First-order Density Research Articles

Hyperbolic sine model: A multidisciplinary stochastic process.

The hyperbolic sine model is obtained by continuing the hierarchy of simple multiplicative stochastic processes beyond the Verhulst model or by transforming some physically important models from different disciplines to their canonical form. The Fokker-Planck equation associated with this Markov process is solved for the stationary first-order probability density function (PDF), and an exact time-dependent solution for the transition PDF is derived in terms of a new type of eigenfunction. As an example, eigenvalues and eigenfunctions are numerically evaluated for the symmetrical case. It is shown that some new models, of possible mathematical or physical importance, can be derived from the hyperbolic sine model.

Generalised 1/ f shot noise

We define a generalised form of 1/f shot noise, which can serve as a source of 1/fα noise for α in the range 0<α<2, and present its first-order amplitude probability density function. For some parameters of the power-law-decaying impulse-response function, the amplitude has a Levy-stable probability density function.

Some properties of symmetrized transition density matrices based on configuration interaction wavefunctions

The discrete, spinless first-order transition density matrix, Γ QR associated with two electronic wavefunctions, Ψ Q and Ψ R, is normally not Hermitian (or symmetric). In the real case a symmetric transition density matrix, Δ QR is defined by Δ kl QR= 12 (Ψ kl QR + Γ lk QR. In the present work some properties of this symmetric matrix are discussed. It is proven that in some cases the eigenvalues of Δ QR occur in pairs such that λ (≠0) is an eigenvalue than - λ is also an eigenvalue. When Ψ Q and Ψ R belong to different irreducible representations (IRs) of an Abelian point group the relation is proven to be exact. If the functions belong to the same IR it is demonstrated by numerical computations that the correspondence is only approximately fulfilled.

Electron Correlation and the Hall Effect in Liquid Metals

Abstract A formula is proposed for the deviation δRH of the Hall coefficient RH from 1/nec. Knowledge of the first-order density matrix characterizes the formula. With strong electron correlation, δRH will depend on the discontinuity of the electron momentum distribution at the Fermi surface, as well as on the strength of the electron-ion interaction in the liquid metal.

Thermal properties of many-electron systems: An integral formulation of density-functional theory.

A new approach for the calculation of thermal properties of many-electron systems is proposed via an integral formulation of the Mermin-Kohn-Sham finite-temperature density-functional theory. The electron density of a thermal-equilibrium state can be determined by solving self-consistently equations for the electron density without using orbitals. Exchange and correlation effects are incorporated. In place of the set of the single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. The formulation can also be applied to general fermions.

Ab initio approach for many-electron systems without invoking orbitals: An integral formulation of density-functional theory.

A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.

Exact relation between density-matrix and density-functional theory for K plus L closed shells in a bare Coulomb field.

The explicit form of the first-order density matrix \ensuremath{\rho}${(\mathrm{r}}_{1}$,${\mathrm{r}}_{2}$) for independent electrons moving in a bare Coulomb field is first set up for the case of K plus L closed shells. The density matrix has a simple separable form in terms of the variables ${r}_{1}$+${r}_{2}$ and \ensuremath{\Vert}${\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$\ensuremath{\Vert}, a property exclusive to the Coulomb potential. Also, the off-diagonal dependence is simply quadratic in \ensuremath{\Vert}${\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$\ensuremath{\Vert}. These two properties allow \ensuremath{\rho}${(\mathrm{r}}_{1}$,${\mathrm{r}}_{2}$) to be written solely in terms of electron and kinetic energy densities. Other implications for density-functional theory are briefly referred to.

Differential force law and related integral theorems for a system of N identical interacting particles. II. Spherical symmetry

The differential force law (DFL) and related integral theorems, derived in a previous paper for general geometries, are applied to spherical systems of identical interacting particles, e.g., electrons. From the special functional form of the first-order density matrix, induced solely by symmetry, the DFL occurs as a scalar, pointwise equation relating radial and rotational parts, trad and trot, of the kinetic energy density t to the force density f and to derivatives of the particle density n. Furthermore, an exact connection between the pressure p, trad, and n is established. Finally some theorems are derived which relate integrals, extending over an arbitrary concentric part of the system and involving t, p, and f, to values of p, n, and n′ at the surface of the sphere with radius R. One of these theorems is a generalized virial theorem tending to the usual well-known virial theorem in the limit R→∞.

