Correlation Energy Functionals Research Articles

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.

Non-local exchange correlation functionals impact on the structural, electronic and optical properties of III–V arsenides

Exchange correlation (XC) energy functionals play a vital role in the efficiency of density functional theory (DFT) calculations, more soundly in the calculation of fundamental electronic energy bandgap. In the present DFT study of III-arsenides, we investigate the implications of XC-energy functional and corresponding potential on the structural, electronic and optical properties of XAs (X = B, Al, Ga, In). Firstly we report and discuss the optimized structural lattice parameters and the band gap calculations performed within different non-local XC functionals as implemented in the DFT-packages: WIEN2k, CASTEP and SIESTA. These packages are representative of the available code in ab initio studies. We employed the LDA, GGA-PBE, GGA-WC and mBJ-LDA using WIEN2k. In CASTEP, we employed the hybrid functional, sX-LDA. Furthermore LDA, GGA-PBE and meta-GGA were employed using SIESTA code. Our results point to GGA-WC as a more appropriate approximation for the calculations of structural parameters. However our electronic bandstructure calculations at the level of mBJ-LDA potential show considerable improvements over the other XC functionals, even the sX-LDA hybrid functional. We report also the optical properties within mBJ potential, which show a nice agreement with the experimental measurements in addition to other theoretical results.

Van der Waals Coefficients for Nanostructures: Fullerenes Defy Conventional Wisdom

The van der Waals coefficients between quasispherical nanostructures can be modeled accurately and analytically by those of classical solid spheres (for nanoclusters) or spherical shells (for fullerenes) of uniform valence electron density, with the true static dipole polarizability. Here, we derive analytically and confirm numerically from this model the size dependencies of the van der Waals coefficients of all orders, showing, for example, that the asymptotic dependence for C(6) is the expected n(2) for pairs of nanoclusters A(n)-A(n), each containing n atoms, but n(2.75) for pairs of single-walled fullerenes C(n)-C(n). Large fullerenes are argued to have much larger polarizabilities and dispersion coefficients than those predicted by either the standard atom pair-potential model or widely used nonlocal van der Waals correlation energy functionals.

Effective homogeneity of the exchange–correlation and non-interacting kinetic energy functionals under density scaling

Correlated electron densities, experimental ionisation potentials, and experimental electron affinities are used to investigate the homogeneity of the exchange-correlation and non-interacting kinetic energy functionals of Kohn-Sham density functional theory under density scaling. Results are presented for atoms and small molecules, paying attention to the influence of the integer discontinuity and the choice of the electron affinity. For the exchange-correlation functional, effective homogeneities are highly system-dependent on either side of the integer discontinuity. By contrast, the average homogeneity-associated with the potential that averages over the discontinuity-is generally close to 4/3 when the discontinuity is computed using positive affinities for systems that do bind an excess electron and negative affinities for those that do not. The proximity to 4/3 becomes increasingly pronounced with increasing atomic number. Evaluating the discontinuity using a zero affinity in systems that do not bind an excess electron instead leads to effective homogeneities on the electron abundant side that are close to 4/3. For the non-interacting kinetic energy functional, the effective homogeneities are less system-dependent and the effect of the integer discontinuity is less pronounced. Average values are uniformly below 5/3. The study provides information that may aid the development of improved exchange-correlation and non-interacting kinetic energy functionals.

Robust Classification Method of Tumor Subtype by Using Correlation Filters

Tumor classification based on gene expression profiles, which is of great benefit to the accurate diagnosis and personalized treatment for different types of tumor, has drawn a great attention in recent years. This paper proposes a novel tumor classification method based on correlation filters to identify the overall pattern of tumor subtype hidden in differentially expressed genes. Concretely, two correlation filters, i.e., Minimum Average Correlation Energy (MACE) and Optimal Tradeoff Synthetic Discriminant Function (OTSDF), are introduced to determine whether a test sample matches the templates synthesized for each subclass. The experiments on six publicly available datasets indicate that the proposed method is robust to noise, and can more effectively avoid the effects of dimensionality curse. Compared with many model-based methods, the correlation filter based method can achieve better performance when balanced training sets are exploited to synthesize the templates. Particularly, the proposed method can detect the similarity of overall pattern while ignoring small mismatches between test sample and the synthesized template. And it performs well even if only few training samples are available. More importantly, the experimental results can be visually represented, which is helpful for the further analysis of results.

