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Electromagnetic properties of medium-mass nuclei from coupled-cluster theory Miorelli, Mirko 2017

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Electromagnetic properties ofmedium-mass nuclei fromcoupled-cluster theorybyMirko MiorelliM.Sc. in Theoretical and Computational Physics, University of Trento, 2013B.Sc. in Physics, University of Trento, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2017© Mirko Miorelli 2017AbstractElectromagnetic probes represent a fundamental tool to study nuclear structure and dynamics.The perturbative nature of the electromagnetic interaction allows for a clean connection be-tween calculated nuclear structure properties and measured cross sections. Ab initio methodshave long represented the gold standard for calculations of nuclear structure observables in lightnuclei. Thanks to recent developments in the scientific community, ab initio calculations havefinally reached the medium- and heavy-mass region of the nuclear chart. However, the chal-lenges modern nuclear structure calculations face are multiple, ranging from the construction ofnuclear forces from chiral effective field theory (χEFT) and the solution of the highly correlatedquantum many-body problem, to a quantitative description of observables with solid treatmentof uncertainties.The work presented in this thesis aims to contribute addressing some of these challenges,using the ab initio coupled-cluster (CC) theory formulation of the Lorentz integral transform(LIT) method. We combine the CC and LIT methods for the computation of electromagnetic in-elastic reactions into the continuum. We show that the bound-state-like equation characterizingthe LIT method can be reformulated based on extensions of the coupled-cluster equation-of-motion (EOM) method, and we discuss strategies for viable numerical solutions. We then focuson the calculation of the electric dipole polarizability (αD), which quantifies the low-energybehaviour of the dipole strength and is related to critical observables such as the radii of theproton and neutron distributions. Using a variety of chiral interactions, and singles and dou-bles excitations, we study 4He, 16,22O and 40,48Ca. Exploiting correlations between αD and thecharge radius, we predict the neutron-skin radius and the polarizability for the double-magic48Ca, the latter recently measured by the Osaka-Darmstadt collaboration. Finally, we studythe impact of triples excitations on the dipole strength in 4He and 16O.iiLay SummaryFrom the infinitesimal abyss of the atomic nucleus to the vastness of stars, a fine thread connectsthese two worlds: nuclear physics. On a journey that began over one-hundred years ago, modernnuclear physics still aspires to unveil the unique properties of the atomic nucleus. In this workwe tag along on this journey, and study the complex interplay between the basic componentsof the nucleus – protons, neutrons, and the fundamental forces between them. With complexmathematical tools and theoretical models, we study the behaviour of nuclei when immersedin magnetic and electric fields. Using sophisticated computer algorithms we build the nucleusfrom scratch and calculate how it deforms, oscillates and eventually breaks as we increase theenergy of these fields. By comparing our results with experimental evidence, we validate ourmodels, a small but impactful step to deepen our knowledge of the heart of matter, the atomicnucleus.iiiPrefaceThis work is based on published and unpublished material. Chapter 1 briefly outlines thepurpose of this thesis. Chapters 2, and 3 are based on common literature in nuclear physicsand have an introductory purpose. Chapter 4 is based on previous work of collaborators, withthe last section being partly original expository material. The first five sections of Chapter 5are based on the book “Many-Body Methods in Chemistry and Physics” by I. Shavitt and R. J.Bartlett, and serve as introductory material for the following topics. The remaining sections ofChapter 5 and Chapter 6 are based on the theoretical parts of Refs. [1–4]. Finally, Chapters 7and 8 and Appendix B, as well as figures and tables, are all based on original results publishedin Refs. [1–7], with the exception of Section 7.4 which is original unpublished material.For quick reference, the published parts of this work have appeared in:[1] – S. Bacca, N. Barnea, G. Hagen, M. Miorelli, G. Orlandini and T. Papenbrock, “Giantand pygmy dipole resonances in 4He, 16,22O, and 40Ca from chiral nucleon-nucleon inter-actions”, Phys. Rev. C 90, 064619 (2014).S.B. and G.H. initiated the project, developed the methods and performed most of thecalculations for 4He and 16,22O. The other co-authors, N.B., G.O. and T.P. participated indrafting the manuscript and analyzing the results. I provided results for the 40Ca nucleus,for which I performed all the numerical calculations. I contributed in the development ofthe method for the calculation of the electric dipole polarizability and helped reviewingthe initial draft of the manuscript, as well as with the analysis of the results presented.[2] – M. Miorelli, S. Bacca, N. Barnea, G. Hagen, G. R. Jansen, G. Orlandini, T. Papenbrock,“Electric dipole polarizability from first principles calculations”, Phys. Rev. C 94, 034317(2016).I was the leading author for this publication. Together with S.B. and G.H., I developedpart of the theory for the calculation of the electric dipole polarizability and performedall the numerical calculations. G.R.J. provided the interactions used in the publication.I personally obtained the results presented and wrote the first draft of the manuscript.All the co-authors participated in the analysis of the results and reviewed the manuscriptbefore submission.[3] – G. Orlandini, S. Bacca, N. Barnea, G. Hagen, M. Miorelli and T. Papenbrock, “Couplingthe Lorentz Integral Transform (LIT) and the Coupled Cluster (CC) Methods: A waytowards continuum spectra of ”not-so-few-body” systems”, Few-Body Syst. 55, 907-911ivPreface(2014).Contribution presented at the 22nd European Conference on Few Body Problems inPhysics, Krakow, Poland, 9 - 13 September 2013. S.B., G.H. and T.P. performed cal-culations for 16O. I obtained the results for 40Ca. All the authors helped interpreting theresults. The manuscript was written by G.O..[4] – T. Xu, M. Miorelli, S. Bacca and G. Hagen, “A theoretical approach to electromagneticreactions in light nuclei”, EPJ Web of Conferences 113, 04016 (2016).Proceedings of the 21st International Conference on Few-Body Problems in Physics (FB21),May 2015, Chicago. S.B., G.H. and I contributed to develop the theoretical frameworkpresented in the proceeding. T.X. and S.B. performed most of the calculations for theresults on the Coulomb sum rule.[5] – M. Miorelli, S. Bacca, N. Barnea, G. Hagen, G. Orlandini and T. Papenbrock, “Electricdipole polarizability: from few- to many-body systems”, EPJ Web of Conferences 113,04007 (2016).Proceedings of the 21st International Conference on Few-Body Problems in Physics (FB21),May 2015, Chicago, where I presented a contributed talk. I performed all the numericalcalculations for the material presented. I personally wrote the manuscript. S.B. reviewedthe manuscript and N.B., G.H., G.O. and T.P. helped interpreting the results.[6] – G. Hagen, A. Ekstro¨m, C. Forsse´n, G. R. Jansen, W. Nazarewicz, T. Papenbrock, K. A.Wendt, S. Bacca, N. Barnea, B. Carlsson, C. Drischler, K. Hebeler, M. Hjorth-Jensen, M.Miorelli, G. Orlandini, A. Schwenk and J. Simonis, “Neutron and weak-charge distribu-tions of the 48Ca nucleus”, Nature Physics 12, 186190 (2016).I was fully responsible for the calculation of the dipole polarizability. I participated inthe development of computational tools utilized in this study and in the discussion andinterpretation of the results. For a full description of the authors contributions, see thesection “contributions” of the publication.[7] – J. Birkhan, M. Miorelli, S. Bacca, S. Bassauer, C.A. Bertulani, G. Hagen, H. Matsubara,P. von Neumann-Cosel, T. Papenbrock, N. Pietralla, V. Yu. Ponomarev, A. Richter, A.Schwenk and A. Tamii, “Electric dipole polarizability of 48Ca and implications for theneutron skin”, Phys. Rev. Lett. 118, 252501 (2017).This is a project in collaboration with the Osaka-Darmstadt experimental group. J.B.,S.B., H.M., P.N-C., N.P., V.Y.P., A.R. and A.T. conducted the experiment and analyzedthe experimental data, as well as wrote the experimental related part of the manuscript.C.A.B. helped in the data analysis. I conducted calculations for the running sum of thepolarizability. Together with A.S., I drafted the theory part of the manuscript. S.B.,G.H., T.P., A.S. and myself analyzed and interpreted the theoretical results. All theco-authors reviewed the manuscript prior to submission.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The nuclear Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Separation of scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Chiral effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Perturbative expansion of χEFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Nuclear electromagnetic reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1 Photoabsorption cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Multipole decomposition of the electromagnetic current . . . . . . . . . . . . . . 153.3 Electric dipole contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Integral transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 The Lorentz integral transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 The Stieltjes integral transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Lanczos numerical calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Inversion methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33viTable of Contents4.4.1 Truncated SVD regularization . . . . . . . . . . . . . . . . . . . . . . . . . 364.4.2 Tikhonov regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4.3 Basis expansion deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . 395 Coupled-cluster theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1.1 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1.2 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1.3 Particle-hole formalism and normal-ordered operators . . . . . . . . . . . 445.1.4 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 The exponential ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 The CC energy functional and the Λ–equations . . . . . . . . . . . . . . . . . . . 525.4 Coupled-cluster equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 545.5 The integral transforms coupled-cluster method . . . . . . . . . . . . . . . . . . . 565.5.1 The similarity transformed LIT . . . . . . . . . . . . . . . . . . . . . . . . 565.5.2 The LIT-CC equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.5.3 Non-symmetric Lanczos algorithm . . . . . . . . . . . . . . . . . . . . . . . 595.5.4 The similarity transformed Stieltjes for the polarizability . . . . . . . . . 615.6 Diagrammatic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.7 Coupled-cluster singles and doubles approximation . . . . . . . . . . . . . . . . . 645.7.1 The m0 sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.7.2 The continued fraction and the Hamiltonian tridiagonalization . . . . . 705.8 Linear triples corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 Spherical coupled-cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.1 Two- and three-body states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Matrix elements and the Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . 796.3 Permutation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.4 J-coupled coupled-cluster diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 827 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1 Benchmarking the new method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1.1 Convergence and Lanczos coefficients . . . . . . . . . . . . . . . . . . . . . 867.1.2 Model-space convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.1.3 4He test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.2 Oxygen and Calcium isotopes with chiral NN forces . . . . . . . . . . . . . . . . 927.2.1 Oxygen isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2.2 Calcium isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.3 Chiral three-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.3.1 4He revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101viiTable of Contents7.3.2 Medium-mass nuclei: Oxygen and Calcium . . . . . . . . . . . . . . . . . 1037.3.3 Neutron skin and polarizability in 48Ca . . . . . . . . . . . . . . . . . . . 1147.4 Inclusion of triples effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Appendix A Vector spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 145Appendix B Coupled-cluster rules and diagrams . . . . . . . . . . . . . . . . . . . 147B.1 Coupled-cluster rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147B.2 Clebsch-Gordan coefficients relations . . . . . . . . . . . . . . . . . . . . . . . . . 148B.3 m0 diagrams in J−coupled scheme (CCSD) . . . . . . . . . . . . . . . . . . . . . 149B.3.1 Diagram R1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150B.3.2 Diagram R2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150B.3.3 Diagram L1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152B.3.4 Diagram L1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152B.3.5 Diagram L1C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153B.3.6 Diagram L1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153B.3.7 Diagram L2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154B.3.8 Diagram L2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154B.4 m0 diagrams in J−coupled scheme (CCSDT-1) . . . . . . . . . . . . . . . . . . . 