Nonlocal Functionals Research Articles

Universality and symmetry breaking in conformally reduced quantum gravity

The scaling properties of quantum gravity are discussed by employing a class of proper-time regulators in the functional flow equation for the conformal factor within the formalism of the background field method. Renormalization group trajectories obtained by projecting the flow on a flat topology are more stable than those obtained from a projection on a spherical topology. In the latter case the ultraviolet flow can be characterized by a Hopf bifurcation with an ultraviolet attractive limiting cycle. Although the possibility of determining the infrared flow for an extended theory space can be severely hampered due to the conformal factor instability, we present a robust numerical approach to study the flow structure around the non-gaussian fixed point as an inverse-problem strategy. In particular it is shown the possibility of having a spontaneous breaking of the diffeomorphism invariance can be realized with non-local functionals of the volume operator.

Ab Initio Calculations of the Interaction between CO2 and the Acetate Ion

A series of ab initio calculations designed to investigate the interaction of CO(2) with acetate are presented. The lowest energy structure, AC-CO(2)-η(2), is predicted by CCSD(T)/aVTZ to be bound by -10.6 kcal/mol. Six of the bound complexes have binding energies on the order of -8 kcal/mol, but analysis shows that the η(1)-CT complex is fundamentally different from the others. The η(1)-CT complex is characterized by geometric distortion, large polarization and induction effects and charge transfer whereas the other five complexes have little geometric distortion and negligible charge transfer. The amount of charge that is transferred from the anion to the CO(2) in the η(1)-CT complex is estimated to be about half an electron by NPA, DMA, CHELPG, and Mulliken analyses, whereas the EDA-ALMO-CTA (B3LYP) approach predicts a charge transfer of 75 me(-). However, the transfer of this small amount of charge leads to an energy lowering of -56 kcal/mol, without which the complex would not be bound. The RI-MP2 geometries closely approximate those resulting from the CCSD optimizations, and the optimized second-order opposite spin (O2) method performs well for all the complexes except for the η(1)-CT complex. DFT methods do not reproduce all the ab initio geometries, binding energies and/or energy ordering of these complexes although the range-separated hybrid meta-GGA (M11) and nonlocal (VV10 and vdwDF10) functionals are shown to yield results significantly better than other functionals considered for this system. The fact that there is such variation among DFT methods has implications for DFT-based ab initio molecular dynamics simulations and for the parametrization of classical force fields based on DFT calculations.

Symmetric Nonlocal Weighted Density Approximations from the Exchange-Correlation Hole of the Uniform Electron Gas.

Nonlocal exchange-correlation energy functionals are constructed using the accurate model exchange-correlation hole for the uniform electron gas developed by Gori-Giorgi and Perdew. The exchange-correlation hole is constrained to be symmetric and normalized, so the resulting functionals can be viewed as symmetrized versions of the weighted density approximation; we call them two-point weighted density approximations. Even without optimization of parameters or functional forms, the exchange-correlation energies for small molecules are competitive with those of the best generalized gradient approximation functionals. Two-point weighted density approximations seem to be an interesting new direction for functional development. A more general version of the conditions that define the energy for fractional electron number and fractional spin are presented. These "generalized flat-planes" conditions are closely linked to the normalization of the spin-resolved exchange-correlation hole at noninteger electron number. This and many other properties of the exact exchange-correlation functional can be imposed straightforwardly and directly in symmetrized weighted density approximation.

