John D. Morgan

Rates of convergence of variational calculations and of expectation values

We present a mathematical and numerical analysis of the rates of convergence of variational calculations and their impact on the issue of the convergence or divergence of expectation values obtained from variational wave functions. The rate of convergence of a variational calculation is critically dependent on the ability of finite linear combinations of basis functions to simulate the nonanalyticities (cusps) in the exact wave function being approximated. A slow rate of convergence of the variational energy can imply that the corresponding variational wave functions will yield divergent expectation values of physical operators not relatively bounded by the Hamiltonian. We illustrate the sorts of problems which can arise by examining Gauss‐type approximations to hydrogenic orbitals. Since all many‐electron wave functions have cusps similar to those in hydrogenic wave functions, this simple example is relevant to variational calculations performed on atoms and molecules. Finally, we offer suggestions on wh...

Large‐order dimensional perturbation theory for two‐electron atoms

An asymptotic expansion for the electronic energy of two‐electron atoms is developed in powers of δ=1/D, the reciprocal of the Cartesian dimensionality of space. The expansion coefficients are calculated to high order (∼20 to 30) by an efficient recursive procedure. Analysis of the coefficients elucidates the singularity structure in the D→∞ limit, which exhibits aspects of both an essential singularity and a square‐root branch point. Pade–Borel summation incorporating results of the singularity analysis yields highly accurate energies; the quality improves substantially with increase in either D or the nuclear charge Z. For He, we obtain 9 significant figures for the ground state and 11 for the 2p2 3Pe doubly excited state, which is isomorphic with the ground state at D=5 by virtue of interdimensional degeneracy. The maximum accuracy obtainable appears to be limited only by accumulation of roundoff error in the expansion coefficients. The method invites application to systems with many electrons or subje...

Resolution of a paradox in the calculation of exchange forces for H+2

Abstract Tang, Toennies and Yiu have shown that despite the inherent symmetry of H+2, wavefunctions obtained from a combination of the unsymmetrized polarization expansion and the 1βR expansion can be used in the Holstein—Herring formula to calculate for large internuclear distances R the leading O (e−R) terms in the exchange energy between the lowest pair of states. However, the associated claim by Tang and Toennies that the polarization expansion of the wavefunction converges not to the gerade molecular wavefunction, but to an asymmetric function localized about a single nucleus, conflicts with other numerical and analytical results. We show by a limiting procedure that use of the infinite polarization expansion for the wavefunction in the Holstein—Herring formula provides a result that is not equal to the exact exchange energy, although it has the correct leading O (e−R) behavior and is impressively close to the exact exchange energy for large R.

Electronic tunneling and exchange energy in the D‐dimensional hydrogen‐molecule ion

Dimensional scaling generates an effective potential for the electronic structure of atoms and molecules, but this potential may acquire multiple minima for certain ranges of nuclear charges or geometries that produce symmetry breaking. Tunneling among such minima is akin to resonance among valence bond structures. Here we treat the D‐dimensional H+2 molecule ion as a prototype test case. In spheroidal coordinates it offers a separable double‐minimum potential and tunneling occurs in only one coordinate; in cylindrical coordinates the potential is nonseparable and tunneling occurs in two coordinates. We determine for both cases the ground state energy splitting ΔED as a function of the internuclear distance R. By virtue of exact interdimensional degeneracies, this yields the exchange energy for all pairs of g, u states of the D=3 molecule that stem from separated atom states with m=l=n−1, for n=1→∞. We evaluate ΔED by two semiclassical techniques, the asymptotic and instanton methods, and obtain good agre...

The Analytic Structure of Atomic and Molecular Wavefunctions and its Impact on the Rate of Convergence of Variational Calculations

In these lectures I would like to present an overview of the analytic structure of atomic and molecular wavefunctions and how that structure determines the rate of convergence of a variational (Rayleigh-Ritz) calculation as the number of basis functions is increased. I shall present some illustrative examples of variational basis-set calculations on both simple model systems and more complicated systems of greater physical relevance, such as the helium atom and small multi-center molecules.