155B.4.1 Diagram R2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155B.4.2 Diagram R3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156B.4.3 Diagram R3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156B.4.4 Diagram R3C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157B.4.5 Diagram L1E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158B.4.6 Diagram L2C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159B.4.7 Diagram L2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159B.4.8 Diagram L3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.4.9 Diagram L3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.4.10 Diagram L3C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Appendix C Radius operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163viiiList of TablesTable 7.1: Theoretical values of αD for different nuclei calculated with the NNLOsatinteraction in comparison to experimental data . . . . . . . . . . . . . . 113Table 7.2: Summary of the results obtained in 16O and 40Ca with three-bodyHamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113ixList of FiguresFigure 2.1: Hierarchy of nuclear forces in χEFT . . . . . . . . . . . . . . . . . . . . 9Figure 3.1: Example of absorption reaction . . . . . . . . . . . . . . . . . . . . . . . 10Figure 3.2: Example of a typical photoabsorption response function R(ω) . . . . . 11Figure 4.1: Schematic representation of the spectrum of a nucleus . . . . . . . . . . 23Figure 4.2: Gaussian integral transform of an analytical response function . . . . . 24Figure 4.3: Stieltjes integral transform in 4He . . . . . . . . . . . . . . . . . . . . . . 29Figure 4.4: Illustration of the ill-posed problem for the inversion of an analyticalfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 4.5: The singular values of the Toeplitz matrix built from a Lorentz kernel 35Figure 4.6: Regularized inversion with truncated singular value decomposition . . 37Figure 4.7: L–curve Tikhonov regularized inversion . . . . . . . . . . . . . . . . . . . 38Figure 4.8: Numerical calculation of the LIT and comparison of its inversion withdifferent methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 5.1: Many-body states in the particle-hole formalism . . . . . . . . . . . . . 45Figure 7.1: Convergence of the LIT vs the number of Lanczos coefficients in 4Heand 16O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 7.2: Convergence of the polarizability vs the number of Lanczos coefficientsin 4He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 7.3: Convergence of the LIT vs model-space size in 4He . . . . . . . . . . . . 88Figure 7.4: Comparison of CCSD and EIHH calculations of the LIT in 4He . . . . 89Figure 7.5: Dipole response function in 4He with two-body interactions . . . . . . 91Figure 7.6: Model-space convergence of the dipole polarizability in 4He with two-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 7.7: Convergence of the LIT vs model-space size in 16,22O . . . . . . . . . . 93Figure 7.8: Comparison of the LITs in 16O and 22O . . . . . . . . . . . . . . . . . . 94Figure 7.9: Response functions in 16,22O with two-body interactions . . . . . . . . 95Figure 7.10: Model-space convergence of the dipole polarizability in 16O with two-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 7.11: Dipole response function in 40Ca with two-body interactions . . . . . . 98xList of FiguresFigure 7.12: Convergence of the electric dipole polarizability in 40,48Ca vs model-space size with two-body interactions . . . . . . . . . . . . . . . . . . . . 99Figure 7.13: The electric dipole polarizability and its running sum in 4He usingthree-body forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Figure 7.14: The dipole response function in 4He with three-body forces . . . . . . 103Figure 7.15: Convergence of the electric dipole polarizability in 16,22O vs model-space size with three-body interactions . . . . . . . . . . . . . . . . . . . 104Figure 7.16: Convergence of the charge radius in 16,22O vs model-space size withthree-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure 7.17: The running sum of the polarizability in 16,22O with three-body inter-actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 7.18: The dipole response function in 16O with three-body forces . . . . . . . 109Figure 7.19: Correlations between the polarizability and the charge radius in 16Oand 40Ca using two- and three-body interactions . . . . . . . . . . . . . 110Figure 7.20: Running sum of the electric dipole polarizability in 16O and 40Ca withthree-body forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 7.21: Electric dipole polarizability αD and neutron skin rskin plotted versusthe charge radius rch in48Ca . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 7.22: Experimental and theoretical photoabsorption cross sections and thepolarizability running sums in 48Ca . . . . . . . . . . . . . . . . . . . . . 116Figure 7.23: Experimental electric dipole polarizability in 48Ca and predictions fromχEFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Figure 7.24: Number of 2p–2h and 3p–3h configurations as a function of the model-space size in 4He and 16O. . . . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 7.25: Convergence of the m0 sum rule in4He in the CCSDT-1 approximationwith the NN-N3LO interaction as a function of the three-body cutsE3max and L3max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Figure 7.26: Convergence of αD in4He with a chiral NN interaction with respect tothe angular momentum cut L3max at h̵Ω = 26 MeV . . . . . . . . . . . . 120Figure 7.27: Convergence of αD in4He with the chiral NN-N3LO interaction withrespect to the model-space size Nmax for different values of h̵Ω and atfixed L3max = 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 7.28: Comparison of m0 and αD in the CCSD and CCSDT-1 approximationswith exact EIHH calculation in 4He. . . . . . . . . . . . . . . . . . . . . 121Figure 7.29: Convergence of αD in16O with the NNLOsat interaction with respectto the angular momentum cut L3max at h̵Ω = 22 MeV . . . . . . . . . . 122Figure 7.30: Comparison of αD in the CCSD and CCSDT-1 approximations withexperimental data by Ahrens et al. in 16O. . . . . . . . . . . . . . . . . 123xiList of Acronyms3NF Three-nucleon forces/ Three-body forcesCC Coupled-clusterCCSD Coupled-cluster singles and doublesCCSDT-1 Coupled-cluster singles, doubles and linear triplesχEFT Chiral effective field theoryCREX Calcium radius experimentDFT Density functional theoryEFT Effective field theoryEIHH Effective interaction hyperspherical harmonicsEOM Equation of motionEOM-CCSDT-1 Equation of motion coupled-cluster singles, doubles and lin-ear triplesEOS Equation of stateGDR Giant dipole resonanceHO Harmonic oscillatorΛ-CCSD(T) Coupled-cluster Λ–equations approximation with singles,doubles and perturbative triplesLEC low-energy constantLIT Lorentz integral transformLIT-CC Lorentz integral transform coupled-cluster methodLO Leading orderMREX Mainz radius experimentxiiList of AcronymsN3LO Next-to-next-to-next-to-leading orderNLO Next-to-leading orderNN Nucleon-nucleon forces/ Two-body forcesNNLO Next-to-next-to-leading orderPDR Pygmy dipole resonanceQCD Quantum chromo-dynamicsSVD Singular value decompositionTSVD Truncated singular value decompositionxiiiAcknowledgementsMy deepest gratitude goes to Dr. Sonia Bacca, who guided me in these four years of challengesby her ever-present guidance and making me strive for perfection. A special thank goes toDr. Gaute Hagen, without whom this project would not have been possible. I would also liketo thank, Prof. Thomas Papenbrock and Prof. Nir Barnea, for the enlightning discussions andinsights. Many thanks go to my former MSc supervisor Prof. Giuseppina Orlandini, who mademe passionate about nuclear physics and presented me with the opportunity to come to Canada.I also want to acknowledge all the members – past and present – of the TRIUMF theory de-partment for the useful discussions, help and support, as well as the following funding agencies:TRIUMF, the National Research Council of Canada, the Natural Sciences and Engineering Re-search Council, the US-Israel Binational Science Foundation, the U.S. Department of Energy,Oak Ridge National Laboratory, and the University of British Columbia.A very special thank you goes to my fiance´e Angela, who shared the experience of thisPhD, being beside me through the successes and – most importantly – enduring my sour moodthrough the stressful moments and setbacks.Finally, I owe this thesis to my parents, who supported me and never let me down since myjourney in physics began in 2007. Without them I would not be who I am.xivDedicationAi miei genitori, Angela e Mauro, per il loro supportoincondizionato, e per avermi reso la persona che sono oggi.xvChapter 1IntroductionSince its discovery, more than one century ago, the atomic nucleus has been at the center oftheoretical and experimental studies, playing a fundamental role in the development of modernphysics. The nucleus is a strongly-correlated, quantum many-body system. Its constituents,protons and neutrons, interact between themselves mainly by the strong force, giving rise toan ample variety of collective phenomena and excitation spectra. The main goal of modern nu-clear structure studies is to explain the unifying mechanisms by which these collective nuclearbehaviours emerge from the underlying strong nuclear interaction between nucleons.In the past, simple and intuitive models and theories where built to describe experimentalobservations, e.g., the nuclear shell model for which M. Goeppert Mayer and J. Hans D. Jensenwere awarded the Nobel prize in 1963. Since those early stages, our knowledge of the nucleushas constantly evolved and so have the theoretical models. Nowadays, the so-called ab initiomethods – such as coupled-cluster theory [8], no-core-shell-model [9], quantum Monte Carlo [10]and in-medium similarity renormalization group [11], to name a few – represent the state-of-the-art approach to study nuclear structure in the light- and medium-mass regime. Their aimis an exact description of nuclear properties starting from the fundamental interaction amongnucleons. While the first models relied on simple ideas to describe the nuclear interaction, e.g.,the one-boson-exchange models, in the last decade nuclear interactions have been systemati-cally derived in the framework of chiral effective field theory (χEFT) [12–14]. The quantumchromo-dynamics (QCD) Lagrangian is expanded at low energy using protons, neutrons andpions as effective degrees of freedom, and rearranged in a momentum power series where thehierarchy between two- and three-body forces emerges naturally. The high-energy physics isincluded in the theory via renormalization and absorbed into coefficients called low-energy con-stants (LECs) which, in turn, are fitted to experimental data, usually nucleon-nucleon scatteringphase shifts and few-body data [13–19]. Since the pioneering work of Weinberg [12, 20], χEFThas been widely used in nuclear physics and has developed into an intense field of research. Inparticular, this new class of realistic interactions, combined with the ever increasing computa-tional power of modern high performance computers, has lead to a recent renaissance of nuclearphysics.From an experimental point of view, nuclei have been and are largely studied via scatteringexperiments with an extensive variety of different probes: hadronic, weak and electromagnetic,1Chapter 1. Introductioneach one giving us insights in different aspects of nuclear dynamics and structure. Electromag-netic probes in particular are of great interest and can be used for a clean and clear comparisonbetween theory and experiment [21]. In fact, the perturbative nature of electromagnetic probesallows for a direct connection between measured cross sections and structure calculations ofnuclear targets, and the contribution of the external probe can be easily disentangled from thestrong force dynamics. Many experiments have been devoted to the study of neutron-rich nu-clei, the exotic properties of these unstable systems being of great help in understanding nuclearmatter and reactions relevant for nucleosynthesis processes taking place in stars. Moreover, inaddition to advances in fundamental science, studying rare-isotopes is expected to yield applica-tions in materials science and medicine. To this end, rare-isotope beam facilities have been builtor upgraded worldwide in the last decades, some of them being TRIUMF (British Columbia,Canada), FRIB (Michigan State, US), RIKEN (Japan), FAIR (Germany) and GANIL (France).In the last years, the huge development of nuclear many-body techniques and the consequentextension of the calculations from light- to medium-mass nuclei, have brought to attention theneed of a more solid treatment of uncertainties in the calculations [22–30]. In fact, despite thegreat success of χEFT potentials, first principle calculations often predict over bound and toocompact nuclei, especially in the medium-mass region [1, 31–38]. This has lead to a plethora ofdifferent interactions that differ in regularization schemes and parametrization of the LECs, see,e.g., the interactions discussed in Refs. [13, 14, 16–18, 39]. As one of the main ingredients in abinitio calculations is the nuclear Hamiltonian, it is obvious that an incorrect knowledge of thenuclear interaction can greatly affect the prediction of observables. Moreover, when performingcomputations, one also has to take into account the uncertainties arising from approximationsin the methods used in the calculations. In order to better test our knowledge of the nucleus interms of fundamental interaction between its components, it is clear that discerning betweenuncertainties from the computational method itself and the Hamiltonian model is of crucialimportance.For an ab initio calculation, the uncertainty σth associated to the prediction of an observablecan be separated into several contributionsσ2th = σ2b + σ2MB + σ2H ,where σb represents the error arising from expansions on a basis set that should be infinite but,in practical cases, is not; σMB is the uncertainty from the approximations in the many-bodymethod used; σH encodes the uncertainty from the input Hamiltonian model1. The estimationof σb is deduced by performing larger model space calculations, i.e., using larger basis sets.The evaluation of σMB often requires extended benchmarks between different methods and the1Note that sometimes σb and σMB are the same or largely intertwined.2Chapter 1. Introductionstudy of the different approximations both from an analytical and