Rationale for switching to nonlocal functionals in density functional theory

Density functional theory (DFT) has been steadily improving over the past few decades, becoming the standard tool for electronic structure calculations. The early local functionals (LDA) were eventually replaced by more accurate semilocal functionals (GGA) which are in use today. A major persisting drawback is the lack of the nonlocal correlation which is at the core of dispersive (van der Waals) forces, so that a large and important class of systems remains outside the scope of DFT. The vdW-DF correlation functional of Langreth and Lundqvist, published in 2004, was the first nonlocal functional which could be easily implemented. Beyond expectations, the nonlocal functional has brought significant improvement to systems that were believed not to be sensitive to nonlocal correlations. In this paper, we use the example of graphene nanodomes growing on the Ir(111) surface, where with an increase of the size of the graphene islands the character of the bonding changes from strong chemisorption towards almost pure physisorption. We demonstrate how the seamless character of the vdW-DF functionals makes it possible to treat all regimes self-consistently, proving to be a systematic and consistent improvement of DFT regardless of the nature of bonding. We also discuss the typical surface science example of CO adsorption on (111) surfaces of metals, which shows that the nonlocal correlation may also be crucial for strongly chemisorbed systems. We briefly discuss open questions, in particular the choice of the most appropriate exchange part of the functional. As the vdW-DF begins to appear implemented self-consistently in a number of popular DFT codes, with numerical costs close to the GGA calculations, we draw the attention of the DFT community to the advantages and benefits of the adoption of this new class of functionals.

Are we van der Waals ready?

We apply a range of density-functional-theory-based methods capable of describing van der Waals interactions with weakly bonded layered solids in order to investigate their accuracy for extended systems. The methods under investigation are the local-density approximation, semi-empirical force fields, non-local van der Waals density functionals and the random-phase approximation. We investigate the equilibrium geometries, elastic constants and binding energies of a large and diverse set of compounds and arrive at conclusions about the reliability of the different methods. The study also points to some directions of further development for the non-local van der Waals density functionals.

Ab initio and semi-empirical van der Waals study of graphene–boron nitride interaction from a molecular point of view

We have performed a systematic semi-empirical and ab initio van der Waals study to investigate the bonding mechanism of benzene (C6H6), triazine (C3N3H3) and borazine (B3N3H6) adsorbed on graphene and a single boron nitride (BN) sheet. The two semi-empirical approaches used to include the van der Waals (vdW) interactions in our density functional theory (DFT) calculations suggest that the strength of the molecule–surface interaction corresponds to a strong physisorption with no net charge transfer between the molecules and the corresponding substrates. This observation is strengthened by the use of first-principles non-local correlation vdW-DF functionals which provide a sound physical basis to include vdW interactions in DFT calculations. In particular we have employed two flavors of vdW-DF functionals which enabled us to determine the role of the non-local correlation effects in the molecule–surface bonding mechanism which cannot be assessed by using only semi-empirical vdW methods. Our study also reveals that the strength of the molecule–surface interaction can be influenced by the electronegativity of the B, C and N atoms.

A benchmark for non-covalent interactions in solids

A benchmark for non-covalent interactions in solids (C21) based on the experimental sublimation enthalpies and geometries of 21 molecular crystals is presented. Thermal and zero-point effects are carefully accounted for and reference lattice energies and thermal pressures are provided, which allow dispersion-corrected density functionals to be assessed in a straightforward way. Other thermal corrections to the sublimation enthalpy (the 2RT term) are reexamined. We compare the recently implemented exchange-hole dipole moment (XDM) model with other approaches in the literature to find that XDM roughly doubles the accuracy of DFT-D2 and non-local functionals in computed lattice energies (4.8 kJ/mol mean absolute error) while, at the same time, predicting cell geometries within less than 2% of the experimental result on average. The XDM model of dispersion interactions is confirmed as a very promising approach in solid-state applications.

Hydrogen migration on the C60 fullerene

We investigate the migration barriers, transition states, and optimum migration paths of hydrogen on the C60 fullerene using the ab initio density functional theory. Calculated energy barriers tend to be higher for non-local exchange-correlation functionals. We find that the migration between adsorption sites with the same energy has high migration barriers, and thus the initial stages of the H nucleation requires desorption–adsorption cycles. The migration barrier is reduced near another H adsorbate. The migration may be involved more in the later stages of the regio-selective hydrogenation of C60. We find that the migration barriers are reduced by the presence of hydrogen sources, and the hydrogenating agents in the environment are required in order to reach the regio-selective hydrogenation of the C60 fullerene.

