# S. P. GoldmanRyerson University · Department of Physics

S. P. Goldman

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49

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Introduction

**Skills and Expertise**

## Publications

Publications (49)

A new method is presented for the calculation of the mean excitation energy (Bethe logarithm) for helium and other two-electron systems. The method requires only a single matrix diagonalization within a correlated Hylleraas basis set extended to contain a wide range of distance scales. High-precision results are obtained for the ground states of Ps...

Accurate relativistic calculations of the oscillator strength densities and photoeffect cross sections for neutral hydrogen and hydrogenic lead (Z = 82) are performed up to the high energy region. Relativistically induced Cooper minima are found in the partial wave contributions at high energies. The results are in good agreement with calculations...

A fast, accurate and stable optimization algorithm is very important for inverse planning of intensity-modulated radiation therapy (IMRT), and for implementing dose-adaptive radiotherapy in the future. Conventional numerical search algorithms with positive beam weight constraints generally require numerous iterations and may produce suboptimal dose...

A brief review of Configuration Interaction (CI) and Correlated calculations is given. New techniques are presented that enhance the precision of CI calculations by several orders of magnitude and match or surpass the best available correlated calculations. These new techniques are based on the uncoupling of all the multi-dimensional integrations t...

Interest in the effect that electric and magnetic fields have on the
internal structure of atoms is as old as quantum mechanics itself. In
practical terms, an atom's spectrum acts as its signature, and so it is
important to understand how electric and magnetic fields alter this
characteristic. In this chapter, a summary of the basic nonrelativistic...

A fast optimization algorithm is very important for inverse planning of intensity modulated radiation therapy (IMRT), and for adaptive radiotherapy of the future. Conventional numerical search algorithms such as the conjugate gradient search, with positive beam weight constraints, generally require numerous iterations and may produce suboptimal dos...

A method for the calculation of logarithmic sums that yields very high accuracy even for small basis set dimensions is introduced. The best values achieved are accurate to 23 significant figures without extrapolation. The sums are performed directly on variational intermediate sets. The method automatically rejects any basis functions that could in...

A new method is presented for the calculation of the mean exictation energy (Bethe logarithm) for helium and other two-electron systems [1]. The method requires only a single matrix diagonalization within a correlated Hylleraas basis set extended to contain a wide range of distance scales. High precision results are obtained for the ground states o...

A method for the calculation of logarithmic sums that yields very high accuracy even for small basis-set dimensions is introduced. The best values achieved are accurate to 23 significant figures without extrapolation. The sums are performed directly on variational intermediate sets. The method automatically rejects any basis functions that could in...

The variational approach to the Dirac-Coulomb Hamiltonian is extended to include simultaneously real electron and positron states. The spurious root previously obtained using finite-basis sets for states with kappa >0 is now clearly identified as the variational ground state of a negative-energy positron, resulting in a variational spectrum that is...

The leading terms in the 1/Z expansion of the two-electron Bethe logarithm are calculated for the states 1s2 1S0, 1s2s 1S1, 1s2s 3S1, 1s2p 1P1 and 1s2p 3PJ by the use of a novel finite basis set method. The resulting QED terms are combined with other relativistic and mass polarisation corrections to obtain total transition frequencies. The results...

The leading two terms in the 1/Z expansion of the two-electron Bethe logarithm are calculated by the application of a new finite basis set method. The results can be expressed in the form ln epsilon (1s2 1S)=ln(19.77(Z-0.0063)2). The high-Z behaviour appears to differ from that of a previous variational calculation by Aashamar and Austvik (1976).

An accurate single centered method for molecular calculations, based on the modified configuration interaction method with infinite angular expansions, is introduced. A calculation of the ground state energy of the hydrogen molecular ion in the Born–Oppenheimer approximation is presented as an example yielding an accuracy for the energy of 7×10−8 a...

An uncoupled correlated variational method (i.e., explicitly using the interelectronic coordinates) for calculations involving several electrons is introduced. All the overlap and Hamiltonian matrix elements, including those involving the electron-electron interaction, are simple products of one-dimensional integrations. Although simple to implemen...

A method of significance to correlated calculations involving several electrons is introduced. All the radial matrix elements in Modified Configuration Interaction (MCI) or correlated calculations ar e formulated as products of simple, independent one-dimensional integrals: one integration between 0 and ∞ and all the others from 0 to 1 (there are n...

A variational approach to molecular calculations is introduced that easily yields accurate non-perturbative values for the electronic-vibrational spectrum of diatomic molecules, avoiding multi-center ed or correlated integrations. With the origin of coordinates at the nuclear center of mass, multi-centered calculations are avoided by expanding the...