numerical point of view. Atlast, the uncertainty coming from the input Hamiltonian model is the most difficult uncertaintyto quantify and can be largely affected by the poor knowledge of σb and σMB. Only recentlythe ab initio community has started addressing the Hamiltonian model uncertainty by lookingat, for example, correlations between different observables [2, 6, 25, 40].The challenges ab initio nuclear structure calculations face are multiple, ranging from theconstruction of nuclear forces from χEFT and the solution of the highly correlated quantummany-body problem to a quantitative description of observables with solid treatment of un-certainties. The work presented in this thesis aims to contribute addressing σb, σMB and σHusing a specific ab initio method, the coupled-cluster (CC) theory formulation of the Lorentzintegral transform (LIT) method, which can be used to study electromagnetic observables innuclei. In the study of σH , this work is not concerned with the development and optimization ofnew Hamiltonians, but rather with the use of those already present in the literature [14, 16, 18].In this work we will introduce the theoretical framework for the study of some electro-magnetic observables of interest, namely the dipole response function (photoabsorption crosssection) and the electric dipole polarizability. Both these quantities can be described in terms ofnuclear collective modes [41, 42], and only recently ab initio methods have been applied for thecalculation of such observables in medium-mass nuclei. On the other hand, extensive studiesof collective modes in medium- and heavy-mass nuclei have been ongoing for many decades,with energy-density functionals methods playing a lead role in the field. The gold standardof phenomenological calculations of nuclear collective modes is the random phase approxima-tion [43] and its various extensions [44], which in the past have been applied in conjunctionwith density-functional approaches [45–47] as well as calculations employing two- and three-body phenomenological potentials [48–50]. While these studies certainly play an importantrole on our understanding of nuclear collective modes and dynamics, this work focuses on novelab initio techniques and so in the following Chapters we will concentrate our attention on thedescription of the dipole response function and the electric dipole polarizability using first prin-ciples approaches.In Chapter 2 we briefly summarize the fundamental concepts of χEFT, and then proceedin deriving the photoabsorption cross section in the non-relativistic limit in Chapter 3. Subse-quently, in Chapter 4, we talk about the problem of the continuum which is a major challengein the calculation of scattering observables and introduce the integral transforms formalism.Chapter 5 introduces CC theory and Chapter 6 its spherical version. The latter is used inChapter 7 to perform calculations of αD and the dipole response function and related observ-ables in medium-mass nuclei using a variety of different chiral interactions. Finally, Chapter 8concludes with a brief summary and outlook of the results presented.3Chapter 2The nuclear HamiltonianThe nuclear Hamiltonian is a fundamental ingredient for any ab initio calculation of nuclearproperties. Nuclear forces have been at the center of theoretical studies since the pioneeringwork of H. Yukawa in 1935. The first phenomenological meson-exchange models were built inthe early 1940s and 50s, and later extended to more complex models in the 1970s and 80s [51–53]. In the late 1980s new phenomenological potentials were introduced which exploited thesymmetries of the nuclear force and were fitted to scattering data [54, 55]. In the early 1990s, amajor breakthrough occurred when S. Weinberg applied the concept of effective field theories tolow-energy QCD [12, 20, 56]. Since then, χEFT has developed into an intense field of research(see Refs. [14] and [13] for a detailed summary) and gave birth to today’s widely used familyof nuclear Hamiltonians.In what follows, we briefly review the fundamental concepts of χEFT, following the review byR. Machleidt and D. R. Entem [14].Chiral effective field theory is based on an expansion of the QCD Lagrangian, the theory ofthe strong interaction between quarks and gluons, and consequently between protons and neu-trons. A peculiarity of QCD is quark confinement, where the strength of the interaction growsas two quarks are pulled apart. On the other hand, the coupling constant is strongly suppressedat high energies (small distances) and in this regime one can treat QCD in a perturbative fash-ion. Unfortunately, nuclear physics phenomena manifest themselves at very low-energy scales,thus making a perturbative treatment of QCD practically impossible2. Effective field theoriescome to help in this matter as they allow us to separate the short-range (high energy) physicsfrom the one at larger scales (low energy), provided that there exists a separation of scales anda gap between the two physics regimes. To build a nuclear EFT one must follow five steps:1. Identify the soft- and hard-scales and the appropriate degrees of freedom for the theory.2. Identify the relevant symmetries of low-energy QCD.3. Build the most general Lagrangian consistent with the above symmetries and eventualsymmetry breaking mechanisms.4. Develop a (low-momentum) expansion to rearrange the contributions of the Lagrangianin a hierarchical scheme.2Calculations of nuclear observables using the QCD Lagrangian are possible [57], but the applicability of thesemethods is currently limited to very small nuclei and to large and nonphysical masses of the nuclear constituents.4Chapter 2. The nuclear Hamiltonian5. Use the expansion to rewrite the theory in terms of Feynman diagrams.2.1 Separation of scalesThe first step consists in identifying the different energy scales at which we want to apply ourtheory. The idea is to separate the low-energy dynamics from the high-energy ones. In thisway, we can determine the relevant degrees of freedom of the theory. For nuclear physics, mostphenomena take place at energy scales of the order of tens of MeV or less. The hadron spectrumis characterized by a large gap between the masses of the lightest mesons mpi ≈ 140 MeV andthe masses of other vector mesons like mρ ≈ 770 MeV and mω ≈ 782 MeV. Thus, it is natural toassume that the pion and the rho masses set the soft and hard scales of the theory. The energyseparation between the light mesons and the heavier ones is much larger than the energies atwhich most nuclear phenomena take place, thus the use of an EFT is justified. From now onwe will denote with Q ≈ mpi and Λχ ≈ mρ the soft and hard scales, respectively. Nucleonswill be the degrees of freedom of the theory as we want to describe nuclear phenomena interms of interactions between neutrons and protons. On the other hand, we know we must alsoinclude pions in our theory as the nuclear force has finite range and phenomenological potentialsshowed how terms such as the tensor force, which is fundamental to describe observables likethe deuteron quadrupole moment [58], arise from pion-exchange contributions [59].2.2 Chiral symmetryA low-energy expansion of QCD should of course start from the QCD Lagrangian, which reads3LQCD = q¯(iγµDµ −M)q − 14GaµνGµνa , (2.1)where q and q¯ are the quark fields, M is the quark mass matrix andDµ = ∂µ − igλa2Aaµ, (2.2)Gaµν = ∂µAaν − ∂νAaµ + gfabcAbµAcν , (2.3)are the covariant derivative and the gluon field tensor. In particular, g is the strong couplingconstant, the λa are Gell-Mann matrices, fabc are the structure constants of the SU(3)color Liealgebra and Aaµ are the gluon fields. The most interesting feature is the last term of Eq. (2.3),which is a gluon-gluon interaction vertex, the peculiarity of the strong force: the gauge bosonsinteract among themselves. Since the masses of the up, down and strange quarks are smallcompared to a typical hadronic scale, i.e., mu,d,s ≪ Λχ ≈ 1 GeV, we can set M ≈ 0. Then, if we3We use Einstein summation where repeated indexes are implicitly summed over.5Chapter 2. The nuclear Hamiltonianredefine the quark fields introducing right-handed and left-handed spinorsqR = 12(1 + γ5)q,qL = 12(1 − γ5)q, (2.4)the Lagrangian becomesL0QCD = ∑j=R,L q¯jiγµDµqj − 14GaµνGµνa . (2.5)By further restricting ourselves only to up and down quarks (which are the only ones rele-vant for standard nuclear physics), one finds that L0QCD is invariant under a global unitarytransformation of the spinorsqR = ⎛⎜⎜⎜⎝uRdR⎞⎟⎟⎟⎠→ e−iΘRi τi/2 ⎛⎜⎜⎜⎝uRdR⎞⎟⎟⎟⎠ ,qL = ⎛⎜⎜⎜⎝uLdL⎞⎟⎟⎟⎠→ e−iΘLi τi/2 ⎛⎜⎜⎜⎝uLdL⎞⎟⎟⎟⎠ ,(2.6)where τi (i = 1,2,3) are the generators of SU(2)flavour. Then, right- and left-handed compo-nents of massless quarks do not mix, a characteristic also known as chiral symmetry. There isevidence that chiral symmetry is spontaneously broken, i.e. the symmetry of the Lagrangianand that of its ground-state are not the same. One would in fact expect the existence of degen-erate hadron 1− and 1+ multiplets, but none is observed. Associated to a spontaneously brokenglobal symmetry, we have massless Goldstone bosons which, in the case of chiral symmetry, areidentified as the isospin triplet of the pions. This also explains why pions are so light comparedto other hadrons. However, because the mass term in Eq. (2.1) is not zero (although we ne-glected it in deriving Eq. (2.5), chiral symmetry is also broken explicitly, as obviously in realitywe have mpi ≠ 0.2.3 Chiral effective LagrangiansWith our energy scales and effective degrees of freedom fixed, we can follow the formalismdeveloped by Callan, Coleman, Wess and Zumino [60, 61] to build the most general Lagrangianconsistent with nuclear physics symmetries and chiral symmetry. A low-energy expansion canbe done in terms of pion masses and powers of derivatives of the pionic fields. The effectiveLagrangian looks like Leff = Lpipi +LpiN +LNN + ... , (2.7)6Chapter 2. The nuclear Hamiltonianwhere the ellipses stand for higher-order terms with more pionic and nucleonic fields. InEq. (2.7), Lpipi encompasses the pionic dynamics, LpiN is the interaction Lagrangian betweenpions and nucleons and LNN provides interactions between two nucleons (and, in general, morenucleons).Furthermore, the individual Lagrangians can be organized asLpipi = L(2)pipi +L(4)pipi + ...LpiN = L(1)piN +L(2)piN +L(3)piN + ...LNN = L(0)NN +L(2)NN +L(4)NN + ...(2.8)where the superscripts denote the so-called chiral dimension [14], i.e., the number of derivativesor pion masses appearing in the Lagrangian.2.4 Perturbative expansion of χEFTOne can perturbatively expand the Lagrangians in Eq. (2.8) by using an expansion param-eter (Q/Λχ)ν . Then χEFT can be rewritten in terms of irreducible Feynman diagrams. Atany given expansion order ν, there exists only a finite number of these diagrams. Within thisframework, nuclear forces emerge with a natural hierarchy controlled by the power ν, as de-picted in Fig. 2.1. At leading-order (LO), only nucleon-nucleon (NN) forces are present and arecharacterized by two terms: a contact term and a pion-exchange term. While truncating theexpansion at LO would clearly be a crude approximation, these two terms already do a goodjob in the description of deuteron observables. The tensor potential, which is key ingredientto describe the deuteron’s quadrupole moment, naturally arises from the one-pion-exchangeterm. For ν = 1 all contributions vanish and so the next-to-leading-order (NLO) contribution isrepresented by ν = 2, where two-pion-exchange terms emerge. Both LO and NLO contributionsaccount for short- and medium-range interactions: most spin and isospin structures includedin phenomenological NN forces are already present at NLO. At next-to-next-to-leading order(NNLO) relativistic corrections come into play and we see the first appearance of three-nucleoninteractions which account for medium-range many-body effects. Again, looking at Fig. 2.1, itis stunning to see the natural emergence of the already empirically known hierarchy of nuclearforces that indicates how two-body forces are dominant with respect to three-body forces.In Fig. 2.1 we have different kinds of interaction vertices represented by solid dots, filled circlesand filled squares, which refer to the leading, sub-leading and sub-sub-leading vertices, respec-tively. The crossed square denotes the NN contact interaction with four derivatives. Thesevertices depend on a number of LECs. The latter embody the short-range physics which hasbeen integrated out in the low-energy expansion. The LECs must be fitted on experimentaldata, usually nucleon-nucleon phase shifts and the binding energy of 3H and other few-bodyobservables for the three-body parts of the interaction.7Chapter 2. The nuclear HamiltonianNN force 3NF 4NFLONLON2LON3LOFigure 2.1: Figure adapted from Ref. [62]. Hierarchy of nuclear forces in χEFT. Solid and dashed linesdenote nucleons and pions, respectively. Dots, circles and squares represent the interaction vertices.In summary, the χEFT framework has a few noteworthy advantages over a pure phenomeno-logical approach, i.e.,(i) An expansion in powers ν of (Q/Λχ)ν , where Q is the momentum associated with theobservable and Λχ ≈ 1 GeV represents the chiral-symmetry breaking scale, which allowsin principle to evaluate nuclear observables to any degree of desired accuracy.(ii) Two- and three-body forces appear naturally and consistently with each other in theexpansion, where three-body forces are sub-leading with respect to two-body forces.