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The structural properties and radial distributions of the valence electron in different excited levels of Na atom (n = 3–5, l = 0–4; n and l being the principal and orbital angular momentum quantum numbers, respectively) under impenetrable spherical confinement have been studied, where the interaction between the frozen core and the valence electron is mimicked by a model potential available in the literature. The effect of the core on the valence electron has been investigated by estimating the structural properties of Na10+ ion under similar confinement. Scaled radial densities at the nucleus and related ratios are presented for a few excited states of the valence electron of Na atom and the corresponding analytic results have been tested numerically.

The statistical complexity and its constituent information measures of Shannon information entropy and disequilibrium for HCl and H $$_2$$ molecules under q-deformed Morse potential are reported in the position and momentum space. A strong dependence of the statistical complexity on the potential deformation parameter q is established.

Atomic systems placed in external potentials manifest various characteristic features which provide useful knowledge about the surroundings. We have studied the structural properties of the ground state of different two-electron systems under isotropic harmonic confinement (IHC). In particular, we have considered negative hydrogen ion, neutral helium atom and positive singly ionized lithium ion to cover all types of charge states. In addition, we have also studied the system of two electrons inside IHC. The wave function is expanded in Hylleraas basis to incorporate the effect of electron correlation in an explicit manner and Ritz variational calculations are performed to obtain the energy eigenvalues and the relative wave functions. The energy levels become more positive with increasing strength of the confining potential. The results show that for ionic systems, the two-electron energy level crosses the respective one-electron threshold at a certain value of the potential beyond which the two-electron level becomes quasi-bound. In order to get deeper insight into such threshold ionization phenomenon, we have examined the contribution of the correlated energy $$E_\mathrm{corr}$$ [sum of radial and angular correlation energy] and radial correlation energy $$E_\mathrm{rad.corr}$$ to the total energy for different two-electron systems under IHC. The Hellmann–Feynman theorem and the virial theorem have been verified as a quantitative validation of the accuracy of our results. The one- and two-electron radial densities have also been analyzed to gain a physical insight into the structural changes of the two-electron systems under IHC. Moreover, the expectation values of different radial and angular variables are also reported which are important to estimate different geometrical and spectral properties.

We consider the $$1s2s\; {^{1,3}\! S}$$ states of the two-electron three-dimensional quantum dot with a Gaussian one-body potential, $$-V_0\exp (-\lambda r^2)$$ . For a single electron, a simple scaling relation allows the reduction into a one-parameter problem in terms of $$\frac{V_0}{\lambda }$$ . However, for the two-electron system, the interelectronic repulsion term, $$\frac{1}{r_{12}}$$ , frustrates this simple scaling transformation, so we face a genuine two-parameter system. We pay particular attention to the location and nature of the critical well-depths, at which the binding energy of the second electron vanishes. Several observations are noteworthy: For all $$\lambda $$ , the triplet critical well-depth is lower than that in the singly excited singlet state. Hence, there exists a finite range of well-depths for which the triplet is bound and the singlet is not, a feature that can possibly be applied in some device. Above its critical well-depth, the triplet state energy is always lower than that of the singly excited singlet. Both well-depths are considerably higher than the critical well-depth in the ground state. The expectation value of the interelectronic repulsion is always lower in the triplet, like the harmonic quantum dot but unlike He-like atoms, the two-particle Debye (Yukawa) atom, or the confined He atom. In the infinite well-depth ( $$V_0$$ ) limit, keeping the well-width $$\left( \frac{1}{\lambda }\right) $$ constant, the energies and other expectation values of the bound states of the two-electron Gaussian quantum dot approach those of a non-interacting harmonic two-electron system.

Accurate restricted Hartree-Fock (RHF) wave-functions are used to investigate information theoretic properties of the model problem of two interacting electrons confined within an infinite spherical potential of radius R. Benchmark quality calculations are performed to characterise this system via a range of information measures as a function of the tunable parameter R, across the full electron correlation regime (low to high correlation; small R to large R). Both the Shannon information entropy and a statistical complexity measure provide quantitative insight regarding the onset of the formation of a ‘Wigner molecule’ state for this system.