A method is introduced that drastically simplifies modified configuration interaction (MCI) calculations involving several electrons. All the radial matrix elements in MCI or correlated calculations are formulated as products of simple, independent one-dimensional integrals: one from 0 to ∞, all the others from 0 to 1. The method is easily applicab...

A simplification of the modified configuration-interaction (MCI) method [S. P. Goldman, Phys. Rev. A 52, 3718 (1995)] is introduced. In its original form, the MCI method improves dramatically the convergence of standard CI calculations by a modification of the radial representation and an a priori mixing of a large number of angular configurations....

A detailed presentation of the modified configuration-interaction (CI) method [S. P. Goldman, Phys. Rev. Lett. 73, 2541 (1994)] is given. The standard two-electron test case is used to introduce the concepts, as well as to show a sharp increase in the convergence of the calculations. Space-ordered radial coordinates and generalized angular function...

A very efficient modified configuration interaction method is introduced for accurate atomic and molecular calculations with basis sets that are substantially smaller than those used by conventional configuration interaction (C). The energy of the ground state of [ital helium], for example, is obtained with a relative error of 3[times]10[sup [minus...

A variational method based on results for self-adjoint operators due to T. Kato [Proc. Phys. Soc. Jpn. 4, 334 (1949)] is developed to calculate upper and lower bounds on the energy eigenvalues for the one-electron Dirac Hamiltonian. The method avoids relativistic variational collapse for any one-electron potential. This result is confirmed analytic...

A very efficient technique for the finite-basis-set calculation of logarithmic sums is introduced. The basis sets contain sequences of nonlinear parameters determined by the zeros of Laguerre polynomials. It is shown that one can easily obtain convergence to 12 digits in fast calculations involving small basis sets. The method is then applied to ca...

A finite-basis-set method is introduced for calculations involving an electron in the presence of a finite-size nucleus. It is found that the inclusion of powers of the form r-gamma+n in the basis set, where gamma is a real number and n an integer, is of fundamental importance, increasing the convergence in the energy eigenvalues by several orders...

A global formula for the asymptotic Lamb shifts of the Rydberg states of helium is obtained by calculating the electric field perturbation due to the Rydberg electron on the He{sup +} (1{ital s}) Lamb shift. The result substantially reduces theoretical uncertainties in calculated transition frequencies. A comparison with experiment for transitions...

A finite-basis-set method is used to calculate relativistic and nonrelativistic binding energies of an electron in a static Coulomb field and in magnetic fields of arbitrary strength (0 < B less-than-or-equal-to 10(13) G). The basis set is composed of products of Slater- and Landau-type functions, and it contains the exact solutions at both the Cou...

The ground-state energy of hydrogenic atoms in strong magnetic fields with B up to 10 exp 12 G is calculated for several values of the nuclear charge Z using a modified Slater-type basis set with different values of the total angular momentum. Both accurate relativistic and nonrelativistic binding energies of hydrogen can be obtained through this s...

A relativistic basis set composed of products of Slater- and Landau-type functions is introduced and applied to the accurate calculation of the ground-state energy of an electron in a static Coulomb field and a magnetic field of arbitrary strength. The relativistic corrections for strong magnetic fields differ from previous relativistic adiabatic a...

A Relativistic finite‐basis‐set method is used to calculate the ground‐state energy of hydrogenic atoms in a strong magnetic field with 10⁹G≤B≤10¹²G, for several values of the nucelar charge Z. A modified Slater‐type basis set with different values of the total angular momentum is used. Even though an upper bound on the energy can not be obtained w...

In a recent paper [A. B. Volkov, Phys. Rev. A 39, 4406 (1989)] the square of the Dirac Hamiltonian matrix built within a finite-basis-set representation is used to obtain variational solutions to the Dirac equation. We show in this Comment that no advantages are gained by the use of this method that is in fact equivalent to the diagonalization of t...

Publisher Summary Finite basis set variational techniques are used widely in nonrelativistic atomic physics for the calculation of energy levels, transition rates, processes involving the absorption and emission of radiation, and scattering cross sections. Therefore, the best choice of a trial function to approximate the ground state in the spectru...

Slater-type basis sets are commonly used to perform nonrelativistic and relativistic variational calculations. A major limitation of this method is that no more than about 14 powers per nonlinear parameter can be used while working in double-precision arithmetic without entering into numerical difficulties. Even for moderate-size basis sets, large...

