Physical Review A

Correlation of two weakly attractive particles on a sphere

Robert J. Hinde, R. Stephen Berry

Published Aug 1, 1990• 14 citations

Abstract

We simulate the behavior of two weakly attractive atoms, either spin-polarized hydrogen atoms or $^{3}\mathrm{He}$ atoms, on a $^{4}\mathrm{He}$ cluster by examining the angular correlation of the atoms as they move on the surface of a sphere. We have calculated converged energy levels, wave functions, and reduced densities ${\mathrm{\ensuremath{\rho}}}^{(\mathit{J})}$(${\mathrm{\ensuremath{\theta}}}_{12}$) for the low-lying states of these two systems. We also give the oscillator strengths for several transitions between low-lying states for the case of spin-polarized hydrogen. The energy levels can be arranged in patterns reminiscent of the rovibrational levels of a linear triatomic molecule, but the reduced densities differ significantly from those predicted by the molecular model. The lowest state for two $^{3}\mathrm{He}$ atoms on a sphere of radius 13.0 bohr is bound, with E=-0.031 90 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.

DOI: 10.1103/physreva.42.2259

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Physical Review A

Correlation of two weakly attractive particles on a sphere

Robert J. Hinde, R. Stephen Berry

Published Aug 1, 1990• 14 citations

Publisher

Abstract

We simulate the behavior of two weakly attractive atoms, either spin-polarized hydrogen atoms or

^{3}\mathrm{He}

atoms, on a

^{4}\mathrm{He}

cluster by examining the angular correlation of the atoms as they move on the surface of a sphere. We have calculated converged energy levels, wave functions, and reduced densities

{\mathrm{\ensuremath{\rho}}}^{(\mathit{J})}

(

{\mathrm{\ensuremath{\theta}}}_{12}

) for the low-lying states of these two systems. We also give the oscillator strengths for several transitions between low-lying states for the case of spin-polarized hydrogen. The energy levels can be arranged in patterns reminiscent of the rovibrational levels of a linear triatomic molecule, but the reduced densities differ significantly from those predicted by the molecular model. The lowest state for two

^{3}\mathrm{He}

atoms on a sphere of radius 13.0 bohr is bound, with E=-0.031 90

{\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}

DOI: 10.1103/physreva.42.2259

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Ground and excited-rovibrational states of weakly bound triatomic molecules containing two helium and one silver atoms are investigated. We consider three systems, $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}\mathrm{Ag}$, $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}\mathrm{Ag}$, and $^{3}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{Ag}$. The three-atom Schr\"odinger equation is solved in hyperspherical coordinates using the potential-energy surface represented as the addition of He-He and He-Ag pair interaction potentials. We compute the energy levels and analyze the structural properties by considering averaged pair distances and bond angles, one-dimensional pair and angle distribution functions, and two-dimensional pair-pair and angle-angle distribution functions. All the He-He-Ag bound states are found to possess similar characteristics, displaying obtuse isosceles ( $^{4}\mathrm{He}_{2}\mathrm{Ag}$ and $^{3}\mathrm{He}_{2}\mathrm{Ag}$ ) or scalene ( $^{3}\mathrm{He}^{4}\mathrm{He}\mathrm{Ag}$ ) triangle configurations where the He-Ag distances are nearly fixed and the Ag bond angle is variable.Ground and excited-rovibrational states of weakly bound triatomic molecules containing two helium and one silver atoms are investigated. We consider three systems, $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}\mathrm{Ag}$, $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}\mathrm{Ag}$, and $^{3}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{Ag}$. The three-atom Schr\"odinger equation is solved in hyperspherical coordinates using the potential-energy surface represented as the addition of He-He and He-Ag pair interaction potentials. We compute the energy levels and analyze the structural properties by considering averaged pair distances and bond angles, one-dimensional pair and angle distribution functions, and two-dimensional pair-pair and angle-angle distribution functions. All the He-He-Ag bound states are found to possess similar characteristics, displaying obtuse isosceles ($^{4}\mathrm{He}_{2}\mathrm{Ag}$ and $^{3}\mathrm{He}_{2}\mathrm{Ag}$) or scalene ($^{3}\mathrm{He}^{4}\mathrm{He}\mathrm{Ag}$) triangle configurations where the He-Ag distances are nearly fixed and the Ag bond angle is variable.

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Osnovno stanje klastera helija i spin-polariziranog vodika

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The ground-state of the helium and spin-polarized hydrogen quantum clusters is investigated in this work using Quantum Monte Carlo methods, as well as the spin-polarized tritium (T↓) liquid. For small mixed helium clusters consisting of up to eight helium atoms the ground-state energy and structure have been obtained. Stability of the 3He4 4He3 cluster, which was previously considered unstable, is predicted. It is confirmed that tritium trimer (T↓)3 is an example of Borromean state. For pure large spin-polarized tritium clusters consisting of up to 320 atoms the ground-state energy is obtained. Stability limits have been investigated for small mixed (T↓)n(D↓)m clusters, for m ≥ 2 i n + m ≤ 10. Especially, it has been shown that all clusters (T↓)ND↓ are stable for N ≥ 3. It is also shown that even 60 T↓ atoms are not enough to form stable cluster (T↓)60H↓. Equation of state, spinodal density, energy per particle at the equilibrium density, unit radius and surface tension of the T↓ liquid have been determined using results obtained in the simulations of liquid phase.The ground-state of the helium and spin-polarized hydrogen quantum clusters is investigated in this work using Quantum Monte Carlo methods, as well as the spin-polarized tritium (T↓) liquid. For small mixed helium clusters consisting of up to eight helium atoms the ground-state energy and structure have been obtained. Stability of the 3He4 4He3 cluster, which was previously considered unstable, is predicted. It is confirmed that tritium trimer (T↓)3 is an example of Borromean state. For pure large spin-polarized tritium clusters consisting of up to 320 atoms the ground-state energy is obtained. Stability limits have been investigated for small mixed (T↓)n(D↓)m clusters, for m ≥ 2 i n + m ≤ 10. Especially, it has been shown that all clusters (T↓)ND↓ are stable for N ≥ 3. It is also shown that even 60 T↓ atoms are not enough to form stable cluster (T↓)60H↓. Equation of state, spinodal density, energy per particle at the equilibrium density, unit radius and surface tension of the T↓ liquid have been determined using results obtained in the simulations of liquid phase.

