Hylleraas Trial Wavefunction where

1. The binding energy of PsHD. 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 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. 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 . 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. Results Table 2. Comparison with other work 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. Hylleraas Trial Wavefunction where and ki, li, mi, ni, pi, qi are non-negative integers. r12 e e+ r2 r13 r23 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. r1 e p r3 Figure 1. Eigenvalues for 1S PsH 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. 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 LudwikAdamowicz, Phys. Rev. A 74, 052502 (2006). [5] P. Van Reethand 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) Table 1. Binding energy as a function of ω