Electron density near an impurity

The behavior of the electron density n(r) [and potential energy V(r)] near an impurity of charge Z is studied by using the linear response theory of an electron gas at finite temperature and with exchange and correlation effects included. The odd powers series in the expansion of n(r) [and V(r)] are calculated exactly by using asymptotic methods, and the coefficients in the series are given in terms of moments taken over the Fermi–Dirac distribution function. In all linear response theories and at all temperatures, the derivative n'(0) = -2Zn₀/a₀, where n₀ is the unperturbed electron density and a₀ is the Bohr radius. The effects of exchange and correlation appear in the fifth- and higher-order terms in n_odd(r).     

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.     

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.     

Exchange energy for two closed shells generated by a bare Coulomb potential energy −<Emphasis Type="Italic">Ze</Emphasis><Superscript>2</Superscript>/<Emphasis Type="Italic">r</Emphasis> in the limit of large <Emphasis Type="Italic">Z</Emphasis>, in two dimensions

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze ²/r and (ii) the exchange energy density e_x(r){\varepsilon_x({\bf r})} . Some progress is also possible concerning the exchange potential V _x (r), one non-local approximation being the Slater potential 2e_x(r)/n(r){2\varepsilon_x(r)/n(r)} , with n(r) the ground state electron density. However, to complete the theory of V _x (r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.     

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.     

On the inner and outer radial density functions

Starting from the two-electron radial density D ₂(r ₁,r ₂), a generalized partitioning of the one-electron radial density function D(r) into two component densities D _a (r) and D _b (r) is discussed for many-electron systems. The literature partitioning (Koga and Matsuyama Theor Chem Acc 115:59, 2006) of D(r) into the inner D _<(r) and outer D _>(r) radial densities is shown to minimize the average variance of the two component density functions D _a (r) and D _b (r). It is also found that the average radial separation halved, , constitutes a lower bound to the standard deviation σ of D(r).     

Topological studies on IRC paths of X+H2→XH+H reactions

Topological properties of potential energy and electronic density distribution on five reaction paths X+H₂→XH+H (X=H, N, HN, H₂C, NC) are investigated at the level of UMP2/6–311G(d,p). It has been found that in the region of the reaction paths studied, where B(r_c)|_s>0 [B(r_c)|_s is the product of ρ(r_c) and ∇²ρ(r_c) at the point of reaction process, i.e., B(r_c)|_s=ρ(r_c)∇² ρ(r_c)] is basically the same as the region of V′(s)<0[V′(s) is the second derivative of potential energy with respect to the reaction coordinate, i.e., V′(s)=d²V/ds²], and the point with maximum B(r_c)|_s is almost coincident with the point of minimum V′(s). It can be concluded from the calculated results that there is a good correlation between the topological properties of potential energy and electronic density distribution along the reaction path. The structure transition state of such collinear reactions may be determined by topological analysis of electronic density. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1167–1174     

Schrödinger Equation Solutions for the Central Field Power Potential Energy I. V(r) = V 0(r/a 0)2ν−2, ν ≥ 1

The solution of a generalized non-relativistic Schrödinger equation with radial potential energy V(r)=V ₀(r/a ₀)^2–2 is presented. After reviewing the general properties of the radial ordinary differential equation, power series solutions are developed. The Green's function is constructed, its trace and the trace of its first iteration are calculated, and the ability of the traces to provide upper and lower bounds for the ground eigenvalue is examined. In addition, WKB-like solutions for the eigenvalues and eigenfunctions are derived. The approximation method yields valid eigenvalues for large quantum numbers (Rydberg states).     

