Equation of motion of the correlated first-order density matrix for the ground-state of the Hookean atom with two electrons

The ground-state wave function Ψ for a given force constant k = 1/4 a.u. of the two-electron Hookean atom is known in exact analytical form. Here the corresponding first-order density matrix γ(r, r′) is studied, particular attention being focussed on its equation of motion. The exact form which results from the known Ψ is displayed, and given a physical interpretation. Harmonic confined model two-electron atoms with arbitrary interaction u(r ₁₂) are also briefly referred to in the present context.     

Analytic inhomogeneous electron liquid and its density for model spin-compensated two-electron atomic ions with Coulomb confinement: an exact nonrelativistic Hamiltonian

In earlier work, Howard and March (HM) proposed an analytic ground-state electron density ρ( r ) starting from the s-wave model of the He atom. Subsequently Ancarani has constructed by numerical methods a variational approach for this s-wave model with a lower energy than the HM result. This clearly means that the HM ρ( r ) is not the ground-state electron density of the He s-wave model. Therefore, we derive here an exact nonrelativistic Hamiltonian, with strong radial correlation plus Coulomb confinement, for which the HM ρ( r ) is indeed the ground-state electron density.     

Analytical Hartree-Fock electron densities for singly charged cations and anions

The Hartree-Fock (HF) electron density has an important property that it is identical to the unknown exact density to the first order in the perturbation theory. We generate the spherically averaged HF electron density ρ(r) by using the numerical HF method for the singly charged 53 cations from Li⁺ to Cs⁺ and 43 anions from H⁻ to I⁻ in their ground state. The resultant density is then accurately fitted into an analytical function F(r), which is expressed by a linear combination of basis functions r ⁿⁱ exp(−ζ _i r). The present analytical approximation F(r) has the following properties: (1) F(r) is nonnegative, (2) F(r) is normalized, (3) F(r) reproduces the HF moments <r ^k > (k=−2 to +6), (4) F(0) is equal to ρ(0), (5) F ^′(0) satisfies the cusp condition and (6) F(r) has the correct exponential decay in the long-range asymptotic region. The present results together with our previous ones for neutral atoms provide a compilation of accurate analytical approximations of the HF electron densities for all the neutral and singly charged atoms with the number of electrons N≤54. Received: 11 July 1997 / Accepted: 27 August 1997     

Analytic structure of ground-state energies and wave functions for the inhomogeneous electron liquid in non-relativistic He-like atomic ions with nuclear charge Ze

The ground state of the He atom for fixed nucleus remains intractable so far as regards exact analytic solutions. However, some important results already exist pertaining to its ground-state wave function Ψ and corresponding electron density n(r). Here, we extend the existing studies by focussing attention on the non-relativistic series of He-like atomic ions with nuclear charge Z. We then find it instructive to start from the energy E(Z) of such a two-electron spin-compensated problem. This is known to have non-analytic behaviour at a critical Z, say Z _c , equal to 0.911028. A form of Darboux transformation going back at very least to Brändas and Goscinski [E. Brändas and O. Goscinski, Int. J. Quantum Chem. 6, 59 (1972)] is refined somewhat here, and compared with a more intuitive approach of Callan [E. Callan, Int. J. Quantum Chem. 6, 431 (1972)]. The important 1/Z expansion of E(Z) is also invoked. The electron density n(r) and the ground-state wave function Ψ are then treated in turn, in a related manner; especially their asymptotic behaviour far from the nucleus. Finally, two exact wave functions for analytically solvable two-Fermion models are shown to sum the infinite series proposed by Fock [V. Fock, Izv. Akad. Nauk SSSR, Ser. Fiz. 18, 161 (1954)].     

