Ground-state Electron Density Research Articles

Nonlocal Wigner-like correlation energy density functional: Parametrization and tests on two-electron systems

Reparametrization of Wigner's correlation energy density functional yields a very close fit to the correlation energies of the helium isoelectronic sequence. However, a quite different reparametrization is required to obtain an equally close fit to the isoelectronic sequence of Hooke's atom. In an attempt to avoid having to reparametrize the functional for different choices of the one-body potential, we propose a parametrization that depends on global characteristics of the ground-state electron density as quantified by scale-invariant combinations of expectation values of local one-body operators. This should be viewed as an alternative to the density-gradient paradigm, allowing one to introduce the nonlocal dependence of the density functional on the density in a possibly more effective way. Encouraging results are obtained for two-electron systems with one-body potentials of the form r(zeta) with zeta=-12,+12,1, which span the range between the Coulomb potential (zeta=-1) and the Hooke potential (zeta=2).

Exact correlated kinetic energy related to the electron density for two-electron model atoms with harmonic confinement

Abstract For arbitrary interparticle interaction u ( r 12 ) , the model two-electron atom in the title is shown to be such that the ground-state electron density ρ ( r ) is determined uniquely by the correlated kinetic energy density t R ( r ) of the relative motion. Explicit results for t R ( r ) are presented for the Hookean atom with force constant k = 1 / 4 , and also for u ( r 12 ) = λ r 12 2 . Possible relevance of the Hookean atom treatment to the ground state of the helium atom itself is briefly discussed.

Two-parameter partially correlated ground-state electron density of some light spherical atoms from Hartree-Fock theory with nonintegral nuclear charge

Partially correlated ground-state electron densities for some spherical light atoms are calculated, into which nonrelativistic ionization potentials represent essential input data. The nuclear cusp condition of Kato is satisfied precisely. The basic theoretical starting point, however, is Hartree-Fock (HF) theory for the N electrons under consideration but with nonintegral nuclear charge Z′ slightly different from the atomic number Z (=N). This HF density is scaled with a parameter λ, near to unity, to preserve normalization. Finally, some tests are performed on the densities for the atoms Ne and Ar, as well as for Be and Mg. © 2007 The American Physical Society.

Correlated analytic ground-state densities for two-electron atomic ions from Be to Ne

Abstract In earlier work, we demonstrated that the three-parameter correlated variational ground-state wave function of Chandrasekhar for two-electron atomic ions led to a closed analytic ground-state electron density ρ ( r ) . Here, we determine the three parameters by imposing (i) Kato's cusp condition at the nucleus, (ii) the correct large-r behavior of ρ ( r ) involving the ionization potential, and (iii) the electrostatic potential created by ρ ( r ) at the nucleus, as given by the 1 / Z expansion of atomic theory. It is then found, from results obtained for Be through Ne, that the electron–electron coalescence condition is increasingly well accounted for as the atomic number Z increases. Finally, the one-body potential of density functional theory is extracted from ρ ( r ) .

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.

Convergence of density functional iterative procedures with a Newton-Raphson algorithm

State of the art first-principles calculations of electronic structures aim at finding the ground state electronic density distribution. The performance of such methodologies is determined by the effectiveness of the iterative solution of the nonlinear density functional Kohn-Sham equations. We first outline a solution strategy based on the Newton-Raphson method. A form of the algorithm is then applied to the simplest and earliest density functional model, i.e., the atomic Thomas-Fermi model. For the neutral atom, we demonstrate the effectiveness of a charge conserving Newton-Raphson iterative method for the computation, which is independent of the starting guess; it converges rapidly, even for a randomly selected normalized starting density.

Can density functional theory make use of experimentally determined ground-state electron densities via the Dirac density matrix for molecules of biological interest?

Abstract More than four decades ago, March and Murray gave a perturbation theory of the single-particle(s) Dirac density matrix γ s ( r , r ′ ) to all orders in a given one-body potential energy V ( r ) . However, for density functional theory in orbital-free form, one requires the functional γ s [ ρ ] where ρ ( r ) is the ground-state electron density. Therefore, in the present study, a first-order non-linear differential equation is proposed for γ s in terms of ρ ( r ) and ∇ ρ ( r ) , plus the single-particle kinetic energy. Since this latter quantity is itself known to be a functional of ρ, the existence of such an equation for γ s would be a significant step along the road to determining the desired functional γ s [ ρ ] . As yet, we have succeeded in giving a rigorous proof of the proposed differential equation for γ s ( r , r ′ ) only for one- and two-level molecules. If it is subsequently proved for an arbitrary number of levels, which we believe should be possible, it would then allow γ s to be calculated for molecules of biological interest, from experimentally measured ground-state densities ρ ( r ) , as the approach is entirely orbital-free.