(iii) Current operators that couple nucleons to external electroweak probes can be in principleconstructed consistently with the potentials at different orders.In this work we will extensively use Hamiltonians derived from χEFT. No unique prescrip-tion exists for the fitting of the LECs and so a broad range of Hamiltonians, which differ inthe fitting procedure and LECs values, have been developed. This thesis work is not concernedwith the fitting procedure, but rather with the use of nuclear potentials and the analysis ofdata obtained with these forces for finite light- and medium-mass nuclei. In Chapter 7, forexample, we will make use of these different interactions [16, 18, 39] to study the dependenceof observables with respect to the employed Hamiltonian and to gauge the uncertainty σH inab initio calculations due to the different LECs parametrizations.8Chapter 3Nuclear electromagnetic reactionsElectromagnetic induced nuclear reactions are an extremely useful tool to investigate the roleof nuclear dynamics. Because of the perturbative nature of the electromagnetic interaction, thenuclear response function can be directly connected to the transition matrix elements of thenuclear Hamiltonian. Furthermore, nuclear structure effects must be taken into account onlyin the target nucleus, as opposed to hadronic reactions where one also has to worry about thestructure of the probe. Because of these features, electromagnetic reactions provide us witha clean way to compare theory and experimental data and represent the ideal tool to studynuclear dynamics [21].Current experiments use a variety of probes and reactions, but here we focus on the pho-toabsorption cross section, which allows us to study the response of the nucleus when interactingwith a photon as depicted in Fig. 3.1.Figure 3.1: Example of a photoabsorption reaction. The nucleus absorbs a photon and then breaksup into fragments.Figure 3.2 shows a typical sketch of a photoabsorption response function obtained studyingreactions such as the photoabsorption (Fig. 3.1). At very low energy one can identify discretestates representing the excited bound-states of the nucleus. Above the break-up thresholdenergy, i.e., the energy needed to break-up the nucleus in pieces, one has a continuum spectrum.This region is dominated by broad resonant peaks whose origin lays in collective modes of thenucleus and for which the major contributions are due to dipole excitations [63].9Chapter 3. Nuclear electromagnetic reactionsFigure 3.2: Example of a typical photoabsorption response function R(ω) as a function of the energytransfer ω.The nuclear electric dipole polarizability is especially sensitive to the low-energy regionof the spectrum and has been subject of intense studies, both from the experimental and thetheoretical side. Classically, it is defined as the proportionality constant αD between the inducedelectric dipole moment d⃗ acquired by a compound system when placed in an external electricfield E⃗ and the field itself, so that d⃗ = αDE⃗. For a quantum system it is defined as an inverseenergy weighted sum rule of the dipole response function R(ω)αD = 2α∫ ∞ωexdωR(ω)ω,where ω is the excitation energy and ωex is the energy of the first state excited by the dipolerelative to the ground-state4.Traditional studies performed with photoabsorption methods [64] have deduced the electricdipole polarizability as an inverse energy weighted sum rule of the photoabsorption cross sec-tion. These studies have focused on the determination of the giant dipole resonances (GDRs)in stable nuclei, originally interpreted as collective modes of all protons oscillating against allneutrons. More recently, better resolution could be achieved in measurements of the GDR basedon inelastic scattering experiments of polarized protons at forward angles. Measurements with(p, p′) were performed at the Research Center for Nuclear Physics in Osaka on heavy nucleisuch as 208Pb [65], 120Sn [66], and very recently on medium-mass nuclei such as 48Ca by theOsaka-Darmstadt collaboration [7].Besides higher resolution, this experimental technique enables one to also access the dipolestrengths at low energy, even below particle emission threshold, allowing thus to study low-lying pygmy dipole resonances (PDRs) in stable, but neutron-rich nuclei.4For very light nuclei, ωex coincides with the break-up threshold energy.10Chapter 3. Nuclear electromagnetic reactionsBeing the polarizability an inverse energy weighted sum rule of the dipole response function, thedetermination of the low-energy strength is of crucial importance for this observable. PDRs canalso be investigated in neutron-rich and unstable nuclei with Coulomb excitation experiments.Only scarce data exist on unstable systems, but recent activity was devoted, e.g., to 22,24O [67]and 68Ni [68]. PDRs are often interpreted as due to oscillations of the excess neutrons againstall other nucleons. In an isotopic chain it is expected that neutron-rich nuclei that exhibit aPDR will have a larger polarizability than other isotopes. To both interpret recent data andguide new experiments, it is important to theoretically map the evolution of αD as a function ofneutron number. Theories that can reliably address exotic nuclei far from the valley of stabilityare needed and ab initio methods are best positioned to deliver both predictive power [21] andestimates of the theoretical accuracy [25, 69].In this chapter we focus on the reaction of Fig. 3.1, where the nucleus absorbs a low-energy photon. The final state of the nucleus can be a bound excited state or any unbound orfragmented configuration. In an inclusive cross section, where no specific final state is measured,all these possible configurations need to be taken into account. In the rest of the Chapter, wewill provide a thorough theoretical derivation for R(ω) shown in Fig. Photoabsorption cross sectionConsidering a general reaction, the unpolarized differential cross section is a function of thetarget density ρt and the incoming flux Jin. Using Fermi’s golden rule, it can be defined asdσ(ω) = 12(2Ji + 1) ∑i,f,λ′ 1ρt∣Jin∣ ∣Sfi∣2V TV dpf(2pi)3 , (3.1)where Sfi is the scattering matrix element between the initial (i) and final (f) states of thesystem and it depends on the photon polarization λ′. The factor V dpf /(2pi)3 is the phase spacevolume of the final configuration. The sums and the factor 1/2(2Ji+1) arise from averaging overthe photon polarization λ′ and the initial (Ji) angular momentum of the target. Let us furtherassume that the photon γ of Fig. 3.1 carries 4-momentum q = (ω,q) and denote the initial andfinal 4-momenta of the nuclear system as pi = (Ei,pi) and pf = (Ef ,pf), respectively.The scattering matrix element Sfi depends on the interaction Hamiltonian between thephoton and the nucleus and on the initial and final states of the system under study. Bydenoting x ≡ (t,x), we can imagine that the photon field A(x) will interact with the nuclearcurrents J(x), so that one can express the interaction Hamiltonian asHint = ∫ dx A(x) ⋅ J(x). (3.2)11Chapter 3. Nuclear electromagnetic reactionsThen Sfi is found by calculating the expectation of Hint asSfi = −i⟨F ; 0∣∫ dt Hint∣I;q, λ⟩, (3.3)where the initial state ∣I;q, λ⟩ is a combination of the photon and nucleus states, while the∣F ; 0⟩ state must take into account that the photon has been absorbed. Thus, we can define∣I;q, λ⟩ = ∣I⟩∣q, λ⟩,∣F ; 0⟩ = ∣F ⟩∣0⟩, (3.4)where ∣I⟩, ∣F ⟩ are the nuclear initial and final states and ∣q, λ⟩ represents the incoming photonof momentum q and polarization λ, while ∣0⟩ is the vacuum state of the photons space. Thephoton field and the nuclear currents act on different spaces so that, with the above definitions,the scattering matrix element can be factorized asSfi = −i∫ d4x⟨0∣A(x)∣q, λ⟩⟨F ∣J(x)∣I⟩, (3.5)where we have introduced the 4-vector notation in the integral. We define creation and annihi-lation operators for the photon field, so that ∣q, λ⟩ = aˆ„λ(q)∣0⟩, and we expand the photon fieldas [70]A(x) = ∫ dk√2ωV∑λ′=±1 (eλ′(k)eikxaˆ„λ′(k) + eλ′(k)e−ikxaˆλ′(k)) , (3.6)where eλ′ is the polarization vector. Making use of Eqs. (3.5) and (3.6), one finally obtainsSfi = −i∫ d4x√2ωVeλ(q)e−iqx⟨F ∣J(x)∣I⟩. (3.7)Since the nuclear current operator J(x) is required to be translational invariant, we can rewritethe current operator J(x) as J(0) translated by x by using the translation operator Tˆ (x) =exp(−ipx), explicitly⟨F ∣J(x)∣I⟩ = ⟨F ∣eipxJ(0)e−ipx∣I⟩ = ei(pf−pi)x⟨F ∣J(0)∣I⟩. (3.8)For the same reason, we can define intrinsic states (i.e., they depend only on the internalcoordinates) for our system ∣i⟩ and ∣f⟩, which are related to ∣I⟩ and ∣F ⟩ by∣I⟩ = 1√Veipi⋅Rcm ∣i⟩,∣F ⟩ = 1√Veipf ⋅Rcm ∣f⟩, (3.9)where 1/√V is a normalization factor and Rcm is the center of mass coordinate. Denoting by rthe intrinsic coordinates, and explicitly writing the expectation value of Eq. (3.8) in coordinate12Chapter 3. Nuclear electromagnetic reactionsspace, one has⟨F ∣J(0)∣I⟩ = ∫ dr∫ dRcmΦ∗F (Rcm, r)J(0)ΦI(Rcm, r)= 1V∫ drφ∗f(r) [∫ dRcme−i(pf−pi)⋅RcmJ(Rcm)]φi(r)= 1V⟨f ∣J(q)∣i⟩,(3.10)where in the last equality J(q) is the Fourier transform of the nuclear current operator withrespect to the center of mass coordinate Rcm, and we used the fact that J(0) = J(Rcm) becauseof translational invariance. Putting all together, Eq. (3.7) becomesSfi = −i 1√2ωV∫ d4xei(pf−pi−q)xeλ(q) ⋅ ⟨F ∣J(0)∣I⟩= −i (2pi)4√2ωVeλ(q) ⋅ ⟨F ∣J(0)∣I⟩δ(4)(pf − pi − q)= −i (2pi)4√2ωVeλ(q) ⋅ 1V⟨f ∣J(q)∣i⟩δ(4)(pf − pi − q).(3.11)Finally, since we are considering a single photon (Jin = 1/V ) scattering on a single nucleus(ρt = 1/V ), the differential cross section of Eq. (3.1) is given bydσ(ω) = pi2(2Ji + 1)ω ∑Ji,Jf ,λ ∣⟨f ∣eλ(q) ⋅ J(q)∣i⟩∣2δ(4)(pf − pi − q), (3.12)where we made use of the following property of the Dirac delta function[δ(4)(x)]2 = V T(2pi)4 δ(4)(x), (3.13)with V T being the space-time volume. The total photoabsorption cross section is found byintegrating Eq. (3.12), asσ(ω) = 4pi2αω12(2Ji + 1) ∑Ji,Jf ,λ ∣⟨f ∣eλ(q) ⋅ J(q)∣i⟩∣2δ(Ef −Ei − ω), (3.14)where we extracted the unit charge e from the current operator and included it in α = e2/4pi,the fine structure constant.In Eq. (3.14) one can identify two terms: a kinematic piece depending only on the momentaand energy of the projectile and target (the delta function), and a dynamical term which dependson the matrix element of the nuclear current operator, ⟨f ∣eλ(q) ⋅ J(q)∣i⟩. It is clear that thismatrix element is the only term containing information about nuclear structure. The initial ∣i⟩and final ∣f⟩ states depend on the nuclear Hamiltonian H since they are obtained by solving13Chapter 3. Nuclear electromagnetic reactionsthe Schro¨dinger equation. The final state also depends on the nuclear and electromagneticcurrents; in the next section we will study this dependence in further detail.3.2 Multipole decomposition of the electromagnetic currentBecause the nuclear Hamiltonian is invariant under rotations, which means it commutes withthe total angular momentum operator, it is convenient to exploit this symmetry and expand thematrix element of the nuclear current operator in terms of eigenfunctions of the total angularmomentum J . Since the current operator is a vector, we must use vector spherical harmonicswhich are defined as [71]YlJM(Ω) = ∑m,λCJMlm1λYlm(Ω) eλ, (3.15)where Ω is the solid angle, Y lm(Ω) are standard spherical harmonics, CJMlm1λ is a Clebsch-Gordancoefficient and eλ is the usual polarization vector introduced before, where we dropped theq-dependence (see Appendix A for details). The current operator is then expanded asJ(q) = ∑lJMJ lJM(q)Y∗lJM(Ωq), (3.16)where J lJM(q) are the coefficients of the expansion. Because λ = −1,0,+1, then the orbitalangular momentum quantum number l is forced by the Clebsch-Gordan coefficient to assumeonly the three allowed values J − 1, J, J + 1. Performing the sum over l givesJ(q) = ∑JM[JJ−1JM (q)Y∗J−1JM (Ωq) + JJJM(q)Y∗JJM(Ωq) + JJ+1JM (q)Y∗J+1JM (Ωq)]= ∑JM[JelJM(q) + JmgJM(q)] , (3.17)where one can identify the electric and magnetic current operatorsJelJM(q) = JJ−1JM (q)Y∗J−1JM (Ωq) + JJ+1JM (q)Y∗J+1JM (Ωq),JmgJM(q) = JJJM(q)Y∗JJM(Ωq), (3.18)depending on the way they transform under parity.Multiplying Eq. (3.16) by Yl′J ′M ′(Ωq), integrating over the solid angle Ωq and using the orthog-onality properties of the vector spherical harmonics, the expansion coefficients of Eq. (3.16) canbe expressed asJ lJM(q) = ∫ dΩqJ(q) ⋅YlJM(Ωq). (3.19)14Chapter 3. Nuclear electromagnetic reactionsUsing Eq. (A.6) to replace the vector spherical harmonics we obtain for the coefficientsJJ−1JM (q) = ∫ dΩqJ(q) ⋅ ⎡⎢⎢⎢⎢⎣√J2J + 1 qˆY JM(Ωq) − i√J + 12J + 1 qˆ ×YJJM(Ωq)⎤⎥⎥⎥⎥⎦ ,JJ+1JM (q) = ∫ dΩqJ(q) ⋅ ⎡⎢⎢⎢⎢⎣−√J + 12J + 1 qˆY JM(Ωq) − i√J2J + 1 qˆ ×YJJM(Ωq)⎤⎥⎥⎥⎥⎦ .(3.20)Substituting Eqs. (3.19) and (3.20) in the electric and magnetic current operators of Eq. (3.18),one obtainsJelJM(q) = qˆY ∗JM (Ωq)∫ dΩq [qˆ ⋅ J(q)]Y JM(Ωq)+ [qˆ ×Y∗JJM(Ωq)]∫ dΩq [qˆ ×YJJM(Ωq)] ⋅ J(q),JmgJM(q) = Y∗JJM(Ωq)∫ dΩq [J(q) ⋅YJJM(Ωq)] .(3.21)For later purposes it is convenient to introduce the electric longitudinal, electric transverse andmagnetic multipole tensorsTel∥JM(q) = 14pi ∫ dΩq [qˆ ⋅ J(q)]Y JM(Ωq),T el⊥JM(q) = i4pi ∫ dΩq [qˆ ×YJJM(Ωq)] ⋅ J(q),TmgJM(q) = 14pi ∫ dΩq [J(q) ⋅YJJM(Ωq)] ,(3.22)and rewrite the current operator in Eq. (3.17) asJ(q) = 4pi∑JM[T el∥JM(q)Y ∗JM (Ωq)qˆ − iT el⊥JM(q) (qˆ ×Y∗JJM(Ωq))+ TmgJM(q)Y∗JJM(Ωq)] . (3.23)We can now choose qˆ ∥ ez and make use of Eqs. (A.8), (A.9) and (A.10) to findJ(q) = ∑JM√4pi(2J + 1) [T el∥JM(q)e0 +MCJMJ01MT el⊥JM(q)e∗M +CJMJ01MTmgJM(q)e∗M]= ∑JM√2pi(2J + 1) [√2T el∥JM(q)e0 −M2T el⊥JM(q)e∗M −MTmgJM(q)e∗M] , (3.24)where we used CJMJ01M = −M/√2.Because in the photoabsorption reaction depicted in Fig. 3.1 the photon is real, only λ = ±1polarization is allowed. The matrix element in Eq. (3.14) becomes then⟨f ∣eλ(q) ⋅ J(q)∣i⟩ = −⟨f ∣∑J√2pi(2J + 1) [T el⊥Jλ (q) + λTmgJλ (q)] ∣i⟩, (3.25)where there is no longitudinal component since the polarization vectors eλ are orthogonal to qˆ.15Chapter 3. Nuclear electromagnetic reactionsTo further manipulate Eq. (3.25) we need to rewrite the transverse electric tensor in a moremanageable way. To do so we substitute Eq. (A.7) into Eq. (3.22), obtainingT el⊥Jλ (q) = i4pi ∫ dΩq [qˆ ×YJJλ(Ωq)] ⋅ J(q)= − 14pi∫ dΩq ⎡⎢⎢⎢⎢⎣√J + 1J(qˆ ⋅ J(q))Y Jλ (Ωq) +√2J + 1J YJ+1Jλ (Ωq) ⋅ J(q)⎤⎥⎥⎥⎥⎦ .