The critical charge, at which the “first ionization energy” of the ground-state of the two-electron atom vanishes, is evaluated for the radial limit, as well as for the s + p , s + p + d , s + p + d + f and s + p + d + f + g partially angularly correlated variational limits, getting closer and closer to the fully-correlated critical charge. The (counterintuitive) square integrability of the various correlated wave functions at their respective critical charges is deduced by noting that the derivatives of the corresponding binding energies with respect to Z (hence, the expectation values of the interelectronic repulsion) remain non-vanishing upon approaching these critical nuclear charges.

The stability of spatially confined two-electron system (Zee) has been studied as a function of continuously varying nuclear charge (Z) by adopting multi-exponent Hylleraas type basis set under Ritz variational framework. The critical nuclear charges ( Z c ) for different radius (R) of the impenetrable spherical cavity along with the variations of correlation energy w.r.t. Z c and the R have been predicted. An analysis of the confined Zee wave function points toward its smooth transition from a hydrogen like to the particle-in-a-box like nature as the system transits through E = 0 state from the sufficiently negative to the positive energy regions.

A diatomic molecule modeled by Shifted Deng-Fan (SDF) oscillator potential and restricted to a small region of space has been considered. Energy spectra and radial matrix elements have been calculated using an accurate nine-point finite difference method. Orientation and alignment is generally studied by taking into account only the permanent dipole moment of the molecule. However, in this work, dependence of these properties on the actual set of matrix elements has been explored. A comparative study of the two has been presented. Effect of boundary radius and applied field strength has also been studied.

The exact solutions of Schr\"odinger's equation with the deformed Hulth\'en potential $V_q(x)=-{\mu\, e^{-\delta\,x }}/({1-q\,e^{-\delta\,x}}),~ \delta,\mu, q>0$ are given, along with a closed--form formula for the normalization constants of the eigenfunctions for arbitrary $q>0$. The Crum-Darboux transformation is then used to derive the corresponding exact solutions for the extended Hulth\'en potentials $V(x)= -{\mu\, e^{-\delta\,x }}/({1-q\,e^{-\delta\,x}})+ {q\,j(j+1)\, e^{-\delta\,x }}/({1-q\,e^{-\delta\,x}})^2, j=0,1,2,\dots.$ A general formula for the new normalization condition is also provided., Comment: 14 pages, two figures

In this study, we obtained the position–momentum uncertainties and some uncertainty relations for the Poschl–Teller-type potential for any $$\ell $$ . The radial expectation values of $$r^{-2}$$ , $$r^{2}$$ and $$p^{2}$$ are obtained from which the Heisenberg Uncertainty principle holds for the potential model under consideration. The Fisher information is then obtained and it is observed that the Fisher-information-based uncertainty relation and the Cramer–Rao inequality hold for this even power potential. Some numerical and graphical results are displayed.

The spectrum of a particle confined in Hulthen plus ring-shaped potential is obtained by solving the time-independent Schrodinger equation numerically. The effect of potential parameters on various properties of the particle have been investigated in detail. The energy levels, radial matrix elements, oscillator strengths and polarizabilities of the particle have been found to show strong dependence on the confining potential parameters. The presence of the ring potential is found to appreciably alter the angular part of dipole matrix elements. Also, it is shown that the comparison theorem of Quantum Mechanics for energy eigenvalues for four different potentials, viz., Coulomb, Hulthen, Yukawa and Hulthen2 is independent of the presence of ring potential.

Using some basic inequalities, the spherical Hellmann potential is assigned its place in the hierarchy of relative ordering, within the set of standard screened Coulomb potentials at all radial distances. The previously known comparison theorem applicable to the screened Coulomb potentials is thereby expanded to ascertain the relative ordering of energy levels for nℓ-states of Hellmann potential vis-a-vis those corresponding to the screened Coulomb potentials. The analytic results are supported by numerical calculation.

