Correlation Function Hyperspherical Harmonic Method Research Articles

Photoionization accompanied by excitation at intermediate photon energies

We calculate the photoionization-with-excitation to photoionization ratios ${R}_{nl}$ and ${R}_{n}={\ensuremath{\Sigma}}_{l}{R}_{nl}$ for atomic helium and positive heliumlike ions at intermediate values of the photon energies. The final-state interactions between the electrons are included in the lowest order of their Sommerfeld parameter. This enables us, in contrast to purely numerical calculations, to investigate the roles of various mechanisms contributing beyond the high-energy limit. The system of two bound electrons is described by the functions obtained by the correlation function hyperspherical harmonic method. For the case of heliumlike ions we present the high-energy limits as a power expansion in the inverse charge of the nucleus. We analyze the contribution of excitation of states with nonzero orbital momenta to the ratios ${R}_{n}$. In the case of helium our results for ${R}_{n}$ are in good agreement with those of experiments and of previous calculations.

CALCULATION OF THE GROUND STATE HYPERFINE STRUCTURE FOR MUONIC HELIUM ATOMS WITH THE USE OF SOME LOCAL PROPERTIES OF THE WAVE FUNCTIONS

The ground-state hyperfine structure and other properties are calculated for muonic helium atoms (3 He +2μ-e- and 4 He +2μ-e-), with the use of some local properties of the wave functions in the domains where two particles are close to each other or far away. Simple wave functions incorporating these properties with one variational parameter is developed. Also, the electron-muon correlation function is considered in proposed wave functions. It has a correct behavior for r12 tending to zero and infinity. The calculated values for hyperfine structure, energy and expectation values of r2n in ground-state, are compared with the multibox variational approach and the correlation function hyperspherical harmonic method. Our obtained results are very close to the values calculated by mentioned methods and giving strong indications that the proposed wave functions in addition to being very simple, provide relatively accurate values for hyperfine structure, energy and expectation values of r2n, emphasizing on the importance of the local properties of the wave functions.

Calculation of the ground-state energy and average distance between particles for the nonsymmetric muonicHe3atom

A calculation of the ground-state energy and average distance between particles in the nonsymmetric muonic $^{3}\mathrm{He}$ atom is given. We have used a wave function with one free parameter, which satisfies boundary conditions such as the behavior of the wave function when two particles are close to each other or far away. In the proposed wave function, the electron-muon correlation function is also considered. It has a correct behavior for ${r}_{12}$ tending to zero and infinity. The calculated values for the energy and expectation values of ${r}^{2n}$ are compared with the multibox variational approach and the correlation function hyperspherical harmonic method. In addition, to show the importance and accuracy of approach used, the method is applied to evaluate the ground-state energy and average distance between the particles of nonsymmetric muonic $^{4}\mathrm{He}$ atom. Our obtained results are very close to the values calculated by the mentioned methods and giving strong indications that the proposed wave functions, in addition to being very simple, provide relatively accurate values for the energy and expectation values of ${r}^{2n}$, emphasizing the importance of the local properties of the wave function.

Shape variation of the two-electron photoionization spectrum with photon energy growth

We trace the evolution of the shape of the two-electron photoionization spectrum with photon energy growth using quite a precise two-electron wave function, obtained by the correlation function hyperspherical harmonic method. We obtain the values of {omega}{sub 1} and {omega}{sub 2} at which the spectrum curve changes its shape. At {omega}={omega}{sub 1} the U shape changes to a W shape. At {omega}={omega}{sub 2} the central W peak splits into two. We consider ground states of the helium atom and of heliumlike ions with the nuclear charge Z, the negative ion of hydrogen H{sup -}, and the excited n{sup 1}S state of helium. The limiting laws for Z>>1 and n>>1 are obtained. The analysis is carried out without calculations of the particular energy distributions themselves.

Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three-Body Problem

A powerful approach to solve the Coulombic quantum three-body problem is proposed. The approach is exponentially convergent and more efficient than the hyperspherical coordinate method and the correlation-function hyperspherical harmonic method. This approach is numerically competitive with the variational methods, such as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate the efficiency of this approach, by calculating the nonrelativistic and infinite-nuclear-mass limit of the ground state energy of the helium atom. The exponential convergency of this approach is due to the full matching between the analytical structure of the basis functions that are used in this paper and the true wavefunction. This full matching was not reached by most other methods. For example, the variational method using the Hylleraas-type basis does not reflects the logarithmic singularity of the true wavefunction at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic series method. Besides these, this work makes use of the least square method to substitute complicated numerical integrations in solving the Schrödinger equation without much loss of accuracy, which is routinely used by people to fit a theoretical curve with discrete experimental data, but here is used to simplify the computation.