Asymptotically exact calculation of the exchange energies of one-active-electron diatomic ions with the surface integral method

We present a general procedure, based on the Holstein–Herring method, for calculating exactly the leading term in the exponentially small exchange energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulae we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Σ and Π states of H+2. We then apply our general formulae to calculate the exchange energy splittings between the lowest states of the diatomic alkali cations K+2, Rb+2 and Cs+2, which are isovalent to H+2. Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.

Exchange Energy for Two-Active-Electron Diatomic Systems Within the Surface Integral Method

Abstract.We have analyzed and reduced a general (quantum-mechanical) expression for the atom-atom exchange energy formulated as a five-dimensional surface integral, which arises in studying the charge exchange processes in diatomic molecules. It is shown that this five-dimensional surface integral can be decoupled into a three-dimensional integral and a two-dimensional angular integral which can be solved analytically using a special decomposition. Exact solutions of the two-dimensional angular integrals are presented and generalized. Algebraic aspects, invariance properties and exact solutions of integrals involving Legendre and Chebyshev polynomials are also discussed.

A statistical analysis of the Stanford fractional charge experiment

Abstract We report arguments indicating that the residual fractional charges observed on niobium spheres in the experiments at Stanford are consistent with a statistical distribution of one kind of fractionally-charged object among the spheres, with a large average number of charges per sphere. Our statistical analysis enables us to conclude with 96% confidence that the spheres annealed on a tungsten substrate have a substantially higher concentration of fractional charges than those annealed on a niobium substrate. We draw attention to circumstantial evidence that such fractional charges were transferred from the tungsten substrate to the surfaces of the niobium spheres. We point out that surface and grain-boundary segregation effects, which previously have not been considered in assessing this experiment, could play a major role in enhancing the concentration of fractionally charged impurities on the surface of the tungsten substrate in contact with the niobium spheres. We conclude that the Stanford experiment would be consistent with a concentration of 1 fractional charge in 1013 to 1014 tungsten atoms, which would not be contradicted by other searches for fractional charges in tungsten.

High precision calculation of helium and atom energy levels

This work is concerned with the high‐presicion calculation of the energies of the gound and excited states of the helium atom (or other light helium‐like ions) to match the recent advances in experimental laser spectroscopic studies of transitions between these states with a precision of better than 10−4 cm−1. At this level of accuracy it is essential to include mass‐polarization effects through 2nd order, relativistic effects properly scaled by appropriate powers of the reduced mass, and quantum electrodynamic (QED) effects. In recent work on excited states of helium we have attained this level of accuracy for all the non‐QED effects. Our results are in good agreement with those obtained independently by G. W. F. Drake. We are proceeding with a high‐precision evaluation of the Bethe logarithm, which is the principal source of uncertainty in the theoretically determined QED effects. The refinement of these calculations at the 10−5 cm−1 level and beyond is expectd to stimulate further advances both theoret...

General Computational Strategies

The poor convergence of the partial sums of the 1/D expansion for atomic and molecular energies is due to the fact that the energy function E(δ), δ ≡ 1/D, is not a polynomial. The large-D limit appears to be an excellent qualitative model, but in order to accurately continue the solution from δ = 0 to δ = 1/3 it is necessary to take into account the functional form of E(δ) in that general region of the comples plane. The most important feature in this functional form is a second-order pole at δ = 1. Incorporating this pole into the form of the summation approximates, using either of three methods that we describe, accounts for over 99% of the solution at δ = 1/3 for ground-state energies of 2 + and helium. Almost all the remaining error can be accounted for by including a rather complicated singularity at δ = 0. Through an analysis of the large-order behavior of the δ expansion, we construct a functional form for this singularity. The functional form suggests that the energy expansion can be summed most effectively using Borel summation, with the integrand of the Borel sum approximated by Pade approximants or, better yet, by approximants in the form of Darboux functions. We present numerical results that support this conclusion.