Compact natural orbital expansions for the helium ground state

Using the natural orbital representation and optimizing the exponents in the Slater-type orbital basis, configuration interaction type wave functions for the helium atom are given which combine compactness and high accuracy.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

Angular correlation of electrons in the ground state of helium

The spatial angular correlation of electrons in the ground state of the helium atom has been examined using configuration interaction and Hylleraas wave-functions. It was found that, in general, the average angle between the electrons is not a maximum when the two electrons are at the same distance from the nucleus. For configuration interaction wave-functions there is a position of the electrons for which the average value of the angle between the electrons is a maximum. Hylleraas wave-functions do not show this behavior.     

Path integral ground state with a fourth-order propagator: application to condensed helium

Ground state properties of condensed helium are calculated using the path integral ground state (PIGS) method. A fourth-order approximation is used as short (imaginary) time propagator. We compare our results with those obtained with other quantum Monte Carlo (QMC) techniques and different propagators. For this particular application, we find that the fourth-order propagator performs comparably to the pair product approximation, and is far superior to the primitive approximation. Results obtained for the equation of state of condensed helium show that PIGS compares favorably to other QMC methods traditionally utilized for this type of calculation.     

Explicitly correlated Gaussian functions with r factors for calculations of the ground state of the helium atom

Explicitly correlated Gaussian functions with r exp( ? β r ) factors have been used in variational calculations of the ground state of the helium atom. Additional correlation factors in the form of even powers of r _ij were introduced to the Gaussian functions with exponential correlation components by differentiating these functions with respect to the correlation exponent β. The algorithm of this method and its computational implementation is described. A number of calculations were performed for the ground state of helium atom to test the performance of the basis sets comparising different numbers of the Gaussians with the exp( ? β r ) and r exp( ? β r ) correlation factors. The numerical results indicate that including functions with r factors does not lead to improved results, contrary to what was anticipated initially. © 1994 by John Wiley & Sons, Inc.     

Correlation holes for the helium dimer

We have investigated the radial electron pair probability distributions (REPPDs) of the helium dimer within the Piris natural orbital functional (PNOF) theory. The analytical formulas to evaluate intracule densities, Fermi, Coulomb, and total correlation holes using our reconstruction functional PNOF-2 [J. Chem. Phys. 126, 214103 (2007)] are derived. The L?wdin's Coulomb holes from PNOF-2 and full configuration interaction calculations are analyzed showing a very similar behavior. New definitions of the Coulomb and Fermi holes based on the cumulant expansion of the two-particle reduced density matrix are presented. The holes are defined in terms of the exact one-particle reduced density matrix and the two-particle cumulant without any reference to the Hartree-Fock state. Through these definitions, we analyze separately the contribution of each component to the total REPPD at several values of the internuclear distance. A straight connection between the Coulomb hole and dispersion interactions is observed.     

23S state of helium

A rapidly convergent method, which has previously been applied to the ground state of the helium atom, has been extended to excited S states. This method is based on an expansion of the wave function in powers of \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {r_1^2 + r_2^2} $\end{document}, ln (r + r), \documentclass{article}\pagestyle{empty}\begin{document}$ w = r_{12} /\sqrt {r_1^2 + r_2^2} $\end{document}. Different effective nuclear charges are used for the inner and the outer electrons. Very satisfactory results are obtained for expectation values of various operators.     

Variational calculations in a halfspace for the interaction of a ground and excited state helium atom with its mirror image

Correlated trial wave-functions (depending explicitly upon the inter-electronic separation in the helium atom) subjected to boundary conditions excluding the helium electrons from the solid surface are used in half space, to evaluate in a variational approach the holding potential between an helium atom in its ground and metastable³S state with a perfectly imaging metallic conductor limited by a plane surface. Inter-electronic correlation effects are pointed out, and are seen to be non negligible. The holding potential for the helium³S displays a well of nearly an order of magnitude deeper than for the ground state. These results could be useful to determine the distance of closest approach of thermal atoms, and an approximate wave function for an helium atom interacting with a surface.     

Newton–Raphson optimization of the explicitly correlated Gaussian functions for calculations of the ground state of the helium atom

Explicitly correlated Gaussian functions have been used in variational calculations on the ground state of the helium atom. The major problem of this application, as well as in other applications of the explicitly correlated Gaussian functions to compute electronic energies of atoms and molecules, is the optimization of the nonlinear parameters involved in the variational wave function. An effective Newton–Raphson optimization procedure is proposed based on analytic first and second derivatives of the variational functional with respect to the Gaussian exponents. The algorithm of the method and its computational implementation is described. The application of the method to the helium atom shows that the Newton–Raphson procedure leads to a good convergence of the optimization process. © 1994 by John Wiley & Sons, Inc.     