(3.26)We then make use of the continuity equation for the nuclear current, which in momentum spacereadsωρ(q) − q ⋅ J(q) = 0 ⇒ qˆ ⋅ J(q) = q ⋅ J(q)q= ωqρ(q). (3.27)Using the above continuity equation to rewrite the first term on the right-hand side of Eq. (3.26)we finally obtainT el⊥Jλ (q) = − 14pi ∫ dΩq⎡⎢⎢⎢⎢⎣√J + 1Jρ(q)Y Jλ (Ωq) +√2J + 1J YJ+1Jλ (Ωq) ⋅ J(q)⎤⎥⎥⎥⎥⎦= T el⊥ρJλ (q) + T el⊥JJλ (q)(3.28)where ρ(q) is the nuclear charge operator and in the last equality we separated the transverseelectric tensor in charge and current transverse components. Also recall that for real photonsω = q. Notice that the transverse charge component T el⊥ρJλ (q) is a Coulomb multipole similar tothe longitudinal Tel∥JM(q) of Eq. (3.22).We would now like to relate the matrix element of Eq. (3.25) to charge and nuclear currents incoordinate space. This is easily done recalling that ρ(q) and J(q) are the Fourier transformsof ρ(r) and J(r) where r represents the intrinsic nuclear coordinates. Making use of the wellknown plane wave expansion [71]e−iq⋅r = 4pi∑lm(−i)ljl(qr)Y lm(Ωr)Y ∗lm (Ωq), (3.29)where jl(qr) are spherical Bessel functions, and orthogonality properties of the spherical andvector spherical harmonics, the transverse electric tensor components becomeT el⊥ρJλ (q) = − 14pi√J + 1J∫ dΩqY Jλ (Ωq)∫ drρ(r)e−iq⋅r= −√J + 1J∑l,m(−i)l ∫ dΩqY Jλ (Ωq)Y ∗lm (Ωq)∫ drρ(r)jl(qr)Y lm(Ωr)= − (−i)J√J + 1J∫ drρ(r)jJ(qr)Y Jλ (Ωr),(3.30)16Chapter 3. Nuclear electromagnetic reactionsT el⊥JJλ (q) = − 14pi√2J + 1J∫ dΩqYJ+1Jλ (Ωq) ⋅ ∫ drJ(r)e−iq⋅r= −√2J + 1J∑m,µ∑l,s(−i)lCJλ(J+1)m1µeµ∫ dΩqY J+1m (Ωq)Y ∗ls (Ωq)⋅ ∫ drJ(r)jl(qr)Y ls (Ωr)= − (−i)J+1√2J + 1J∫ drjJ+1(qr)J(r) ⋅YJ+1Jλ (Ωr).(3.31)Notice that the magnetic tensor of Eq. (3.22) has a structure similar to the electric currentcomponent in Eq. (3.31), but for the different parity. Thus it is straightforward to obtainTmgJλ (q) = − (−i)J ∫ drjJ(qr)J(r) ⋅YJJλ(Ωr). (3.32)In this thesis we are interested in the GDR and PDR regions, which are at low energy, wellbelow the pion-photo production threshold. It is convenient to expand Eqs. (3.30)–(3.32) inthe low-energy limit qr ≪ 1. In this regime, the Bessel functions appearing in the electric andmagnetic tensors can be expanded asjl(qr) qr→0≈ (qr)l(2l + 1)!! . (3.33)Equations (3.30), (3.31) and (3.32) then becomeT el⊥ρJλ (q) ≈ − (−i)J(2J + 1)!!√J + 1J∫ dr(qr)Jρ(r)Y Jλ (Ωr), (3.34)T el⊥JJλ (q) ≈ − (−i)J+1((2J + 3)!!√2J + 1J∫ dr(qr)J+1J(r) ⋅YJ+1Jλ (Ωr), (3.35)TmgJλ (q) ≈ − (−i)J(2J + 1)!! ∫ dr(qr)JJ(r) ⋅YJJλ(Ωr). (3.36)From the above expressions it is evident that for low energies, the magnetic and electric chargetensor go to zero with ∼ (qr)J and thus dominate against the electric current tensor whichgoes to zero as ∼ (qr)J+1. In the following we will focus on the electric charge tensor and inparticular on the J = 1 dipole contribution which is the dominant one at low energies and inthe PDR and GDR region [63].17Chapter 3. Nuclear electromagnetic reactions3.3 Electric dipole contributionsSince for real photons q = ω, the J = 1 multipole dominates at low energies. Hence, consideringthe J = 1 term of the electric charge tensor, the matrix element of Eq. (3.25) is⟨f ∣eλ(q) ⋅ J(q)∣i⟩ = − ⟨f ∣√6piT el⊥ρ1λ (q)∣i⟩ (3.37)where the electric dipole transition operator is√6piT el⊥ρ1λ (q) = − iq√4pi3 ∫ drρ(r)rY 1λ (Ωr). (3.38)To proceed further we need the explicit form of the charge density. In our picture of the nucleuscomposed of A interacting point-like nucleons, where only protons are charged, a reasonablechoice isρ(r) = Z∑i=1Qiδ(3)(r − ri) (3.39)where Z is the total number of protons in the nucleus and Qi is the charge operator. Noticethat the unit charge e has already been factorized out of the density when we calculated thephotoabsorption cross section in Eq. (3.14).The above expression considers only point-like charges, while in reality nucleons have a finitesize. Finite-size effects and internal dynamics of nucleons are typically taken into account byelectromagnetic form factors. In the low-energy regime of interest (q = ω < 100 MeV), theelectric form factor of the proton is 1 and that of the neutron is 0.Protons ∣pi⟩ and neutrons ∣ni⟩ are eigenstates of the charge operator Qi, but are also character-ized by the isospin projection quantum number τ z. Thus, we can express the charge operatormaking use of the isospin projection operator asQi = 12(1 + τ zi ), (3.40)and it is easy to check that, given τ z ∣p⟩ = ∣p⟩ and τ z ∣n⟩ = −∣n⟩, one has Qi∣p⟩ = ∣p⟩ and Qi∣n⟩ = 0.Thus, the charge density operator of Eq. (3.39) becomesρ(r) = 12A∑i=1 (1 + τ zi ) δ(3)(r − ri), (3.41)18Chapter 3. Nuclear electromagnetic reactionsand the dipole excitation operator now is√6piT el⊥ρ1λ (q) = − iq√4pi3 12 A∑i=1∫ drY 1λ (Ωr)r(1 + τ zi )δ(3)(r − ri)= − iq√4pi312A∑i=1∫ dr√34pirλrr(1 + τ zi )δ(3)(r − ri)= − iq2A∑i=1(1 + τ zi )rλi ,(3.42)where we have used the analytic expression for Y 1λ (Ωr) = √3rλ/√4pir [71], integrated over thespace and used [71]rλ=±1 = ∓ 1√2(x ± iy). (3.43)The isoscalar part of Eq. (3.42), i.e., the isospin independent term, is identically zero when westudy the intrinsic nucleus, i.e., when we refer coordinates to the center of mass frame, sinceA∑i=1 rλi = 0. (3.44)The final expression for the dipole excitation operator thus becomes√6piT el⊥ρ1λ (q) = − iq2 A∑i=1 τ zi rλi = −iqΘλ, (3.45)and the total photoabsorption cross section isσ(ω) = 4pi2αω 1(2Ji + 1) ∑Ji,Jf 12 ∑λ=±1 ∣⟨f ∣Θλ∣i⟩∣2δ(Ef −Ei − ω), (3.46)where we set q = ω (real photons) and introduced the compact notationΘλ = 12A∑i=1 τ zi rλi . (3.47)The operator Θλ is a product of two rank one tensor operators acting on the isospin (τ) andcoordinate (rλ) space, respectively. Selection rules on the total angular momentum impose∣Ji − 1∣ ≤ Jf ≤ Ji + 1, (3.48)and because Θλ does not carry spin, one has that∣Li − 1∣ ≤ Lf ≤ Li + 1 (3.49)19Chapter 3. Nuclear electromagnetic reactionsmust be true. Equivalently, selection rules on the isospin impose∣Ti − 1∣ ≤ Tf ≤ Ti + 1. (3.50)Thus, Θλ is also a rank one tensor since it connects states whose quantum numbers differ onlyby ∆T = ±1 and ∆L = ±1.We can now use the Wigner-Eckart theorem (see Section 6.2) to express the matrix element inEq. (3.46) in a different way. In particular, focusing on the rλ operator one has⟨f ∣rλ∣i⟩ = CJfMfJiMiJλCJfMfJiMiJ0⟨f ∣r0∣i⟩, (3.51)where J is the angular momentum carried by the operator. From the considerations leading toEqs. (3.48) and (3.49), we know that J = 1. Furthermore, if we assume a spherical nucleus inits ground state, we have Ji =Mi = 0. Then, Eq. (3.51) becomes⟨f ∣rλ∣i⟩ = C1λ001λC100010⟨f ∣r0∣i⟩ = C1λ001λ⟨f ∣z∣i⟩, (3.52)where we used r0 = z, and so that12∑λ=±1 ∣⟨f ∣Θλ∣i⟩∣2 = 12 ∑λ=±1C1λ001λ∣⟨f ∣Θ∣i⟩∣2 = ∣⟨f ∣Θ∣i⟩∣2. (3.53)In the above expression we used Θ to represent the z component of the operator Θλ. The crosssection of Eq. (3.46) can be rewritten asσ(ω) = 4pi2αωR(ω), (3.54)where we have introduced the dipole response function of the nucleusR(ω) = ⨋f∣⟨f ∣Θ∣i⟩∣2δ (Ef −Ei − ω) . (3.55)In the above equation we introduced the symbol ⨋f , where f represents all the quantum numbersof the final state, and ⨋ represents a sum over all bound final states and an integral over thecontinuum final states. In fact, since we are considering inclusive reactions, the sum has toaccount for any possible configuration that might arise from the initial nucleus, being that abound and/or a fragmented system. For later reference, the full formula for the cross sectionof Eq. (3.54) is thenσ(ω) = 4pi2αω⨋f∣⟨f ∣Θ∣i⟩∣2δ (Ef −Ei − ω) . (3.56)20Chapter 3. Nuclear electromagnetic reactionsThe above result is the most important formula derived in this chapter, which will be used lateron, and whose calculations will be presented in Chapter 7.21Chapter 4Integral transformsA direct calculation of the dipole response function (Eq. (3.55)) is typically not feasible withoutapproximations since it requires the knowledge of the full spectrum of the nucleus and involvesthe explicit calculation of wave functions in the continuum where the implementation of theboundary conditions is a major issue. Another difficulty is presented by the fact that at a givenenergy the nucleus can break up in different channels, corresponding to different fragments ofvarying sizes, as shown graphically in Fig. 4.1. In a direct and exact calculation, each and allof these break-up channels have to be accounted for in the sum of Eq. (3.55).Excitation Energyground state““bound excited statecontinuum2-body break-up 3-body break-up ... A-body break-upFigure 4.1: Schematic representation of the spectrum of a nucleus. Above particle emission threshold,typically first the two-body break-up channel opens up, then the three-body break-up channel up to theA-body break-up channel appear.The idea behind integral transform methods [72] is to avoid the explicit calculation of thesecontinuum states by mapping R(ω) to another function which is more easily computable, i.e.,the integral transform I(σ) = ∫ dωK(σ,ω)R(ω). (4.1)The function K(σ,ω) is known as the integral kernel and defines the type of integral transform.As we will prove in the next section, the calculation of the response function is reduced to abound-state like problem [72–77] and the only step left is to recover R(ω) from Eq. (4.1) via aninversion or deconvolution procedure [76, 78]. Different from, e.g., the Fourier transform, wherethe inverse kernel is analytically defined, the deconvolution of Eq. (4.1) is an ill-posed problem.While many choices for the functional form of the kernel are possible, the retrieval of theresponse function from the integral transform can be made easier if one chooses an appropriatekernel, i.e., a kernel that is reasonably localized and preserves the information of the responsefunction. In fact, the convolution in Eq. (4.1) has the effect of blurring the response functionand spreading the information of R(ω) in the σ space by a degree which is controlled by the22Chapter 4. Integral transformswidth of the integral kernel. To illustrate this concept, in Fig. 4.2 we show in the left panel anexample of a response function, and in the right panel its integral transform obtained with theGaussian kernelKG(σ,ω) = 1√2piΓ2exp(−(ω − σ)22Γ2) . (4.2)The parameter Γ controls the width of the kernel: for very small Γ, the Gaussian will tendto a Dirac delta, while for larger Γ it will be a very wide function. It is evident that for verylarge widths, the information on the structure of the response function is totally lost whencalculating the integral transform. This behaviour is shown in Fig. 4.2: panel (a) shows atest response function (solid blue) obtained analytically, and panel (b) compares the Gaussianintegral transforms of the latter using different values for the parameter Γ.0 20 40 60 80 100ω02468101214R(ω)(a)0 20 40 60 80 100σ02468101214I L(σ)(b)Γ = 0.1Γ = 10.0Γ = 100.0Figure 4.2: Gaussian integral transform of an analytical response function. The test response functionis shown in the left panel and its integral transform with the Gaussian kernel of Eq. (4.2) in the rightpanel. The different curves represent the transform calculated with different values of the width, Γ = 0.1(solid red), Γ = 10.0 (dashed green) and Γ = 100.0 (dotted orange).To preserve the information, one would like to have a kernel which resembles a delta function.However, we cannot choose a kernel which is the delta function itself, as we would not makeany progress in solving Eq. (4.1) since the integral transform would basically be the responsefunction itself, i.e., Iδ(σ) = ∫ dωR(ω)δ(ω − σ) = R(σ). (4.3)A variety of kernels, all representations of the delta function, have been proposed in the past.Here we will focus on two particular ones: the Lorentz kernel [74, 77]KL(σ,ω) = Γpi1(ω − σ)2 + Γ2 , (4.4)23Chapter 4. Integral transformswhich can be used to calculate the response function, and the Stieltjes kernel [73]KS(σ,ω) = 1ω + σ , (4.5)which will be used to obtain the electric dipole polarizability αD. The dipole polarizability is asum rule of the photoabsorption cross section of Eq. (3.56) and, together with other momentsof σ(ω), i.e.,mk = ∫ dωωkσ(ω), (4.6)it can be used to access information about nuclear dynamics when the full response functioncannot be obtained explicitly. The polarizability is defined asαD = m−22pi2, (4.7)and using Eqs. (3.54) and (3.55), one getsαD = 2α∫ dωR(ω)ω= 2α⨋f∣⟨f ∣Θ∣i⟩∣2Ef −Ei . (4.8)From Eq. (4.8) it is evident that the dipole polarizability probes the low-energy part of the re-sponse function, i.e., the excitation contributions at small ω are enhanced by the inverse energyweight, while the ones at large ω are strongly suppressed. Again, the calculation of Eq. (4.6)requires the explicit knowledge of the full spectrum in the continuum and thus presents a prob-lem similar to the one faced in the calculation of the response function.In Section 4.1 we will show how one can use the Lorentz kernel to calculate the responsefunction and in Section 4.2 how to obtain the electric dipole polarizability from the Stielt-jes kernel. The choice of the kernel is made based on a compromise between computationalsophistication and ability to obtain a stable deconvolution.4.1 The Lorentz integral transformThe LIT of the response function defined by Eq. (4.1) with the integral kernel of Eq. (4.4) isgiven by IL(σ) = Γpi∫ dω R(ω)(ω − σ)2 + Γ2 . (4.9)24Chapter 4. Integral transformsUsing the definition of the response function of Eq. (3.55), the LIT becomesIL(σ) = Γpi⨋f∫ dω ∣⟨f ∣Θ∣i⟩∣2(ω − σ)2 + Γ2 δ (Ef −Ei − ω)= Γpi⨋f∣⟨f ∣Θ∣i⟩∣2(Ef −Ei − σ)2 + Γ2= Γpi⨋f⟨i∣Θ„∣f⟩⟨f ∣Θ∣i⟩(Ef −Ei − σ + iΓ) (Ef −Ei − σ − iΓ)= Γpi⨋f⟨i∣Θ„ (Ef −Ei − σ + iΓ)−1 ∣f⟩⟨f ∣ (Ef −Ei − σ − iΓ)−1 Θ∣i⟩= Γpi⟨i∣Θ„ (H −Ei − σ + iΓ)−1 (H −Ei − σ − iΓ)−1 Θ∣i⟩,(4.10)where H is the nuclear Hamiltonian. In Eq. (4.10) we first integrated out the delta functionand then used the Schro¨dinger equation H ∣f⟩ = Ef ∣f⟩ together with the completeness relation⨋f ∣f⟩⟨f ∣ = 1ˆ. Note that the final result of Eq. (4.10) does not depend on the final states ∣f⟩anymore, and so in order to calculate IL(σ) one needs only the initial (ground) state ∣i⟩ of thenucleus and the nuclear Hamiltonian H. Indeed, by defining⟨Ψ˜∣ = ⟨i∣Θ„ (H −Ei − σ + iΓ)−1 ,∣Ψ˜⟩ = (H −Ei − σ − iΓ)−1 Θ∣i⟩, (4.11)the calculation of the integral transform in Eq. (4.10) consists in solving a bound state Schro¨dinger-like equation with a source term(H −Ei − σ − iΓ) ∣Ψ˜⟩ = Θ∣i⟩. (4.12)It is worth noting that the state ∣Ψ˜⟩, although solution of the above equation, is not associatedwith any physical eigenstate of the system, but it is solely a mathematical tool.Because the right-hand side of Eq. (4.12) contains the ground-state ∣i⟩, which is exponentiallysuppressed for r → ∞, the state ∣Ψ˜⟩ on the left-hand side must have the same asymptoticbehaviour of ∣i⟩ and thus is also bound-state-like.Another form of Eq. (4.10), which will turn useful in Section 4.3, can be obtained making25Chapter 4. Integral transformsuse ofIm [⟨i∣Θ„ (H −Ei − σ + iΓ)−1 Θ∣i⟩] = Im⎡⎢⎢⎢⎣⨋f ⟨i∣Θ„∣f⟩⟨f ∣Θ∣i⟩(Ef −Ei − σ + iΓ)⎤⎥⎥⎥⎦= Im⎡⎢⎢⎢⎢⎣⟨i∣Θ„ (Ef −Ei − σ + iΓ)Θ∣i⟩(Ef −Ei − σ)2 + Γ2⎤⎥⎥⎥⎥⎦= Γ⨋f⟨i∣Θ„ 1(Ef −Ei − σ + iΓ) ∣f⟩⟨f ∣ 1(Ef −Ei − σ − iΓ) Θ∣i⟩= Γ⟨i∣Θ„ (H −Ei − σ + iΓ)−1 (H −Ei − σ − iΓ)−1 Θ∣i⟩= piIL(σ).