Statistical–mechanical models with separable many-body interactions: especially partition functions and thermodynamic consequences

We start from a classical statistical–mechanical theory for the internal energy in terms of three- and four-body correlation functions g ₃ and g ₄ for homogeneous atomic liquids like argon, with assumed central pair interactions f(r_ij){\phi(r_{ij})} . The importance of constructing the partition function (pf) as spatial integrals over g ₃, g ₄ and f{\phi} is stressed, together with some basic thermodynamic consequences of such a pf. A second classical example taken for two-body interactions is the so-called one-component plasma in two dimensions, for a particular coupling strength treated by Alastuey and Jancovici (J Phys (France) 42:1, 1981) and by Fantoni and Tellez (J Stat Phys 133:449, 2008). Again thermodynamic consequences provide a particular focus. Then quantum–mechanical assemblies are treated, again with separable many-body interactions. The example chosen is that of an N-body inhomogeneous extended system generated by a one-body potential energy V(r). The focus here is on the diagonal element of the canonical density matrix: the so-called Slater sum S(r, β), related to the pf by pf(b) = òS(r, b)d[(r)\vec]{{\rm pf}(\beta) = \int {S({\bf r}, \beta)}d\vec {r}}, β = (k _B T)⁻¹. The Slater sum S(r, β) can be related exactly, via a partial differential equation, to the one-body potential V(r), for specific choices of V which are cited. The work of Green (J Chem Phys 18:1123, 1950), is referred to for a generalization, but now perturbative, to two-body forces. Finally, to avoid perturbation series, the work concludes with some proposals to allow the treatment of extended assemblies in which regions of long-range ordered magnetism exist in the phase diagram. One of us (Z.D.Z.) has recently proposed a putative pf for a three-dimensional (3D) Ising model, based on two, as yet unproved, conjectures and has pointed out some important thermodynamic consequences of this pf. It would obviously be of considerable interest if such a pf, together with conjectures, could be rigorously proved.     

Bosonised DFT potential estimated from QMC calculations of the ground-state density for the inhomogeneous electron liquid in Be

To avoid the solution of numerous Kohn–Sham one-body potential equations for wave functions in density functional theory, various groups independently proposed the use of Pauli potential to bosonise the customary one-body potential theory. Here, we utilise our recent quantum Monte Carlo calculations of the ground-state electron density of the Be atom to estimate the bosonised one-body potential V_B(r) and hence extract the Pauli potential for this atom.     

Comment on Dual-Hard-Sphere Models for Liquid Semiconductors Si And Ge

Abstract

Various generalized dual-hard-sphere (DHS) models are reviewed on calculating the liquid structure factor for semiconductor elements Si and Ge. It is found that the model generalized by Canessa, Mariani and Vignolo gives the best fitting of experimental structure factor S _exp(k) in the range k > 2k_F , (k_F , the Fermi wave vector), and all previous models including a new generalized model by the author fail to reproduce the experimental structure factor S _exp(k) of Si and Ge in the whole range of k vector.     

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-pair radial density functions

We introduce and discuss a generalized electron-pair radial density function G(q; a) that represents the probability density for the electron-pair radius |r ₁+ar ₂| to be q, where a is a real-valued parameter. The density function G(q; a) is a projection of the two-electron radial density D ₂(r ₁, r ₂) along lines r ₁ + ar ₂ ± q = 0 in the r ₁ r ₂ plane onto a point in the qa plane, and connects three densities S(s), D(r), and T(t), defined independently in the literature, as a smooth function of a: For an N-electron (N ≥ 2) system, S(s) = G(s; + 1), D(r) = 2G(r; 0)/(N − 1), and T(t) = G(|t|;−1)/2, where S(s) and T(t) are the electron-pair radial sum and difference densities, respectively, and D(r) is the single-electron radial density. Simple illustrations are given for the helium atom in the ground 1s² and the first excited 1s2s ³S states.     

