The variational method complemented with the use of explicitly correlated Gaussian basis functions is one of the most powerful approaches currently used for calculating the properties of few-body systems. Despite its conceptual simplicity, the method offers great flexibility, high accuracy, and can be used to study diverse quantum systems, ranging from small atoms and molecules to light nuclei, hadrons, quantum dots, and Efimov systems. The basic theoretical foundations are discussed, recent advances in the applications of explicitly correlated Gaussians in physics and chemistry are reviewed, and the strengths and weaknesses of the explicitly correlated Gaussians approach are compared with other few-body techniques.

Theory and application of explicitly correlated Gaussians

Multiconfiguration wave function expansions combined with configuration interaction methods are a method of choice for complex atoms where atomic state functions are expanded in a basis of configuration state functions. Combined with a variational method such as the multiconfiguration Hartree-Fock (MCHF) or multiconfiguration Dirac-Hartree-Fock (MCDHF), the associated set of radial functions can be optimized for the levels of interest. The present review updates the variational MCHF theory to include MCDHF, describes the multireference single and double process for generating expansions and the systematic procedure of a computational scheme for monitoring convergence. It focuses on the calculations of energies and wave functions from which other atomic properties can be predicted such as transition rates, hyperfine structures and isotope shifts, for example.

Advanced multiconfiguration methods for complex atoms : I. Energies and wave functions

This paper is part of the 2012 Quantum Chemistry thematic issue. Sergiy Bubin,*,† Michele Pavanello,*,‡ Wei-Cheng Tung, Keeper L. Sharkey, and Ludwik Adamowicz* †Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, United States ‡Department of Chemistry, Rutgers University Newark, Newark, New Jersey 07102, United States Department of Chemistry and Biochemistry and Department of Physics, University of Arizona, Tucson, Arizona 85721, United States

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Born-Oppenheimer and non-Born-Oppenheimer, atomic and molecular calculations with explicitly correlated Gaussians.

Long‐Range Casimir Forces: Theory and Recent Experiments on Atomic Systems

Sources of uncertainty are reviewed for calculated atomic and molecular data that are important for plasma modeling: atomic and molecular structure and cross sections for electron-atom, electron-molecule, and heavy particle collisions. We concentrate on model uncertainties due to approximations to the fundamental many-body quantum mechanical equations and we aim to provide guidelines to estimate uncertainties as a routine part of computations of data for structure and scattering.

Uncertainty Estimates for Theoretical Atomic and Molecular Data

Nonrelativistic energies for the low-lying states of lithium are calculated using the variational method in Hylleraas coordinates Variational eigenvalues for the infinite nuclear mass case with up to 34 020 terms are $\ensuremath{-}7478\phantom{\rule{016em}{0ex}}060\phantom{\rule{016em}{0ex}}323\phantom{\rule{016em}{0ex}}910\phantom{\rule{016em}{0ex}}147(1)$ au for $1{s}^{2}2s{\phantom{\rule{016em}{0ex}}}^{2}\phantom{\rule{-016em}{0ex}}S$, $\ensuremath{-}7354\phantom{\rule{016em}{0ex}}098\phantom{\rule{016em}{0ex}}421\phantom{\rule{016em}{0ex}}444\phantom{\rule{016em}{0ex}}37(1)$ au for $1{s}^{2}3s{\phantom{\rule{016em}{0ex}}}^{2}\phantom{\rule{-016em}{0ex}}S$, $\ensuremath{-}7318\phantom{\rule{016em}{0ex}}530\phantom{\rule{016em}{0ex}}845\phantom{\rule{016em}{0ex}}998\phantom{\rule{016em}{0ex}}91(1)$ au for $1{s}^{2}4s{\phantom{\rule{016em}{0ex}}}^{2}\phantom{\rule{-016em}{0ex}}S$, $\ensuremath{-}7410\phantom{\rule{016em}{0ex}}156\phantom{\rule{016em}{0ex}}532\phantom{\rule{016em}{0ex}}652\phantom{\rule{016em}{0ex}}41(4)$ au for $1{s}^{2}2p{\phantom{\rule{016em}{0ex}}}^{2}\phantom{\rule{-016em}{0ex}}P$, and $\ensuremath{-}7335\phantom{\rule{016em}{0ex}}523\phantom{\rule{016em}{0ex}}543\phantom{\rule{016em}{0ex}}524\phantom{\rule{016em}{0ex}}688(3)$ au for $1{s}^{2}3d{\phantom{\rule{016em}{0ex}}}^{2}\phantom{\rule{-016em}{0ex}}D$ The selection of the minimum set of angular momentum configurations is discussed, with the $2P$ and $3D$ states as examples to demonstrate the impact of various configurations on the variational energies It is shown by numerical example that the second spin function (ie, coupled to form a triplet intermediate state) has no significant effect on either the variational energies or the spin-dependent Fermi contact term Results of greatly improved accuracy for the Fermi contact term are presented for all the states considered

/pdf/variational-energies-and-the-fermi-contact-term-for-the-low-2319bynd3l.pdf

Variational energies and the Fermi contact term for the low-lying states of lithium: Basis-set completeness

The method of complex-coordinate rotation is applied to investigate resonances in positron scattering by helium ions. Here we present results for high-angular-momentum states $(L=2--6).$ Together with the earlier values for the $L=0$ and 1 states, these stabilized complex eigenvalues exhibit a regular pattern. We also calculate some narrow S-wave resonances below the excited positronium $(N=2)$ threshold. Our converged results for the first S state lying above the ${\mathrm{He}}^{+}$ $(n=5)$ threshold differ substantially from the calculation by Igarashi and Shimamura [Phys. Rev. A 56, 4733 (1997)].

High-angular-momentum resonances in positron scattering by a He + ion

In this work, we perform a study on high-lying $S$-wave resonances in Ps-H scattering below the ${e}^{+}$-H${}^{\ensuremath{-}}$ threshold. The method of complex-coordinate rotation is applied together with highly correlated wave functions containing all six interparticle distances. Using such Hylleraas bases up to 6412 terms, the energy positions up to the 8$S$ state and the widths up to the 7$S$ state in the Rydberg series converging to the H${}^{\ensuremath{-}}$ threshold are determined. The energies of 4$S$ to 7$S$ states from the present calculations are fitted into the quantum defect formula, from which the energies of higher members in the Rydberg series can be deduced.

Higher-lying resonances in Ps-H scattering below the e + -H - threshold

The static dipole polarizabilities of the first three lowest states of hydrogen molecules ${\mathrm{H}}_{2}$, HD, and ${\mathrm{D}}_{2}$ are calculated using variationally generated wave functions in Hylleraas coordinates without using the Born-Oppenheimer approximation. Our results for ${\mathrm{H}}_{2}$ and HD are two orders of magnitude more accurate than the best literature values. We also improve the previous best results for ${\mathrm{D}}_{2}$.

Polarizabilities of hydrogen molecules calculated without using the Born-Oppenheimer approximation

Energies and Oscillator Strengths for Lithium-Like Ions

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