Hylleraas Functions Research Articles

Shannon information entropy in position space for doubly excited states of helium with finite confinements

Quantifying electron localization in quantum confined systems remains challenging, especially for excited states. A quantum dot (QD) is represented by a helium atom in a finite oscillator potential. The effect of dot width variation on the electron localization in QD is systematically examined via Shannon entropy for low-lying doubly excited states (2s21Se, 2p21Se, 2s3s 1Se) obtained using highly correlated Hylleraas functions. In particular, the most effective dot width where the electron density is the most localized is determined successfully and justified by the electron density plot for all three states.

Computation of two-electron screened Coulomb potential integrals in Hylleraas basis sets

The Gegenbauer expansion and Taylor expansion methods are developed to accurately and efficiently calculate the two-electron screened Coulomb potential integrals in Hylleraas basis sets. The combination of these two methods covers the entire parameter space of the integrals, including arbitrary total angular momenta, two-electron configurations, powers of inter-electronic coordinate, and complex screening parameters. Numerical examples are given and comparisons with other computational methods in some restricted situations are made. The present methods can be easily applied to calculate the bound and resonant states of two-electron atoms or exotic three-body systems embedded in the screening environment by using the Hylleraas or Hylleraas-CI basis functions.

Spherically compressed helium atom described by perturbative and variational methods

We analyze the energy spectrum of the three lowest-lying S symmetry states for the spherically confined helium atom as a function of the box radius by using an approach based on a time independent perturbation theory and two variational methods. The first treatment depends on exact solutions for confined one-electron atoms, whereas in the latter two methods exponents and linear coefficients are variationally optimized via { s , t , u } -Hylleraas functions and Generalized { r 1 , r 2 , r 12 } -Hylleraas basis sets that fulfill appropriate boundary conditions. Although it is found that throughout most of the box radii here analyzed the variational energies for the three states lie below those perturbatively obtained, an opposite trend occurs toward the weak and strong confinement regions for the singlet excited and triplet states, respectively.

Ground state of Li andBe+using explicitly correlated functions

We compare the explicitly correlated Hylleraas and exponential basis sets in the evaluations of ground state of Li and ${\text{Be}}^{+}$. Calculations with Hylleraas functions are numerically stable and can be performed with the large number of basis functions. Our results for ground-state energies $\ensuremath{-}7.478\text{ }060\text{ }323\text{ }910\text{ }10(32)$ and $\ensuremath{-}14.324\text{ }763\text{ }176\text{ }790\text{ }43(22)$ of Li and ${\text{Be}}^{+}$, correspondingly, are the most accurate to date. When small basis set is considered, explicitly correlated exponential functions are much more effective. With only 128 functions we obtained about ${10}^{\ensuremath{-}9}$ relative accuracy, but the severe numerical instabilities make this basis costly in the evaluation.

Electron affinity of Li7 calculated with the inclusion of nuclear motion and relativistic corrections

Explicitly correlated Gaussian functions have been used to perform very accurate variational calculations for the ground states of (7)Li and (7)Li(-). The nuclear motion has been explicitly included in the calculations (i.e., they have been done without assuming the Born-Oppenheimer (BO) approximation). An approach based on the analytical energy gradient calculated with respect to the Gaussian exponential parameters was employed. This led to a noticeable improvement of the previously determined variational upper bound to the nonrelativistic energy of Li(-). The Li energy obtained in the calculations matches those of the most accurate results obtained with Hylleraas functions. The finite-mass (non-BO) wave functions were used to calculate the alpha(2) relativistic corrections (alpha=1c). With those corrections and the alpha(3) and alpha(4) corrections taken from Pachucki and Komasa [J. Chem. Phys. 125, 204304 (2006)], the electron affinity (EA) of (7)Li was determined. It agrees very well with the most recent experimental EA.

Approximating the target wave function in positronium-helium scattering

We consider the effect of using necessarily inexact target wave functions in a Kohn variational calculation of the $\mathrm{Ps}\mathrm{He}$ scattering length. We use the simplest closed-shell or one-term Hylleraas function in the static-exchange approximation and examine two plausible ways of handling such an inexact function. Both of these methods have been employed in the past, and our results are compared with these previous calculations. Significant differences are found which may persist in more elaborate calculations.

Singular and nonsingular three-body integrals for exponential wave functions.

Integrals which are individually singular, but which may be combined to yield convergent expressions, are needed for computations of relativistic effects and various properties of atomic and quasiatomic systems. As computations become more detailed and precise, more such integrals are required. This paper presents general formulas for the radial parts of the singular and nonsingular (regular) integrals that occur when three-body systems are described using wave functions that include exponentials in all three interparticle coordinates. Our results are compared with those found in the literature for some of the integrals, and are also shown to be consistent with previously reported results for Hylleraas functions (a limiting case in which one of the exponential parameters is set to zero).

One-range addition theorems for derivatives of Slater-type orbitals.

