Exchange energy for two closed shells generated by a bare Coulomb potential energy −<Emphasis Type="Italic">Ze</Emphasis><Superscript>2</Superscript>/<Emphasis Type="Italic">r</Emphasis> in the limit of large <Emphasis Type="Italic">Z</Emphasis>, 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 −Ze ²/r and (ii) the exchange energy density e_x(r){\varepsilon_x({\bf r})} . Some progress is also possible concerning the exchange potential V _x (r), one non-local approximation being the Slater potential 2e_x(r)/n(r){2\varepsilon_x(r)/n(r)} , with n(r) the ground state electron density. However, to complete the theory of V _x (r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.     

Study of the kinetic energy densities of electrons as applied to quantum dots in a magnetic field

There are three expressions for the kinetic energy density t( r ) expressed in terms of its quantal source, the single-particle density matrix: t _A( r ) , the integrand of the kinetic energy expectation value; t _B( r ) , the trace of the kinetic energy tensor; t _C( r ) , a virial form in terms of the ‘classical’ kinetic field. These kinetic energy densities are studied by application to ‘artificial atoms‘ or quantum dots in a magnetic field in a ground and excited singlet state. A comparison with the densities for natural atoms and molecules in their ground state is made. The near nucleus structure of these densities for natural atoms is explained. We suggest that in theoretical frameworks which employ the kinetic energy density such as molecular fragmentation, density functional theory, and information-entropic theories, one use all three expressions on application to quantum dots, and the virial expression for natural atoms and molecules. New physics could thereby be gleaned.     

Kinetic energy density in terms of electron density for closed-shell atoms in a bare Coulomb field

A long-term aim in density functional theory is to obtain the kinetic energy density t(r) in terms of the ground-state electron density ρ(r). Here, t(r) is written explicitly in terms of ρ(r) for an arbitrary number 𝒩 of closed shells in a bare Coulomb field. In the limit as 𝒩→∞, closed results for t(r) follow from the earlier analysis of ρ(r) by Heilmann and Lieb. [Phys. Rev. A 52 , 3628 (1995)]. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 281–283, 1998     

Electron density near an impurity

The behavior of the electron density n(r) [and potential energy V(r)] near an impurity of charge Z is studied by using the linear response theory of an electron gas at finite temperature and with exchange and correlation effects included. The odd powers series in the expansion of n(r) [and V(r)] are calculated exactly by using asymptotic methods, and the coefficients in the series are given in terms of moments taken over the Fermi–Dirac distribution function. In all linear response theories and at all temperatures, the derivative n'(0) = -2Zn₀/a₀, where n₀ is the unperturbed electron density and a₀ is the Bohr radius. The effects of exchange and correlation appear in the fifth- and higher-order terms in n_odd(r).     

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.     

Ten-Electron Central Field Problem: An Inhomogeneous Electron Liquid

Abstract

Recent work has been carried out on the exchange energy density epsive;_x(r) of a ten-electron atomic ion in the (bare Coulomb) limit of large atomic number Z [Howard, I. A. et al (2000). Phys. Rev. A, 62, 062512]. This analytical study of epsive; x(r) was made possible by the existence of a closed form of the first-order (idempotent) density matrix (IDM).

Here, some generalizations are effected to a central potential energy V(r) which (a) localizes the ten electrons and (b) yields closed K and L shells for these ten electrons occupying the lowest eigenstates with spin compensation. In particular, it is shown that p-shell properties alone determine the IDM in this example of a confined inhomogeneous electron liquid.     

Direct minimization of the energy in density functional theory

The minimization of the energy functional of the first-order density matrix γ( r , r ') is achieved using unitary transformations applied to γ. Equivalently, such transformations can be carried out also on one-electron orbitals (natural orbitals) and their occupation (integer or non-integer) numbers. The conventional local density approximation based on the electron density p( r ) is then considered as a special case. The direct minimization of the energy functional of p with respect to the parameters of the unitary transformation leads to stationary conditions that are all equivalent to the Kohn–Sham equations. Preliminary numerical tests show that the proposed algorithms for the direct minimization of the energy work in a satisfactory manner. © John Wiley & Sons, Inc.     