The Kohn-Sham formalism: A critique and an extension

The kohn-Sham formalism is critically reviewed for the purpose of pointing out some of its difficulties. In particular, it is shown that the equality between the one-electron density ϱ Φ (r) given in terms of N single-particle functions and ϱ Ψ(r) expressed in terms of m > N orbitals, does not guarantee that the variational space remains unmodified. An extension of the Kohn-Sham equations is discussed in the context of a two-step variation of the reduced first-order density operator expressed in an orbital expansion.

Hartree-Fock exchange energy from the application of generalized Padé approximants to the first-order density matrix.

Hartree-Fock exchange energy from the application of generalized Padé approximants to the first-order density matrix.

Asymptotic formula far from nucleus for exchange energy density in Hartree-Fock theory of closed-shell atoms.

In Hartree-Fock theory, the exchange energy density can be expressed solely in terms of the first-order density matrix. Far from the nucleus of a closed-shell atom, idempotency of the density matrix yields the exchange energy density as the magnitude of the Coulomb energy ${e}^{2}$/r times the electron density \ensuremath{\rho}. Thus two lengths enter the asymptotic form in contrast to ${\ensuremath{\rho}}^{\mathrm{\ensuremath{-}}1/3}$ alone of local-density theory.

Strong consistency and rates for recursive probability density estimators of stationary processes

Let { X j} j = − ∞ ∞ be a vector-valued stationary process with a first-order univariate probability density f on R d . We consider the recursive estimation of f( x) from n observations { X j} j=1 n which need not be independent. For processes { X j} j = − ∞ ∞ which are asymptotically uncorrelated, we establish sharp rates for the almost sure convergence of kernel-type estimators f n ( x).

Almost sure convergence of recursive density estimators for stationary mixing processes

Let { X j } be a vector-valued stationary strong mixing process with a first-order probability density f on R d . We establish rates of almost sure convergence for recursive density estimators n ( x) of f( x) of the kernel type.

Asymptotic form of first-order density matrix in the Hartree-Fock approximation for doubly occupied levels

By using asymptotic results on the first- and second-order density matrices, γ( r, r′) and Г( r 1 r 2; r′ 1 r′ 2 re spectively, it is shown that γ( r, r′) for large r′ and all r can be expressed, in the Hartree-Fock approximation for doubly occupied levels, solely through the electron densities of the N- and ( N − 1)-particle systems.

Energies, electron densities, and first-order density matrices from one-body external potentials. II.

Energies, electron densities, and first-order density matrices from one-body external potentials. II.

Gaussian and other approximations to the first-order density matrix of electronic systems, and the derivation of various local-density-functional theories

Various approximations and models for Hartree-Fock kinetic energy T and Hartree-Fock exchange energy K are systematically derived with a minimum of assumptions and comparatively studied by testing on atoms. A Gaussian ansatz for the spherical average of the first-order density matrix and the uniform-gas ansatz are shown both to lead to the same formulas for T and K except for the values of certain numerical coefficients. Without any assumption one gets in all cases T=(3/2) F \ensuremath{\rho}(r)[1/\ensuremath{\beta}(r)]dr, where \ensuremath{\beta}(r) is the local temperature parameter of Ghosh, Berkowitz, and Parr [Proc. Natl. Acad. Sci. USA 81, 8028 (1984)]. The Gaussian ansatz gives K=(\ensuremath{\pi}/2) F ${\ensuremath{\rho}}^{2}$(r)\ensuremath{\beta}(r)dr, while the uniform-gas ansatz gives K=(9\ensuremath{\pi}/20) F ${\ensuremath{\rho}}^{2}$(r)\ensuremath{\beta}(r)dr. Assuming further only the existence of a local equation of state of the form \ensuremath{\beta}=\ensuremath{\beta}(\ensuremath{\rho}), one then gets, by imposing the exact normalization condition for the first-order density matrix, from the Gaussian ansatz T=(3\ensuremath{\pi}${/2}^{5/3}$) F ${\ensuremath{\rho}}^{5/3}$(r)dr and K${=2}^{\mathrm{\ensuremath{-}}1/3}$ F ${\ensuremath{\rho}}^{4/3}$(r)dr, respectively, 3.4% and 7.5% larger than the Thomas-Fermi-Dirac T and K which result from the uniform-gas ansatz. Numerical evidence is presented that shows preference for the Gaussian ansatz. A modified Gaussian ansatz also is examined.