Exchange force for two-level systems such as LiH and [formula omitted

Using the Dawson–March transformation, the Dirac density matrix γ( r, r′) can be written in terms of the density amplitude ρ ( r ) and a phase θ( r), where ρ( r) ≡ γ( r, r) is the ground-state electron density. Here, for systems such as LiH or H 3 - , it is shown how the force −∂ V( r)/∂ r corresponding to the one-body potential V( r) can be written, given the ground-state density ρ( r) from a high-level ab initio calculation or a diffraction experiment. The Hartree–Fock ground-state densities for LiH and H 3 - are utilized to calculate approximate phase angles θ( r). Finally, for H 3 - the force −∂ V( r)/∂ r corresponding to V( r) is calculated.

The relationship between the molecular structure and the chemical properties: what about the topological analysis of the electron density distribution? An application to hydrogen bonds

Molecular structure is determined by nuclei positions. From the Hellmann-Feynman theorem we know that the exact ground state electron density distribution ρ(r) depends on the nuclei positions only. Furthermore, the Hohenberg-Kohn theorem states that the total energy of a system can be written in terms of its ρ(r) distribution. In this context, the relationship between the structure of a molecular system and its physical and chemical properties should be reflected by the electron distribution. As far as the molecular structure is the straightforward consequence of interatomic interactions, a correspondence between these interactions and the observed ρ(r) applies. Accordingly, the interatomic interactions described in terms of ρ(r) can be considered as a fundamental subject of study to get insight on the structure-properties relationship, as the former is a conceptual bridge between the latter. The topological analysis of ρ(r) developed by Bader and co-workers [1] is a useful tool for characterizing atomic interactions in internuclear regions. It permits to obtain the molecular space partition into atomic basins that are separated by zero-flux surfaces S(r )o f the electron distribution and behave as quantum objects. Along the bond path directions ρ(r) is minimum at the surfaces S(r), where topological bond critical points rBCP appear and ρ(r) exhibits saddle distributions. Analysis of topological and energetic magnitudes of ρ(r )a trBCP permits a deep characterization of interactions. Following this description we have analyzed experimental and theoretically calculated energetic properties of hydrogen bonded systems in terms of the ρ(r) distribution at their rBCP’s [2-7].

Comment on “Towards a differential equation for the nonrelativistic ground-state electron density of the He-like sequence of atomic ions”

In a recent paper [Phys. Rev. A 71, 042501 (2005)], Howard and March presented the exact ground state wave function of the spherical He-like atom, and many physical aspects of this solution were analyzed. We show that this function is not the exact solution of the model.

Just how good is DFT?

Density functional theory (DFT) has become a ‘‘household tool,’’ so to speak, for the average chemist. We now take it for granted: we expect DFT to provide a reasonably good description of almost any molecule we can make—or rather, we can think of—and justifiably so. For organic molecules, this is a perfectly valid expectation. However, for transition metal complexes and transition states, the performance of DFT, while generally very good, is less certain, while van der Waals complexes are largely beyond the range of applicability of current DFT methods. Given DFT’s broad and increasing popularity among bioinorganic chemists, a need has been felt for a critical evaluation of the performance of DFT, from a bioinorganic perspective. This series of four Commentaries is intended to address that need. Modern DFT began in 1964, with a famous paper by Hohenberg and Kohn [1] which contained the following two rigorous theoretical results. First, the groundstate electron density uniquely determines the electronic wavefunction and hence all ground-state properties of an electronic system. Second, the energy of an electron distribution is a function—more technically, a functional—of the electron density and this functional is a minimum for ground-state densities. The Hohenberg–Kohn theorem assures us that the density functional is a universal quantity, but it does not specify its form. These results reduce the problem

Long-range asymptotic behaviour of the ground-state electron density in He-like ions as a function of atomic number

From the early work of Hoffmann–Ostenhof, the electron density ρ(r) for He-like ions at sufficiently large r has the form Ar n exp(−2 √ 2 Ir) where I is the ionization potential. A derivation of the relation between n and I is briefly presented. The non-relativistic ionization potential I is then calculated quantitatively from diffusion quantum Monte Carlo (DMC) calculations, as is the range of validity of the above asymptotic form for Z 2. The above arguments are valid away from Z = Zc, the critical atomic number 0.911 028 at which the ionization potential I tends to zero. Again analytic theory plus DMC is used at Z = Zc to quantify ρ(r).

Kinetic energy and ground-state electron densities as fingerprints of wavefunction entanglement in two-electron spin-compensated atomic models

We take as a starting point the ground-state electron density in two-electron model atoms in which Coulomb confinement in the He atom is first replaced by harmonic restoring forces. Switching off electron–electron interactions, one readily constructs a third-order differential equation for the ground-state electron density, as in the recent work of March and Ludena (2004 Phys. Lett. A 330 16). We then switch on two different model interactions, first in the so-called Hookean atom going back to Kestner and Sinanoglu (1962 Phys. Rev. 128 2687), in which e2/r12 is retained as in He, and secondly in the Moshinsky (1968 Am. J. Phys. 36 52) atom in which Kr212/2 is switched on. Some analyticity properties of the low-order linear homogeneous differential equations which result are next studied. He-like atomic ions are then treated in the limit of large atomic number Z. In this latter case, one identifies both the electron–nuclear cusp, or equivalently Kato's theorem, and the corresponding electron–electron cusp in the ground-state spatial wavefunction Ψ(r1, r2). A final comment concerns quantum information and entanglement in relation to the recent work of Amovilli and March (2004 Phys. Rev. A 69 054302).