Hyper-generalized-gradient functionals constructed from the Lieb-Oxford bound: Implementation via local hybrids and thermochemical assessment

In 2009 Odashima and Capelle (OC) showed a way to design a correlation-only density functional that satisfies a Lieb-Oxford bound on the correlation energy, without empirical parameters and even without additional theoretical parameters. However, they were only able to test a size-inconsistent version of it that employs total energies. Here, we show that their alternative size-consistent form that employs energy densities, when combined with exact or semilocal exchange, is a local hybrid (lh) functional. We test several variants of this nonempirical OC-lh functional on standard molecular test sets. Although no variant yields enthalpies of formation with the accuracy of the semilocal Tao-Perdew-Staroverov-Scuseria (TPSS) exchange-correlation, OC-lh correlation with exact exchange yields rather accurate energy barriers for chemical reactions. Our purpose here is not to advocate for a new density functional, but to explore a previously published idea. We also discuss the importance of near-self-consistency for fully nonlocal functionals.

Entanglement Rényi entropies in conformal field theories and holography

An entanglement Renyi entropy for a spatial partition of a system is studied in conformal theories which admit a dual description in terms of an anti-de Sitter gravity. The divergent part of the Renyi entropy is computed in 4D conformal N=4 super Yang-Mills theory at a weak coupling. This result is used to suggest a holographic formula which reproduces the Renyi entropy at least in the leading approximation. The holographic Renyi entropy is an invariant functional set on a codimension 2 minimal hypersurface in the bulk geometry. The bulk space does not depend on order $n$ of the Renyi entropy. The holographic Renyi entropy is a sum of local and non-local functionals multiplied by polynomials of $1/n$.

Addressing the Coulomb potential singularity in exchange-correlation energy integrals with one-electron and two-electron basis sets

Nonlocal functionals for the exchange-correlation energy like the weighted density approximation require evaluating six-dimensional integrals with a Coulomb singularity. The convergence of a straightforward grid-based approach is linear in the number of grid points, because grid points where the integrand’s magnitude exceeds a threshold must be neglected. This slow convergence makes extrapolation to the infinite-grid limit problematic. We introduce an alternative approach, based on basis-set expansion using either conventional three-dimensional basis functions or explicitly-correlated basis functions. The approach using explicitly-correlated gaussian geminal basis functions converges particularly rapidly.

Strategy for the optical property studies in ultraviolet nonlinear optical crystals from density functional theory

The predictive capabilities of the density functional theory with various exchange–correlation functionals on the energy band gaps and optical properties in ultraviolet nonlinear optical (UV NLO) crystals are systematically investigated. We find that the energy band gaps of these crystals can be accurately predicted by the non-local hybrid functionals and sX-LDA functionals, but their calculated optical properties are not consistent with the experimental results compared to the usually adopted LDA or GGA functionals with the scissors corrections. Accordingly, we propose that a precise and effective strategy to study the optical properties in the UV NLO crystals without any experimental parameter should be the combination of different functional calculations.

Performance of the M11-L density functional for bandgaps and lattice constants of unary and binary semiconductors

The recently developed SOGGA11 and M11-L density functionals have been tested for the prediction of bandgaps and lattice constants by comparing to databases containing 31 bandgaps and 34 lattice constants. To make a comparative assessment we also test several other density functionals against the same databases; in particular, we test the local spin density approximation, PBE, PBEsol, SOGGA, TPSS, revTPSS, and M06-L local density functionals and the HSE screened-exchange hybrid nonlocal density functional; and for a subset of 13 lattice constants we also compare the mean errors to those of the AM05 and WC local density functionals and the HISS and HSEsol nonlocal density functionals. The tests show that, of the ten functionals tested against all 65 data, the SOGGA, PBEsol, and HSE functionals are the most accurate for lattice constants, whereas the HSE, M11-L, and M06-L density functionals are the most accurate for bandgaps. However, the SOGGA11 density functional is the most accurate generalized gradient approximation for bandgaps.