Singly-excited states of the two-electron atom cease being bound when Z → 1 (from above), the outer orbital becoming infinitely diffuse. The asymptotic relations lim Z → 1 ⁡ ( Z − 1 ) k 〈 ( 1 s n s ) 1 , 3 S | r 12 k | ( 1 s n s ) 1 , 3 S 〉 = 〈 ( n − 1 ) s ( 0 ) | r k | ( n − 1 ) s ( 0 ) 〉 , where k = − 1 , 1 , 2 , 3 , ⋯ , are demonstrated to hold. Here, ( n − 1 ) s ( 0 ) is a hydrogenic s orbital with principal quantum number ( n − 1 ) . New, more nuanced light is shed on the already challenged dogma that the Pauli principle keeps the electrons further apart in the triplet than in the corresponding singlet.

The one- and two-electron Yukawa atoms, also referred to as the Debye-H\"uckel or screened Coulomb atoms, have been topics of considerable interest both for intrinsic reasons and because of their relevance to terrestrial and astrophysical plasmas. At sufficiently high screening the one-electron Yukawa atom ceases to be bound. Some calculations appeared to suggest that as the screening increases in the ground state of the two-electron Yukawa atom (in which both the one-particle attraction and the interparticle repulsion are screened) the two electrons are detached simultaneously, at the same screening constant at which the one-electron atom becomes unbound. Our results rule this scenario out, offering an alternative that is not less interesting. In particular, it is found that for $Zl1$ a mild amount of screening actually increases the binding energy of the second electron. At the nuclear charge ${Z}_{c}\ensuremath{\approx}0.911028...$, at which the bare Coulomb two-electron atom becomes unbound, and even over a range of lower nuclear charges, an appropriate amount of screening gives rise to a bound two-electron system.

In this study, the information-theoretic measures in both the position and momentum spaces for the pseudoharmonic potential using Fisher information, Shannon entropy, Renyi entropy, Tsallis entropy, and Onicescu information energy are investigated analytically and numerically. The results obtained are applied to some diatomic molecules. The Renyi and Tsallis entropies are analytically obtained in position space using Srivastava–Niukkanen linearization formula in terms of the Lauricella hypergeometric function. Also, they are obtained in the momentum space in terms of the multivariate Bell polynomials of Combinatorics. We observed that the Fisher information increases with n in both the position and momentum spaces, but decreases with l for all the diatomic molecules considered. The Shannon entropy also increases with increasing n in the position space and decreases with increasing l. The variations of the Renyi and Tsallis entropies with l are also discussed. The exact and numerical values of the Onicescu information energy are also obtained, after which the ratio of information-theoretic impetuses to lengths for Fisher, Shannon, and Renyi are obtained. © 2015 Wiley Periodicals, Inc.

Energy eigenvalues of Harmonium atom are reported for the first time under spherically confined Debye and spherically confined exponentially cosine screened coulomb potential. Energy of different states of Harmonium is analyzed as a function of confinement radius and Debye screening length. Comparison of radial matrix elements of Harmonium atom under spherically confined Debye and spherically confined exponentially cosine screened coulomb potential is done. Interesting results are obtained.

By employing the Pekeris-type approximation to deal with the centrifugal term, we solve the Schrodinger equation with the Deng–Fan molecular potential for all values of $$l$$ (orbital angular momentum quantum number). Using the Nikiforov–Uvarov (N–U) method, the approximate analytical bound state energy eigenvalues and the corresponding wave functions are obtained. The results obtained are in good agreement with those ones found in the literature. The bound state energy eigenvalues for a set of diatomic molecules (HCl, LiH, H $$_{2}$$ , ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF) corresponding to the Deng–Fan molecular potential for arbitrary values of n and $$l$$ quantum numbers are reported.

Benchmark numerical results on the ground and excited state eigenvalues and the ground state static and dynamic dipole polarizabilities are reported for a hydrogen atom confined at the center of a spherical box with penetrable walls. The dynamic polarizabilities are negative except when the frequency of incident radiation is below the 1s–2p transition frequency or in the frequencies immediately below a 1s–np transition.

