Exchange Energy Density Research Articles

A simple nonlocal model for exchange

This work presents a new nonlocal model for the exchange energy density. The model is obtained from the product of the Kohn-Sham one-particle density matrix used to construct exact [Hartree-Fock-like (HF)] exchange, and an approximate density matrix used to construct local spin-density approximation (LSDA) exchange. The proposed exchange energy density has useful formal properties, including correct spin and coordinate scaling and the correct uniform limit. It can readily be evaluated in finite basis sets, with a computational scaling intermediate between HF exchange and semilocal quantities such as the noninteracting kinetic energy density. Applications to representative systems indicate that its properties are typically intermediate between HF and LSDA exchange, and often similar to global hybrids of HF and LSDA exchange. The model is proposed as a novel "Rung 3.5" ingredient for constructing approximate exchange-correlation functionals.

Exchange energy for two closed shells generated by a bare Coulomb potential energy −Ze 2/r in the limit of large Z, in two dimensions

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze2/r and (ii) the exchange energy density \({\varepsilon_x({\bf r})}\) . Some progress is also possible concerning the exchange potential Vx(r), one non-local approximation being the Slater potential \({2\varepsilon_x(r)/n(r)}\) , with n(r) the ground state electron density. However, to complete the theory of Vx(r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.

First-principles study on structural properties and phase stability of III-phosphide (BP, GaP, AlP and InP)

We have performed first principles total-energy calculations based on the full-potential augmented plane-wave plus local orbitals (FP-LAPW+lo) method in order to investigate the phase transformations under high pressure of III-phosphide (BP, GaP, AlP, InP) in the zinc-blende, NaCl, CsCl, sc16, cmcm, d-β-tin, NiAs, Immm, and Imm2 structures. The generalized gradient approximation (GGA) was used for the exchange and correlation energy density functional. Our work relates directly to recent experimental work on III-phosphide compounds. The ground-state properties such as lattice parameter, bulk modulus and its pressure derivative as well as the structural phase stability. Along with previous work, we now have enough theoretical results to support a different systematics of the high-pressure phases of III-phosphide compounds.

Non-empirical derivation of the parameter in the B88 exchange functional

The B88 exchange energy density functional (created by Becke in 1988) is a crucial part of the most popular density functional in use today, B3LYP. B88 contains one empirical parameter which was fitted to Hartree–Fock exchange energies for the noble gas atoms. We show how local approximations to exchange become relatively exact under a very specific approach to the limit of large numbers, but the usual gradient expansion does not. The leading corrections can be captured by generalized gradient approximations, producing a non-empirical derivation of the parameter in B88.

Investigation of Exchange Energy Density Functional Accuracy for Interacting Molecules.

We present a comparison of exchange-only interaction energies obtained using several standard exchange functionals in the generalized gradient approximation to Hartree-Fock results for interacting molecules. We observe that functionals with an enhancement factor using a 2/5 power dependence on the gradient of the density for large density gradients offer consistently better agreement with Hartree-Fock calculations than that of alternative functionals. We revisit the functional offering the closest agreement and recalculate it to include its exact large gradient dependence.

Role of exchange in density-functional theory for weakly interacting systems: Quantum Monte Carlo analysis of electron density and interaction energy

We analyze the density-functional theory (DFT) description of weak interactions by employing diffusion and reptation quantum Monte Carlo (QMC) calculations, for a set of benzene-molecule complexes. While the binding energies depend significantly on the exchange-correlation approximation employed for DFT calculations, QMC calculations show that the electron density is accurately described within DFT, including the quantitative features in the reduced density gradient. We elucidate how the enhancement of the exchange-energy density at a large reduced density gradient plays a critical role in obtaining accurate DFT description of weakly interacting systems.