Al enhances the H2storage capacity of graphene at nanoribbon borders but not at central sites: A study using nonlocal van der Waals density functionals

We performed ab initio density-functional-theory calculations to investigate the adsorption of molecular hydrogen on pristine and Al-doped hydrogen-passivated zigzag graphene nanoribbons (GNRs) using nonlocal van der Waals functionals that have recently been proposed for an accurate description of exchange and correlation effects in weakly bound systems. Our results, which are compared with those obtained using the standard local density and generalized gradient approximations, show that neither pristine GNRs nor substitutionally Al-doped GNRs with an Al atom occupying a central, lateral, or subedge site satisfy the binding-energy criterion specified by the U.S. Department of Energy for novel hydrogen-storage materials. However, the criterion is satisfied by a GNR doped with an Al atom at an edge site and also by zigzag GNRs with adsorbed Al at lateral hole sites.

Generalized nonlocal kinetic energy density functionals based on the von Weizsäcker functional

We generalize the ideas behind the procedure for the construction of kinetic energy density functionals with a nonlocal term based on the structure of the von Weizsäcker functional, and present several types of nonlocal terms. In all cases, the functionals are constructed such that they reproduce the linear response function of the homogeneous electron gas. These functionals are designed by rewriting the von Weizsäcker functional with the help of a parameter β that determines the power of the electron density in the expression, a strategy we have previously used in the generalization of Thomas-Fermi nonlocal functionals. Benchmark calculations in localized systems have been performed with these functionals to test both their relative errors and the quality of their local behavior. We have obtained competitive results when compared to semilocal and previous nonlocal functionals, the generalized nonlocal von Weizsäcker functionals giving very good results for the total kinetic energies and improving the local behavior of the kinetic energy density. In addition, all the functionals discussed in this paper, when using an adequate reference density, can be evaluated as a single integral in momentum space, resulting in a quasilinear scaling for the computational cost.

Performance of dispersion-corrected density functional theory for the interactions in ionic liquids

Potential energy curves for the dissociation of cation-anion associates representing the building units of ionic liquids have been computed with dispersion corrected DFT methods. Non-local van der Waals density functionals (DFT-NL) for the first time as well as an atom pair-wise correction method (DFT-D3) have been tested. Reference data have been computed at the extrapolated MP2/CBS and estimated CCSD(T)/CBS levels of theory. The investigated systems are combined from two cations (1-butyl-3-methylimidazolium and tributyl(methyl)posphonium) and three anions (chloride, dicyanamide, acetate). We find substantial stabilization from London dispersion energy near equilibrium of 5-7 kcal mol(-1) (about 5-6% of the interaction energy). Equilibrium distances are shortened by 0.03-0.09 Å and fundamental (inter-fragment) vibrational frequencies (which are in the range 140-180 cm(-1)) are increased by typically 10 cm(-1) when dispersion corrections are made. The dispersion-corrected hybrid functional potentials are in general in excellent agreement with the corresponding CCSD(T) reference data (typical deviations of about 1-2%). The DFT-D3 method performs unexpectedly well presumably because of cancellation of errors between the dispersion coefficients of the cations and anions. Due to self-interaction error, semi-local density functionals exhibit severe SCF convergence problems, and provide artificial charge-transfer and inaccurate interaction energies for larger inter-fragment distances. Although these problems may be alleviated in condensed phase simulations by effective Coulomb screening, only dispersion-corrected hybrid functionals with larger amounts of Fock-exchange can in general be recommended for such ionic systems.

Dispersion Interactions with Density-Functional Theory: Benchmarking Semiempirical and Interatomic Pairwise Corrected Density Functionals.

