Kinetic-energy systems, density scaling, and homogeneity relations in density-functional theory

We examine the behavior of the Kohn-Sham kinetic energy T_s[ρ] and the interacting kinetic energy T[ρ] under homogeneous density scaling, ρ(r)→ζρ(r). Using convexity arguments, we derive simple inequalities and scaling constraints for the kinetic energy. We also demonstrate that a recently derived homogeneity relation for the kinetic energy [S. B. Liu and R. G. Parr, Chem. Phys. Lett. 278, 341 (1997)] does not hold in real systems, due to nonsmoothness of the kinetic-energy functional. We carry out a numerical study of the density scaling of T_s[ρ] using ab initio densities, and find it exhibits an effective homogeneity close to 5/3. We also explore alternative reference systems for the kinetic energy which have fewer particles than the true N-particle interacting system. However, we conclude that the Kohn-Sham reference system is the only viable choice for accurate calculation, as it contains the necessary physics

C_8H_8: a density functional theory study of molecular geometries introducing the localised bond density

In this paper we use density functional theory with all the common exchange-correlation functionals to investigate the structures of three isomers of C_8H_8 found in F. A. Cotton's text, barrelene, cyclooctatetraene, tetramethylenecyclobutane and also ethane and ethene. All calculations were performed with TZ2P basis sets and large quadrature. The results are compared with experiment and those obtained with Hartree–Fock theory. Delocalisation in the three molecules is discussed. A localised bond density in introduced to explain the transferability of the trends in the predictions of the functionals between different molecules. Three-parameter adiabatic connection functionals are examined and their usefulness in geometry prediction questioned. Finally a physical picture of the correlation as modelled by density functional theory is presented and used to explain trends in the overestimation or underestimation of bond lengths

Optimized Lieb-Oxford bound for the exchange-correlation energy

Using the ideas of Lieb and Oxford [Int. J. Quantum Chem. 19, 427 (19810], we show that the exchange-correlation energy, and indirect part of the Coulomb energy, are bounded from below by -1.6358∫ρ^(4/3) (x)dx, where ρ(x) is the single-particle density

A new chemical concept: Shape chemical potentials

Within the density functional formalism, we introduce the shape chemical potential μ_i(^n) for subsystems, which in the limiting case of point subsystems, is a local chemical potential μ^n(r). It describes the electron withdrawing/donating ability of specified density fragments. The shape chemical potential does not equalize between subsystems, and provides a powerful new method to identify and describe local features of molecular systems. We explore the formal properties of μ_i(^n) especially with respect to discontinuities, and reconcile our results with Sanderson’s principle. We also perform preliminary calculations on model systems of atoms in molecules, and atomic shell structure, demonstrating how μ_i(^n) and μ^n(r), identify and characterize chemical features as regions of different shape chemical potential. We present arguments that shell structure, and other chemical features, are not ever obtainable within Thomas–Fermi-type theories

Thomas–Fermi–Dirac–von Weizsäcker models in finite systems

To gain an understanding of the variational behavior of kinetic energy functionals, we perform a numerical study of the Thomas–Fermi–Dirac–von Weizsäcker theory in finite systems. A general purpose Gaussian-based code is constructed to perform energy and geometry optimizations on polyatomic systems to high accuracy. We carry out benchmark studies on atomic and diatomic systems. Our results indicate that the Thomas–Fermi–Dirac–von Weizsäcker theory can give an approximate description of matter, with atomic energies, binding energies, and bond lengths of the correct order of magnitude, though not to the accuracy required of a qualitative chemical theory. We discuss the implications for the development of new kinetic functionals

Correlation potentials and functionals in Hartree-Fock-Kohn-Sham theory

We compute molecular Hartree-Fock-Kohn-Sham correlation potentials from ab initiocoupled-cluster densities via a modified Zhao, Morrison and Parr [Phys. Rev. A, 50, (1994) 2138] scheme involving exact exchange. We examine the potential for several small systems, and observe complex structure. By fitting a functional expansion to our potentials we obtain a closed-shell functional which is an improvement over other pure correlationfunctionals in Hartree-Fock-Kohn-Sham calculations. The leading term in our functional is dependent on the number of electrons. Our results lead us to question the utility of correlation defined within the Hartree-Fock-Kohn-Sham scheme, and to consider alternative partitionings of the exchange-correlation energy

An extensive study of gradient approximations to the exchange-correlation and kinetic energy functionals

We formalize the procedure of functional development, in a general theoretical framework. Expansion in a functional basis set, and fitting via an error functional to a data set, casts functional development as a variational problem to obtain the functional basis-set and data-set limits. Overfitting is avoided by defining the optimum number of parameters. We implement our theory for an investigation of first- and second-order generalized gradient approximations (GGA) to the exchange-correlation and kinetic energy functionals, within an ab initio model. A variety of functional basis sets, including a general finite-element representation, is constructed to represent both one-dimensional and multidimensional GGA enhancement factors. An extensible data set consisting of 429 atomic and diatomic, neutral and cationic species, at stretched and equilibrium geometries, is constructed from Moller–Plesset level exchange-correlation energies, and Hartree–Fock kinetic energies. The range of chemically relevant density and gradient variables is examined. Exhaustive fitting investigations are carried out, to determine the accuracy of the GGA representation of the ab initio models. In the exchange-correlation case we demonstrate that we can reach the functional basis-set and data-set limit, which correspond to a root-mean-square (rms) error of ∼10∼10 mH (6.3 kcal/mol). Changing the functional basis set, higher-order density variables such as the kinetic energy density, multidimensional enhancement factors, and exact exchange yield no significant improvement, and our fits represent an effective solution of the GGA problem for exchange-correlation, at the Møller–Plesset level. In the kinetic energy case, accurate functionals with rms errors of ∼80∼80 mH (50 kcal/mol) are developed. These exhibit a beautifully simple kinetic energy enhancement factor, and are a step towards orbital-free calculations

C_8H_8: a density functional theory study of molecular geometries introducing the localised bond density

In this paper we use density functional theory with all the common exchange-correlation functionals to investigate the structures of three isomers of C_8H_8 found in F. A. Cotton's text, barrelene, cyclooctatetraene, tetramethylenecyclobutane and also ethane and ethene. All calculations were performed with TZ2P basis sets and large quadrature. The results are compared with experiment and those obtained with Hartree–Fock theory. Delocalisation in the three molecules is discussed. A localised bond density in introduced to explain the transferability of the trends in the predictions of the functionals between different molecules. Three-parameter adiabatic connection functionals are examined and their usefulness in geometry prediction questioned. Finally a physical picture of the correlation as modelled by density functional theory is presented and used to explain trends in the overestimation or underestimation of bond lengths