The experimental and theoretical QTAIMC study of the atomic and molecular interactions in dinitrogen tetroxide

The atomic and molecular interactions in a crystal of dinitrogen tetraoxide, alpha-N2O4, have been studied in terms of the quantum topological theory of molecular structure using high-resolution, low-temperature X-ray diffraction data. The experimental electron density and electrostatic potential have been reconstructed with the Hansen-Coppens multipole model. In addition, the three-dimensional periodic electron density of crystalline alpha-N2O4 has been calculated at the B3LYP/cc-pVDZ level of theory with and without the geometry optimization. The application of the quantum theory of atoms in molecules and crystals (QTAIMC) recovered the two types of intermolecular bond paths between O atoms in crystalline alpha-N2O4, one measuring 3.094, the other 3.116 A. The three-dimensional distribution of the Laplacian of the electron density around the O atoms showed that the lumps in the negative Laplacian fit the holes on the O atoms in the adjacent molecules, both atoms being linked by the intermolecular bond paths. This shows that the Lewis-type molecular complementarity contributes significantly to intermolecular bonding in crystalline N2O4. Partial overlap of atomic-like basins created by zero-flux surfaces in both the electron density and the electrostatic potential show that attractive electrostatic interaction exists between O atoms even though they carry the same net formal charge. The exchange and correlation contributions to the potential energy density were also computed by means of the model functionals, which use the experimental electron density and its derivatives. It was found that the intermolecular interactions in alpha-N2O4 are accompanied by the correlation energy-density ;bridges' lowering the local potential energy along the intermolecular O...O bond paths in the electron density, while the exchange energy density governs the shape of bounded molecules.

Local Hybrid Functionals with an Explicit Dependence on Spin Polarization

A new family of local hybrid functionals with a position-dependent mixture of local and exact exchange has been constructed. On the basis of conceptual similarities between local hybrids and explicit nondynamical correlation functionals, the relative spin polarization has been introduced as an additional variable into the local mixing function (LMF) that determines the position dependence of the exact exchange admixture. This leads to two new two-parameter LMFs, one based on the ratio of the local kinetic energy density to the von Weizsacker kinetic energy density and the other on the dimensionless density gradient. After optimization of the two free parameters for small test sets, significant improvements of atomization energies of the full G3 set are obtained (mean absolute errors of 2.96 and 3.18 kcal mol(-1), respectively), with minor deterioration for classical reaction barriers (HTBH38 and NHTBH38 test sets). The simultaneous description of two-center three-electron dimer cations remains a challenge. It is shown that the exact and local spin density exchange energy densities have closely related gauge definitions, so that gauge mismatch is only a minor problem for local hybrids based on local exchange.

Assessment of a density functional with full exact exchange and balanced non-locality of correlation

The recently proposed non-local functional of Perdew, Staroverov, Tao, and Scuseria (PSTS) [Phys. Rev. A 78, 052513 (2008)] belongs to the general family of local hybrids and hyper-generalized gradient approximations for the ground-state exchange-correlation energy. It is distinguished from other functionals of this family by the use of formally full exact exchange, balanced non-locality of correlation, and satisfaction of many exact constraints. We report here a self-consistent, generalized Kohn–Sham implementation of PSTS using the exact exchange energy density in the less-than-optimal conventional gauge. We extensively benchmark this version of PSTS for a series of properties against other well-established density functional approximations. It is shown that PSTS in the conventional gauge performs well for most of the properties under study.

Exchange energy density in density-functional theory via the Dirac density matrix for a nonrelativistic 10-electron atomic ion compared with Becke’s proposal for a gradient-corrected local-density-approximation result

Especially in atomic systems, it is now well established that exchange energy generally dominates correlation effects. Therefore we focus here on the exchange energy density ${\ensuremath{\epsilon}}_{x}(r)$ as given in terms of the idempotent Dirac density matrix. This is then brought into contact with the one-parameter form of Becke's functional, which corrects the local-density-approximation form $\ensuremath{-}{c}_{x}{[n(r)]}^{4/3}$ with $n(r)$ as the ground-state electron density, ${c}_{x}=(3/4){e}^{2}{(3/\ensuremath{\pi})}^{1/3}$, by terms involving the dimensionless gradient ratio $|\ensuremath{\nabla}n(r)|/{n}^{4/3}(r)$. A particular nonrelativistic model of the 10-electron Ne-like atomic ions, with large atomic number $Z$, is then compared to Becke's approximation to ${\ensuremath{\epsilon}}_{x}(r)$.

Some questions on the exchange contribution to the effective potential of the Kohn–Sham theory

The current success of Density Functional Theory applications hinges upon the availability of explicitly density-dependent functionals to self-consistently solve a set of one-electron equations, the Kohn–Sham (KS) equations, which determine the occupied orbitals and its associated electronic density. In KS theory, a local exchange potential is proposed as part of an effective potential. This potential is compared to the exchange operator of the Hartree–Fock theory, which is of a non-local nature. The present paper discusses the variational framework of the KS equations, and the equivalence between both exchange potentials within a correlation-free theory. The common difficulties of current local exchange functionals to correctly simulate the non-locality of the exchange energy density in chemical systems are also analyzed and explained through an exactly solvable model. We give then numerical arguments and conclude by analyzing the performance of various commonly used approximations to exchange functionals.

Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfaction

We construct a nonlocal density functional approximation with full exact exchange, while preserving the constraint-satisfaction approach and justified error cancellations of simpler semilocal functionals. This is achieved by interpolating between different approximations suitable for two extreme regions of the electron density. In a ``normal'' region, the exact exchange-correlation hole density around an electron is semilocal because its spatial range is reduced by correlation and because it integrates over a narrow range to $\ensuremath{-}1$. These regions are well described by popular semilocal approximations (many of which have been constructed nonempirically), because of proper accuracy for a slowly varying density or because of error cancellation between exchange and correlation. ``Abnormal'' regions, where nonlocality is unveiled, include those in which exchange can dominate correlation (one-electron, nonuniform high density, and rapidly varying limits), and those open subsystems of fluctuating electron number over which the exact exchange-correlation hole integrates to a value greater than $\ensuremath{-}1$. Regions between these extremes are described by a hybrid functional mixing exact and semilocal exchange energy densities locally, i.e., with a mixing fraction that is a function of position $\mathbf{r}$ and a functional of the density. Because our mixing fraction tends to 1 in the high-density limit, we employ full exact exchange according to the rigorous definition of the exchange component of any exchange-correlation energy functional. Use of full exact exchange permits the satisfaction of many exact constraints, but the nonlocality of exchange also requires balanced nonlocality of correlation. We find that this nonlocality can demand at least five empirical parameters, corresponding roughly to the four kinds of abnormal regions. Our local hybrid functional is perhaps the first accurate fourth-rung density functional or hyper-generalized gradient approximation, with full exact exchange, that is size-consistent in the way that simpler functionals are. It satisfies other known exact constraints, including exactness for all one-electron densities, and provides an excellent fit to the 223 molecular enthalpies of formation of the G3/99 set and the 42 reaction barrier heights of the BH42/03 set, improving both (but especially the latter) over most semilocal functionals and global hybrids. Exact constraints, physical insights, and paradigm examples hopefully suppress ``overfitting.''

TREATMENTS OF THE EXCHANGE ENERGY IN DENSITY-FUNCTIONAL THEORY

Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.

Exchange energy density and exchange potential via a hartree–fock plus mp2 study of the electron liquid in the ground-state conformer of glycine

Very recent criticisms of existing exchange-correlation functionals by Wanko et al. applied to systems of biological interest have led us to reopen the question of the ground-state conformer of glycine: the simplest amino acid. We immediately show that the global minimum of the Hartree–Fock (HF) ground-state leads to a planar structure of the five non-hydrogenic nuclei, in the non-ionized form NH2–CH2–COOH. This is shown to lie lower in energy than the zwitterion structure NHB3 +–CH2–COO−, as required by experiment. Refinement of the nuclear geometry using second-order Møller–Plesset perturbation theory (MP2) is also carried out, and bond lengths are found to accord satisfactorily with experimentally determined values. The ground-state electron density for the MP2 geometry is then redetermined by HF theory and equidensity contours are displayed. The HF first-order density matrix γ( r , r ′) is then used to obtain similar exchange-energy density (ε x ( r )) contours for the lowest conformer of glycine. At first sight, their shape looks almost the same as for the density ρ( r ), which seems to vindicate the LDA proportional to ρ( r )3/4. However, by way of an analytically soluble model for an atomic ion, it is shown that this has to be corrected to obtain an accurate HF exchange energy Ex as the volume integral of ε x ( r ). Finally, recognizing that for larger amino acids, the use of HF plus MP2 perturbation corrections will become prohibitive, we have used the HF information for ε x ( r ) and ρ( r ) to plot the truly non-local exchange potential proposed by Slater, from the density matrix γ( r , r ′). This latter calculation should be practicable for large amino acids, but there adopting Becke's one-parameter form of ε x ( r ) correcting LDA exchange. Some future directions are suggested.

Local hybrid functionals based on density matrix products.