Using addition theorems for STOs introduced by the author with the help of complete orthonormal sets of psi(alpha)-ETOs (Guseinov II (2003) J Mol Model 9:190-194), where alpha=1, 0, -1, -2, ..., a large number of one-range addition theorems for first and second derivatives of STOs are established. These addition theorems are especially useful for computation of multicenter-multielectron integrals over STOs that arise in the Hartree-Fock-Roothaan approximation and also in the Hylleraas function method, which play a significant role for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules, and solids. The relationships obtained are valid for arbitrary quantum numbers, screening constants and location of STOs.

Polarizabilities and other properties of the tdµ molecular ion

Wave functions of the Hylleraas type were used earlier to calculate energy levels of muonic systems. Recently, we found in the case of the molecular ions H2+, D2+, and HD+ that it was necessary to include high powers of the internuclear distance in the Hylleraas functions to localize the nuclear motion when treating the ions as three-body systems without invoking the Born–Oppenheimer approximation. We tried the same approach in a muonic system, tdµ– (triton, deuteron, and muon). Improved convergence was obtained for J = 0 and 1 states for shorter expansions when we used this type of generalized Hylleraas function, but as the expansion length increased the high powers were no longer useful. We obtained good energy values for the two lowest J = 0 and 1 states and compared them with the best earlier calculations. Expectation values were obtained for various operators, the Fermi contact parameters, and the permanent quadrupole moment. The cusp conditions were also calculated. The polarizability of the ground state was then calculated using second-order perturbation theory with intermediate J = 1 pseudostates. (It should be possible to measure the polarizability by observing Rydberg states of atoms with tdµ– acting as the nucleus.) In addition, the initial sticking probability (an essential quantity in the analysis of muon catalyzed fusion) was calculated and compared with earlier results. PACS Nos.: 30.00, 36.10-k, 02.70-c

Calculation of theHe− 1s2s22Sresonance using fully correlated Hylleraas functions

Calculation of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">−</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mi> </mml:mi><mml:mrow><mml:msup><mml:mrow><mml:mn>1</mml:mn><mml:mi>s</mml:mi><mml:mn>2</mml:mn><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi /></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi>S</mml:mi></mml:math>resonance using fully correlated Hylleraas functions

Calculations of doubly excited states in helium-like systems using Hylleraas functions

Doubly excited or Feshbach-type resonances in helium-like systems have been calculated using the complex-rotation method. Using Hylleraas-type wavefunctions of up to 1632 terms, we calculate resonance parameters (both resonance positions and widths) below the N = 3 and 4 thresholds of the hydrogenic systems with nuclear charge from Z = 2 to Z = 10. Comparisons are made with the other calculations for in the literature.

High-order nonlinear susceptibilities of helium

With the goal of providing critically evaluated atomic data for modeling high harmonic generation processes in noble gases, we present calculations of frequency-dependent nonlinear susceptibilities of the ground state of helium, within the framework of Rayleigh-Schr\"odinger perturbation theory at lowest applicable order. The nonrelativistic, infinite-nuclear-mass, atomic Hamiltonian is decomposed in terms of Hylleraas coordinates and spherical harmonics, and the hierarchy of inhomogeneous equations of perturbation theory is solved iteratively. The mixed Hylleraas and Frankowski basis functions are employed to represent accurately the ground-state and perturbed wave functions. We believe our results for nonlinear susceptibilities of helium are the most accurate (and in many cases, the only) available to date, and they are offered as benchmark data for design of future multiphoton experiments.

Variational calculation of ground-state energy of positronium negative ions.

The ground-state energy of positronium negative ions, ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$, is calculated using double-basis-set Hylleraas functions in which four nonlinear parameters are used. The optimized result of E=-0.524 010 140 465 7 Ry obtained by using 744-term function represents the lowest variational energy for ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ in the literature to date to our knowledge. The wave function is then used to calculate the positron annihilation rate for ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$.

Complex-coordinate calculation of 1,3D resonances in two-electron systems.

Feshbach-type $^{1,3}$D resonances in two-electron systems, Z=2--10, have been investigated using the method of complex rotation. These states lie below the n=2 and 3 thresholds of hydrogenic systems. Wave functions containing up to 1230 Hylleraas functions have been used, giving accurate results for positions and widths. Comparisons between various calculations are given.

Magnetically dressed one-electron molecular orbitals.

A general method for solving the stationary one-electron, two-center Coulomb problem with a superimposed (uniform) strong magnetic field is described and applied. For arbitrary orientation of the field with respect to the line connecting the centers, the pertinent Schroedinger equation is solved by evaluating analytically the Hamiltonian matrix in a basis of (nonorthogonal) Hylleraas functions and solving numerically the generalized eigenvalue problem for this matrix. A detailed study of the properties of ''magnetically dressed'' (diatomic) one-electron molecular orbitals is performed by calculating energies and wave functions for the H/sub 2//sup +/ and (H-He)/sup 2+/ systems for field strengths up to about 10/sup 8/ T. Molecular-orbital correlation diagrams are presented and discussed, in which dressed-orbital energies are displayed as a function of internuclear distance R at fixed angle theta between field direction and internuclear axis, and as a function of theta at fixed R. Equilibrium internuclear distances and total binding energies are calculated as functions of field strength for the magnetically dressed H/sub 2//sup +/ system in its lowest gerade and ungerade states at theta = 0 and theta = 90/sup 0/. The influence of the magnetic field on molecular binding properties as well as on the separation behavior of molecularmore » orbitals at large internuclear distances is illustrated by means of wave-function plots. Whenever possible, our results are compared to those of previous investigations. The convergence properties of our method are discussed.« less

Quantum size effects in semiconductor crystallites: Calculation of the energy spectrum for the confined exciton

For the exciton (e −, h +) confined to a sphere (a simple model for the light absorption of a small semiconductor particle) we calculate some of the electronic states. We use procedures similar to those for the two-electron atom. For radii R smaller than the “exciton radius” simple one-configuration wavefunctions suffice. For larger R a more accurate description involving Hylleraas functions is required.