Landé factor measurement for the Te2 electronic ground state

Non-linear ground-state quantum beat resonance under harmonically modulated excitation has been studied for a single rovibrational level of a diatomic molecule. Excitation of tellurium dimers by means of the 5145 nm line of an argon ion laser was used. The Landé factor for the rovibrational (ν″ = 6.J″ = 52) X_0g⁺ state of ¹³⁰Te₂ molecules has been determined equalling (1.68 ± 0.05) × 10⁻⁴. The result is close to the theoretically predicted one.     

An investigation of the reliability of the Galerkin-Petrov method with a special study of the helium atom ground state

A convergence characterization of the Galerkin-Petrov method by means of quantities characterizing pairs of subspaces is presented. The usefulness of our approach for setting up reliable computational schemes is demonstrated for the He atom ground state. Several methods of constructing pairs of subspaces for use in quantum chemistry are suggested.     

Variational functions of the padé type—application to the ground state of the helium atom

It is shown that trial functions involving Padé approximants yield satisfactory results for the ground state of the helium atom. In particular, the five-parameter form reproduces the best variational function of the type Ψ = e^–zs?(u) obtained numerically, to a remarkable extent.     

Configuration interaction benchmark for Be ground state

A set of 432 energy-optimized Slater-type radial orbitals together with spherical harmonics up to ? = 30 is used to approximate the corresponding full configuration interaction (CI) expansion for Be ground state. An analysis of radial and angular patterns of convergence for the energy yields a basis set incompleteness error of 8.7 μhartree of which 85% comes from radial basis truncations for ? ≤ 30. Select-divide-and-conquer CI (Bunge in J Chem Phys 125:014107, 2006; Bunge and Carbó-Dorca in J Chem Phys 125:014108, 2006) produces an energy upper bound 0.02(1) μhartree above the full CI limit. The energy upper bound E = ?14.6673473 corrected with these two truncation energy errors yields E = ?14.6673560 a.u. (Be) in fair agreement with the latest explicitly correlated Gaussian results of E = ?14.66735646 a.u. (Be). The new methods employed are discussed. It is acknowledged that at this level of accuracy traditional atomic CI has reached a point of diminishing returns. Modifications of conventional (orbital) CI to seek for significantly higher accuracy without altering a strict one-electron orbital formalism are proposed.     

High‐precision Hy–CI variational calculations for the ground state of neutral helium and helium‐like ions

Hylleraas–configuration interaction (Hy–CI) method variational calculations with up to 4648 expansion terms are reported for the ground ¹S state of neutral helium. Convergence arguments are presented to obtain estimates for the exact nonrelativistic energy of this state. The nonrelativistic energy is calculated to be ?2.9037 2437 7034 1195 9829 99 a.u. Comparisons with other calculations and an energy extrapolation give an estimated nonrelativistic energy of ?2.9037 2437 7034 1195 9830(2) a.u., which agrees well with the best previous variational energy, ?2.9037 2437 7034 1195 9829 55 a.u., of Korobov (Phys Rev A 2000, 61, 64503), obtained using the universal (exponential) variational expansion method with complex exponents (Frolov, A. M.; Smith, V. H. Jr. J Phys B Atom Mol Opt Phys 1995, 28, L449). In addition to He, results are also included for the ground ¹S states of H^?, Li⁺, Be⁺⁺, and B⁺³. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

An accurate calculation of the ground state energy of the helium atom and the hydrogen negative ion using hyperspherical coordinates

The method of hyperspherical harmonics using an adiabatic separation between the hyper-radial coordinate and the hyperspherical angles is applied to the calculation of the ground state of the helium atom and the hydrogen negative ion. For each system the simple adiabatic separation provides much of the electron correlation energy. The non-adiabatic coupling is included by means of a perturbation treatment using both the Rayleigh—Schrödinger and Brillouin—Wigner approaches. These provide rapid convergence to the exact energy of −2.90372 au.     

Application of a H 2 + -like model to helium atom. Solution of the wave equation at the ground state

The wave equation of helium atom is solved based on a “H ₂ ⁺ -like” model. A good agreement between the theoretical and experimental ionization energies is obtained.     

Lifetime Measurement of the 33 P state of helium

The electron-photon delayed coincidence technique has been used to measure the lifetime of the 3³ P excited state in helium from exponential decay of the time coincidence spectrum. The present results are in good agreement with corresponding results of other workers.     

Correlation of ground and excited state dissociation constants of trans -para- and ortho-substituted cinnamic acids

The ground and excited state dissociation constants oftrans- para- and ortho-substituted cinnamic acids have been determined in 50% (v/v) dioxan-water mixture at 30°C using the Forster cycle. The measured dissociation constants are analysed in the light of single and dual substituent parameter (DSP) equations. Excited state dissociation constants ofp-substituted cinnamic acids correlate well with the exalted substituent constants. The DSP method of analysis shows that resonance effect is predominant relative to inductive effect in the excited state than in the ground state for the para-substituted acids. The single parameter equation gives poor correlation for the ortho-substituted acids. However, the DSP analysis shows fairly good correlation. The inductive effect is predominant relative to the resonance effect in the excited state than in the ground state