University of Windsor Scholarship at UWindsor Electronic Theses and Dissertations Theses, Dissertations, and Major Papers 10-5-2017 Analysis of the Radius of Convergence of the Perturbation Expansion for the Ground State Energy of Two-Electron Atoms Ryan Louis Peck University of Windsor Follow this and additional works at: https://scholar.uwindsor.ca/etd Recommended Citation Peck, Ryan Louis, "Analysis of the Radius of Convergence of the Perturbation Expansion for the Ground State Energy of Two-Electron Atoms" (2017). Electronic Theses and Dissertations. 7289. https://scholar.uwindsor.ca/etd/7289 This online database contains the full-text of PhD dissertations and Masters’ theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license—CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via email ([email protected]) or by telephone at 519-253-3000ext. 3208. Analysis of the Perturbation Expansion for the Ground State Energy of Two-Electron Atoms by Ryan Peck A Thesis Submitted to the Faculty of Graduate Studies through the Department of Physics in Partial Fulfillment of the Requirements for the Degree of Master of Science at the University of Windsor Windsor, Ontario, Canada c 2017 Ryan Peck Analysis of the Perturbation Expansion for the Ground State Energy of Two-Electron Atoms by Ryan Peck APPROVED BY: ______________________________________________ M. Hlynka Department of Mathematics & Statistics ______________________________________________ C. Rangan Department of Physics ______________________________________________ G. W. F. Drake, Advisor Department of Physics Aug. 17, 2017 Author's Declaration of Originality I hereby certify that I am the sole author of this thesis and that no part of this thesis has been published or submitted for publication. I certify that, to the best of my knowledge, my thesis does not infringe upon anyones copyright nor violate any proprietary rights and that any ideas, tech- niques, quotations, or any other material from the work of other people included in my thesis, published or otherwise, are fully acknowledged in accordance with the standard referencing practices. Furthermore, to the extent that I have in- cluded copyrighted material that surpasses the bounds of fair dealing within the meaning of the Canada Copyright Act, I certify that I have obtained a written permission from the copyright owner(s) to include such material(s) in my thesis and have included copies of such copyright clearances to my Appendix. I declare that this is a true copy of my thesis, including any final revisions, as approved by my thesis committee and the Graduate Studies office, and that this thesis has not been submitted for a higher degree to any other University or Institution. iii Abstract This thesis solves a controversial physics problem that has existed in the litera- ture for nearly a century { finding the radius of convergence of the perturbation expansion for the ground state energy of the two-electron atom. This problem is important to study because it makes progress towards finding the possible structures that can exist in the quantum mechanical three-body problem. This perturbation expansion is a convergent series and in physics these are rare to work with. We usually refer to this perturbation expansion as the \1=Z expan- sion". There is still much to learn about finding effective methods of determining the radii of convergence for convergent series. The first 1000 coefficients of the 1=Z expansion are calculated with very high precision and are compared to pre- vious values in the literature. These coefficients are determined by using a new type of basis set that is introduced in this work, the pyramidal basis set, which is very useful in describing high-order wave functions generated by perturbation theory. Using the series of ratios of the resulting coefficients along with a series acceleration technique, the radius of convergence of the 1=Z expansion is found to be λ∗ = 1:0975(2). iv Dedication Dedicated to my family. v Acknowledgments I would like to thank my supervisor Dr.Drake for giving me the opportunity to work for him, for funding my research, and for sharing his knowledge of atomic physics with me. I would also like to thank the Drake research group for making the office a fun place to be. Thank you to my family in Windsor for the countless meals we shared. I will definitely miss those! Last but not least, thanks to my mom, dad and sister for all of the long-distance chats that always brightened my days. This work would not be possible without the support I received from all of these amazing people. vi Table of Contents Author's Declaration of Originality iii Abstract iv Dedication v Acknowledgments vi List of Figures ix List of Tables xi 1 Introduction 1 2 Literature Review 6 3 Theory 14 3.1 The Hamiltonian . 