Slater's nonlocal exchange potential and beyond

The local density approximation (LDA) to the exchange potential V_x( r ), namely the ρ^1/3 electron gas form, was already transcended in Slater's 1951 paper. Here, using Dirac's 1930 form for the exchange energy density ? _x( r ), the Slater (Sl) nonlocal exchange potential V( r ) is defined by 2? _x( r )/ρ( r ). In spherical atomic ions, say the Be or Ne‐like series, this form V( r ) already has the correct behavior in both r → 0 and r → ∞ limits when known properties of the exchange energy density ? _x( r ) and the ground‐state electron density ρ( r ) are invoked. As examples, some emphasis will first be given to the use of the so‐called 1/Z expansion in such spherical atomic ions, for which analytic results can be obtained for both ? _x( r ) and ρ( r ) as the atomic number Z becomes large. The usefulness of the 1/Z expansion is directly demonstrated for the U atomic ion with 18 electrons by comparison with the optimized effective potential prediction. A rather general integral equation for the exchange potential is then proposed. Finally, without appeal to large Z, two‐level systems are considered, with specific reference to the Be atom and to the LiH molecule. In all cases treated, the Slater potential V( r ) is a valuable starting point, even though it needs appreciable quantitative corrections reflecting directly atomic shell structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

On-top pair-density interpretation of spin density functional theory,with applications to magnetism

The on-top pair density P(r, r) gives the probability that one electron will be found on top of another at position r. We find that the local spin density (LSD) and generalized gradient (GGA) approximations for exchange and correlation predict this quantity with remarkable accuracy. We show how this fact and the usual sum-rule arguments explain the success of these approximations for real atoms, molecules, and solids, where the electron spin densities do not vary slowly over space. Self-consistent LSD or GGA calculations make realistic predictions for the total energy E, the total density n(r), and the on-top pair density P(r,r), even in those strongly “abnormal” systems (such as stretched H₂) where these approximations break symmetries and yield unrealistic spin magnetization densities m(r). We then suggest that ground-state ferromagnetic iron is a “normal” system, for which for LSD or GGA m(r) and the related local spin moment are trustworthy, but that iron above the Curie temperature and antiferromagnetic clusters at all temperatures are abnormal system for which the on-top pair density interpretation is more viable than the standard physical interpretation. As an example of a weakly abnormal system, we consider the four-electron ion with nuclear charge Z → ∞ © 1997 John Wiley & Sons, Inc.     

A complete characterization for <Emphasis Type="Italic">k</Emphasis>-resonant Klein-bottle polyhexes

A hexagonal tessellation K(p, q, t) on Klein bottle, a non-orientable surface with cross-cap number 2, is a finite-sized elemental benzenoid which can be produced from a p × q-parallelogram of hexagonal lattice with usual identifications of sides and with torsion t. Unlike torus, Klein bottle polyhex K(p, q, t) is not transitive except for some degenerated cases. We shall show, however, that K(p, q, t) does not depend on t. Accordingly, criteria for K(p, q, t) to be k-resonant for every positive integer k will be given. Moreover, we shall show that K(3, q, t) of 3-resonance are fully-benzenoid.      

On the conjoint gradient correction to the Hartree—Fock kinetic and exchange energy density functionals

The kinetic and the exchange energy functionals are expressed in the form T[ρ] = C_TF∫ drρ^5/3(r)f_t(s) and K[ρ] = C_D∫ drρ^4/3(r)f_K(s), where C_TF = (3/10)(3π²)^2/3 and C_D = −(3/4)(3/π)^4/3 are the Thomas-Fermi and the Dirac coefficients, respectively, and s = |∇ρ(r)|/C_sρ^4/3(r), with C_s = 2(3π²)^1/3. These expressions are used to perform a comparison of f_T(s) and f_K(s) in terms of their generalized gradient expansion approximations. It is shown that f_κ(s) and is congruent to f_T(s) in the range characteristic of the interior regions of atoms and many solids and that the second-order gradient expansion of the kinetic energy provides a rather reasonable approximation to the generalized gradient expansion approximation of both the kinetic and the exchange energy functionals. © 1996 John Wiley & Sons, Inc.     