Structural properties of some liquid transition metals

This article addresses the computation of structural properties of liquid transition metals, namely, 3d (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn), 4d (Zr, Pd, Ag and Cd) and 5d (Pt, Au and Hg). We have calculated the structure factor S(q), pair distribution function g(r), interatomic distance r ₁, coordination number n ₁, long wavelength limit of structure factor S(0) and isothermal compressibility χ_T for liquid transition metals. To describe electron–ion interaction, we have used our own model potential along with one component plasma reference system. To see the influence of exchange and correlation effect, Sarkar et al.'s [Mod. Phys. Lett. B12, 639 (1998)] local field correlation function is used. Thus, our newly constructed model potential has successfully generated the structural properties (structure factor S(q), pair distribution function g(r), interatomic distance r ₁, coordination number n ₁, long wavelength limit of structure factor S(0) and isothermal compressibility χ_T ) for liquid transition metals.     

Optimal Hylleraas wave functions

Hylleraas wave functions composed of the optimally combinedN terms (2 N 20) are presented for two-electron atoms with nuclear chargesZ = 1 (H^–), 2(He), 3(Li⁺), 5(B³⁺), and 10(Ne⁸⁺). The spherically-averaged electron density (r) and electron-pair densityh(r ₁₂) are constructed in a simple and analytical functional form from the 20-term functions. Comparison of several one- and two-electron moments r ^k and r ₁₂ ^k shows that the present density functions have near-exact accuracy.     

Use of biopartitioning micellar chromatography and RP-HPLC for the determination of blood–brain barrier penetration of α-adrenergic/imidazoline receptor ligands,and QSPR analysis

For this study, 31 compounds, including 16 imidazoline/α-adrenergic receptor (IRs/α-ARs) ligands and 15 central nervous system (CNS) drugs, were characterized in terms of the retention factors (k) obtained using biopartitioning micellar and classical reversed phase chromatography (log k_BMC and log k_wRP, respectively). Based on the retention factor (log k_wRP) and slope of the linear curve (S) the isocratic parameter (φ₀) was calculated. Obtained retention factors were correlated with experimental log BB values for the group of examined compounds. High correlations were obtained between logarithm of biopartitioning micellar chromatography (BMC) retention factor and effective permeability (r(log k_BMC/log BB): 0.77), while for RP-HPLC system the correlations were lower (r(log k_wRP/log BB): 0.58; r(S/log BB): –0.50; r(φ₀/P_e): 0.61). Based on the log k_BMC retention data and calculated molecular parameters of the examined compounds, quantitative structure–permeability relationship (QSPR) models were developed using partial least squares, stepwise multiple linear regression, support vector machine and artificial neural network methodologies. A high degree of structural diversity of the analysed IRs/α-ARs ligands and CNS drugs provides wide applicability domain of the QSPR models for estimation of blood–brain barrier penetration of the related compounds.     

Determination of the ionic radii by means of the Kohn–Sham potential: Identification of the chemical potential

Under the Kohn–Sham theory, we examine solutions for the equations δT_S/δρ(r) = 0 and δT_S/δρ(r) = ν_KS(r) that link the chemical potential of the electronic system with the effective Kohn–Sham potential through μ = ν_KS(r) + δT_S/δρ. For single ions, we identify the chemical potential with the eigenvalue of the frontier orbital when the atom is in the limit of full ionization. For the case of cations, the chemical potential is found above ?(I + A)/2 and has the property of grouping ions with the same chemical characteristics. For the anion instead, the chemical potential is fixed at the ionization energy. By solving the above equations numerically, two radial points called r_? and r₊ are obtained and compared with the Shannon–Prewitt ionic radius. Moreover, we found for the halide series, that r_? is numerically equivalent to rm, the radii where the electrostatic potential has its minimum, but shows different behavior upon charge variation. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

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.     