Dependence of Structural and Electronic Properties of Uranium Monochalcogenides on Exchange–Correlation Energy Functionals

We study the dependence of the structural properties of uranium monochalcogenides, U X where X = S, Se, and Te, as well as their electronic ones on the exchange–correlation energy functionals within the spin density functional theory, carrying out all electron calculations by the fully relativistic full-potential linear-combination-of-atomic-orbitals method. We employ two functionals of the local spin density approximation (LSDA) and two functionals of the generalized gradient approximations (GGA); the former two are the Perdew–Zunger and Perdew–Wang functionals and the latter two are the Perdew–Burke–Ernzerhof (PBE) and PBEsol functionals. We also examine the effects of the relativistic correction to the LSDA exchange part of each functional. We find that, for lattice constants, bulk moduli, and cohesive energies, the results of the calculations using the PBE functional are in the best agreement with the experimental results. On the contrary, we find that calculated total magnetic moments and one-electro...

Comparative study of structural and electronic properties of Cu-based multinary semiconductors

We present a systematic and comparative study of the structural and electronic properties of Cu-based ternary and quaternary semiconductors using first-principles electronic structure approaches. The important role that Cu $d$ electrons play in determining their properties is illustrated by comparing results calculated with different exchange--correlation energy functionals. We show that systematic improvement of the calculated anion displacement can be achieved by using the Heyd-Scuseria-Ernzerhof (HSE06) functional compared with the Perdew-Burke-Ernzerhof (PBE) functional. Quasiparticle band structures are then calculated within the ${G}^{0}$${W}^{0}$ approximation using the crystal structures optimized within the HSE06 functional and starting from the PBE+$U$ mean-field solution. Both the calculated quasiparticle band gaps and their systematic variation with chemical constituents agree well with experiments. We also predict that the quasiparticle band gap of the prototypical semiconductor Cu${}_{2}$ZnSnS${}_{4}$ in the kesterite phase is 1.65 eV and that the gap of the stannite phase is 1.40 eV. These results are consistent with available experimental values, which vary from 1.45 to 1.6 eV.

First-principles accurate total energy surfaces for polar structural distortions ofBaTiO3,PbTiO3, andSrTiO3: Consequences for structural transition temperatures

Specific forms of the exchange correlation energy functionals in first-principles density functional theory-based calculations, such as the local density approximation (LDA) and generalized-gradient approximations (GGA), give rise to structural lattice parameters with typical errors of -2% and 2%. Due to a strong coupling between structure and polarization, the order parameter of ferroelectric transitions, they result in large errors in estimation of temperature dependent ferroelectric structural transition properties. Here, we employ a recently developed GGA functional of Wu and Cohen [Phys. Rev. B 73, 235116 (2006)] and determine total-energy surfaces for zone-center distortions of BaTiO3, PbTiO3, and SrTiO3, and compare them with the ones obtained with calculations based on standard LDA and GGA. Confirming that the Wu and Cohen functional allows better estimation of structural properties at 0 K, we determine a new set of parameters defining the effective Hamiltonian for ferroelectric transition in BaTiO3. Using the new set of parameters, we perform molecular-dynamics (MD) simulations under effective pressures p=0.0 GPa, p=-2.0 GPa, and p=-0.005T GPa. The simulations under p=-0.005T GPa, which is for simulating thermal expansion, show a clear improvement in the cubic to tetragonal transition temperature and c/a parameter of its ferroelectric tetragonal phase, while the description of transitions at lower temperatures to orthorhombic and rhombohedral phases is marginally improved. Our findings augur well for use of Wu-Cohen functional in studies of ferroelectrics at nano-scale, particularly in the form of epitaxial films where the properties depend crucially on the lattice mismatch.

