On the proof by reductio ad absurdum of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states of Coulomb systems

It is demonstrated that the original reductio ad absurdum proof of the generalization of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states for isolated many‐electron Coulomb systems with Coulomb‐type external potentials by Gross and colleagues is self‐contradictory, since the to‐be‐refuted assumption (negation) regarding the ensemble one‐electron densities and the assumption regarding the external potentials are logically incompatible to each other due to the Kato electron‐nuclear cusp theorem. It is proved, however, that the Kato theorem itself provides a satisfactory proof of this theorem. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Comment on “On the original proof by reductio ad absurdum of the Hohenberg–Kohn theorem for many‐electron Coulomb systems”

We argue with Kryachko's criticism [Int J Quantum Chem 2005, 103, 818] of the original proof of the second Hohenberg‐Kohn theorem. The Kato cusp condition can be used to refute a “to‐be‐refuted” statement as an alternative to the original proof by Hohenberg and Kohn applicable for Coulombic systems. Since alternative ways to prove falseness of the “to‐be‐refuted” statement in a reduction ad absurdum proof do not exclude each other, Kryachko's criticism is not justified. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

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

Decomposition 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     

Proof of finiteness of Kohn–Sham theory electron interaction potential at the nucleus of atoms

We employ Kato's theorem to prove that the electron interaction potential of Kohn–Sham density functional theory is finite at the nucleus of spherically symmetric and sphericalized atoms and ions. Therefore, this finiteness is a direct consequence of the electron–nucleus cusp condition for the density. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 205–208, 2000     

Hohenberg–Kohn theorems in the presence of magnetic field

In this article, we examine Hohenberg–Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg–Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble‐representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble‐representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians. © 2014 Wiley Periodicals, Inc.     

Pair density related to one‐electron information for the ground state of spin‐compensated two‐electron systems

The recent study by Joubert on effects of Coulomb repulsions in a many‐electron system has focused attention on an integral identity involving the pair density. This has motivated the derivation presented here of a vectorial differential form related to this integral result. Our differential identity is then illustrated explicitly by using (i) an exact ground‐state wave function for the so‐called Hookean atom having external potential energy (1/2)kr², with k = 1/4, and (ii) Moshinsky's model in which both the interparticle interaction and the external potential are of harmonic type. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Necessary conditions for the N‐representability of pair distribution functions

A necessary condition for the N‐representability of the electron pair density proposed by one of the authors (E. R. D.) is generalized. This shows a link between this necessary condition and other, more widely known, N‐representability conditions for the second‐order density matrix. The extension to spin‐resolved electron pair densities is considered, as is the extension to higher‐order distribution functions. Although quantum mechanical systems are our primary focus, the results are also applicable to classical systems, where they reduce to an inequality originally derived by Garrod and Percus. As a simple application, bounds to the average angle between an electron pair are derived. It is shown that computational methods based on variational minimization of the energy with respect to the electron pair density can give extremely poor results unless robust N‐representability constraints are considered. For reference, constraints for the N‐representability of the pair density are summarized. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Spectral/structural data of helium atoms with exponential‐cosine‐screened coulomb potentials

Atomic systems with exponential‐cosine‐screened Coulomb (ECSC) or static screened Coulomb (SSC) potentials have drawn considerable attention recently due to the possible applications to atoms in plasma environments. In this work, we develop a computing scheme to deal with the electron–electron correlation terms with the screened Coulomb interactions instead of the conventional expansion method using the Gegenbauer's addition theorem. Based on this approach, we investigate the helium atom with the ECSC potentials. Bypassing the complex expansion functions for electron–electron interactions, the proposed approach simplifies the calculations greatly and provides an advantage on programming. The results are found to be in good agreement with the existing data. Bound‐state energies, oscillator strengths, and multipole polarizabilities varying with the screening parameters are presented. Comparisons of the ECSC potentials are made with the SSC potentials. The influence of screening effect on the energies, oscillator strengths, and polarizabilities is discussed. © 2015 Wiley Periodicals, Inc.     

Hellmann–Feynman theorem near the threshold

The features of the spectrum structure are considered for situations where some parameter λ of the N‐electron Hamiltonian reaches the threshold value η under which the discrete energy level falls into the continuous spectrum. The electron density properties are also studied. It is proved that for a sequence of the wave functions converging in energy to the lower bound of the continuous spectrum as λ approaches η the corresponding sequence of the electron densities converges to the density of the (N ? 1)‐electron ground state. The results generalize the Hellmann–Feynman theorem for the cases where only the one‐side energy derivatives exist or there is no limiting wave function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Perturbational approach to the electron correlation cusp applied to helium‐like atoms

