Scaling in Isoelectronic Molecules

A modification of a scaling method introduced for atoms by Ellison enables one to use expectation values calculated for one molecule in calculations of the energy of a second molecule isoelectronic to the first. In going from Hz to Hez2+t,h e results are only fair, but in going from LiF to BeO, the results are suffeciently good to allow prediction of equilibrium distance and several expectation values as well as energy. The dipole moment is a notable exception, which reveals one basic dissimilarity between the two molecules, the ionic character. LiF dissociates to ions, Be0 to neutral atoms, causing our method to break down at large internuclear distance. The inverse scaling transformation, from Be0 to LiF, is also accomplished, with similar results. I

Calculated Electronic Profiles for Liquid-Metal Surfaces

The electronic density profile for a liquid-metal surface can be calculated by solving the self-consistent Lang-Kohn equations for the electronic wave functions. One requires a surface density profile for the ion cores, which enters the electrostatic and pseudopotential parts of the electronic Hamiltonian. We use oscillatory profiles, suggested by those found by molecular-dynamics simulations on a pseudoatom model. Calculating surface potentials and work functions, we obtain excellent agreement with experiment (within 0.2 eV). It is shown that use of either step-function ion profiles or a simple variational method leads to serious errors (12 eV) for these quantities

Statistical Mechanical Derivation of the Lippmann Equation. The Dielectric Constant

We consider the polarizable electrochemical interface with spherical symmetry, and show that the common assumption of an invariant dielectric constant violates the mechanical equilibrium condition, unless its value is that of vacuum. The polarizable particles must be taken into account explicitly, which we do by deriving distribution functions for interacting charged and polarizable particles, neglecting short-range forces and short-range correlations, Calculating the change in surface tension when the distributions change so as to keep constant the temperature and the pressure inside and outside the interface, we obtain the Lippmann equation

Surface Tension of a Charged and Polarized System

Usually, the formula for the surface tension of a planar charged and polarized interface is obtained from that for a system involving only short-range forces, y = - dz [p - px(z)] by replacing the tangential pressure p , by p , + E2/8u. Problems with this include (a) p, is no longer explicitly defined, (b) the electrostatic stress term E2/8 pi is not correct in general but only if polarization is proportional to density of polarizable species, (c) the derivation of the formula in terms of p and p, involves calculating the work to expand a volume containing the interface, and this work cannot be written in terms of the pressure of the surroundings when there are long-range forces. To derive a formula free from these objections, we consider the spherical system contained between r = R, and r = R2 and containing charged and dipolar particles, the orientation of the latter giving rise to the electrical polarization. There is no electric field, electric polarization, or local charge density for r \u3c R, or for r \u3e R2. If this system is expanded keeping the ratios of all radii fixed, the work done by the surroundings is 4u1R,2bR-l p2R&3R2),which is set equal to the change in free energy, calculated from the canonical partition function. The surface tension is defined as (R,,/2)(pl -p2), where R,, is the surface of tension. When R, becomes infinite (plane interface), the value of R,, becomes irrelevant. Both long-range and short-range terms in the surface tension are shown to behave properly for R, - m, the long-range terms being proportional to l d r [-p(r) V(r) + 3P(r) E(?)] (P = polarization). If only charged particles are present (no polarization), correlations and short-range forces are neglected, and the distribution of each charged species ni follows the Boltzmann law with energy qiV, it is shown that kmini - E2/8r is independent of z. Using this fact with our surface tension formula, we prove the Lippmann equation. If dipolar particles are present as well as charged particles, the former must be included in Cini. Then the quantity k E i n , - E2/8u - EP is shown independent of z, and our surface tension formula again leads to the Lippmann equation

Hellmann-Feynman Theorem in Thomas-Fermi and Related Theories

The general Hellmann-Feynman theorem (derivative of energy with respect to a parameter = expectation value of derivative of Hamiltonian) is proved for theories in which the electron density is determined by making the energy functional stationary. Some simple applications are given

Dipole Moments in Thomas-Fermi-Dirac and Thomas-Fermi Theories.

It is shown that the electronic contribution to the dipole moment, calculated from a solution to the Thomas—Fermi—Dirac or Thomas—Fermi equations, should be equal and opposite to the nuclear contribution. Thus, the Thomas—Fermi—Dirac and Thomas—Fermi theories predict vanishing dipole moments for all molecular systems

General Correction for Electrode Sphericity in Voltammetry of Nernstian Systems

The current is considered at a stationary reversible spherical electrode whose potential E(t) as a function of time is given, such that E(t) determines the ratio of oxidized to reduced species at the electrode surface. Writing the current as that for planar geometry, I0, plus corrections for sphericity, we derive formulae for the corrections. The first two are expressed as integrals over I0, with no explicit dependence on the potential, for any form of E(t), and whether the reduced species diffuses into the electrode or into the solution. If the ratio of diffusion constants for oxidized and reduced species is taken as unity, and if oxidized and reduced species are both in the solution, the corrections become extremely simple (no integrations or differentiations), regardless of the form of E(t). The first correction is evaluated in several cases where this does not obtain

The Isoelectronic Principle and the Accuracy of Binding Energies in the Hückel Method

In the Hükkel and other methods, binding energies are calculated by subtracting the sum of orbital electronic energies for the molecule from the sum of orbital electronic energies for the separated atoms, and not considering the internuclear repulsion. Since this last may be several orders of magnitude greater than the binding energy, reasonable results could not be obtained without an approximate cancellation with another neglected term. It is shown that such a cancellation is a consequence of the isoelectronic principle (invariance of binding energy to change in atomic number of constituent atom). Numerical examples are given

Electron Densities for Homonuclear Diatomic Molecules from the Thomas-Fermi-Dirac Theory

Electron densities are calculated from approximate solutions to the Thomas-Fermi-Dirac equation for homonuclear diatomic molecules. The accuracy of expectation values calculated from these densities is assessed. In general, one obtains fair agreement with self-consistent-field and experimental results, but this is insufficient when a property is a difference between electronic and nuclear contributions. An important example is the Hellmann-Feynman force on a nucleus, the net force necessarily being repulsive, as for closed-shell atoms. Forc and energy results for such a situation are compared with experiment. Finally, it is shown that a modified theory, previously applied to atoms, gives improvement in expectation values depending on the electron density near the nuclei

Surface Tensions of Molten Salt Mixtures

It is always found that the surface tension of a mixture of molten salts lies below the mean of the surface tensions of the pure components, weighted by mole fractions. Part of this effect is certainly due to the well-known fact that the composition near the surface is enriched in the component of lower surface tension. We show that an effect of the long-range forces, present for fluids involving Coulombic interactions, is of greater importance. The effect is due to the electroneutrality constraint on the distribution functions. As we show by separate calculations, both effects must be considered to obtain good agreement with experimental results for alkali halide melts