Localized versus extended systems in density functional theory: Some lessons from the Kohn-Sham exact exchange potential

A long-standing puzzle in density functional theory is the issue of the long-range behavior of the Kohn-Sham exchange-correlation potential at metal surfaces. As an important step toward its solution, it is proven here, through a rigorous asymptotic analysis and an accurate numerical solution of the optimized-effective-potential integral equation, that the Kohn-Sham exact exchange potential decays as $\text{ln}(z)/z$ far into the vacuum side of an extended semi-infinite jellium. In contrast with the situation in localized systems, such as atoms, molecules, and slabs, this dominant contribution does not arise from the so-called Slater potential. This exact exchange result provides a strong constraint on the suitability of approximate correlation-energy functionals.

Exchange and correlation energy functionals for two-dimensional open-shell systems

We consider density functionals for exchange and correlation energies in two-dimensional systems. The functionals are constructed by making use of exact constraints for the angular averages of the corresponding exchange and correlation holes, respectively, and assuming proportionality between their characteristic sizes. The electron current and spin are explicitly taken into account, so that the resulting functionals are suitable to deal with systems exhibiting orbital currents and/or spin polarization. Our numerical results show that in finite systems the proposed functionals outperform the standard two-dimensional local spin-density approximation, still performing well also in the important limit of the homogeneous two-dimensional electron gas.

Force calculation for orbital-dependent potentials with FP-(L)APW + lo basis sets

Within the linearized augmented plane-wave method for electronic structure calculations, a force expression was derived for such exchange–correlation energy functionals that lead to orbital-dependent potentials (e.g., LDA + U or hybrid methods). The forces were implemented into the WIEN2k code and were tested on systems containing strongly correlated d and f electrons. The results show that the expression leads to accurate atomic forces.

Random-phase-approximation-based correlation energy functionals: Benchmark results for atoms

The random phase approximation for the correlation energy functional of the density functional theory has recently attracted renewed interest. Formulated in terms of the Kohn-Sham orbitals and eigenvalues, it promises to resolve some of the fundamental limitations of the local density and generalized gradient approximations, as, for instance, their inability to account for dispersion forces. First results for atoms, however, indicate that the random phase approximation overestimates correlation effects as much as the orbital-dependent functional obtained by a second order perturbation expansion on the basis of the Kohn-Sham Hamiltonian. In this contribution, three simple extensions of the random phase approximation are examined; (a) its augmentation by a local density approximation for short-range correlation, (b) its combination with the second order exchange term, and (c) its combination with a partial resummation of the perturbation series including the second order exchange. It is found that the ground state and correlation energies as well as the ionization potentials resulting from the extensions (a) and (c) for closed subshell atoms are clearly superior to those obtained with the unmodified random phase approximation. Quite some effort is made to ensure highly converged data, so that the results may serve as benchmark data. The numerical techniques developed in this context, in particular, for the inherent frequency integration, should also be useful for applications of random phase approximation-type functionals to more complex systems.

Spin-dependent localized Hartree-Fock density-functional approach for the accurate treatment of inner-shell excitation of closed-shell atoms

We present a spin-dependent localized Hartree-Fock (SLHF) density-functional approach for the treatment of the inner-shell excited-state calculation of atomic systems. In this approach, the electron spin-orbitals in an electronic configuration are obtained first by solving Kohn-Sham (KS) equation with SLHF exchange potential. Then a single-Slater-determinant energy of the electronic configuration is calculated by using these electron spin-orbitals. Finally, a multiplet energy of an inner-shell excited state is evaluated from the single-Slater-determinant energies of the electronic configurations involved in terms of Slater's diagonal sum rule. This procedure has been used to calculate the total and excitation energies of inner-shell excited states of close-shell atomic systems: Be, B^+, Ne, and Mg. The correlation effect is taken into account by incorporating the correlation potentials and energy functionals of Perdew and Wang's (PW) or Lee, Yang, and Parr's (LYP) into calculation. The calculated results with the PW and LYP energy functionals are in overall good agreement with each other and also with available experimental and other ab initio theoretical data. In addition, we present some new results for highly excited inner-shell states.

Vorticity expansion approximation of the exchange-correlation energy functional in current density functional theory

We propose the vorticity expansion approximation (VEA) formulas of the exchange and correlation energy functionals of the current density functional theory (CDFT). They have a form of the second-order expansion with respect to the vorticity. Expansion coefficients are determined by requiring them to satisfy exact relations that have been derived from scaling properties of exchange and correlation energy functionals. Resultant VEA formulas satisfy a larger number of exact relations than those of the local-density approximation of the CDFT. Due to the well-behaved forms, the VEA formulas can be in quite good agreement with the exchange and correlation energies of the homogeneous electron liquid under a uniform magnetic field.