Nonlocal energy density functionals: Models plus some exact general results

Holas and March (Phys Rev 1995 A51, 2040) gave a formally exact expression for the force ??V_xc( r )/? r associated with the exchange‐correlation potential V_xc( r ) of density functional theory. This forged a precise link between first‐ and second‐order density matrices and V_xc( r ). Here models are presented in which these low‐order matrices can be related to the ground‐state electron density. This allows nonlocal energy density functionals to be constructed within the framework of such models. Finally, results emerging from these models have led to the derivation of some exact “nuclear cusp” relations for exchange and correlation energy densities in molecules, clusters, and condensed phases. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Towards an exact orbital-free single-particle kinetic energy density for the inhomogeneous electron liquid in the Be atom

Holas and March [Phys. Rev. A51, 2040 (1995)] wrote the gradient of the one-body potential V(r) in terms of low-order derivatives of the idempotent Dirac density matrix built from a single Slater determinant of Kohn–Sham orbitals. Here, this is first combined with the study of Dawson and March [J. Chem. Phys. 81, 5850 (1984)] to express the single-particle kinetic energy density of the Be atom ground-state in terms of both the electron density n(r) and potential V(r). While this is the more compact formulation, we then, by removing V(r), demonstrate that the ratio t(r)/n(r) depends, though non-locally, only on the single variable n′(r)/n(r), no high-order gradients entering for the spherical Be atom.     

DFT studies and AIM analysis of AlN-polycycles

The structure and properties of AlN-polycycles were studied by DFT (density functional theory) method. The results of calculations were obtained at B3LYP/6-311G(d, p) level on model species. Topological parameters such as electron density, its Laplacian, kinetic electron energy density, potential electron energy density, and total electron energy density at the ring critical points (RCP) from Bader’s ‘Atoms in molecules’ (AIM) theory were analyzed in detail. These results indicate a good correlation between ρ(3, +1), G(r), H(r), and V(r) averaged values and hardness of AlN-polycycles. The aromaticity of all molecules has been studied by nucleus-independent chemical shift. There is a linear correlation between ΣNICS(0.0)_molecule values and polarizability.     

Correlated and idempotent Dirac first-order density matrices with identical diagonal Fermion density: a route to extract a one-body potential energy in TDDFT

After a brief introduction to the use of the idempotent Dirac first-order density matrix (DM), its time-dependent generalization is considered. Special attention is focused on the equation of motion for the time-dependent DM, which is characterized by the one-body potential V(r, t) of time-dependent density functional theory. It is then shown how the force –∇ V(r, t) can be extracted explicitly from this equation of motion. Following a linear-response treatment in which a weak potential V(r, t) is switched on to an initially uniform electron gas, the non-linear example of the two-electron spin-compensated Moshinsky atom is a further focal point. We demonstrate explicitly how the correlated DM for this model can be constructed from the idempotent Dirac DM, in this time-dependent example.     

Charge densities for singlet and triplet electron pairs

Analytic properties of charge densities associated with singlet and triplet electron pairs, ρ₀( r ) and ρ₁( r ), are presented. In an N‐electron system with total spin S, distributions ρ₀( r ) and ρ₁( r ) are independent of the spin projection quantum number M (spin rotation invariance), as opposed to the usual spin‐up and spin‐down electron densities, ρ_α( r ) and ρ_β( r ). We derive equations showing that in the case of a wave function which is a spin‐eigenfunction, ρ₀( r ) and ρ₁( r ) are linear combinations of the total charge density ρ( r ) and the uncompensated spin density ρ_s( r )=[ρ_α( r )−ρ_β( r )]/2M. For a wave function which is not an eigenfunction of $\mathcal{S}^{2}$, no such relationship exists. In a related discussion, a definition of the high‐spin solution corresponding to a given spin‐unrestricted Hartree–Fock wave function is proposed, and a notion of effectively unpaired electrons is introduced. The distributions ρ₀( r ) and ρ₁( r ) are shown not to be invariant under spin coupling between isolated systems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 651–660, 2000     