For arbitrary values n and l quantum numbers, we present the solutions of the 3-dimensional Schrodinger wave equation with the pseudoharmonic potential via the SU(1, 1) Spectrum Generating Algebra (SGA) approach. The explicit bound state energies and eigenfunctions are obtained. The matrix elements r 2 and $${r\frac{d}{dr}}$$ are obtained (in a closed form) directly from the creation and annihilation operators. In addition, by applying the Hellmann–Feynman theorem, the expectation values of r 2 and p 2 are obtained. The energy states, the expectation values of r 2 and p 2 and the Heisenberg uncertainty products (HUP) for set of diatomic molecules (CO, NO, O2, N2, CH, H2, ScH) for arbitrary values of n and l quantum numbers are obtained. The results obtained are in excellent agreement with the available results in the literature. It is also shown that the HUP is obeyed for all diatomic molecules considered.

Using dimensional analyses, the scaling properties of the Heisenberg uncertainty relationship as well as the various information theoretical uncertainty-like relationships are derived for the bound states corresponding to the superposition of the power potential of the form V(r) = Zr-n + Sigma(i) Z(i)r(ni), where Z, Z(i), n, n(i) are parameters, in the free state as well as in the additional presence of a spherical penetrable boundary wall located at radius R The uncertainty product and all other net information measures are shown here to depend only on the parameters [s(i)] defined by the ratios Z(i)/Z((ni+2)/(n+2)) Introduction of a finite potential, V-c at the radial distance r >= R results in a complete set of scaling parameters given by [s(i), t(1), t(2)], where t(1) is given by RZ(1(n+2)) and t(2) = V-c/(Z)(2/(n+2)).

Clarifications about the source of error in [Phys. Lett. A 371 (2008) 2271] of momentum-space information measures are given along with the revised estimates which satisfy all the points commented upon by Astashkevich. We note that the position-space quantities calculated previously remain unchanged. Our main conclusion, that the inter-nuclear distance, R, dependence of statistical complexity and Fisher–Shannon measure derived from the simple Coulson wave function contain significant information on molecular bonding in H 2 + , remains qualitatively valid in the light of calculations using more accurate wave functions as reported by Astashkevich.

A statistical relative complexity measure, based on the Kullback–Leibler distance measure defining the relative information and the Carbo quantum similarity index defining the relative disequilibrium is proposed. It is shown that with the specific choice of prior density corresponding to the atom at the beginning of the subshell, this measure reveals the diversity of atoms as the subshells are filled across the periodic table. Numerical tests are reported using the non-relativistic Hartree–Fock as well as the relativistic Dirac–Fock density for all atoms in the periodic table.

We report accurate computations of static and dynamic dipole polarizabilities for the ground and excited states of hydrogen atom confined at the center of either a single and/or multiwalled fullerene cage, using the Galerkin variational method developed on a B-spline basis set. The results obtained in the cage-free case are in good agreement with other numerical techniques such as the mapped Fourier grid method. It is shown that the addition of new walls does not modify the polarizability of the ground state, but changes significantly those of the excited states. A condition on the ratio of the scaled second derivative of density to the scaled density evaluated at the nuclear position is tested in case of the cage confined atoms for the first time, and the significance of such numerical results are discussed. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