Some properties of the structure factor S(q) in two-dimensional classical liquids near freezing

We explore general properties of the main peak of the structure factor S(q) near the melting temperature T _melt in liquids confined in two dimensions, especially for the one component plasma model and for monatomic liquids interacting through inverse twelfth-power potentials. Those properties are the height of the peak, S(q _m), where q _m is the position of maximum in the peak, and the ratio between S(q _m) and q _m/Δq, where 2Δq is the width of the peak. The results obtained are then compared with those for similar systems in three dimensions. Other magnitude that we use to compare two-dimensional and three-dimensional simple liquids is r _m/Δr, where r _m is the position of the main peak in the pair distribution function g(r) and 2Δr is the width of that peak.     

Scaling in the S and P states of the helium isoelectronic sequence

Mean values of r ₁ ^r and r _r ¹² for the ground and several excited states of the helium isoelectronic sequence are used to demonstrate that a simple scaling which superimposes the distribution function f(r ₁₂) as a function of the atomic number leads to a similar result for the electron density distribution D(r₁). On the basis of a screening interpretation of the scaling parameter , it is concluded that screening is greater in the singlet than the triplet state of a particular configuration, that screening is greater in the P states than the corresponding S states, and that the screening approaches the limiting value of 1 for the highly excited states. The perturbation expansions of Scherr and Knight are used to determine the limiting value of when Z and the relationship between the scaling parameter and the scale factor, chosen so that a trial wave function satisfies the virial theorem, is discussed. A brief discussion of the scaling of the Coulomb hole is presented.     

The exchange-correlation potential of DFT obtained from a semiempirically fine-tuned Hartree–Fock density for inhomogeneous electron liquids

The present authors have given an exact theory of the exchange-correlation potential V _xc (r) in terms of (i) the exact ground-state electron density n(r) and (ii) the idempotent Dirac density matrix γ(r,?r′) generated by the DFT one-body potential V(r), having n(r) as its diagonal element. Here, we display two approximate consequences: (a) a form of V _xc (r) generated by the semiempirically fine-tuned HF density of Cordero et al. (N.A. Cordero, N.H. March, and J.A. Alonso, Phys. Rev. A 75, 052502 (2007)) and (b) the exchange-only potential V _x (r) determined solely by the HF ground state density for the Be atom.     

Recombination of HCO and DCO ions with electrons

Recombination of HCO⁺ and DCO⁺ ions with electrons was studied in afterglow plasma. The flowing afterglow with Langmuir probe (FALP) apparatus was used to measure the recombination rate coefficients and their temperature dependencies in the range 150–270 K. To obtain a recombination rate coefficient for a particular ion, the dependencies on partial pressures of gases used in the ion formation were measured. The variations of α_HCO⁺(T) and α_DCO⁺(T) seem to obey the power law: α_HCO⁺(T) = (2.0 ± 0.6) × 10⁻⁷ (T/300)^−1.3 cm³ s⁻¹ and α_DCO⁺(T) = (1.7 ± 0.5) × 10⁻⁷ (T/300)^−1.1 cm³ s⁻¹ over the studied temperature range.     

Effect of counterions on photoprocesses of thiacarbocyanine in a binary solvent blend

A model describing the effect of counterion X⁻ (X = Cl, I) on the deactivation kinetics of the S ₁ state of thiacarbocyanine Cy⁺X⁻ is presented. According to the model, the ion pair Cy⁺X⁻ in a binary solution is characterized by a distribution function f(r) over interatomic distances r, which depends on the composition of the mixture. The assumption of kinetically independent local states of the ion pair, which decay with the rate constants k _i(r)(i = 1–4 is the index of the decay channel), is made. The statistic analysis of the experimental data in terms of the model permitted us to find the functions f(r) and to estimate the parameters of the constants k _i(r).     

On the complete set of functions in the Heitler–London method for two-electron problem

It is shown that under certain restrictions the system of determinants φ_i(x₁)φ_k(x₂) ? φ_k(x₁)ψ_i(x₂) constructed from two different sets of orbitals ψ_k and φ_k will be the complete set of functions for antisymmetrical two-electron wave functions if the condition i < k is imposed.