Electrostatic Interpretation Research Articles

Monotonicity, interlacing and electrostatic interpretation of zeros of exceptional Jacobi polynomials

Denote by P̂n(α,β)(x) the X1-Jacobi polynomial of degree n. These polynomials were introduced and studied recently by Gómez-Ullate, Kamran and Milson in a series of papers. In this note we establish some properties of the zeros of P̂n(α,β)(x), such as interlacing and monotonicity with respect to the parameters α and β. They turn out to possess an electrostatic interpretation. The vector, whose components are the zeros, is a saddle point of the energy of the corresponding logarithmic field.

Heine-Stieltjes correspondence and a new angular momentum projection for many-particle systems

A new angular momentum projection for systems of particles with arbitrary spins is formulated based on the Heine-Stieltjes correspondence, which can be regarded as the solutions of the mean-field-plus -pairing model in the strong-pairing interaction $G\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. Properties of the Stieltjes zeros of the extended Heine-Stieltjes polynomials, whose roots determine the projected states, and the related Van Vleck zeros are discussed. An electrostatic interpretation of these zeros is presented. As examples, applications to $n$ nonidentical particles of spin $1/2$ and to identical bosons or fermions are made to elucidate the procedure and properties of the Stieltjes zeros and the related Van Vleck zeros. It is shown that the new angular momentum projection for $n$ identical bosons or fermions can be simplified with the branching multiplicity formula of U$(N)\ensuremath{\downarrow}\mathrm{O}(3)$ and the special choices of the parameters used in the projection. Especially, it is shown that the solutions for identical bosons can always be expressed in terms of zeros of Jacobi polynomials. However, unlike nonidentical particle systems, the $n$-coupled states of identical particles are nonorthogonal with respect to the multiplicity label after the projection.

Jacobi–Sobolev-type orthogonal polynomials: holonomic equation and electrostatic interpretation – a non-diagonal case

Consider the Sobolev-type inner product where p and q are polynomials with real coefficients, α, β>−1, ℙ(x)=(p(x), p′(x)) t , and is a positive semidefinite matrix, with M 0, M 1≥0, and λ∈ℝ. We obtain an expression for the family of polynomials , orthogonal with respect to the above inner product, a connection formula that relates with some family of Jacobi polynomials and the holonomic equation that they satisfy, as well as an electrostatic interpretation of their zeros.

Zeros of orthogonal polynomials generated by canonical perturbations of measures

In this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the Uvarov perturbation of a positive Borel measure dμ(x). When the measure is semiclassical, then its electrostatic interpretation is given.

Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

We find approximate expressionsx̃(k,n,a)andỹ(k,n,a)for the real and imaginary parts of thekth zerozk=xk+iykof the Bessel polynomialyn(x;a). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions ofk,nandais obtained. It is shown that the resulting complex numberx̃(k,n,a)+iỹ(k,n,a)isO(1/n2)-convergent tozkfor fixedk.

The Laguerre-Sobolev-type orthogonal polynomials. Holonomic equation and electrostatic interpretation

The Laguerre-Sobolev-type orthogonal polynomials. Holonomic equation and electrostatic interpretation

Composite‐system models

AbstractComposite many‐electron systems are considered in terms of the wave‐functions for the different component subsystems (e.g., atoms). In this weak‐interaction limit, there may be degrees of freedom, which arise from that of the degeneracies of the wave‐functions for each of the isolated subsystems, and which then are describable in terms of an effective Hamiltonian for the couplings and liftings of these degeneracies in the composite system. The representation of an ordinary Schrödinger Hamiltonian in such a 0‐order model space is developed in the context of an example of a (possibly macroscopically large) set of H atoms interacting with one another, to obtain a Heisenberg spin‐Hamiltonian, which is formally “complete” in not neglecting any higher‐order permutations. At the same time, the interactions are hierarchically ordered to manifest the more important ones first, all in a “linked” size‐consistent manner, avoiding various earlier‐encountered “catastrophes.” A classical electrostatic interpretation of the higher order exchange matrix elements is made, and some novel spin‐coupling dependences are found with a decay like an inverse power‐law range (rather than exponential). © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

On a class of equilibrium problems in the real axis

In a series of seminal papers, Thomas J. Stieltjes (1856–1894) gave an elegant electrostatic interpretation for the zeros of classical families of orthogonal polynomials, such as Jacobi, Hermite and Laguerre polynomials. More generally, he extended this approach to the zeros of polynomial solutions of certain second-order linear differential equations (Lamé equations), the so-called Heine–Stieltjes polynomials. In this paper, a class of electrostatic equilibrium problems in R , where the free unit charges x 1 , … , x n ∈ R are in presence of a finite family of “attractors” (i.e., negative charges) z 1 , … , z m ∈ C ∖ R , is considered and its connection with certain class of Lamé-type equations is shown. In addition, we study the situation when both n → ∞ and m → ∞ , by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields.