We present an investigation of the (1s2s)1,3S excited states of the two-electron atom immersed in a plasma modeled by the Debye or screened Coulomb potential. Three variants of the Debye atom are considered. The validity of Hund's multiplicity rule is confirmed, and the contribution of the interparticle repulsion energy to the singlet-triplet splitting is examined. The feature that this system shares with the unscreened two-electron atom as well as with the confined two-electron atom and the two-electron quantum dot is that the triplet wave function is contracted relative to that of the singlet. This feature affects both the behavior of the 2s-electron ionization energies and the relative magnitudes of the interparticle repulsion energies in the singlet vs. the triplet. Debye screening of the one-body attraction effectively reduces the nuclear charge, enhancing the reversal of the relative magnitudes of the triplet vs. singlet interparticle repulsion energies. Debye screening of the interparticle repulsion acts in an opposite way.We present an investigation of the (1s2s)1,3S excited states of the two-electron atom immersed in a plasma modeled by the Debye or screened Coulomb potential. Three variants of the Debye atom are considered. The validity of Hund's multiplicity rule is confirmed, and the contribution of the interparticle repulsion energy to the singlet-triplet splitting is examined. The feature that this system shares with the unscreened two-electron atom as well as with the confined two-electron atom and the two-electron quantum dot is that the triplet wave function is contracted relative to that of the singlet. This feature affects both the behavior of the 2s-electron ionization energies and the relative magnitudes of the interparticle repulsion energies in the singlet vs. the triplet. Debye screening of the one-body attraction effectively reduces the nuclear charge, enhancing the reversal of the relative magnitudes of the triplet vs. singlet interparticle repulsion energies. Debye screening of the interparticle repulsio...

Theoretic measures of information entropies like Shannon entropy and Fisher information are studied for multiple quantum well systems(MQWS). The effect of shape and number of wells in the MQWS is explored in detail. The shapes taken are: rectangular, parabolic and V-shape. Onicescu energy is an important tool to study the information content stored in the system, which is also found to depend on shape and number of wells of heterostructures. Statistical measure of complexity also shows noticeable dependence on these parameters.

Information-theory measures, in particular the Shannon entropy, Fisher information and statistical complexity, are used to discuss the variations among several commonly encountered model two-electron correlated wavefunctions. The Hookean, Moshinsky, and three-parameter Chandrasekhar wavefunctions are considered in real and momentum space, with further comparisons to the Hookean-Hartree-Fock (HF) wavefunction of Ragot, the numerical HF limit, and the hydrogenic (pure Coulomb) limit. The purpose of the study is to quantitatively analyze the effect of different models for inclusion of electron-electron correlation on information-theoretical measures, including statistical complexity, which characterize the electron distribution in position and momentum space.

Lower bound for the shape complexity measure of Lopez-Ruiz–Mancini–Calbet (LMC), C LMC is studied. Analytical relations for simple examples of the harmonic oscillator, the hydrogen atom and two-electron ‘entangled artificial’ atom proposed by Moshinsky are derived. Several numerical examples of the spherically confined model systems are presented as the test cases. For the homogeneous potential, C LMC is found to be independent of the parameters in the potential which is not the case for the non-homogeneous potentials.

It is shown that the energy of a hydrogen-like atom confined inside a spherical cavity of radius, R, and potential barrier, V0, is quantitatively defined by the ratio . Here, the conventional spherical density (r) is scaled as ηl(r) = and the ratio of the second derivative η(r) to ηl(r) is evaluated at the nucleus. Numerical results of the ratios are presented for 1s, 2s, 2p, and 3d states at several values of V0. For such states, the characteristic radii of confinement leading to the well-defined values of energy are identified. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Numerical calculations on the gradient and Laplacian forms of the position space Fisher information measure are reported using the 1-normalised Dirac–Fock densities (shape function), σ ( r ) , for atoms H–Lr. It is shown that the difference in effective electrostatic potentials, corresponding to the gradient and the Laplacian form of Fishers' information, is completely defined by the shape function (the density per particle) at the nucleus, σ ( r = 0 ) . The influence of relativistic effects on the Fisher information is recovered for the first time.

Using the Hartree–Fock non-relativistic wave functions in the position and momentum spaces, the statistical measure of complexity C, due to Lopez-Ruiz, Mancini, and Calbet for the neutral atoms as well as their monopositive and mononegative ions with atomic number Z = 1–54 are reported. In C, given by the product of exponential power Shannon entropy and the average density, the latter is then replaced by the Fisher measure to obtain the Fisher–Shannon plane. Our numerical results suggest that in overall the Fisher–Shannon plane reproduces the trends given by C, with significantly enhanced sensitivity in the position, momentum and the product spaces in all neutral atoms and ions considered. © 2007 Elsevier B.V. All rights reserved.