Rayleigh–Schrödinger perturbation expansions for a hydrogen atom in a polynomial perturbation

Rayleigh–Schrouml;dinger (RS ) perturbation expansions for the eigenvalues E(λ) of a hydrogen atom in the general polynomial perturbation V(r) = aλr + b>²r², a, b > 0, are studied. When a² = 2b, the ground state energy is exactly E(λ) = -(1/2) + (3/2)a>, i.e., the RS series is truncated. In the case a² > 2b, the RS series is negative Stieltjes. In general, when λ < 0, a well of depth ω ≈ -a²/(4b²) (note the λ independence) is situated at r_ω = a/(2b|λ|). When a² > 2b/N², and interaction between this well and the hydrogenic state ψ_NLM(λ) is possible, thus creating a pair of asymptotically degenerate eigenstates separated by a “gap” δE(λ). The large order behavior of the RS coefficients E may be computed from the asymptotics of δE(λ), which is, in turn, related to the tunnelling integral. For excited states, stricter inequalities must be obeyed for Stieltjes behavior. The E⁽ⁿ⁾_NLM may be calculated either numerically or in closed form via the “so(4, 2) Lie algebra technology” for such hydrogenic problems.     

Correlated and idempotent Dirac first-order density matrices with identical diagonal Fermion density: a route to extract a one-body potential energy in TDDFT

After a brief introduction to the use of the idempotent Dirac first-order density matrix (DM), its time-dependent generalization is considered. Special attention is focused on the equation of motion for the time-dependent DM, which is characterized by the one-body potential V(r, t) of time-dependent density functional theory. It is then shown how the force –∇ V(r, t) can be extracted explicitly from this equation of motion. Following a linear-response treatment in which a weak potential V(r, t) is switched on to an initially uniform electron gas, the non-linear example of the two-electron spin-compensated Moshinsky atom is a further focal point. We demonstrate explicitly how the correlated DM for this model can be constructed from the idempotent Dirac DM, in this time-dependent example.     

Inner and outer radial density functions in many-electron atoms

When any two electrons are considered simultaneously, the radial density function D(r) in many-electron atoms is shown to be rigorously separated into inner D _<(r) and outer D _>(r) radial densities. Accordingly, radial properties such as the electron–nucleus attraction energy V _en and the diamagnetic susceptibility χ _d are the sum of the inner and outer contributions. The electron–electron repulsion energy V _ee has an approximate relation with the minus first moment of the outer density D _>(r). For the 102 atoms He through Lr in their ground states, different characteristics of local maxima in the radial densities D _<(r), D _>(r), and D(r) are reported based on the numerical Hartree-Fock wave functions. Relative contributions of the inner and outer components to V _en and are also discussed for these atoms.     

On the use of multipole expansion of the Coulomb potential in quantum chemistry

Some features of the multipole expansion of the Coulomb potential V for a system of point charges are studied. It is shown that multipole expansion is convergent both locally in L₂(R³) and weakly on some classes of functions. One-particle Hamiltonians H_n = H₀ + V_n, where H₀ is the kinetic energy operator and V_n is the n-th partial sum of the multipole expansion of V, are discussed, and the convergence of their eigenvalues to those of H = H₀ + V with increasing n is proved. It is also shown that the discrete spectrum eigenfunctions of H_n converge to those of H both in L₂(R³) (together with their first and second derivatives) and uniformly on R³. © 1996 John Wiley & Sons, Inc.     

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x₁,...,x _r be closed points in general position in projective spacePn, then the linear subspaceV ofH ⁰ (⨑ⁿ,O(d)) (the space of homogeneous polynomials of degreed on ⨑ⁿ) formed by those polynomials which are singular at eachx _i, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x₁,...,x _r. As such, the “expected” value for the dimension ofV is max(0,h ⁰(O(d))−r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.     