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R. Guardiola, Jesús Navarro, M. Portesi

We study low-lying excited states of ${}^{4}\mathrm{He}$ clusters up to a cluster size of 40 atoms in a variational framework. The ansatz wave function combines two- and three-body correlations, coming from a translationally invariant configuration interaction description, and Jastrow-type short-range correlation. We have previously used this scheme to determine the ground-state energies of ${}^{4}\mathrm{He}$ and ${}^{3}\mathrm{He}$ clusters. Here we present an extension of this ansatz wave function having a good quantum angular momentum L. The variational procedure is applied independently to the cases with $L=0,2,4,$ and upper bounds for the corresponding energies are thus obtained. Moreover, centroid energies for L excitations are calculated through the use of sum rules. A comparison with previous calculations is also made.We study low-lying excited states of ${}^{4}\mathrm{He}$ clusters up to a cluster size of 40 atoms in a variational framework. The ansatz wave function combines two- and three-body correlations, coming from a translationally invariant configuration interaction description, and Jastrow-type short-range correlation. We have previously used this scheme to determine the ground-state energies of ${}^{4}\mathrm{He}$ and ${}^{3}\mathrm{He}$ clusters. Here we present an extension of this ansatz wave function having a good quantum angular momentum L. The variational procedure is applied independently to the cases with $L=0,2,4,$ and upper bounds for the corresponding energies are thus obtained. Moreover, centroid energies for L excitations are calculated through the use of sum rules. A comparison with previous calculations is also made.

Physical review. B, Condensed matterMay 24, 20016 citations

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Gregory S. Ezra, R. Stephen Berry

The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell $n=3$ in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\ensuremath{\rho}({\ensuremath{\theta}}_{12})$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell $n=3$ in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\ensuremath{\rho}({\ensuremath{\theta}}_{12})$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.

Physical Review AOct 1, 198357 citations

We simulate the behavior of two weakly attractive atoms, either spin-polarized hydrogen atoms or

^{3}\mathrm{He}

atoms, on a

^{4}\mathrm{He}

cluster by examining the angular correlation of the atoms as they move on the surface of a sphere. We have calculated converged energy levels, wave functions, and reduced densities

{\mathrm{\ensuremath{\rho}}}^{(\mathit{J})}

(

{\mathrm{\ensuremath{\theta}}}_{12}

^{3}\mathrm{He}

atoms on a sphere of radius 13.0 bohr is bound, with E=-0.031 90

{\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}

^{4}\mathrm{He}_{2}\mathrm{Ag}

and

^{3}\mathrm{He}_{2}\mathrm{Ag}

) or scalene (

^{3}\mathrm{He}^{4}\mathrm{He}\mathrm{Ag}

) triangle configurations where the He-Ag distances are nearly fixed and the Ag bond angle is variable.

k)

arising from a uniform background smeared throughout the sphere. The convergence of the ground state energy of the singlet spin state of the system is investigated as a function of sphere size (essentially

{r}_{s},

the Wigner--Seitz density parameter) for cases where there is no background potential

(k=0)

and for when $k\ensuremath{\ne}0.$ With

k=0

and small

{r}_{s},

we observe a maximum in the ground state density at the origin of the sphere. At ${r}_{s}\ensuremath{\approx}8\mathrm{a}.\mathrm{u}.,$ the ground state density acquires a minimum at the origin. For this and larger systems we identify the formation of a ``Wigner'' molecule state. We further investigate the ground state density as a function of k and also the correlation hole density as a function of

{r}_{s}

and k. We invert the Kohn--Sham equation for a two electron system and calculate the local effective potential and correlation potential (to within an additive constant) as functions of the radial coordinate for a number of values of

{r}_{s}

and k.

We study low-lying excited states of

{}^{4}\mathrm{He}

clusters up to a cluster size of 40 atoms in a variational framework. The ansatz wave function combines two- and three-body correlations, coming from a translationally invariant configuration interaction description, and Jastrow-type short-range correlation. We have previously used this scheme to determine the ground-state energies of

{}^{4}\mathrm{He}

and

{}^{3}\mathrm{He}

clusters. Here we present an extension of this ansatz wave function having a good quantum angular momentum L. The variational procedure is applied independently to the cases with

L=0,2,4,

and upper bounds for the corresponding energies are thus obtained. Moreover, centroid energies for L excitations are calculated through the use of sum rules. A comparison with previous calculations is also made.

n=3

in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\ensuremath{\rho}({\ensuremath{\theta}}_{12})$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.