Automatic simulation of electrochemical transients by the adaptive Huber method for Volterra integral equations involving Kernel terms exp[?α(t ? τ)]erex{[β(t ? τ)]1/2} and exp[?α(t ? τ)]daw {[β(t ? τ)]1/2}

Development of fully automatic methods for the simulation of transient experiments in electroanalytical chemistry is a desirable element of the contemporary trends of laboratory automation in electrochemistry. In accord with this idea, the adaptive Huber method, elaborated by the present author, is intended to solve automatically integral equations of Volterra type, encountered in the theory of controlled-potential transients. The coefficients of the method have been recently obtained for integral transformation kernels involving terms K(t, τ) = exp[−α(t − τ)]erex{[β(t − τ)]^1/2} and K(t, τ) = exp[−α(t − τ)]daw{[β(t − τ)]^1/2} with α ≥ 0 and β ≥ 0, which are known to occur in the above integral equations. In this work the validity of the resulting method, for electrochemical simulations, is examined using representative examples of electroanalytical models involving integral equations with various special cases of such kernel terms. The performance of the method is found similar to that previously reported for integral equations involving exclusively kernels K(t, τ) = 1, K(t, τ) = (t − τ)^−1/2, and K(t, τ) = exp[−λ (t − τ)](t − τ)^−1/2 with λ > 0.     

Electron Beam Induced Transformation of MoO3 to MoO2 and a New Phase MoO

The transformation of MoO₃ induced by electron beam irradiation was studied by electron energy‐loss spectroscopy (EELS) in combination with electron diffraction and high‐resolution transmission electron microscopy (HRTEM) techniques. The routes of structure transformation were dependent on the applied electron current density. In case of low current density, MoO₂ was obtained. In case of high current density, MoO with a rock‐salt structure is suggested to be the final phase. The change in oxidation states of the Mo oxides was deduced from the features in energy‐loss near edge structure (ELNES) of the O K‐edge. Quantitative analysis was successfully employed on Mo M₃‐edge and O K‐edge to obtain the O/Mo ratio of the reduced phases. The mechanisms of different structure transformation behaviors were suggested in the frame of radiolysis enhanced diffusion.     

Direct evaluation of the electron density correlation function of partially crystalline polymers

A discussion of the general properties of the one-dimensional electron density correlation function K(z) of a partially crystalline polymer with lamellar structure shows that application of a graphical extrapolation procedure permits direct determination of the crystallinity, the specific inner surface, and the electron density difference η_c ? η_a. The procedure is based upon the occurrence of a straight section in the “self-correlation” range of K(z). Curved and nonparallel lamellae do not invalidate the concept. In the case of heterogeneous samples composed of partially crystalline and totally amorphous regions, some of the parameters of the experimentally obtained correlation function, as for example the invariant K(0), are affected and may lose their definiteness. Use of the method is demonstrated in a detailed discussion of the correlation functions measured for a sample of lowdensity polyethylene at 25 and 100°C.     

Theory of variational calculation with a scaling correct moment functional to solve the electronic schrödinger equation directly for ground state one‐electron density and electronic energy

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a nonlinear, approximate density functional of a three spatial dimension one‐electron density for an N electron system which is tractable in practice, is a long‐desired goal in electronic structure calculation. In a seminal work, Parr et al. (Phys. Rev. A 1997, 55, 1792) suggested a well behaving density functional in power series with respect to density scaling within the orbital‐free framework for kinetic and repulsion energy of electrons. The updated literature on this subject is listed, reviewed, and summarized. Using this series with some modifications, a good density functional approximation is analyzed and solved via the Lagrange multiplier device. (We call the attention that the introduction of a Lagrangian multiplier to ensure normalization is a new element in this part of the related, general theory.) Its relation to Hartree–Fock (HF) and Kohn–Sham (KS) formalism is also analyzed for the goal to replace all the analytical Gaussian based two and four center integrals (∫g_i( r ₁)g_k( r ₂)rd r ₁d r ₂, etc.) to estimate electron‐electron interactions with cheaper numerical integration. The KS method needs the numerical integration anyway for correlation estimation. © 2012 Wiley Periodicals, Inc.