Local properties of three-body atomic wave functions

The local properties and accuracy of the positronium negative-ion $({\mathrm{Ps}}^{\ensuremath{-}})$ ground-state wave functions obtained by the stochastic variational method (SVM) and by direct solution of the Schr\odinger equation with the help of the correlation-function hyperspherical-harmonic method (CFHHM) are studied and compared. Though the energy, calculated by both methods, agrees to up to ten digits, the amplitudes of the values of the operator $D=H\ensuremath{\Psi}/E\ensuremath{\Psi}\ensuremath{-}1,$ characterizing local deviation of the wave function from its true value, in all of the coordinate space in the SVM are consistently larger (by up to five orders of magnitude) than in the CFHHM, despite the fact that the SVM observables except $〈\ensuremath{\delta}({\mathbf{r}}_{k})〉$ converge to significantly more digits than the CFHHM observables for their respective selected bases.

FBS Colloquium Hyperspherical-Harmonics Methods for Few-Body Problems

A review of hyperspherical-harmonics (HH) methods from the standpoint of their applications is given. In the first lecture, the symmetrized and unsymmetrized HH bases and symmetrization methods are presented. The physical obstacles to the straightforward application of the HH expansion are discussed, and expansion acceleration methods are described. In the second lecture, the main HH methods are described, including the correlation function hyperspherical harmonic method (CFHHM), the potential harmonic (PH) methods, and the correlated HH methods (PHH, CHH). The third lecture discusses the advantages and limitations of different HH methods in applications, and compares the results for specific few-body problems obtained by HH methods as well as non-HH methods.

Nonvariational calculation of the hyperfine splitting and other properties of the ground state of the muonic3He atom

The properties of the ground state of the muonic helium atom $e\ensuremath{\mu}{}^{3}{\mathrm{He}}^{2+}$ have been calculated nonvariationally, using the correlation function hyperspherical harmonic method utilizing a nonlinear parametrization of the correlation function. The parametrization is similar to the one used in an earlier paper for $e\ensuremath{\mu}{}^{4}{\mathrm{He}}^{2+}$ but the differences in the convergence were found to be important for the choice of optimal parameters. The parametrization is especially suited to accelerate the convergence of singular operators. As a result, the obtained expectation values of the $\ensuremath{\delta}({\mathbf{r}}_{k})$ operators have error margins smaller than the differences in the literature. The lowest-order hyperfine splitting, which depends on the fine structure constant and on the magnetic moment of the ${}^{3}{\mathrm{He}}^{2+}$ nucleus, is compared with values in the literature.

Nonvariational calculation of the hyperfine splitting and other properties of the ground state of the muonic helium atom

The properties of the ground state of the muonic Helium atom $e\ensuremath{\mu}{ }^{4}{\mathrm{He}}^{2+}$ have been calculated nonvariationally. The correlation function hyperspherical harmonic method utilizing a nonlinear parametrization of the correlation function has been used. Up to $N=561$ coupled second-order differential equations were taken into account. Although all parametrizations of the correlation function accelerate the convergence with respect to linear parametrizations by several orders of magnitude, an especially fast converging parametrization was found. All parametrizations make the observables converge to the same values in the limit of large $N$. The lowest-order hyperfine splitting obtained, 4454.206(3) MHz, has error margins smaller than the differences in the literature. One variational value is 0.023 MHz lower and another 0.013 MHz higher, after the adjustment for the different masses used. The expectation value of the distance between the electron and the muon also differs slightly from that in the literature, while the energy obtained was below the variational values.

Nonvariational calculation of the sticking probability and fusion rate for the microdt molecular ion.

The sticking probability for the \ensuremath{\mu}dt molecular ion is estimated with the help of directly calculated nonvariational wave functions obtained by the correlation function hyperspherical harmonic method. The method generates an accurate, locally correct \ensuremath{\mu}dt wave function which leads to precise estimates of the partial-wave sticking coefficients and the fusion rate. Our calculations are compared with other precision computations.

Nonvariational calculation of the relativistic, finite-size, and QED corrections for the 2 (1)S excited state of the helium atom.

Relativistic and QED corrections are calculated by using a direct solution of the Schroedinger equation for the 2 [sup 1][ital S] excited state of the helium atom obtained with the correlation-function hyperspherical-harmonic method. Our extremely accurate nonvariational results for relativistic, QED, and finite-size corrections coincide exactly (up to 0.000 03 cm[sup [minus]1]) with the values obtained in precision variational calculations of Drake [Nucl. Instrum. Methods Phys. Res. B 5, 2207 (1988)] and Baker, Hill, and Morgan [in [ital Relativistic], [ital Quantum] [ital Electrodynamic] [ital and] [ital Weak] [ital Interaction] [ital Effects] [ital in] [ital Atoms], edited by Walter Johnson, Peter Mohr, and Joseph Sucher, AIP Conf. Proc. No. 189 (AIP, New York, 1989), p. 123] for both infinite and finite nuclear masses. This confirms that a discrepancy of 0.0033 cm[sup [minus]1] between theory and experiment is not a result of an inaccuracy of variational wave functions, but is rooted in our inadequate knowledge of the QED operators. A better understanding of the different QED contributions to the operators (such as, for example, a more precise estimate of the Bethe logarithm) is therefore needed to explain the discrepancy.