Nearest-neighbour spacing distributions of the β-Hermite ensemble of random matrices

The evolution with β of the distributions of the spacing ‘ s’ between nearest-neighbour levels of unfolded spectra of random matrices from the β-Hermite ensemble ( β-HE) is investigated by Monte Carlo simulations. The random matrices from the β-HE are real symmetric and tridiagonal where β, which can take any positive value, is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues. The distribution of eigenvalues coincide with those of the three classical Gaussian ensembles for β=1, 2, 4. The use of the β-HE ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices. Generalized gamma distributions are shown to be excellent approximations of the nearest-neighbor spacing (NNS) distributions for any β while being still simple. They account both for the level repulsion in ∼ s β when s→0 and for the whole shape of the NNS distributions in the range of ‘ s’ which is accessible to experiment or to most numerical simulations. The exact NNS distribution of the GOE ( β=1) is in particular significantly better described by a generalized gamma distribution than it is by the Wigner surmise while the best generalized gamma approximation coincides essentially with the Wigner surmise for β>∼2. They describe too the evolution of the level repulsion between that of a Poisson distribution and that of a GOE distribution when β increases from 0 to 1. The distribution of ln ( s), related to the electrostatic interaction energy between neighbouring charges, is accordingly well approximated by a generalized Gumbel distribution for any β⩾0. The distributions of the minimum NN spacing between eigenvalues of matrices from the β-HE, obtained both from as-calculated eigenvalues and from unfolded eigenvalues are Brody distributions which are classically used to characterize the spectral fluctuations of various physical systems.

The fixed-trace β-Hermite ensemble of random matrices and the low temperature distribution of the determinant of an N × N β-Hermite matrix

The β-Hermite ensemble (β-HE) of tridiagonal N × N random matrices of Dumitriu and Edelman (2002 J. Math. Phys. 43 5830) is a continuum of ensembles in which β, the reciprocal of the temperature in the 2D electrostatic interpretation of the eigenvalue characteristics, can take any value. The eigenvalue distributions coincide with those of the classical Gaussian ensembles (GOE, GUE, GSE) for β = 1, 2, 4. A fixed-trace β-Hermite ensemble (β-FTHE) is defined from the β-HE and is used to extend the spherical ensembles of classical symmetries to β-spherical ensembles. At low temperature, when β → ∞, for a fixed value of N, the asymptotic distributions of reduced determinants DN,β of random N × N β-H and β-FTH matrices are shown to be standard Gaussians. Accordingly, the fluctuations of the potential at the origin, −ln |DN,β|, have a generalized Gumbel distribution at low temperature. For large N and large β, a ln(N) variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions.

Electrostatic models for zeros of polynomials: Old, new, and some open problems

We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented.

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.

Electrostatic interpretation of the force − ∂Vxc/ ∂r connected with the exchange-correlation potential: direct relation to single-particle kinetic energy density in Be-atom

In a recent study by one of us [Phys. Rev. A 65 (2002) 034501], an electrostatic interpretation has been proposed for the force − ∂V xc/ ∂r associated with the exchange-correlation potential energy V xc( r) in a spherical atom such as Be or Ne. Here, this proposal is employed to relate − ∂V xc/ ∂r directly to (a) the exact ground-state density ρ( r) (e.g., from diffraction experiments or quantum Monte Carlo simulation) and its low-order derivatives, and (b) the single-particle kinetic energy density t s( r), calculated again from the exact ρ( r), for the example of the Be-atom. It would, of course, be important if this type of relation could be extended beyond a (1s) 2(2s) 2 two-level single-particle configuration.