(4.13)Thus, one can writeIL(σ) = 1piIm [⟨i∣Θ„ 1(H −Ei − σ − iΓ) Θ∣i⟩] = − ipi ⟨i∣Θ„∣Ψ˜⟩. (4.14)Note that the LIT can be used also to calculate the dipole polarizability. One can use Eq. (4.9)and the fact that for Γ→ 0 the Lorentz kernel becomes a Dirac delta to findαD = 2α limΓ→0∫ IL(σ)σ dσ. (4.15)The above expression will be also used in the computation of the results presented in Chapter 7.4.2 The Stieltjes integral transformSimilarly to what we did in Eq. (4.10) for the LIT, using the completeness 1ˆ = ⨋f ∣f⟩⟨f ∣, we canrewrite the polarizability of Eq. (4.8) asαD = 2α⨋f⟨i∣Θ„ 1Ef −Ei ∣f⟩⟨f ∣Θ∣i⟩= 2α⟨i∣Θ„ 1H −EiΘ∣i⟩.(4.16)One way to evaluate the above expression would be to represent the Hamiltonian H on afinite set of square integrable N basis functions ∣n⟩ (for example the harmonic-oscillator basis).One could then diagonalize the Hamiltonian in this basis and obtain its N eigenstates ∣β⟩ andeigenvalues Eβ, and then calculate Eq. (4.16) asαD = 2α ∑β,β′⟨i∣Θ„∣β⟩⟨β∣ 1H −Ei ∣β′⟩⟨β′∣Θ∣i⟩= 2α∑β∣⟨i∣Θ„∣β⟩∣2Eβ −Ei .(4.17)26Chapter 4. Integral transformsThe latter, resembles Eq. (4.8), where the final states have been replaced by the eigenstates ∣β⟩of the Hamiltonian. While the ∣β⟩ eigenstates have been built using a discrete and finite basis,and so have finite norm, the Hamiltonian has, in principle, both continuum and bound states.One may then question the use of such a discrete basis to calculate the polarizability. Again,this procedure can be justified by the use of an integral transform, e.g., the Stieltjes integraltransform. Similarly to Eq. (4.9), one calculatesIS(σ) = 2α∫ dωR(ω)ω + σ , (4.18)which with the definition of R(ω) becomesIS(σ) = 2α⨋f∫ dω ∣⟨i∣Θ∣f⟩∣2ω + σ δ(Ef −Ei − ω)= 2α ⟨i∣Θ„ (H −Ei + σ)−1 Θ∣i⟩= 2α ⟨i∣Θ„∣Φ˜(σ)⟩,(4.19)where ∣Φ˜(σ)⟩ is the solution of(H −Ei + σ)∣Φ˜(σ)⟩ = Θ∣i⟩. (4.20)If we restrict σ > 0 and because Θ∣i⟩ → 0 asymptotically, for large distances ∣Φ˜(σ)⟩ shouldsatisfy a Schro¨dinger equation with eigenvalues smaller than Ei. This implies ∣Φ˜(σ)⟩ → 0asymptotically, i.e., ∣Φ˜(σ)⟩ has bound state properties and one is allowed to use an expansionover a discrete basis to solve Eqs. (4.19) and (4.20).Since Eqs. (4.18) and (4.8) only differ by the presence of σ, the polarizability αD can beretrieved by taking the limit for σ → 0+ of Eq. (4.19) asαD = 2α limσ→0+ IS(σ). (4.21)The one-sided limit follows from the fact that we have restricted σ > 0. In fact, if we were toallow σ < 0, we encounter poles when evaluating IS(σ) as it is shown in Fig. 4.3. Note that theposition of these poles might vary depending on the Hamiltonian and on the truncation of thebasis used in the calculations. They never appear in the positive region σ > 0.27Chapter 4. Integral transforms-30.0 -20.0 -10.0 0.0 10.0σ [MeV]10−210−1100101102103104I S(σ)[fm2 MeV−1]Figure 4.3: Stieltjes integral transform in 4He from Miorelli et al. [2]. The poles appear only in thenegative region of σ, hence the limit in Eq. (4.21) is well defined.4.3 Lanczos numerical calculationThe Lorentz and Stieltjes integral transform have successfully been calculated in the past usingthe Lanczos algorithm [79].Suppose A is an n × n symmetric matrix, then the Lanczos tridiagonal matrix isAtr =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝a0 b1 0 0 0 ... 0b1 a1 b2 0 0 ... 00 b2 a2 b3 0 ... 00 0 b3 a3 b4 ... 0... ... ... ... ... ... 0... ... ... ... ... ... bm−10 0 0 0 0 ... am⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (4.22)where m < n. The elements of Atr are obtained iteratively usingbi+1qi+1 = Aqi − aiqi − biqi−1,ai = q„iAqi, (4.23)bi+1 = ∣∣biqi∣∣,28Chapter 4. Integral transformswhere {qi}, i = 1, ...,m is the set of orthonormal Lanczos vectors and the initial conditions areb0 = 1, q−1 = 0 and q0 is a normalized random vector, i.e., ∣∣q0∣∣ = 1. While the algorithmis straightforward, it may suffer from numerical instability, i.e., increasing the number of it-erations, the Lanczos vectors lose orthogonality. To restore orthogonality, one has to supplyEqs. (4.23) with an additional Gram-Schmidt reorthogonalization step.Equation (4.14) can be solved using the Lanczos algorithm. To this end, let us define q−1 = 0andq0 = Θ∣i⟩√⟨i∣Θ„Θ∣i⟩ , (4.24)A =H −Ei − σ − iΓ. (4.25)Performing the Lanczos procedure one obtains Atr = (H − zL)tr,Atr =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝a0 − zL b1 0 0 0 ... 0b1 a1 − zL b2 0 0 ... 00 b2 a2 − zL b3 0 ... 00 0 b3 a3 − zL b4 ... 0... ... ... ... ... ... 0... ... ... ... ... ... bm−10 0 0 0 0 ... am − zL⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (4.26)where zL = Ei + σ + iΓ. Using the representation of A in the Lanczos basis we can rewriteEq. (4.14) as IL(σ) = ∣∣q0∣∣2piIm [q„0A−1tr q0] . (4.27)From the basic definition of the inverse of a matrix we have(AtrA−1tr )ij =∑k(Atr)ik (A−1tr )kj = δij . (4.28)29Chapter 4. Integral transformsChoosing j = 0 one has then the following system⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝a0 − zL b1 0 0 0 ... ...b1 a1 − zL b2 0 0 ... ...0 b2 a2 − zL b3 0 ... ...0 0 b3 a3 − zL b4 ... ...... ... ... ... ... ... ...... ... ... ... ... ... ...⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝X0X1X2X3......⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1000......⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (4.29)whereXk = (A−1tr )k0 , (4.30)and so Eq. (4.27) becomes IL(σ) = ∣∣q0∣∣2piIm [X0] . (4.31)At this point, to get the elements Xi we can use Cramer’s rule, namelyXi = det [(Atr)i]det [Atr] , (4.32)where (Atr)i represents the matrix Atr for which the i-th column is substituted by the rightmostvector in Eq. (4.29). For our case we haveX0 = det [(Atr)0]det [Atr] = det [B](a0 − zL)det [B] − b1det [B0] = 1(a0 − zL) − b1 det[B0]det[B] , (4.33)whereB =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝a1 − zL b2 0 ...b2 a2 − zL b3 ...0 b3 a3 − zL ...... ... ... ...⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, B0 =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝b1 b2 0 ...0 a2 − zL b3 ...0 b3 a3 − zL ...... ... ... ...⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (4.34)Notice thatdet [B0]det [B] = b1det [C](a1 − zL)det [C] − b2det [C0] = b1(a1 − zL) − b2 det[C0]det[C] , (4.35)30Chapter 4. Integral transformswhereC =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝a2 − zL b3 0 ...b3 a3 − zL b4 ...0 b4 a4 − zL ...... ... ... ...⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, C0 =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝b2 b3 0 ...0 a3 − zL b4 ...0 b4 a4 − zL ...... ... ... ...⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (4.36)From Eqs. (4.33) and (4.35) we can see that each successive term in the denominator has thesame structure. Thus, iterating up to the full size m of Atr we obtainX0 = 1(a0 − zL) − b21(a1−zL)− b21(a2−zL)− b23(a3−zL)−.... (4.37)Finally, we can rewrite the integral transform of Eq. (4.31) asIL(σ) = ⟨i∣Θ„Θ∣i⟩piIm⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1(a0 − zL) − b21(a1−zL)− b21(a2−zL)− b23(a3−zL)−...⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (4.38)where the coefficients ai, bi are obtained from the tridiagonalization of the Hamiltonian matrixusing the Lanczos algorithm with the first vector given in Eq. (4.24).As a consequence, the dipole polarizability of Eq. (4.21) can be calculated in a similar wayusing the Lanczos algorithm on the Stieltjes transform of Eq. (4.19), obtainingαD = 2α⟨i∣Θ„Θ∣i⟩ limzS→−E−i⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1(a0 − zS) − b21(a1−zS)− b21(a2−zS)− b23(a3−zS)−...⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (4.39)where zS = Ei − σ.In this section we showed that both the LIT and the Stieltjes transform can be evaluatedwith the Lanczos algorithm via the solution of the linear system in Eq. (4.29) which allows usto rewrite our integral transforms in terms of easily computable continued fractions. Note thatonce the Lanczos coefficients ai and bi have been calculated, the evaluation of the continuedfraction is extremely fast and one can obtain IL(σ) for different values of σ and Γ very efficiently.31Chapter 4. Integral transforms4.4 Inversion methodsIn Section 4.1 we have shown how the calculation of integral transforms of the response func-tion, in particular the LIT, does not require the explicit knowledge of the continuum states.However, the response function is the physical observable we are interested in and so we needa method to extract it from the integral transform itself. This procedure is called inversion ofthe integral transform.From a numerical point of view, the convolution integral in Eq. (4.1) must be performed ona discrete grid, so that we can rewrite the integral as a sum, i.e.,I(σj) = 1NN∑i=1K(σj , ωi)R(ωi), (4.40)or in a more compact notation I = KR, (4.41)where I ∈ RM and R ∈ RN are column vectors, K ∈ RM×N is a matrix representation of thekernel on a bi-dimensional finite grid of spacing ∆i = 1/N and ∆j = 1/M . The factor 1/N inEq. (4.40) has been absorbed into K =K/N .While a straightforward way of finding the response function R from Eq. (4.41) would be toevaluate R = K−1I, it is not guaranteed the inverse of K exists, especially as in general we canhave M ≠ N and K is not a square matrix. We can, however, use an optimization procedure toobtain R, i.e., we define a cost functionJ(R˜) = 12∣∣I −KR˜∣∣2, (4.42)and we minimize it with respect to the function R˜. The smaller the cost function in Eq. (4.42),the closer the function R˜ will be to the real response function R. The optimal solution is givenby minimizing J(R˜) with respect to R˜. The gradient of the cost function with respect to R˜reads5 ∇J(R˜) = KTKR˜ −KTI, (4.43)and setting it to zero we find the solutionR˜ = (KTK)−1KTI = T −1KTI, (4.44)where we introduced the Toeplitz matrix T = KTK. The matrix T is square and its inverse5The gradient can be easily calculated element-wise if one expresses the cost function of Eq. (4.42) in compo-nents,J(R˜) = 12∑m(Im −∑nKmnR˜n) .Then, the k-th element of the gradient ∇J(R˜)k is found by calculating partial derivative with respect to ∂/∂R˜k.32Chapter 4. Integral transformsmight, in principle, be calculated. However, in practice KTK could be singular or ill-conditioned,making its inverse prone to large numerical errors. This is shown in Fig. 4.4(a) where we showthe LIT (red) with Γ = 20 of the two-peaked response function introduced in Fig. 4.2, whileFig. 4.4(b) shows the inversion (green) using Eq. (4.44) to estimate R˜.0 20 40 60 80 100σ0. L(σ)(a) Γ = 200.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0ω−2.0−˜(ω)[×10+18](b)Figure 4.4: Illustration of the ill-posed problem for the inversion of an analytical function. Panel (a)– The LIT with Γ = 20 of the response function in Fig. 4.2(a). Panel (b) – The inversion of the LIT inpanel (a) obtained using Eq. (4.44).The noise in the inversion of Fig. 4.4(b) can be explained by inspecting the singular valuesof the Toeplitz matrix. We can decompose T using singular value decomposition (SVD), i.e.,T = USVT = N∑i=1uisivTi , (4.45)where S = diag(s1, s2, ..., sN) is the diagonal matrix of the singular values {σi}i of the matrixT , and U and V are the matrices of the left- and right-singular vectors of T , respectively, and33Chapter 4. Integral transformsare unitary matrices, i.e., VT = V−1 and UT = U−1. The inverse of T can be found asT −1 = (USVT )−1 = VS−1UT = N∑i=1uTi visi. (4.46)Figure 4.5(a) shows the singular values of the matrix T obtained from the Lorentz kernelused for the LIT in Fig. 4.4(a) by discretizing the σ and ω spaces on grids of 1000 points.Only about the first 30 singular values are appreciably larger than 10−16 which is the standardfloating point precision on modern computing machines.0 10 20 30 40 50 60 70 80i10−1610−1410−1210−1010−810−610−410−2100|Singularvalues| (a)0 200 400 600 800 1000i−0.06−0.04−||Singularvectors||(b)u0 u5 u10Figure 4.5: Panel (a) – The singular values of the Toeplitz matrix T built from a Lorentz kernel withΓ = 20, where i = 1, ...,N with N the rank of the matrix T . Panel (b) – Three of the left-singular vectorsui of T for i = 0,5,10 in blue, red and green, respectively.Since singular values appear in the denominator of Eq. (4.46), it is clear that even smallnumerical errors will be comparable to the smallest si and so will affect the elements of T −1. Inparticular, in Fig. 4.5(b) we present some of the left-singular vectors of T , showing that higherfrequency modes belong to vectors corresponding to larger i-indexes. The smaller singular valuesamplify the higher frequency modes, thus explaining the nature of the high-frequency noise inFig. 4.4(b). A way to suppress the noise in the estimate of R˜ and thus to the ill-conditioned34Chapter 4. Integral transformsnature of T is to use regularization techniques. In the next sections we will describe two ofthese techniques, namely the truncated SVD (TSVD) regularization [80] and the Tikhonovregularization [81, 82].4.4.1 Truncated SVD regularizationThe TSVD regularization suppresses the high-frequency noise by neglecting singular valuessmaller than a certain threshold t or index it, i.e., the truncated inverse S−1t of the diagonalmatrix S is defined asS−1t =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩1/si if i ≤ it0 otherwise(4.47)The pseudo-inverse of the Toeplitz matrix T is then evaluated viaT −1t = VS−1t UT = it∑i=1uTi visi. (4.48)The optimal value for the truncation threshold can be chosen by repeatedly solving Eq. (4.44)using T −1t and evaluating the cost function in Eq. (4.42) for different values of the thresholdindex it. Figure 4.6(a) shows the cost J(R˜) as a function of the truncation index it. When it istoo small, the cost is large suggesting we do not have enough singular values to fully capture allthe features of the real response function. On the other hand, if we include a large number ofsingular values then the noise is not suppressed anymore and large numerical errors will startagain to dominate our estimate R˜. The solution lies in between these two extrema and we canchoose the value it = 24 which is the one that minimizes the cost function J(R˜). In Fig. 4.6(b)the black dashed line is the real response function, while the solid green line is the result ofthe inversion using a regularized Toeplitz matrix with singular values truncated at it = 24. Theinversion in Fig. 4.6(b) agrees very well with the original response function except for somesmall ripples close to the sharp edges (also knows as Gibbs phenomenon, see Ref. [83]). Ingeneral, however, selecting the value of it that gives the lowest cost is not the best choice. Thisis particularly true if the integral transform is affected by noise, in which case choosing for itthe minimum of the curve in Fig 4.6(a) could result in over fitting R˜ to the noise.35Chapter 4. Integral transforms0 5 10 15 20 25 30 35i10−1110−910−710−5J(R˜)(a)0 20 40 60 80 100ω0.