We present a novel similarity metric comparing exact and semilocal density functional theory (DFT) exchange holes in real space. This metric is obtained from the product of the one-particle density matrix and the uniform electron gas model density matrix. The metric is bound between 0 and 1, 1 in the uniform electron gas, 0 in regions asymptotically far from finite systems, and can detect delocalization of the exact exchange hole and effective fractional occupations. We also present a parameter-free local hybrid functional that uses this similarity metric to locally mix exact and semilocal DFT exchange energy densities. The resulting functional gives better thermochemistry and reaction barrier heights than our original local hybrids [Jaramillo et al., J. Chem. Phys. 118, 1068 (2003)], while retaining moderate accuracy for symmetric radical cation dimers.

Non-locality in the exchange energy density of the inhomogeneous electron liquids in H2O, NH3 and CH4 treated by Hartree–Fock theory

The exchange energy density has been studied by Hartree–Fock (HF) theory in the 10-electron molecules H2O, NH3 and CH4. The local density approximation (LDA) then provides a natural starting point for interpreting the HF data. Though LDA leads to ‘shape similarity’ for the exchange density of these three molecules, there is, (a) no universality and (b) multivalued character at low densities. The low-density regime can, however, be interpreted in terms of a functional , with ρ the ground-state electron density. This functional is here provided with a firm theoretical foundation. Thus, there is a ‘cross-over’ from LDA at high densities to a low-density functional in HF theory.

Structural, electronic and energetic properties of silicon carbon alloys

We studied the influence of alloying on the structural and electronic properties of the unrelaxed and relaxed Si 1− y C y random alloys by means of ab initio theoretical calculations using two methods: (i) a supercell approach in connection with the plane–wave pseudopotential method; (ii) the full-potential augmented plane–wave plus local orbitals (APW+lo) method. The first method is used to obtain the relaxed atomic structure. The relaxed atomic positions obtained by pseudopotential calculations were used to calculate the band structure via the second method. The local density approximation was used for the exchange and correlation energy density functional. We investigated the lattice parameters and band gap energies. We found that a quite smaller gap appears in the neighborhood of y = 0.03125 concentration of C atoms. The band gap shows a large anomalous bowing and is strongly composition dependent. The electron densities of states for the unrelaxed and relaxed Si 1− y C y are also presented. A model structure of 16- and 32-atom supercells is used. The calculated formation enthalpy and individual energy contributions for y = 0.5 show that the instability of Si 1− y C y alloys is dominated by the largest term of the elastic energy.

Accurate and efficient method for the treatment of exchange in a plane-wave basis

We describe an accurate and efficient extension of Chawla and Voth's [J. Chem. Phys. 108, 4697 (1998)] plane-wave based algorithm for calculating exchange energies, exchange energy densities, and exchange energy gradients with respect to wave-function parameters in systems of electrons subject to periodic boundary conditions. The theory and numerical results show that the computational effort scales almost linearly with the number of plane waves and quadratically with the number of k vectors. To obtain high accuracy with relatively few k vectors, we use an adaptation of Gygi and Baldereschi's [Phys. Rev. B 34, 4405 (1986)] method for reducing Brillouin-zone integration errors.

Scaling relations, virial theorem, and energy densities for long‐range and short‐range density functionals

AbstractDecomposition of the Coulomb electron–electron interaction into a long‐range and a short‐range part is described within the framework of density functional theory, deriving some scaling relations and the corresponding virial theorem. We study the behavior of the local density approximation in the high‐density limit for the long‐range and the short‐range functionals by carrying out a detailed analysis of the correlation energy of a uniform electron gas interacting via a long‐range‐only electron–electron repulsion. Possible definitions of exchange and correlation energy densities are discussed and clarified with some examples. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Modern possibilities for calculating some properties of molecules and crystals from the experimental electron density

Methods for calculating some properties of molecules and crystals from the electron density reconstructed from a precise X-ray diffraction experiment using the multipole model are considered. These properties include, on the one hand, the characteristics of the electron density and the inner-crystal electrostatic field and, on the other hand, the local electronic energies (kinetic, potential, total), the exchange energy density, the electron-pair localization function, the localized-orbital locator, the effective crystal potential, and others. It is shown that the integration of these characteristics over pseudoatomic volumes bounded by the surfaces of the zero flux of the electron density gradient makes it possible to characterize directly from an experiment the properties of molecules and crystals in terms of the atomic contributions. The computer program WinXPRO2004, realizing these possibilities, is briefly described.