Channel Modeling and Threshold Signal Processing in Underwater Acoustics: An Analytical Overview

An overview of underwater acoustic channel modeling and threshold signal processing is presented, which emphasizes the inhomogeneous, random, and non-Ganssian nature of the generalized channel, combined with appropriate weak-signal detection and estimation. Principal attention is given to the formal structuring of the scattered and ambient acoustic noise fields, as well as that of the desired signal, including both fading and Doppler "smear" phenomena. The role of general receiving arrays is noted, as well as their impact on spatial and temporal signal processing and beam forming, as indicated by various performance measures in detection and estimation. The emphasis here is on limiting optimum threshold systems, with some attention to suboptimum cases. Specific first-order probability density functions (pdf's) for the non-Ganssian components of typical underwater acoustic noise environments are included along with their field covariances. Several examples incorporating these pdf's are given, to illustrate the applications and general methods involved. The fundamental role of the detector structure in determining the associated optimum estimators is noted: the estimators arc specific linear or nonlinear functionals of the original optimum detector algorithm, depending on the criterion (i.e., minimization of the chosen error or cost function) selected. Results for both coherent and incoherent modes of reception are presented, reflecting the fact that frequently signal epoch is not known initially at the receiver. To supplement the general discussion, a selected list of references is included, to provide direct access to specific detailed problems, techniques, and results, for which the present paper is only a guide.

A CNDO/S study of the importance of electron repulsion parameters and charge density changes on the weakness of the v8(e 2g) vibronic activity in the benzene 260 nm band

We investigate the disparity between the large observed ratio ( ≈100:1 ) of v 6 to v 8 vibrationally induced intensity in the 260 nm ( 1B 2u– 1A 1g) absorption band of benzene and that computed from CNDO/S and INDO/S (typically 4:1). The calculated v 8 intensity is relatively insensitive to small admixtures of the v 6 mode, but is found to be quite sensitive to the nature of the semiempirical electron repulsion integral (γ) distance dependence. A new perturbation theory is applied successfully which relates the vibronic coupling directly to changes in the Fock matrix and the simple first-order transition density.

Energies, electron densities, and first-order density matrices from one-body external potentials.

To help simplify the many-electron problem, a new algorithm is presented which enables the generation of approximate ground-state electron densities and approximate ground-state energy differences of isoelectronic processes for interacting and noninteracting systems. This is accomplished starting exclusively from knowledge of the following: one isoelectronic ground-state ``seed'' density ${\ensuremath{\rho}}_{A}$, the external potential ${v}_{A}$ which generates ${\ensuremath{\rho}}_{A}$, and the external potential ${v}_{B}$ for which the ground-state energy ${E}_{B}$ and density ${\ensuremath{\rho}}_{B}$ are sought. Use is made of a matching theorem in conjunction with trial first-order density matrices (one-matrices), the virial theorem, and a variation of a Legendre-transform idea. Furthermore, knowledge of the exact energy ${E}_{A}$ enables the approximation of ${E}_{B}$ using either one of two energy-difference formulas which are approximations to the exact integrated Hellmann-Feynman energy difference. Initial tests for noninteracting electrons are constructed.

The atom in a molecule: A density matrix approach

A method based on first-order density matrices is proposed to define an atom in a molecule, which is in accord with a previously given definition using density functional theory. The promotion energy Ep is expressed in terms of a defined division function α(r). By minimizing Ep with respect to α(r), one can obtain α(r), by which an atom in a molecule is uniquely determined. It is shown that such an ‘‘atom’’ satisfies a local virial theorem, depicted intuitively, Some other properties of the atom in a molecule are also discussed. Comparison is made with a definition due to Bader, and an approximate but simple approach to determine an atom in a molecule is proposed.