A recently proposed perturbational approach to the electron correlation cusp problem 1 is tested in the context of three spherically symmetrical two‐electron systems: helium atom, hydride anion, and a solvable model system. The interelectronic interaction is partitioned into long‐ and short‐range components. The long‐range interaction, lacking the singularities responsible for the electron correlation cusp, is included in the reference Hamiltonian. Accelerated convergence of orbital‐based methods for this smooth reference Hamiltonian is shown by a detailed partial wave analysis. Contracted orbital basis sets constructed from atomic natural orbitals are shown to be significantly better for the new Hamiltonian than standard basis sets of the same size. The short‐range component becomes the perturbation. The low‐order perturbation equations are solved variationally using basis sets of correlated Gaussian geminals. Variational energies and low‐order perturbation wave functions for the model system are shown to be in excellent agreement with highly accurate numerical solutions for that system. Approximations of the reference wave functions, described by fewer basis functions, are tested for use in the perturbation equations and shown to provide significant computational advantages with tolerable loss of accuracy. Lower bounds for the radius of convergence of the resulting perturbation expansions are estimated. The proposed method is capable of achieving sub‐μHartree accuracy for all systems considered here. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a (nonlinear, approximate) density functional of three spatial dimension one‐electron density for an N‐electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas‐Fermi kinetic energy (～∫ρ^5/3d r ₁) and Parr electron–electron repulsion energy (～∫ρ^4/3d r ₁) main‐term functionals are accepted, and they should, the later described, compact one‐electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg–Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground‐state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one‐electron density models. © 2008 Wiley Periodicals, Inc. J Comput Chem 2009     

A Uniqueness Theorem of Molecular Recognition

Phrased in terms of electron density deformations due to molecular interactions, an optimality condition, and the fundamental holographic properties of molecular electron densities, it is shown that molecular recognition is necessarily unique. A simple proof is given and the connections of this result with the Duality Principle of Molecular Recognition and related Selectivity Measures for molecular recognition are discussed.     

Conditions for accurate description of the electron density of atoms and molecules

Based on the Kato's cusp condition of the electron density and our recent relations for local strongly decaying properties in an electronic system, necessary conditions for trial electron densities of atomic and molecular systems are derived. These conditions take the form of integral‐differential equations, and their validity is verified numerically. The relevance of these conditions to the Thomas–Fermi problem in the orbital‐less density functional approach is discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Exact relations between the electron density and external potential for systems of interacting and noninteracting electrons

The differential virial theorem (DVT) is an explicit relation between the electron density ρ( r ), the external potential, kinetic energy density tensor, and (for interacting electrons) the pair function. The time‐dependent generalization of this relation also involves the paramagnetic current density. We present a detailed unified derivation of all known variants of the DVT starting from a modified equation of motion for the current density. To emphasize the practical significance of the theorem for noninteracting electrons, we cast it in a form best suited for recovering the Kohn–Sham effective potential v_s( r ) from a given electron density. The resulting expression contains only ρ( r ), v_s( r ), kinetic energy density, and a new orbital‐dependent ingredient containing only occupied Kohn–Sham orbitals. Other possible applications of the theorem are also briefly discussed. © 2012 Wiley Periodicals, Inc.     

On the relation between basis set convergence and electron correlation: a critical test for modern ab initio quantum chemistry on a ��mindless�� data set

The correlation of the only two error sources in the solution of the electronic Schrödinger equation is addressed: the basis set convergence (incompleteness) error (BSIE) and the electron correlation effect. The electron correlation effect and basis set incompleteness error are found to be correlated for all of the molecules in Grimme??s ??mindless?? data set (MB08-165). One can use an extrapolation to the HF or MP2 complete basis set (CBS) limit to see with which type of quantum chemical problem (??simple?? and ??hard??) the researcher is dealing. The origin of the slow convergence of the partial wave expansion can be the Kato cusp condition for electron?Celectron coalescence. Such an extrapolation is possible for many large molecular systems and would give the researcher an idea about the expected electron correlation level that would lead to the desired theoretical accuracy. In other words, it is possible to use not only the CBS energy value itself but the speed with which it is reached to get extra information about the molecular system under study.     

Grid‐based density functional calculations of many‐electron systems

Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many‐electron systems in the 3D cartesian coordinate grid (CCG). The atom‐centered localized gaussian basis set, electronic density, and the two‐body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange‐correlation potential, while Hay‐Wadt‐type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, and atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom‐centered grid as well as the grid‐free calculation. Results for three atoms are also given. Combination of CCG and the convolution procedure used for classical Coulomb potential can provide reasonably accurate and reliable results for many‐electron systems. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

The 6,6‐Dicyanopentafulvene Core: A Template for the Design of Electron‐Acceptor Compounds

The electron‐accepting ability of 6,6‐dicyanopentafulvenes (DCFs) can be varied extensively through substitution on the five‐membered ring. The reduction potentials for a set of 2,3,4,5‐tetraphenyl‐substituted DCFs, with varying substituents at the para‐position of the phenyl rings, strongly correlate with their Hammett σ_p‐parameters. By combining cyclic voltammetry with DFT calculations ((U)B3LYP/6‐311+G(d)), using the conductor‐like polarizable continuum model (CPCM) for implicit solvation, the absolute reduction potentials of a set of twenty DCFs were reproduced with a mean absolute deviation of 0.10 eV and a maximum deviation of 0.19 eV. Our experimentally investigated DCFs have reduction potentials within 3.67–4.41 eV, however, the computations reveal that DCFs with experimental reduction potentials as high as 5.3 eV could be achieved, higher than that of F₄‐TCNQ (5.02 eV). Thus, the DCF core is a template that allows variation in the reduction potentials by about 1.6 eV.     

A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces

Bearing in mind the insight into the Hohenberg–Kohn theorem for Coulomb systems provided recently by Kryachko (Int J Quantum Chem 103:818, 2005), we present a re-statement of this theorem through an elaboration on Lieb’s proof as well as an extension of this theorem to finite subspaces. Contribution to the Serafin Fraga Memorial Issue.