Applications of electrostatic interpretation of components of effective Kohn–Sham potential in atoms

The fundamental significance of the components of the electronic Kohn–Sham potential evaluated at the nucleus is highlighted via the numerical studies on atoms He–Lu which suggest their formally similar power-law relationship in expressing the associated components of total electronic energy. Similar studies on the isoelectronic series of closed shell atoms lead to the linear correlations. The proposed static exchange–correlation charge density concept [S. Liu, P. A. Ayers, and R. G. Parr, J. Chem. Phys. 111, 6197 (1999)] is used to interpret these relationships. The maxima in the static integrated radial exchange–correlation charge density function, Qxc(r), in atoms are shown to reflect the shell boundaries. The quantum Monte Carlo density derived exchange–correlation potentials for Be and Ne are used to obtain Qxc(r) that can be used as standards to directly assess the quality of approximate exchange–correlation potentials. For the negative ions, Qxc(r) displays a characterstic outer minimum as a consequence of the Sen–Politzer theorem [K. D. Sen and P. Politzer, J. Chem. Phys. 90, 4370 (1989)]. This minimum is found to be related with the stability of negative ions.

Electrostatic interpretation for the zeros of certain polynomials and the Darboux process

The purpose of this paper is to extend the electrostatic interpretation of Stieltjes to the zeros of a larger class of important polynomials. To keep things simple I consider polynomials that are obtained from those considered by Stieltjes by means of the so-called Darboux process applied to the difference operator featuring in their three term recursion relation. It will turn out that some of the beauty in the results of Stieltjes is preserved in this process. It would be interesting to interpret the Darboux process at the electrostatic level and this paper gives “experimental data” that might help in this direction.

Lamé differential equations and electrostatics

The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation A ( x ) y ′ ′ + 2 B ( x ) y ′ + C ( x ) y = 0 , \begin{equation*} A(x) y^{\prime \prime } + 2 B(x) y’ + C(x) y = 0, \end{equation*} where A ( x ) , B ( x ) A(x), B(x) and C ( x ) C(x) are polynomials of degree p + 1 , p p+1, p and p − 1 p-1 , is under discussion. We concentrate on the case when A ( x ) A(x) has only real zeros a j a_{j} and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients r j r_{j} in the partial fraction decomposition B ( x ) / A ( x ) = ∑ j = 0 p r j / ( x − a j ) B(x)/A(x) = \sum _{j=0}^{p} r_{j}/(x-a_{j}) , we allow the presence of both positive and negative coefficients r j r_{j} . The corresponding electrostatic interpretation of the zeros of the solution y ( x ) y(x) as points of equilibrium in an electrostatic field generated by charges r j r_{j} at a j a_{j} is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.

An Electrostatics Approach to the Determination of Extremal Measures

One of the most important aspects of the minimal energy (or induced equilibrium) prob- lem in the presence of an external field - sometimes referred to as the Gauss variation problem - is the determination of the support of its solution (the so-called extremal measure associated with the field). A simple electrostatic interpretation is presented here, which is apparently new and anyway suggests a novel, rather systematic approach to the solution. By way of illustration, the classical results for Jacobi, Laguerre and Freud weights are explicitly recovered by this alternative method.

Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials

We show a way to adapt the ideas of Stieltjes to obtain an electrostatic interpretation of the zeroes of a large class of orthogonal polynomials.

From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex

The electrostatic interpretation of the Jacobi--Gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex. Moreover, several new estimates, based on extensive numerical studies, for approximation along the line using Jacobi--Gauss--Lobatto quadrature points as the nodal sets are presented. The electrostatic analogy is extended to the two-dimensional case, with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented. The matrices are evaluated by computing the Lebesgue constants and they share the property that the nodes along the edges of the simplex are the Gauss--Lobatto quadrature points of the Chebyshev and Legendre polynomials, respectively. This makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h-p finite element methods.

Nuclear quadrupole coupling constants in alkali halide molecules: an ab initio quantum chemical study

The electric field gradients, and hence nuclear quadrupole coupling constants, plus the dipole moments of the alkali halide molecules LiF, LiCl, LiBr, NaF, NaCl, NaBr, KF, KCl and KBr have been calculated using ab initio quantum chemical methods that range from SCF to coupled cluster techniques with perturbative corrections for triple excitations [CCSD(T)], employing basis sets of quadruple zeta quality with multiple sets of polarization functions. Zero-level vibrational corrections to these quantities and spectroscopic constants also were computed. The level of agreement with experimental data is generally good, especially in the case of the light molecules. The field gradients at the Na and Cl nuclei in NaCl were analysed using the constrained spatial orbital variation technique in order to quantify the importance of polarization, charge transfer and Pauli repulsion contributions, and to understand why a simple electrostatic interpretation of the field gradients fails in the case of alkali halides. The dipole polarizabilities, Sternheimer antishielding constants γ∞ and electric field gradient–electric field hyperpolarizabilities ϵ of the alkali and halide ions also were calculated at a correlated level of theory (ACPF).