L-separated Hylleraas basis functions in the perturbation expansion of atomic states: Accurate partial-wave energies of second order in 1/Z for the 1s2 ground state.

The previously derived method of l-projected Hylleraas basis functions [for calculating the individual partial waves ${R}_{l}$(${r}_{1}$,${r}_{2}$)${P}_{l}$(costheta) of an atomic pair function of first perturbation order in 1/Z] is analyzed again and improved. It suffices to project, onto ${P}_{l}$(costheta), a subset of Hylleraas functions (the powers of u=${r}_{12}$ and s=${r}_{1}$+${r}_{2}$ alone). The l-separated Hylleras functions give rapid convergence since they possess a similar discontinuity at ${r}_{1}$=${r}_{2}$ as the exact partial wave [C. Schwartz, Phys. Rev. 126, 1015 (1962)]. The handling of the l projections is easy. The method is applied to the 1${s}^{2}$ pair to obtain, at relatively low basis-set dimensions and without using any nonlinear parameter, the heretofore most accurate values of the partial-wave energies ${E}_{l}^{2}$ (l=0,1,2,. . .) and of the entire second-order energy ${E}_{\mathrm{He}{}^{2}}$.

Line-shape parameters forP1Feshbach resonances in He andLi+

The line-shape parameter ($q$) for the three lowest $^{1}P$ autoionization states of He and ${\mathrm{Li}}^{+}$ below the first excited target (${\mathrm{He}}^{+}$, ${\mathrm{Li}}^{2+}$) level are accurately calculated. In order to do this we rederive a rigorous expression for $q$ in terms of the Feshbach formalism; the formula contains additional terms representing the effect of other resonances on the resonance being calculated. Hylleraas functions are used to calculate $Q\ensuremath{\Phi}$, and the exchange approximation is used for the nonresonant continuum. Results agree well with the experiment where available. Our previous calculations of positions and widths are not affected; however, it is noted that the energy of the newest experiment of Morgan and Ederer for the lowest $^{1}P$ autoionization state of He (60.151 \ifmmode\pm\else\textpm\fi{} 0.01 eV) is such that when it is combined with the older experimental value of Madden and Codling (60.133 \ifmmode\pm\else\textpm\fi{} 0.015 eV), the weighted mean of the two measurements (60.145 \ifmmode\pm\else\textpm\fi{} 0.008 eV) is in exceptional agreement with our previous value (60.145 eV).

Perturbation theory in1Zfor atoms: First-order pair functions in anl-separated Hylleraas basis set

For the $\frac{1}{Z}$ perturbation theory of atoms, a partial-wave method is presented for determining first-order pair wave functions. It rests on the fact that the Hylleraas variational principle decouples for the individual partial waves ($l=0,1,2,\dots{}$) and that all partial waves for $l\ensuremath{\gg}1$ are easily representable (Schwartz, 1962). The $l\mathrm{th}$ partial wave is approximated in basis functions obtained by projecting the well-known Hylleraas functions (containing the powers of ${r}_{12}$ onto ${P}_{l}(cos{\ensuremath{\theta}}_{12})$. Results for the ${1s}^{2}$ ground state show rapid convergence. The variational value for the (total) second-order ${1s}^{2}$ energy, which would be provided by 45 Hylleraas functions, is achieved with 10, 12, 9, 8, 5, and 5 basis functions for $l=0,1,2,3,4$, and $5$, respectively. For any $1\ensuremath{\ge}6$, one function is sufficient. Also good convergence is found for three-electron integrals (parts of the second-order lithium energy).

CI-Hylleraas variational calculation on the ground state of the neon atom

This calculation demonstrates that the combined configuration-interaction-Hylleraas (CI-Hy) variational method of Sims and Hagstrom can be applied to the $^{1}S$ ground state of the neon atom. Coordinates ${r}_{\mathrm{ij}}$ have been included in the wave function so as to correlate explicitly the motion of all pairs of electrons in the atom. An energy of - 128.8298 hartree has been obtained from an 83-term wave function (73.5% of the correlation energy). An analysis of the pair correlation energies shows that it is more difficult to represent the electronic correlation in $p$-$p$ and $s$-$p$ pairs than $s$-$s$ pairs using Hylleraas functions. More experience with CI-Hy calculations on atoms containing as many as ten electrons is required before the method can compete with and give higher accuracy than the conventional CI calculations.