New practicable forms for exchange and correlation energy functionals of the CDFT

We propose the vorticity expansion approximation (VEA) formulas of both the exchange and correlation energy functionals of the current density functional theory (CDFT). Similarly to the strategy of the generalized gradient approximation of the density functional theory, the coefficients of expansions are determined by requiring the VEA formulas to satisfy exact relations that are derived from uniform and nonuniform coordinate scaling of electrons. As compared with formulas of the local density approximation of the CDFT, the resultant VEA formulas satisfy a large number of exact relations. Due to the well-behaved forms, the VEA formulas can be in quite good agreement with the exchange and correlation energies of the homogeneous electron liquid under a uniform magnetic field.

System-adapted correlation energy density functionals from effective pair interactions

This article discusses some ideas concerning an ‘average-pair-density functional theory’, in which the ground-state energy of a many-electron system is rewritten as a functional of the spherically and system-averaged pair density. These ideas are further clarified with simple physical examples. Then it is shown that the proposed formalism can be combined with density functional theory to build system-adapted correlation energy functionals. A simple approximation for the unknown effective electron–electron interaction that enters in this combined approach is described, and results for the He series and for the uniform electron gas are briefly reviewed.

Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence

The combination of density functional theory with other approaches to the many-electron problem through the separation of the electron-electron interaction into a short-range and a long-range contribution is a promising method, which is raising more and more interest in recent years. In this work some properties of the corresponding correlation energy functionals are derived by studying the electron-electron coalescence condition for a modified (long-range-only) interaction. A general relation for the on-top (zero electron-electron distance) pair density is derived, and its usefulness is discussed with some examples. For the special case of the uniform electron gas, a simple parameterization of the on-top pair density for a long-range only interaction is presented and supported by calculations within the ``extended Overhauser model''. The results of this work can be used to build self-interaction corrected short-range correlation energy functionals.

Phase diagram of harmonically confined one-dimensional fermions with attractive and repulsive interactions

We construct the complete $U\text{\ensuremath{-}}\ensuremath{\mu}$ phase diagram for harmonically confined ultracold fermionic atoms with repulsive and attractive interactions($\ensuremath{\mu}$ is the chemical potential and $U$ the interaction strength). Our approach is based on density-functional theory, and employs analytical expressions for the kinetic and correlation energy functionals, permitting us to obtain closed expressions for all phase boundaries and characteristic lines of the phase diagram, both for repulsive and attractive interactions.

Correlation energy functionals dependent on an effective number of electrons: Charged species and equilibrium geometries

Recently proposed spin-dependent and spin-independent correlation energy functionals [Perez-Jimenez et al., J. Chem. Phys. 116, 10571 (2002)] based on an effective number of electrons N are extended to deal with charged systems. By introducing the concept of an effective atomic number Z analogous to N, the spin-dependent functional in combination with Becke's exchange [Becke, Phys. Rev. A 38, 3098 (1988)] yields a mean absolute error (MAE) of 5.4 kcal/mol for the 88 ionization potentials and 58 electron affinities included in the extended G2 set, and a MAE of 4.1 kcal/mol for the 312 data comprising the above plus the 148 enthalpies of formation of the extended G2 set and the 18 total energies of the neutral atoms H through Ar. Geometry optimizations performed on the 53 molecules of the G2-1 test set with the above combination of exchange and correlation functionals yield MAEs of 0.017 A and 1.5 degrees for the 68 bond lengths and 29 angles analyzed as compared with the experimental estimates.

Open shells in reduced-density-matrix-functional theory

Reduced-density-matrix-functional theory is applied to open-shell systems. We introduce a spin-restricted formulation by appropriately expressing approximate correlation-energy functionals in terms of spin-dependent occupation numbers and spin-independent natural orbitals. We demonstrate that the additional constraint of total-spin conservation is indispensable for the proper treatment of open-shell systems. The formalism is applied to the first-row open-shell atoms. The obtained ground-state energies are in very good agreement with the exact values as well as other state of the art quantum chemistry calculations