An accurate method for variational calculations of atomic properties is constructed using the principle that the results must be invariant under gauge transformations. The technique is used with multiexponential basis sets and provides very fast convergence. The method is applied to the relativistic calculation of the dipole polarizability in hydro...

A detailed presentation of an analytic variational Dirac-Hartree-Fock (VDHF) procedure that avoids the problems of spurious roots and collapse is given. The conditions for the avoidance of spurious roots are discussed, as well as the behavior of the one-electron and total energies as upper bounds under variations of the nonlinear parameters or the...

A relativistic virial theorem is used to construct a new optimization technique for the variational calculation of Dirac-Hartree-Fock energies in multielectron atomic systems. The method provides fast convergence and a natural criterion for the choice of the accuracy that should be required of the calculations.

A variational Dirac-Hartree-Fock procedure is introduced which does not
exhibit problems of spurious roots, variational collapse, or continuum
dissolution. The optimized eigenvalues converge uniformly from above to
the numerical Dirac-Hartree-Fock results as the dimension of the basis
set is increased. Results for the 1s2, 2s2, and
2p21/2 shells ar...

A new approach to the variational representation of the Dirac equation is presented. The method takes advantage of the conditions satisfied by the eigenfunctions at the origin. In this way, a variational representation of the complete Dirac-Coulomb spectrum without spurious roots is obtained. Rigorous proofs of bounds for the positive and negative...

A new variational representation of the Dirac equation is presented. The basis set is required to satisfy some general energy-independent conditions at the origin. The method is applied to the case of a Coulomb potential. The resulting variational solutions form a discrete representation of the complete Dirac spectrum including both positive and ne...

A new variational method is presented which permits a direct and rapidly convergent calculation of almost divergent sums over complete sets of intermediate states, using a finite-basis-set representation. The formalism for first-order perturbation calculations of infinite sums for two-electron systems is presented and the technique is used to calcu...

New and simple extrapolation techniques are introduced, of relevance to variational calculations in atomic and molecular physics. The inverse of the number of independent basis functions usually used as the independent variable in the extrapolation to infinite basis sets is replaced by a measure of the goodness of the representation that a basis se...

Relativistic generalizations are derived for the well-known
nonrelativistic electric-dipole oscillator-strength sum rules. The
relativistic sum rules include both positive- and negative-energy
states. The derivations are valid for a Dirac electron in an arbitrary
local potential. We also present a number of simple integral properties
related to the...

Rates are calculated for the decay of metastable 2s1/2 ions to the ground state by the simultaneous emission of two photons. The calculation includes all relativistic and retardation effects, and all combinations of photon multipoles which make significant contributions up to Z=100. Summations over intermediate states are performed by constructing...

Variational solutions to the Dirac equation in a discrete L2 basis set are investigated. Numerical calculations indicate that for a Coulomb potential, the basis set can be chosen in such a way that the variational eigenvalues satisfy a generalized Hylleraas-Undheim theorem. A number of relativistic sum rules are calculated to demonstrate that the v...

A variational discrete representation of the relativistic energy
spectrum of an electron in a Coulomb field is constructed. It is shown
that by a proper choice of the variational basis set, the eigenvalues
satisfy a generalized Hylleraas-Undheim theorem. A number of
relativistic sum rules which can be evaluated exactly are calculated by
means of th...

The Lamb shift of ${\mathrm{He}}^{+}$ is derived from the measured anisotropy in the electric-field-induced quenching radiation of the metastable $2{s}_{\frac{1}{2}}$ state. The results demonstrate that the anisotropy method can be applied with high precision to one-electron ion beams, as well as neutral beams. We find a Lamb shift of 14040.2\ifmmo...

A method is developed to find physically valid analytic solutions to the
Einstein field equations for a static, spherically symmetric
distribution of matter. A generating function can be chosen that
satisfies physical constraints, and the metric is obtained by
quadratures. Three physically valid solutions within the distribution of
matter are given...

The problem of finding static, spherically symmetric, solutions of Einstein’s equations for a perfect fluid is reformulated. A field equation connecting the pressure and density and free of metric components is obtained. Upon finding a solution of this field equation, the metric components are then obtained by quadrature. A solution-generating tech...

A finite-basis-set method is introduced for calculations involving an electron in the presence of a finite size nucleus. It is found that the inclusion of powers of the form r{sup -Î³+n} in the basis set is of fundamental importance, increasing the convergence in the energy eigenvalues by several orders of magnitude in the case of large nuclear cha...