Influence of the linking chain length on the recombination kinetics of covalently bonded triplet radical pairs and magnetic field effects

Nanosecond laser flash photolysis technique is used to study the formation and decay kinetics of covalently linked triplet radical pairs (RP) formed after photoinduced electron transfer in the series of 21 zinc porphyrin—chain—viologen (Pph—Sp_n—Vi²⁺) dyads, where the number of atoms (n) in the chain increases from 2 to 138. In poorly viscous polar solvents (acetone, CHCl₃—CH₃OH (1 : 1) mixture), the dependence of the rate constant of RP formation on n can be described by the equation k _e = k _e ⁰ n ^–a at k _e ⁰ = 2.95·10⁸ s^–1 anda = 0.8. In the zero magnetic field, the RP recombination rate constant (k _r(B = 0)) is significantly lower than k _e and ranges from 0.7·10⁶ to 8·10⁶ s^–1. The dependence of k _r(B = 0) on n is extreme. The dependence k _r(B = 0) reaches a maximum at n = 20. In the strong magnetic field (B = 0.21 T), the significant retardation of triplet RP recombination is observed. The chain length has an insignificant effect on k _r(B = 0.21 T), which ranges from 0.3·10⁶ to 0.9·10⁶ s^–1. The regularities found are discussed in terms of the interplay of molecular and spin dynamics.     

Volumetric Properties of Some Aliphatic Mono- and Dicarboxylic Acids in Water at 298.15 K

The apparent molar volumes, V _φ , of two series of homologous aliphatic carboxylic acids, H(CH₂) _n COOH [n=0–5] and (CH₂) _n (COOH)₂ [n=0–5], were determined in dilute aqueous solutions by density measurements at T=298.15 K. Densities were measured using a vibrating-tube densimeter (DMA 5000, Anton Paar, Austria) at T=298.15 K. These results were used to calculate the apparent molar volumes of each solute over the concentration range 0.0050≤m/(mol⋅kg⁻¹)≤0.3000. Values of the apparent molar volumes of undissociated acids V_f(u)⁰V_{\phi (u)}^{0} were also calculated. The variation of V_f(u)⁰V_{\phi (u)}^{0} was determined as a function of the aliphatic chain length of the studied carboxylic acids.     

On the connectivity index of trees

The connectivity index χ₁(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T(n, r) be the set of trees on n vertices with diameter r. In this paper, we determine all trees in T(n, r) with the largest and the second largest connectivity index. Also, the trees in T(n, r) with the largest and the second largest connectivity index are characterized. Mei Lu is partially supported by NNSFC (No. 10571105).     

Validation experiment for large-sample prompt-gamma neutron activation analysis

A method is proposed for the implementation of large-sample prompt-gamma neutron activation analysis (LS-PGNAA). The method was tested with four different sample materials at the thermal PGNAA facility at JAERI, Japan. The macroscopic scattering cross section (Σ _s) and absorption cross section (Σ _a) of the samples were determined by monitoring the neutron flux in four positions just outside the sample container. With the Σ _s and Σ _a determined, the spatial neutron density distribution [n(r)] inside the sample material was derived. Taking n(r) and the gamma-ray self-absorption into account simultaneously, the effective geometric gamma-ray detection efficiency for large samples as a function of gamma-ray energy was calculated. Taking silicon as test element, the concentrations found agreed to within 7% with the known concentrations in the four sample materials examined, both when using relative standardization and with absolute standardization.     

Isomorphism in electron-pair densities of atoms

The spherically averaged electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities are a couple of densities which characterize the motion of electron pairs in atomic systems. We study a generalized electron-pair density (q; a, b) that represents the probability density function for the magnitude of two-electron vector a r _j +b r _k of any pair of electrons j and k to be q, where a and b are nonzero real numbers. In particular, h(u)=g(u;1, −1) and d(R) = . It is shown that the scaling property of the Dirac delta function and the inversion symmetry of orbitals in atoms due to the central force field generate several isomorphic relations in the electron-pair density (q; a, b) with respect to the two parameters a and b. The approximate isomorphism d(R)≅8h(2R) known in the literature between the intracule and extracule densities is a special case of the present results. Received: 24 May 2000 / Accepted: 18 July 2000 / Published online: 27 September 2000