˜(ω)(b) Original responseTSVD inversion (it = 24)Figure 4.6: Panel (a) – The cost J(R˜) as a function of the threshold index it which regulates thetruncation of the singular values of T . Panel (b) – The solid green line is the inversion obtained withthe TSVD method with it = 24, the dashed black line is the original response function one would like toreproduce.4.4.2 Tikhonov regularizationThe Tikhonov regularization scheme aims to suppress the high-frequency oscillations due tonoise by using a regularized version of the T matrix of the formTλ = KTK + λ1ˆ, (4.49)where 1ˆ is the identity matrix and λ is a numerical coefficient. The regularized Toeplitz matrixin Eq. (4.49) can also be obtained by minimizing a regularized cost functionJ(R˜) = 12∣∣I −KR˜∣∣2 + λ2∣∣R˜∣∣2. (4.50)The additional term in Eqs. (4.49) and (4.50) has the effect of shifting the small singular valuesso that they are appreciably larger than the errors due to numerical precision, thus suppressingthe high frequency modes.36Chapter 4. Integral transformsAs for the TSVD scheme, the regularization depends on a parameter λ for which the optimalvalue must be determined. To determine the optimal value we can proceed by repeatedlysolving the regularized least-square problem of Eq. (4.50) for different values of λ. We thenplot the regularization term versus the cost term in a graph also knows as L–curve [84, 85] asin Fig. 4.7(a). The best point is chosen to be the one that gives the best trade-off betweensmall regularization and small error, usually located around the “shoulder” of the L–curve. InFig. 4.7(a) the red circle represents the best value λ = 1.7 × 10−9 for the problem at hand. InFig. 4.7(b) the solid green line represents the inversion obtained using the regularized matrixof Eq. (4.49) with λ = 1.7 × 10−9, and the dashed black curve is the original response functionwe want to reproduce via the inversion.2.0 3.0 4.0 5.0 6.0 7.0 8.0||IL−KR˜|| [×104]116.3116.5116.7116.9||R˜||(a) λ = 1.7× 10−90 20 40 60 80 100ω0.˜(ω)(b) Original responseTikhonov regularized inversionFigure 4.7: Panel (a) – L–curve of regularization vs the cost terms; each point corresponds to a differentvalue of the regularization parameter λ and the red circle represents the optimal λ value located at theL–curve shoulder. Panel (b) – The original response function (solid black line) compared to the inversionusing the Tikhonov regularization scheme with λ = 1.7 × 10−9 (dashed green curve).37Chapter 4. Integral transforms4.4.3 Basis expansion deconvolutionWhile the deconvolution methods presented in the previous section have been successfully ap-plied in different fields [86–88], they cannot easily be used to invert LITs obtained from nuclearphysics calculations based on finite basis expansions. In fact, the LIT is usually calculatedby solving Eq. (4.11) by expanding the state ∣Ψ˜⟩ on a discrete and finite basis set, e.g, theharmonic-oscillator basis. The calculated LIT can be viewed as the convolution of a discreteresponse function with a Lorentz kernel. The number of discrete states included in the responsefunction increases proportionally with the number of basis functions included in the expansion,and the continuum is reached when one includes an infinite number of basis.The discrete response function obtained from the LIT for Γ → 0 is shown in Fig. 4.8 withthe solid black line. A deconvolution of the LIT would try to recover the discrete nature ofthe response function. Indeed, the regularized solution with λ = 1 × 10−8 in Fig. 4.8 (solid red)shows oscillations which, however, are not due to the ill-conditioned nature of the inversion. Itis easy to see that the peaks of the oscillations appear in correspondence of the discrete peaksof the LIT calculated at small Γ. Negative oscillations are again due to the Gibbs phenomenonwhich arises in presence of sharp peaks.20 30 40 50 60 70 80 90 100ω0.˜(ω)Tikhonov (λ = 1× 10−8)Tikhonov (λ = 1× 10−4)IL(σ,Γ→ 0)Figure 4.8: Example of a LIT calculated at Γ → 0 (black) compared to the inversion of the sameLIT at Γ = 10 using the Tikhonov method and for two different values of the regularization parameterλ = 1 × 10−8 (red) and λ = 1 × 10−4 (green).Figure 4.8 clearly points out the discrete nature of the calculated LIT. On the other hand,the physical continuum does not show the strong oscillations observed in the red curve. Onecould increase the value of the regularization parameter λ to get rid of the residual oscillationsand obtain a curve like the green one in Fig. 4.8. However, if the oscillations in the red curveare due to physical and not mathematical reasons (as it is in this case), we have no clear way38Chapter 4. Integral transformsto choose which value of λ is the best.To obtain physically meaningful solutions to the inversion problem, one has to perform aconstrained inversion where the constraints come from physical considerations. The idea is toexpress the estimated response function asR˜(ω) = ν∑ncnfn(ω,{αi}i), (4.51)where {αi} and cn is a set of unknown parameters and fn(ω,{αi}i) are known functions. TheLIT of the estimate is thenI(R˜)L (σ) = KR˜ = ν∑ncnI(fn)L (σ), (4.52)where I(fn)L (σ) = Γpi ∫ dω fn(ω,{αi})(ω − σ)2 + Γ2 . (4.53)With the ansatz of Eq. (4.51), the cost function in Eq. (4.42) becomesJ(R˜) = 12∣∣IL(σ) − ν∑ncnI(fn)L (σ)∣∣2 , (4.54)and the optimal values for the parameters are calculated numerically by minimizing Eq. (4.54).The number of basis functions ν used in the expansion of Eq. (4.51) acts as a regularizationparameter: with small ν the resulting R˜(ω) is smoother, while for large ν the estimate of theresponse function generally exhibits oscillatory behaviours. On the other hand, the oscillatorybehaviour also depends on the functional form of fn(ω,{αi}i), which is usually based on somephysical constraints. A common choice for these basis functions is [78]fn(ω,α1, α2, α3) = (ω − α1)α2e− (ω−α1)nα3 . (4.55)However, the particular shape of the basis functions depend on the system under study. Forexample in Ref. [1] we use the following non-linear ansatzfn(ω,β) = ω3/2eG(ω)e− ωnβ , (4.56)where G(ω) is a Gamow factor which is zero in the case the first break-up channel of the nucleusis the emission of a neutron and isG(ω) = −αpiZ1Z2√µω, (4.57)if the first break-up channel involves two fragments of charges Z1 and Z2, respectively. Here,α is the fine structure constant and µ is the reduced mass of the two fragments. Moreover,when minimizing Eq. (4.54) with the ansatz of Eq. (4.56) we also need to force the response39Chapter 4. Integral transformsfunction to be zero below the break-up threshold ωth. This inversion method [76] so far hasbeen proven to be the best one and will be used to invert the LIT calculations performed inthis thesis work and presented in Chapter 7. A very strong check of the inversion procedure isprovided by inverting IL(σ) at different values of Γ, i.e., the resolution scale.40Chapter 5Coupled-cluster theoryWhile the use of integral transforms can help us to overcome the continuum problem, in orderto calculate the LIT and the polarizability, besides solving Eqs. (4.14) and (4.19), we also needthe many-body nuclear ground state ∣i⟩ = ∣Ψ0⟩. Such a state can be computed by solving themany-body Schro¨dinger equationH ∣Ψ0⟩ = E0∣Ψ0⟩, (5.1)where H is the nuclear Hamiltonian and E0 is the ground-state energy of the many-bodysystem. The solution of Eq. (5.1) must take into account the strong short-range correlationscharacteristic of the nuclear interaction, which play a crucial role in the correct description ofthe wave function ∣Ψ0⟩.First introduced in nuclear physics by Coester and Ku¨mmel [89, 90] as a suitable methodto treat the strong nuclear correlations, coupled-cluster theory has been largely used in thedescription of many-electron systems in ab initio and computational chemistry for the study ofatoms and molecules (see Ref. [91] and references therein) and quantum dots [92]. Only recently,with the advent of soft realistic nuclear potentials and the ever increasing computational powerof modern computers, coupled-cluster theory has experienced a renaissance in nuclear physics[93]. With its mild (polynomial) computational scaling with increasing mass number A, coupled-cluster theory is a perfect choice if one wants to extend calculations of break-up observablesfrom the light- to the medium-mass nuclei.In the following sections we will first introduce the basic tools of second quantization [91]and similarity transformations, and then dive into formal coupled-cluster theory showing howone can use it to solve the many-body Schro¨dinger equation. With the use of integral transformswe then further extend coupled-cluster theory to the calculation of break-up electromagneticobservables such as the polarizability and the response function [1, 2].5.1 Preliminaries5.1.1 Second quantizationIn a many-body context, second quantization is a powerful tool which allows us to representthe states of the system in a much simpler way. The idea is to use a basis in which each stateis represented by the occupation number of some single-particle basis states. For example,given a single-particle basis {∣φ1⟩, ∣φ2⟩, ..., ∣φm⟩}, we count the number of particles ni occupying41Chapter 5. Coupled-cluster theoryeach state ∣φi⟩. Then we can define a new basis of vectors labelled by the occupation numbers∣n1, n2, ....., nm⟩, where ∑i ni = A, with A the total number of particles. The space spanned bythe occupation number basis is called Fock space and it is a direct sum of Hilbert spacesF =H0 ⊕H1 ⊕H2 ⊕ ...⊕Hm, (5.2)where ∣n1⟩ ∈ H1, ∣n1, n2⟩ ∈ H2, ∣n1, n2, ...nm⟩ ∈ Hm and ∣0⟩ ∈ H0 is the vacuum state where alloccupation numbers are zero, meaning that no particles exist. In this space one then introducescreation a„i and annihilation ai operators with respect to the vacuum state ∣0⟩. Every operatorof the theory can be represented in the occupation number basis using annihilation and creationoperators. Moreover, all the states belonging to the Fock space can be obtained by acting onthe vacuum via ∣n1, n2, ..., nm⟩ = (a„1)n1 (a„2)n2 ⋅ ⋅ ⋅ (a„m)nm ∣0⟩. (5.3)Nuclear systems are fermionic systems and wave functions must be antisymmetric. In firstquantization formalism, the total wave function is an antisymmetric product of single-particlestates ∣Φ⟩ = A[∣φi⟩∣φj⟩ ⋅ ⋅ ⋅ ∣φk⟩], (5.4)where A is an antisymmetrizing operator and ∣Φ⟩ is the so-called Slater determinant. In secondquantization, the Slater determinant defined in Eq. (5.4) becomes∣Φ⟩ = ∣ij....k⟩ = a„ia„j ⋅ ⋅ ⋅ a„k∣0⟩, (5.5)with the meaning that a„i creates a particle in the single-particle orbital ∣φi⟩. Antisymmetryis then enforced by the following anti-commutation relations for the creation and annihilationoperators{a„i , a„j} = {ai, aj} = 0,{a„i , aj} = {aj , a„i} = δij . (5.6)The above conditions are obtained by noting that a„i ∣i⟩ = 0 and because (a„iaj + aia„j)∣0⟩ = 0and (a„iai + aia„i)∣0⟩ = ∣0⟩. Operators can also be represented using the second quantizationformalism and they are functions of creation and annihilation operators, i.e.,F =∑pq⟨p∣f ∣q⟩a„paq (5.7)V = 12∑pqrs⟨pq∣v∣rs⟩a„pa„qasar = 14 ∑pqrs⟨pq∣∣rs⟩a„pa„qasar (5.8)W = 16∑pqrstu⟨pqr∣w∣stu⟩a„pa„qa„rauatas = 136 ∑pqrstu⟨pqr∣∣stu⟩a„pa„qa„rauatas (5.9)42Chapter 5. Coupled-cluster theorywhere F , V and W are one-, two- and three-body operators respectively, and where for V wedefined the antisymmetric product ⟨pq∣∣rs⟩ = ⟨pq∣v∣rs⟩−⟨pq∣v∣sr⟩ and an analogous one is definedfor the three-body operator W.5.1.2 Wick’s theoremAn important tool for the calculation of matrix elements in second quantization formalism isWick’s theorem, which introduces the concepts of:ˆ normal product N [...] of a string of operators ABC...Z, defined as the rearrangementof these operators in such a way that annihilation operators are at the right of creationoperators, with a phase which depends on the total number of permutations occurred;ˆ contraction between two operators, defined asAB = AB −N [AB]. (5.10)The time-independent Wick’s theorem states that a product of a string of creation and annihi-lation operators is equal to their normal product plus the sum of all possible normal productswith contractions (see Ref. [91] for a quick proof or Ref. [94] for a more complete one). Informulas, this means thatABC...Z = N [ABC...Z] + ∑contrN [ABC...Z]. (5.11)For example, if we consider only three operators, we haveABC = N [ABC] +N [ABC] +N [ABC] +N [ABC], (5.12)where the normal product with contractions is defined asN [ABCD...V Z] = (−1)PCVN [ABD...Z], (5.13)with P being the number of permutations performed to move V next to C. From Eq. (5.11)follows that the vacuum expectation value of any string of operators is zero unless in the lastterm all operators are contracted. In fact, the vacuum expectation value of a normal-orderedproduct is always zero.5.1.3 Particle-hole formalism and normal-ordered operatorsAs it appears from Eq. (5.5), the vacuum state plays a central role in the second quantizationframework and it is the starting point to build many-body states. In nuclear physics – and43Chapter 5. Coupled-cluster theoryquantum chemistry – it is convenient to define a new vacuum state∣0⟩ ≡ ∣Φ0⟩ = ∣ijk...