Using the Dirac–Fock relativistic wave functions in the position space the shape complexity, C LMC s , form of the statistical complexity measure due to Lopez-Ruiz, Mancini, and Calbet and the simple two parameter complexity measure, Γα,β, proposed by Shiner, Davison and Landsberg for atoms H–Lr are calculated. A comparison with the non-relativistic Hartee–Fock results is carried out to ascertain the influence of the relativistic effects on complexity. Unlike the non-relativistic Hartree–Fock results, which lead to the non-increasing C LMC s , the Dirac–Fock calculations lead to both C LMC s and Γα,β clearly displaying an increasing trend in complexity as the atomic number increases.

The dimensional analyses of the position and momentum variance based quantum mechanical Heisenberg uncertainty measure and the other useful net entropic information measures for the bound states of two constrained Coulomb potentials are reported for the first time. The potentials describe an electron moving in the central field due to a nucleus of charge Z with radius R defining the constraints as (a) the truncated potential given by , and (b) the radius of the impenetrable spherical wall. The net information measures for the two potentials are explicitly shown to be independent of the scaling of the set [Z, R] at a fixed value of ZR. Analytic proof is presented, for the first time, showing the presence of a characteristic extremum in the variation of the net information entropy as a function of the radius R with its location scaling as Z−1. Numerical results are presented which support the validity of the scaling properties.

The dimensional analyses of the position and momentum variances-based quantum mechanical Heisenberg uncertainty measure, as well as the entropic information measures given by the Shannon information entropy sum and the product of Fisher information measures are carried out for two widely used nonrelativistic isotropic exponential-cosine screened Coulomb potentials generated by multiplying the superpositions of (i) Yukawa-like, −Z(e−μr/r), and (ii) Hulthen-like, −Zμ(1/(eμr − 1)), potentials by cos(bμr) followed by addition of the term a/r2, where a and b ≥ 0, μ are the screening parameters and Z, in case of atoms, denotes the nuclear charge. Under the spherical symmetry, all the information measures considered are shown to be independent of the scaling of the set [μ, Z] at a fixed value of μ/Z, a, and b and the other parameters defining the superpositions of the potentials. Numerical results are presented, which support the validity of the scaling properties. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

In presence of the spherically confined three-dimensional potentials with impenetrable boundaries, the generalized pseudospectral method is shown to provide accurate eigenvalues, eigenfunctions, and radial expectation values for (a) the isotropic harmonic oscillator, (b) the H atom and (c) the Davidson oscillator. Several novel degeneracy conditions are obtained for (a) when the radius of confinement is suitably chosen at the radial nodes corresponding to the free states.

Following the concept of static exchange-correlation charge density, the total integrated exchange-charge density parameter, Qx, is calculated within the exchange-only, relativistic optimized potential model and employed to examine the atomic shell boundaries in nobelium atom. Six clear maxima in Qx(r) are shown to reflect all the shell boundaries. A comparison of the Qx(r) with the non-relativistic calculations reveal interesting differences in the core and valence regions. The radial behavior of Qx(r) is shown to be extremely sensitive to the details of the relativistic effects.

Using the mathematical properties of the confluent hypergeometric functions, the conditions for the incidental, simultaneous, and interdimensional degeneracy of the confined D-dimensional (D > 1) harmonic oscillator energy levels are derived, assuming that the isotropic confinement is defined by an infinite potential well and a finite radius Rc. Very accurate energy eigenvalues are obtained numerically by finding the roots of the confluent hypergeometric functions that confirm the degeneracy conditions. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

A model physical problem is studied in which a system of two electrons is subject either to soft confinement by means of attractive oscillator potentials or by entrapment within an impenetrable spherical box of finite radius $R.$ When hard confinement is present the oscillators can be switched off. Exact analytical solutions are found for special parameter sets, and highly accurate numerical solutions (18 decimal places) are obtained for general cases. Some interesting degeneracy questions are discussed at length., Comment: 17 pages, two figures. arXiv admin note: substantial text overlap with arXiv:1108.4878