We present a comparative assessment of the accuracy of two different approaches for evaluating dispersion interactions: interatomic pairwise corrections and semiempirical meta-generalized-gradient-approximation (meta-GGA)-based functionals. This is achieved by employing conventional (semi)local and (screened-)hybrid functionals, as well as semiempirical hybrid and nonhybrid meta-GGA functionals of the M06 family, with and without interatomic pairwise Tkatchenko-Scheffler corrections. All of those are tested against the benchmark S22 set of weakly bound systems, a representative larger molecular complex (dimer of NiPc molecules), and a representative dispersively bound solid (hexagonal boron nitride). For the S22 database, we also compare our results with those obtained from the pairwise correction of Grimme (DFT-D3) and nonlocal Langreth-Lundqvist functionals (vdW-DF1 and vdW-DF2). We find that the semiempirical kinetic-energy-density dependence introduced in the M06 functionals mimics some of the nonlocal correlation needed to describe dispersion. However, long-range contributions are still missing. Pair-wise interatomic corrections, applied to conventional semilocal or hybrid functionals, or to M06 functionals, provide for a satisfactory level of accuracy irrespectively of the underlying functional. Specifically, screened-hybrid functionals such as the Heyd-Scuseria-Ernzerhof (HSE) approach reduce self-interaction errors in systems possessing both localized and delocalized orbitals and can be applied to both finite and extended systems. Therefore, they serve as a useful underlying functional for dispersion corrections.

Density functional calculation of many-electron systems in Cartesian coordinate grid

A recently developed density functional method, within Hohenberg-Kohn-Sham framework, is used for faithful description of atoms, molecules in Cartesian coordinate grid, by using an LCAO-MO ansatz. Classical Coulomb potential is obtained by means of a Fourier convolution technique. All two-body potentials (including exchange-correlation (XC)) are constructed directly on real grid, while their corresponding matrix elements are computed from numerical integration. Detailed systematic investigation is made for a representative set of atoms/molecules through a number of properties like total energies, component energies, ionization energies, orbital energies, etc. Two nonlocal XC functionals (FT97 and PBE) are considered for pseudopotential calculation of 35 species while preliminary all-electron results are reported for 6 atoms using the LDA XC density functional. Comparison with literature results, wherever possible, exhibits near-complete agreement. This offers a simple efficient route towards accurate reliable calculation of many-electron systems in the Cartesian grid. Future prospect of this method is also discussed.

Existence of a positive solution to systems of differential equations of fractional order

In this paper, we consider a system of (continuous) fractional boundary value problems given by { − D 0 + ν 1 y 1 ( t ) = λ 1 a 1 ( t ) f ( y 1 ( t ) , y 2 ( t ) ) , − D 0 + ν 2 y 2 ( t ) = λ 2 a 2 ( t ) g ( y 1 ( t ) , y 2 ( t ) ) , where ν 1 , ν 2 ∈ ( n − 1 , n ] for n > 3 and n ∈ N , subject either to the boundary conditions y 1 ( i ) ( 0 ) = 0 = y 2 ( i ) ( 0 ) , for 0 ≤ i ≤ n − 2 , and [ D 0 + α y 1 ( t ) ] t = 1 = 0 = [ D 0 + α y 2 ( t ) ] t = 1 , for 1 ≤ α ≤ n − 2 , or y 1 ( i ) ( 0 ) = 0 = y 2 ( i ) ( 0 ) , for 0 ≤ i ≤ n − 2 , and [ D 0 + α y 1 ( t ) ] t = 1 = ϕ 1 ( y ) , for 1 ≤ α ≤ n − 2 , and [ D 0 + α y 2 ( t ) ] t = 1 = ϕ 2 ( y ) , for 1 ≤ α ≤ n − 2 . In the latter case, the continuous functionals ϕ 1 , ϕ 2 : C ( [ 0 , 1 ] ) → R represent nonlocal boundary conditions. We provide conditions on the nonlinearities f and g , the nonlocal functionals ϕ 1 and ϕ 2 , and the eigenvalues λ 1 and λ 2 such that the system exhibits at least one positive solution. Our results here generalize some recent results on both scalar fractional boundary value problems and systems of fractional boundary value problems, and we provide two explicit numerical examples to illustrate the generalizations that our results afford.

A quantitative isoperimetric inequality for fractional perimeters

Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.