14 3.2 Variational Method . 18 3.3 Basis Sets . 22 3.3.1 Hylleraas Coordinates . 22 3.3.2 Hylleraas Basis Sets . 23 3.3.3 Pyramidal Basis Sets . 27 3.4 Rayleigh-Schr¨odingerPerturbation Theory . 30 3.5 Neville-Richardson Extrapolation . 34 4 Results 37 4.1 The Procedure . 37 4.2 Calculations . 43 4.3 Interesting Findings . 101 4.3.1 Diagonalization Scheme Comparison . 101 4.3.2 Oscillating Coefficients . 101 4.3.3 Decay of Coefficients . 105 vii 5 Conclusion 108 6 Future Work 110 Appendices 111 A Linear Variational Method . 111 B Hylleraas Undheim Macdonald Theorem . 113 C Matrix Diagonalization . 116 C.1 Jacobi Method . 117 C.2 Tridiagonalization . 120 C.3 Power Method and Inverse Iteration Method . 126 D Givens' Method . 129 E Number of terms in a single Hylleraas basis set . 132 Bibliography 135 Vita Auctoris 138 viii List of Figures 3.1 Interleaving of variational energies as basis size is increased . 21 3.2 Image of Hylleraas coordinates taken from reference [26] . 22 3.3 Comparison of the triple and double basis sets' rates of conver- gence to the true energy E for the 1s2p1P state of neutral helium. 25 3.4 Spread of nonlinear parameters with increasing basis size . 28 4.1 Least squares fit to the ratios from n = 100 to 999 . 87 4.2 Least squares fit to the ratios from n = 100 to 400 . 88 4.3 Least squares fit to the sn from n = 100 to 800 . 89 4.4 Least squares fit to the sn from n = 100 to 300 . 90 4.5 Least squares fit to the sn from n = 100 to 400 . 91 4.6 Least squares fit to the sn from n = 100 to 500 . 92 4.7 Least squares fit to the tn from n = 100 to 800 . 93 4.8 Least squares fit to the tn from n = 100 to 300 . 94 4.9 Least squares fit to the tn from n = 100 to 400 . 95 4.10 Least squares fit to the tn from n = 100 to 500 . 96 4.11 Least squares fit to the vn from n = 100 to 400 . 97 4.12 Least squares fit to the vn from n = 100 to 150 . 98 4.13 The vn from n = 100 to 200 . 99 4.14 Runtime for dall2016.f and dtridlz2016.f . 102 4.15 PYC5TG974 vs. PYC5TGC68 . 103 4.16 Baker vs. PYC5TG974 . 104 4.17 Baker vs. PYC5TGC68 . 104 ix 4.18 Helium En rapid decrease . 106 − 4.19 sn decay for different H triple basis set sizes . 107 x List of Tables 2.1 Important estimates of the radius of convergence of the 1=Z ex- pansion . 13 3.1 number of terms N in a typical triple basis set for various Ω. 26 3.2 Construction of a pyramidal basis set . 29 4.1 Highly accurate energies for helium 1P states achieved by double basis sets . 40 4.2 Triple basis set optimized for H− . 45 4.3 Triple basis set optimized for He . 45 4.4 Pyramidal basis with α0 = 1:300, β0 = 1:450 and M = 2:2 . 46 4.5 A special pyramidal basis set with α0 = 2:000, β0 = 2:200 and M = 2:0 containing a double basis set (α1 = 0:500; β1 = 0:500); (α2 = 0:950; β2 = 0:200) . 46 4.6 E2 for the He triple basis set in table 4.3 . 47 4.7 E2 for the optimized pyramidal basis set in table 4.4 . 48 1 4.8 Comparison of different En. D+01 represents 10 . 50 4.9 Accuracy of the series coefficients . 86 5.1 Important estimates of the radius of convergence of the 1=Z ex- pansion . 109 xi Chapter 1 Introduction It is important to know the basics of convergent series in order to understand what a radius of convergence is. A convergent power series is a series of the form 1 X n an(x − x0) (1.1) n=0 in which the coefficients an are getting progressively smaller at a fast enough rate with increasing n, such that when the entire series is added together the sum is a finite number for some range of values a < x < b. x0 is referred to as the point of expansion of the series and choosing different x0 will result in different ranges of convergence for the series.