Hylleraas Trial Wavefunction where
1
Trial Function Binding Energy Present work ω = 10 Extrapolated Hylleraas ω = 10 [3] Hylleraas ω = 12 [3] Extrapolated [3] Explicitly Correlated Gaussians [4] 0.039196324 0.039196654 0.0391966076 0.0391967051 0.0391967147(4 2) 0.039196765251 Hylleraas Trial Wavefunction where and k i , l i , m i , n i , p i , q i are non- negative integers. The binding energy of PsH D. Woods , S. J. Ward and P. Van Reeth † University of North Texas, † University College London Introduction In response to proposed measurements of Ps scattering by the St. Olaf’s positron experimental group [1], we have begun a theoretical investigation of Ps scattering from simple atoms. For our first step of this investigation, we have computed the binding energy of the fundamental four-body Coulomb system, 1 S PsH. We are using a very flexible trial function of Hylleraas form which includes all inter-particle distances. We use atomic units throughout, unless we state otherwise . p e+ e e r 1 r 2 r 3 r 23 r 12 r 13 We use a method given by Todd [6], which finds a subset of the full basis set that has the lowest energy. The remaining terms are added into this subset one by one to see which contributes the most to lowering the energy. This process stops when eigenvalues of the upper and lower matrices differ by 10 -7 or when LAPACK [7] returns an error. Acknowledgements The authors acknowledge the UNT Computing and Information Technology Center's High Performance Computing Initiative for providing resources on the Talon HPC cluster. We appreciate discussions with Prof. J.W. Humberston. References [1] Jason Engbrecht, Private communication, (2008). [2] G. W. F. Drake and Zong-Chao Yan, Phys. Rev. A 52, 3681 (1995). [3] Zong-Chao Yan and Y. K. Ho, Phys. Rev. A 59, 2697 (1999). [4] Sergiy Bubin and Ludwik Adamowicz, Phys. Rev. A 74, 052502 (2006). [5] P. Van Reeth and J. W. Humberston, Nucl. Instrum. Methods B 221, 140 (2004). [6] A. Todd, Ph.D. thesis, The University of Nottingham, Results Conclusion Our most accurate value of the binding energy compares favorably with a previous calculation [3] and the most accurate calculation to-date [4]. We have optimized the nonlinear parameters, however, it is expected that further refinement of the choice of nonlinear parameters should improve the results. We are currently using the Hylleraas trial wavefunction as the short-range terms of the full S-wave scattering wavefunction for Ps-H collisions. Method We calculate the energy of the bound state of 1 S PsH using the Rayleigh-Ritz variational method. We perform the integrations using the asymptotic- expansion method of Drake and Yan [2]. For values of ω > 5, linear dependencies exist in the matrices. Yan and Ho [3] overcame the problem of linear dependence by splitting the basis set into blocks based on Figure 1. Eigenvalues for 1 S PsH ω N(ω) Terms Used Binding Energy Terms Used [3] Binding Energy [3] 6 7 8 9 10 924 1716 3003 5005 8008 916 1585 1925 2155 2205 0.03916950 9 0.03918956 8 0.03919455 9 0.03919586 502 918 1589 2625 3501 0.039158895 6 0.039187922 3 0.039194876 8 0.039196380 The computed eigenvalues are comparable to those given earlier in reference [5]. The stabilized eigenvalues correspond to the resonances computed by Yan and Ho [3], which we denote by dashed lines. Table 1. Binding energy as a function of ω Table 2. Comparison with other work , i i i i i i q p n m l k i i i i i i q p n m l k r r r i i N i i r r r r r r e r r r r r r c P r r r Ψ 23 13 3 12 2 1 ) ( 3 2 1 3 2 1 ) ( 1 23 3 2 1 3 2 1 ) , , ( ) , , ( ) 1 ( ) , , ( | | | H E
description
The binding energy of PsH D. Woods , S. J. Ward and P. Van Reeth † University of North Texas, † University College London. Method We calculate the energy of the bound state of 1 S PsH using the Rayleigh-Ritz variational method. - PowerPoint PPT Presentation
Transcript of Hylleraas Trial Wavefunction where
Trial Function Binding EnergyPresent work ω = 10
ExtrapolatedHylleraas ω = 10 [3]Hylleraas ω = 12 [3]
Extrapolated [3]Explicitly Correlated Gaussians [4]
0.0391963240.0391966540.03919660760.03919670510.0391967147(42)0.039196765251
Hylleraas Trial Wavefunction
where and ki, li, mi, ni, pi, qi are non-negative integers.
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The binding energy of PsHD. Woods, S. J. Ward and P. Van Reeth†
University of North Texas, † University College London
Introduction In response to proposed measurements of Ps scattering by the St. Olaf’s positron experimental group [1], we have begun a theoretical investigation of Ps scattering from simple atoms. For our first step of this investigation, we have computed the binding energy of the fundamental four-body Coulomb system, 1S PsH. We are using a very flexible trial function of Hylleraas form which includes all inter-particle distances. We use atomic units throughout, unless we state otherwise .
p
e+
e
e
r1
r2
r3
r23
r12
r13
We use a method given by Todd [6], which finds a subset of the full basis set that has the lowest energy. The remaining terms are added into this subset one by one to see which contributes the most to lowering the energy. This process stops when eigenvalues of the upper and lower matrices differ by 10-7 or when LAPACK [7] returns an error.
Acknowledgements The authors acknowledge the UNT Computing and Information Technology Center's High Performance Computing Initiative for providing resources on the Talon HPC cluster. We appreciate discussions with Prof. J.W. Humberston.
References[1] Jason Engbrecht, Private communication, (2008).[2] G. W. F. Drake and Zong-Chao Yan, Phys. Rev. A 52, 3681 (1995).[3] Zong-Chao Yan and Y. K. Ho, Phys. Rev. A 59, 2697 (1999).[4] Sergiy Bubin and Ludwik Adamowicz, Phys. Rev. A 74, 052502 (2006).[5] P. Van Reeth and J. W. Humberston, Nucl. Instrum. Methods B 221, 140 (2004).[6] A. Todd, Ph.D. thesis, The University of Nottingham, (2007), unpublished.[7] LAPACK (www.netlib.org/lapack)
Results
Conclusion Our most accurate value of the binding energy compares favorably with a previous calculation [3] and the most accurate calculation to-date [4]. We have optimized the nonlinear parameters, however, it is expected that further refinement of the choice of nonlinear parameters should improve the results. We are currently using the Hylleraas trial wavefunction as the short-range terms of the full S-wave scattering wavefunction for Ps-H collisions.
Method We calculate the energy of the bound state of 1S PsH using the Rayleigh-Ritz variational method.
We perform the integrations using the asymptotic- expansion method of Drake and Yan [2]. For values of ω > 5, linear dependencies exist in the matrices. Yan and Ho [3] overcame the problem of linear dependence by splitting the basis set into blocks based on correlations between the electrons and positron.
|
|| HE
Figure 1. Eigenvalues for 1S PsH
ω N(ω) TermsUsed
BindingEnergy
Terms Used [3]
BindingEnergy [3]
6789
10
9241716300350058008
9161585192521552205
0.039169509 0.039189568 0.039194559 0.039195860 0.039196324
502918158926253501
0.03915889560.03918792230.03919487680.03919638090.0391966076
The computed eigenvalues are comparable to those given earlier in reference [5]. The stabilized eigenvalues correspond to the resonances computed by Yan and Ho [3], which we denote by dashed lines.
Table 1. Binding energy as a function of ω
Table 2. Comparison with other work
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