Precise non-variational calculation of the ?dt molecular ion

Direct solutions of the Schrodinger equation for the ground and excitedS-states of the μdt molecular ion are obtained with the correlation-function hyperspherical-harmonic method. The method generates locally correct wave functions that lead to precise expectation values of the Hamiltonian and of different functions of interparticle distances. The nonlinear correlation factor used in the present work is chosen to provide a proper description of asymptotic behaviour and cusp singularities in all variables. The calculations are compared with our previous CFHH calculations with the linear correlation factor (which accounts only for the μd and μt cusp structure) and with other precision computations. Significant improvements over the linear correlation factor are found, with values of some observables having comparable or better accuracy than in the current literature.

Correlation-function hyperspherical-harmonic calculation of the microdt molecular ion.

Direct solution of the Schrodinger equation for the ground and excited S states of the μdt molecular ion is obtained with the help of the correlation-function hyperspherical-harmonic method. Given the proper correlation function, chosen from physical considerations, the method generates wave functions, accurate in the whole range of interparticle distances, which lead in turn to precise estimates of the expectation values of the Hamiltonian and of different functions of interparticle distances. Our results are compared with those obtained in other precision calculations

Graphical studies of three-body wave functions obtained with the correlation-function hyperspherical-harmonic method.

Graphical representations of three-particle wave functions obtained by direct solution of the Schr\odinger equation with the correlation-function hyperspherical-harmonic (CFHH) method are obtained and analyzed for ground and excited states of the helium atom and for the ground state of the \ensuremath{\mu}dd mesomolecular ion. The inclusion of adequate singular and cluster-correlation behavior is shown to be of crucial importance for a proper description of the wave function. In the CFHH method the wave function is a product of a correlation function and of a smooth factor expanded into hyperspherical-harmonic (HH) functions. While the HH expansion by itself is not able to reproduce a correct form of the wave function, the inclusion of the correlations results in its proper description even for low values of the maximal global momenta ${\mathit{K}}_{\mathit{m}}$.

Precise nonvariational calculation of excited states of helium with the correlation-function hyperspherical-harmonic method.

Direct solution of the Schrodinger equation for resonant S states of the Helium atom is obtained with the help of the complex rotation method (which reduces the resonant problem to that of bound states) and the correlation function hyperspherical harmonic method (CFHHM). In the CFHH method the bound state solution is a product of a correlation function and of a smooth factor expanded into hyperspherical harmonic functions. Given a proper correlation function, chosen from physical considerations, the method generates resonant wave functions, accurate in the whole range of interparticle distances. Calculated energies are compared to those obtained by variational and other precise computations.

Accuracy of three-body wave functions obtained with the correlation-function hyperspherical-harmonic method.

The local convergence and accuracy of wave functions obtained by direct solution of the Schr\"odinger equation with the help of the correlation-function hyperspherical-harmonic method are analyzed for ground and excited states of the helium atom and for the ground state of the positronium negative ion. The inclusion of the cusp conditions into the correlation function is shown to be of crucial importance, not only near the coalescence points, but also away from them. The proper inclusion of all cusps yields for the ground state of the helium atom the local wave-function accuracy of about ${10}^{\mathrm{\ensuremath{-}}7}$ for different interparticle distances. The omission of one of the cusps in the excited helium atom reduces the wave-function precision to ${10}^{\mathrm{\ensuremath{-}}2}$ near the corresponding coalescence point and to ${10}^{\mathrm{\ensuremath{-}}4}$--${10}^{\mathrm{\ensuremath{-}}5}$ away from it.

Matrix elements of potentials in the correlation-function hyperspherical-harmonic method.

Matrix elements of two-body potentials and correlation functions between three- and four-body hyperspherical states, including their velocity-dependent parts, are calculated analytically for any value of the total orbital angular momentum. The resulting formulas contain explicitly written functions of the radial variable, and the Raynal-Revai coefficients. The latter are expressible through finite sums of 3-j and 9-j symbols. The formulas allow precise and fast evaluation of matrix elements of the effective potential in the correlation-function hyperspherical-harmonic method for atomic, molecular, and nuclear three- and four-body problems. The generalization to any number of particles is straightforward.

Correlation-function hyperspherical harmonic calculations of the pp micro, dd micro, and tt micro molecular ions.

The direct solution of the Schr\"odinger equation for the ground state of the pp\ensuremath{\mu}, dd\ensuremath{\mu}, and tt\ensuremath{\mu} molecular ions is obtained with the help of the correlation-function hyperspherical harmonic method. Given the proper correlation function, chosen from physical considerations, the method generates wave functions, accurate in the whole range of interparticle distances, which leads in turn to precise estimates of the expectation values of the Hamiltonian and of different functions of interparticle distances. Our results are compared with those obtained in other precision calculations.

Precise nonvariational calculation of the positronium negative ion.

The three-body Schr\odinger equation is solved directly for the ground state of the positronium negative ion by using a rapidly convergent correlation function hyperspherical harmonic method, which involves no adjustable parameters. The inclusion of 169 hyperspherical functions yields the ground-state energy 0.262 004 86 a.u., which is converged to an error in the seventh significant figure. Expectation values of different functions of interparticle distances as well as the two-photon annihilation rate are calculated and compared with those obtained by variational calculations.