Interacting inhomogeneous electron liquids with harmonic confinement: s-wave model

In an earlier work by Holas et al. [A. Holas, I.A. Howard, N.H. March. Phys. Lett. A, 310, 451 (2003)], the ground-state electron density ρ( r ) for model two-electron spin-compensated atoms was obtained in terms of the relative motion wave function for the case of harmonic confinement and arbitrary interparticle interaction u(r 12). Following recent interest in the s-wave approximation to ρ( r ) for Coulombic potential energy at separation r 12 between the electrons, we here consider both harmonic interaction u(r 12) proportional to and also the inverse square law . The interplay between correlation and confinement clearly emerges in such spin-compensated models of interacting inhomogeneous electron liquids.

Electron densities of the He-like sequence of atomic ions, and associated physical properties

Motivated by the studies of Baker et al. [J.D. Baker, D.E. Freund, R.N. Hill, J.D. Morgan III, Phys. Rev. A 41 (1990) 1247] and of Li and Shakeshaft [T. Li, R. Shakeshaft, Phys. Rev. A 71 (2005) 052505] on the stability of two electrons and an arbitrary positive charge, we here use the Chandrasekhar three-parameter variational wave function to present analytical ground-state electron densities for the He-like sequence of atomic ions. This input is then used to calculate: (i) kinetic energy densities with correlation, (ii) the single-particle von Weizsäcker counterparts of (i), (iii) the electrostatic potentials at the atomic nuclei, and (iv) ground-state energy differences between the lightest members of the He series. We stress that (ii)–(iv) above are calculated solely from knowledge of the analytical ground-state densities.

Density matrices in direct and momentum space for a model of the He-like sequence of atomic ions

The ground-state electron density $\ensuremath{\rho}(r)$ in a model of the He-like sequence of atomic ions, in which the Coulomb repulsion is replaced by the $s$-wave component, has recently been obtained by Howard and March [Phys. Rev. A 71, 042501 (2005)]. Their result is here generalized to derive the one-particle density matrix yielding the Howard and March $\ensuremath{\rho}(r)$ on the diagonal. Its momentum space counterpart is also presented and the momentum density is studied in detail, with particular emphasis on the asymptotic behavior of momentum distribution. Attention is then focussed on the kinetic energy density and on the spectral decomposition of the ground-state wave function. A natural orbital analysis in both direct and momentum space is also presented. Finally, the relevance for density functional theory of this exactly solvable model is briefly considered.

S-Wave component of the ground-state electron density in the Hookean model compared with the helium atom

From the exact ground-state wave function of the two-electron Hookean atom, in which the Coulombic repulsion energy e 2/ r 12 between electrons 1 and 2 at separation r 12 is retained but the confinement is harmonic with external potential kr 2/2, the s-wave component is extracted for force constant k = 1/4. The radial electron density for this l = 0 component is compared with that for the exact ground-state density, and they are quantitatively very similar. Finally, a linear differential equation is found for the s-wave density and this equation is compared and contrasted with the known result for the exact case.

In situpressure study ofRb4C60insulator to metal transition by Compton scattering

Compton scattering has been shown to be a powerful tool for studying the ground state electronic density in real materials. Using synchrotron radiation, we have studied pressure effects on Rb4C60 by measuring the Compton profiles below and above the insulator to metal transition at 0.8 GPa. The experimental results are compared with the corresponding calculated results, obtained from new ab initio energy band structure calculations. These results allow us to quantitatively evaluate contributions to the Compton profiles resulting from the contraction of the unit cell as well as from the contraction of the C60 molecule itself. In this paper, we point out an unexpected contraction of the volume of the C60 molecule, leading to a major effect on the electronic density of the Rb4C60 compound.

Differential equation for the ground-state density of artificial two-electron atoms with harmonic confinement

A long-term aim of density functional theory is to obtain a differential equation for the ground-state electron density ρ(r) in a closed-shell atom, the simplest example being He. Since He remains intractable analytically, artificial atoms with harmonic repulsive potential energy u(r12) have therefore been studied. Here we exploit recent work on ρ(r) for such a two-electron system, with u(r12) = λ/r212, to construct a second-order linear differential equation for ρ(r). This is compared and contrasted with available results for different choices of u(r12).

Force −∂Vxc/∂r associated with the exchange-correlation potential Vxc(r) in the neutral Ne atom

Following an electrostatic interpretation of the force Fxc = −∂Vxc/∂r associated with the exchange-correlation potential Vxc(r), we present both analytical and numerical results for Fxc(r) in the Ne atom. The basic input is an existing quantum Monte Carlo (QMC) calculation of the ground-state electron density ρ(r) in this atom. The analytic form of ∂Vxc/∂r is in terms of the number of electrons Q(r) enclosed in a sphere of radius r centred on the nucleus, plus two phases needed to characterize the radial wavefunctions of density functional theory. Eigenvalue equations are presented for these phases, and used numerically. A brief discussion is added on the result of replacing the QMC ground-state density by its Hartree–Fock counterpart.