The density functional descriptors of chemical reactivity given by electronegativity, global hardness and softness are reported for a representative set of spherically confined atoms of IA, IIA, VA and VIIIA series in the periodic table. The atomic electrons are confined within the impenetrable spherical cavity defined by a given radius of confinement satisfying the Dirichlet boundary condition such that the electron density vanishes at the radius of confinement. With this boundary condition the non-relativistic spin-polarized Kohn-Sham equations were solved. The electronegativity in a confined atom is found to decrease as the radius of confinement is reduced suggesting that relative to the free state the atom loses its capacity to attract electrons under confined conditions. While the global hardness of a confined atom increases as the radius of confinement decreases, due to the accompanying orbital energy level crossing, it does not increase infinitely. At a certain confinement radius, the atomic global hardness is even reduced due to such crossover. General trends of the atomic softness parameter under spherically confined conditions are reported and discussed.

Cusp relations for the kinetic energy density are derived for the ground and excited states of atoms, ions or molecules. The two most frequently used kinetic energy density expressions imply different behaviour at the nuclei.

Using classical electrostatics, the total effective integrated charge-density function is calculated for Be and Ne using the multiplicative potentials derived from (1) Hartree and (2) Hartree–Fock approximation to quantal density functional theory (3)exchange-only optimized effective potential and (4) Kohn–Sham exchange-correlation potential using the quantum Monte Carlo density. The evolution of effective integrated charge-density function for these atoms is examined as the electron correlation is built up stepwise from its absence to the stage of its near complete presence. These results provide a deeper understanding of the Kohn–Sham exchange-correlation potential in terms the correspondingly defined integrated charge-density functions based on the Poisson equation.

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.

In this Letter Quantum Similarity for Atoms (H–Xe) is investigated using electron densities and shape functions, looking for patterns of periodicity as in Mendeleev’s Table. An LS -dependent restricted Hartree–Fock method is used to obtain the wave functions from which the electron densities are calculated. Utilizing the quantum similarity proposed by Carbo a nearest neighbour dominated similarity is retrieved, masking periodicity. Introduction of the information discrimination concept with reference to the noble gas atom of the previous row is found to reveal periodicity, with improved results when densities are replaced by shape functions throughout. This confirms recent literature on the fundamental role of the shape function as carrier of information.

The central role of the shape function sigma(r) from the density functional theory (DFT), the ratio of the electron density rho(r) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential micro of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined.

Using the static exchange-correlation charge density concept, the total integrated exchange-charge density function is calculated within the nonrelativistic spin-restricted exchange-only (i) optimized effective potential model, and (ii) nonvariational local potential derived from the exchange-only work potential within the quantal density functional theory, for the ground-state isoelectronic series: Ga+, Zn, Cu−; In+, Cd, Ag−; and Tl+, Hg, Au−. The difference between the exchange charge density function derived from these potentials is employed to evaluate the first-order correlation-kinetic contribution to the integrated exchange charge density. This contribution is found to be important for both the intra- and inter-shell regions. Screening effects on the contribution due to the nd10 (n = 3–5) subshells are discussed through comparisons with similar calculations on Ca, Sr, and Ba, wherein nd10 electrons are absent. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Over the past decade and a half, many new accurate density functionals, based on the generalized gradient approximation, have been proposed, and they give energies close to chemical accuracy. However, accuracy of the energy functional does not guarantee that its functional derivative, which gives the corresponding potential, is also accurate all over space. For example, although the Becke88 exchange-energy functional gives very good exchange energies, its functional derivative goes as —1/2 m comparison to the correct —1 for r ⇇ where ris the distance of the electron from a finite system. On the other hand, accuracy of the potential is of prime importance if one is interested in properties other than the total energy; properties such as optical response depend crucially on the potential in the outer regions of a system. In this paper we present a different approach, based on the ideas of Harbola and Sahni, to obtain the potential directly from the energy density of a given approximation, without taking recourse to the functional derivative route. This leads to a potential that is as accurate as the functional itself. We demonstrate the accuracy of our approach by presenting some results obtained from the Becke88 functional.