation energy of DFT) as a function of the interaction strength. The structure of the triples-energy contribution is considered in a similar fashion, leading to a quadratic model for the triples correction to the AC curve (AC-T). From a consideration of the structure of a two-level configuration-interaction (CI) energy expression of the hydrogen molecule, a simple two-parameter CI model (AC-CI) is proposed to account for the effects of static correlation on the AC. When parametrized in terms of the same input data, the AC-CI model offers improved performance over the corresponding AC-D model, which is shown to be the lowest-order contribution to the AC-CI model. The utility of the accurately calculated AC curves for the analysis of standard density functionals is demonstrated for the BLYP exchange-correlation functional and the interaction-strength-interpolation (ISI) model AC integrand. From the results of this analysis, we investigate the performance of our proposed two-parameter AC-D and AC-CI models when a simple density functional for the AC at infinite interaction strength is employed in place of information at the fully interacting point. The resulting two-parameter correlation functionals offer a qualitatively correct behavior of the AC integrand with much improved accuracy over previous attempts. The AC integrands in the present work are recommended as a basis for further work, generating functionals that avoid spurious error cancellations between exchange and correlation energies and give good accuracy for the range of densities and types of correlation contained in the systems studied here.

Extension to Negative Values of the Coupling Constant of Adiabatic Connection for Interaction-Strength Interpolation

A new exchange and correlation functional, called ISIN here, of the fifth rung of Jacob’s ladder is presented. It is based on the explicit approximation of Wλ, the integrand in the adiabatic connection (AC) with λ representing the coupling constant. Besides utilizing the two leading terms of each asymptotic expansion of Wλ at λ = 0 and ∞, the ISIN extends the coupling constant λ to negative values (i.e., to attractive electron−electron interactions). For the simple system of two electrons on the surface of a sphere (2ESS), the correlation energies yielded by the ISIN are in excellent agreement with the exact values. However, the ISIN seriously fails to approximate Wλ when W0′, the slope at λ = 0, goes to −∞, which leads to much more negative correlation energies for real systems.

Adiabatic connection in density-functional theory: Two electrons on the surface of a sphere

Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.

Spin-resolved second-order correlation energy of the two-dimensional uniform electron gas

For the two-dimensional electron gas, the exact high-density limit of the correlation energy is evaluated here numerically for all values of the spin polarization. The result is spin-resolved into $\uparrow\uparrow$, $\uparrow\downarrow$, and $\downarrow\downarrow$ contributions and parametrized analytically. Interaction-strength interpolation yields a simple model (LSD) for the correlation energy at finite densities.

Quantum Monte Carlo investigations of density functional theory of the strongly inhomogeneous electron gas

We use a variational quantum Monte Carlo realization of the adiabatic connection technique to calculate the most relevant quantities in Hohenberg-Kohn-Sham density functional theory for several strongly inhomogeneous electron-gas systems. Results for the coupling-constant dependence of the exchange-correlation energy, the pair-correlation function, the exchange-correlation hole, and the exchange and correlation energy densities are presented. Comparisons are made with the interaction strength interpolation ~ISI! approximation, the local density approximation ~LDA!, the gradient expansion approximation ~GEA!, the generalized gradient approximation ~GGA!, and the weighted density approximation ~WDA!. The coupling-constant dependence of the exchange-correlation energy is accurately described by an ISI model that incorporates information on the strong-interaction limit. Unlike either the LDA or GEA, the WDA is successful in describing the nonlocal structure of the exchange-correlation hole. The LDA errors in the exchange-correlation energy density show a remarkable correlation with the Laplacian of the density. The GGA worsens the error in the integrated exchange-correlation energy as the inhomogeneity of the systems increases. This failure is shared by current meta-GGA functionals and is shown to be caused by the inability of these functionals to describe the LDA overestimation ~in absolute value! of the exchange energy density around density maxima. It is suggested that this effect could be taken into account by including Laplacian terms in semilocal density functionals.

Accurate adiabatic connection curve beyond the physical interaction strength

In order to better approximate and understand the exchange-correlation functional in density functional theory, the adiabatic connection curve is accurately calculated beyond the physical interaction strength using a simulated scaling method. This is done for Hooke’s atom, two interacting electrons in a harmonic well potential. Extrapolation of the accurate curve to the infinite-coupling limit agrees well with the strictly correlated electron hypothesis. The interaction strength interpolation is shown to be a good, but not perfect, fit to the adiabatic curve. Arguments about the locality of functionals and convexity of the adiabatic connection curve are examined in this regime.