⟩, (5.14)and redefine the creation and annihilation operators with respect to this new reference state.The state ∣Φ0⟩ is often referred to as the Fermi vacuum, where the indexes i, j, k, ... refer to theoccupied single-particle orbitals of the system which, in turn, define the Fermi surface. Withthe vacuum definition of Eq. (5.14), orbitals corresponding to i, j, k, ... are called hole states,while the other orbitals, which we will denote by a, b, c, ..., and are above the Fermi surface, arecalled particle states. Excited states are then obtained exciting particles out of the referencestate and promoting them to orbitals which lay above the Fermi surface, i.e.,∣Φab...ij... ⟩ = a„aa„b...ajai∣Φ0⟩. (5.15)A visual example of the reference state and some excited states is shown in Fig. 5.1. In theparticle-hole formalism, i.e., where we redefine the vacuum state as in Eq. (5.14), both a„a and aiare pseudo-creation operators: a„a creates a particle in one of the excited (unoccupied) orbitalsabove the Fermi sea, while ai creates a hole in the reference state (occupied) orbitals. On theother hand, both a„i and aa are pseudo-annihilation operators. The operator a„i destroys (fills)a hole in the reference state, while aa annihilates a particle above the Fermi surface.(a) (b) (c)Fermi seafree orbitalsFigure 5.1: Example of excited states in the particle-hole formalism. The blue circles refer to occupiedorbitals below the Fermi level, white circles represent holes in the Fermi sea and red circles representexcited occupied orbitals above the Fermi level. (a) representation of the unexcited reference state ∣Φ0⟩;(b) one-particle-one-hole excited state obtained with a„aai∣Φ0⟩; (c) two-particle-two-hole excited stateobtained by the application of a„aa„bajai on the reference state ∣Φ0⟩.We can extend the normal product definition of Wick’s theorem to take into account ournew vacuum state ∣Φ0⟩: we say that a product of creation and annihilation operators is normal-ordered with respect to the Fermi vacuum ∣Φ0⟩ when all the pseudo-creation operators (a„a andai) are to the left of all pseudo-annihilation (aa and a„i ) operators. This definition ensures44Chapter 5. Coupled-cluster theoryWick’s theorem has the same form as in Eq. (5.11) and that expectation values on the Fermivacuum ∣Φ0⟩ of normal-ordered products are zero.With the help of Wick’s theorem and the anti-commutation relations in Eq. (5.6), the one-body operator from Eq. (5.7) can be rewritten asF = ∑pq⟨p∣f ∣q⟩a„paq= ∑pq⟨p∣f ∣q⟩N [a„paq] +∑pq⟨p∣f ∣q⟩a„paq= ∑pq⟨p∣f ∣q⟩N [a„paq] +∑i⟨i∣f ∣i⟩= FN +∑ifii,(5.16)where a„paq = δpq and since aa∣Φ0⟩ = 0, only if p and q are hole states the term does not vanishwhen calculating the expectation value on the reference vacuum state. In the last equalitywe denote by FN the normal-ordered part of the operator F , and ⟨i∣f ∣i⟩ = fii. Taking theexpectation value on the reference state one has⟨Φ0∣F ∣Φ0⟩ = ⟨Φ0∣FN ∣Φ0⟩ +∑ifii =∑ifii, (5.17)since by construction the expectation value of normal-ordered operators on the reference stateis zero. For the two-body operator of Eq. (5.8) we can follow a similar procedure, i.e.,V = 14∑pqrs⟨pq∣∣rs⟩N [a„pa„qasar + a„pa„qasar + a„pa„qasar+a„pa„qasar + a„pa„qasar + a„pa„qasar + a„pa„qasar⎤⎥⎥⎥⎥⎦= 14∑pqrs⟨pq∣∣rs⟩N [a„pa„qasar] +∑ipq⟨pi∣∣qi⟩N [a„paq] + 12∑ij ⟨ij∣∣ij⟩= V 2bN + V 1bN + ⟨Φ0∣V ∣Φ0⟩,(5.18)where V 2bN represents the normal-ordered two-body part of V and V1bN is the normal-orderedone-body operator. Similarly, also the expression of the three-body operator W followsW = 136∑pqrstu⟨pqr∣∣stu⟩N [a„pa„qa„rauatas] + 14 ∑ipqrs⟨ipq∣∣irs⟩N [a„pa„qasar]+ 12∑ijpq⟨ijp∣∣ijq⟩N [a„paq] + 16∑ijk⟨ijk∣∣ijk⟩= W 3bN +W 2bN +W 1bN + ⟨Φ0∣W ∣Φ0⟩,(5.19)45Chapter 5. Coupled-cluster theorywhere W 1bN and W2bN denote the normal-ordered one- and two-body parts of the operator W .Given a three-body Hamiltonian H = F + V +W , we then haveH = F + V +W= ⟨Φ0∣(F + V +W )∣Φ0⟩ + (FN + V 1bN +W 1bN )+ (V 2bN +W 2bN ) +W 3bN= H0 +H1bN +H2bN +H3bN ,(5.20)where in the last equality we grouped the normal-ordered zero-, one-, two- and three-bodyterms. The normal-ordered Hamiltonian HN is then given byHN =H1bN +H2bN +H3bN =H −H0. (5.21)With Eq. (5.21) we can rewrite the Schro¨dinger equation for a state ∣Ψf ⟩ asHN ∣Ψf ⟩ = ∆Ef ∣Ψf ⟩, (5.22)where ∆Ef = Ef −H0 = Ef − ⟨Φ0∣H ∣Φ0⟩. The zero-body term H0 represents the uncorrelatedenergy of the reference state ∣Φ0⟩. To obtain the results shown in Chapter 7, we use theharmonic-oscillator single-particle basis and we then perform a Hartree-Fock calculation (seeRefs. [95, 96] for details). Then, we choose as reference state the Slater determinant built onthe Hartree-Fock basis, so that H0 represents the Hartree-Fock energy of the nuclear systemunder study. The residual correlation energy ∆Ef of the state ∣Ψf ⟩ is then obtained by solvingEq. (5.22) with a suitable many-body method, in our case coupled-cluster theory.Three-body forces in the nuclear Hamiltonian are crucial to obtain a correct descriptionof nuclear observables. However, a full and explicit treatment of the three-body terms of theinteraction adds substantial technical difficulties such as more complex equations to solve andlarge amounts of matrix elements which need to be stored. On the other hand, the treatment ofnormal-ordered two-body interactions is standard knowledge nowadays as it becomes computa-tionally as easy as doing calculations with at most two-body forces. Thus, an economic way toinclude three-body effects in nuclear structure computations is to normal order the three-bodypart of the Hamiltonian as in Eq. (5.19) and neglect the pure three-body part W 3bN , while in-cluding the zero-, one- and two-body parts in the lower-rank components of the interaction. Inother words, one uses a nuclear Hamiltonian such as the one in Eq. (5.20), where the residualthree-body interaction H3bN is neglected. In Ref. [97], Hagen et. al. showed that this approxi-mation is well under control and the effect of H3bN is negligible. In Chapter 7, when performingcalculations with three-body forces, we will always neglect the normal-ordered three-body partof the Hamiltonian, but keep three-body forces effects via the inclusion of W 1bN and W2bN .46Chapter 5. Coupled-cluster theory5.1.4 Similarity transformationsMany-body methods often rely on the expansion of the many-body wave function in termsof single-particle basis functions (e.g., the harmonic-oscillator basis). While exact results canbe obtained if a full expansion is implemented, from a practical point of view the basis isusually truncated due to limited computational resources. The single-particle basis truncationintroduces a separation of the full Hilbert space H into two sub-spaces HP and HQ, such thatPH =HP QH =HQ,HP ⊥HQ,P +Q = 1ˆ,P2 = P Q2 = Q,(5.23)where P and Q are projection operators on the HP and HQ sub-spaces, respectively. The HPsubspace represents the space containing the single-particle basis functions used to expand themany-body wave function, while HQ is the subspace of the excluded basis functions.The expectation value on a state ∣Ψ⟩ of an operator Θ (for example the Hamiltonian H), canthen be expressed in terms of these two sub-spaces. We first introduce a general transformationeS so that the expectation value of the normal-ordered operator ΘN becomes⟨Θ⟩ = ⟨Ψ∣ΘN ∣Ψ⟩⟨Ψ∣Ψ⟩ = ⟨Ψ∣eSe−SΘNeSe−S ∣Ψ⟩⟨Ψ∣eSe−S ∣Ψ⟩ = ⟨ΦL∣Θ∣ΦR⟩⟨ΦL∣ΦR⟩ , (5.24)where ⟨ΦL∣ = ⟨Ψ∣eS , ∣ΦR⟩ = e−S ∣Ψ⟩ and Θ = e−SΘNeS is the similarity transformed operator.Note that we distinguish between left and right states since in general ⟨ΦL∣ ≠ ∣ΦR⟩„6. Makinguse of the projection operators P and Q we then obtain⟨Θ⟩ = ⟨ΦL∣(P +Q)Θ(P +Q)∣ΦR⟩⟨ΦL∣(P +Q)∣ΦR⟩= ⟨ΦL∣(PΘP +PΘQ +QΘP +QΘQ)∣ΦR⟩⟨ΦL∣(P +Q)∣ΦR⟩ .(5.25)We can now choose the similarity transformation operator S so that ∣ΦR⟩ ∈HP , or equivalentlyQ∣ΦR⟩ = 0. With this choice, Eq. (5.25) becomes⟨Θ⟩ = ⟨ΦL∣(PΘP +QΘP)∣ΦR⟩⟨ΦL∣P ∣ΦR⟩ . (5.26)6If the operator S is anti-hermitian, i.e., S = −S„, then one has ⟨ΦL∣ = ∣ΦR⟩„47Chapter 5. Coupled-cluster theoryIf we further assume the similarity transformation is such that the operator Θ does not connectthe HP and HQ sub-spaces, namely QΘP = 0, then one obtains the set of operator equationsQΘP = 0,⟨Θ⟩P = PΘP. (5.27)The first of the above equations is also known as decoupling condition.While in this specific example we partitioned the Hilbert space based on the single-particlestates included (HP ) or excluded (HQ) from the many-body wave function expansion, in generalone can use different prescriptions to separate H in sub-spaces, e.g., P could be the projectoronto the ground-state of the system and HQ the subspace of the excited states. As we will see,similarity transformations play an important role in CC theory and the above formalism willcome handy throughout all the next sections.5.2 The exponential ansatzGiven a normal-ordered, non-relativistic Hamiltonian, the Schro¨dinger equation for the groundstate ∣Ψ0⟩ of an A-body system follows from Eq. (5.21), henceHN ∣Ψ0⟩ = ∆E0∣Ψ0⟩. (5.28)Coupled-cluster theory expands the many-body, fully correlated ground state wave function∣Ψ0⟩ using an exponential ansatz ∣Ψ0⟩ = eT ∣Φ0⟩, (5.29)where ∣Φ0⟩ is a reference Slater determinant as defined in Eq. (5.14) and T , also known ascluster operator, is an excitation operator defined asT = T1 + T2 + ... + TA = A∑n=1Tn, (5.30)whereTn = 1(n!)2 ∑ij...ab... tab...ij...N [a„aa„b ⋅ ⋅ ⋅ ajai] . (5.31)The tab...ij... are called coupled-cluster amplitudes, and are unknown numerical coefficients whichmust be determined. The Tn are excitation operators since they excite particles and holes ontop of the reference determinant and so we say Tn is an n-particle−n-hole (np−nh) excitationoperator. Equation (5.28) can be rewritten ase−THNeT ∣Φ0⟩ = ∆E0∣Φ0⟩. (5.32)48Chapter 5. Coupled-cluster theoryIntroducing the similarity transformed HamiltonianH = e−THNeT , (5.33)the correlation energy ∆E0 is simply found by solvingH ∣Φ0⟩ = ∆E0∣Φ0⟩. (5.34)Equations (5.33) and (5.34) depend on the amplitudes of the Tn cluster operators. Theseamplitudes can be found by solving the so-called coupled-cluster equations which are obtainedby projecting Eq. (5.34) on the excited states of the similarity transformed Hamiltonian, i.e.,⟨Φab...ij... ∣H ∣Φ0⟩ = ∆E0⟨Φab...ij... ∣Φ0⟩ = 0 (5.35)Equation (5.35) denotes the so called coupled-cluster equations. They are equivalent to thedecoupling condition in Eq. (5.27). In fact, we can divide the many-body Hilbert space intotwo partitions such that ∣Φ0⟩ ∈ P and the excited states ∣Φab...ij... ⟩ ∈ Q. Then, using the propertiesof P and Q in Eq. (5.23) and Eq. (5.35) we have0 = ⟨Φab...ij... ∣ (P +Q)H (P +Q) ∣Φ0⟩ = ⟨Φab...ij... ∣QHP ∣Φ0⟩, (5.36)which resembles the first line of Eq. (5.27). Similarly, rewriting Eq. (5.34) as∆E0P ∣Φ0⟩ = PHP ∣Φ0⟩, (5.37)we notice that it is equivalent to the second line of Eq. (5.27).Notice that the choice of the exponential ansatz, unlike other choices such as, for example,the configuration interaction linear ansatz [98], guarantees the size extensivity of the solu-tion [99–101], i.e., the correlation energy scales, in the limit of a large system, linearly with thesystem size itself. This is an important property of CC as the accuracy of the truncation of thecluster operator T does not deteriorate with increasing number of particles [102] (see Ref. [103]for an excellent proof of this).Next, we focus on the form of the similarity transformed Hamiltonian of Eq. (5.33) and showthat only connected terms (the meaning of which will be clear later in this section) are relevantfor H. We start by making use of the Baker-Campbell-Hausdorff formula [104] to expand the49Chapter 5. Coupled-cluster theoryexponentials of HH =e−THNeT = (1 − T + T 22!− T 33!+ ...)HN (1 + T + T 22!+ T 33!+ ...)=HN + [HN , T ] + 12![[HN , T ] , T ] + 13![[[HN , T ] , T ] , T ] + ... . (5.38)The similarity transformed Hamiltonian can thus be expressed as an infinite sum of commuta-tors between the normal-ordered HN and the cluster operator T . We can use the generalizedWick’s theorem [94] to evaluate the commutators, i.e.,[HN , T ] =HNT − THN = N [HNT ] +HNT −N [THN ] − THN . (5.39)Since bothHN and T contain an even number of annihilation and creation operators, N [HNT ] =N [THN ], the commutator becomes[HN , T ] =HNT − THN =HNT − THN . (5.40)The contractions in the commutator of Eq. (5.40) are intended to be full, i.e., all the annihilationand creation operators in HN and T must be contracted since non-fully contracted termswould give zero contributions to expectation values. Furthermore, the T operators contain onlyparticle creation operators a„a, a„b, ... and hole annihilation operators ai, aj , ..., meaning there’sno non-zero contraction between T operators, i.e., [Tm, Tn] = 0. Using Eq. (5.10) and thedefinition of particle and hole creation and annihilation operators, together with the definitionof ∣Φ0⟩, we havea„aab∣Φ0⟩ = (−N [a„aab] + a„aab) ∣Φ0⟩ = 0, (5.41)ab a„a∣Φ0⟩ = (−N [aba„a] + aba„a) ∣Φ0⟩ = δab∣Φ0⟩, (5.42)a„iaj ∣Φ0⟩ = (−N [a„iaj] + a„iaj) ∣Φ0⟩ = δij ∣Φ0⟩, (5.43)aj a„i ∣Φ0⟩ = (−N [aja„i] + aja„i) ∣Φ0⟩ = 0. (5.44)The above equations show that only contractions in which particle creation (Eq. (5.42)) andhole annihilation (Eq. (5.43)) operators are on the right give non-zero contributions. Thisenforces that only contractions between HN and T , where the T operators are on the rightof the Hamiltonian survive when contracted. The above observations show us the infinitesum in Eq. (5.38) is actually truncated, where the truncation order is given by the form ofthe Hamiltonian. If the Hamiltonian is a two-body operator, we can connect at most four Toperators to HN , while if HN has a three-body part, we will be able to connect at most six T50Chapter 5. Coupled-cluster theoryoperators. For a two-body Hamiltonian, Eq. (5.38) can be simplified asH =HN +HNT + 12!HNTT + 13!HNTTT + 14!HNTTTT = (HNeT )C , (5.45)where we have introduced a new notation to remind the reader that only connected terms,i.e., terms for which each T operator is contracted with HN (also denoted by the multiplecontraction lines in the first equality), are included in the similarity transformed Hamiltonian.5.3 The CC energy functional and the Λ–equationsThe CC equations of Eq. (5.36) for the tab...ij... amplitudes can be obtained also in a variationalframework by defining the energy functional [91]ε(Λ, T ) = P(1 +ΛQ)HP, (5.46)where P and Q satisfy Eq. (5.23) and project on the reference and excited states, respectively,and we introduced a new de-excitation operator Λ. The dependence on T of the energy func-tional is implicit as the similarity transformed Hamiltonian, defined in Eq. (5.33), depends onit. The Λ de-excitation operator can be defined in a similar way as for the T operator, i.e.,Λ = Λ1 +Λ2 +Λ3 + ... +ΛA = A∑n=1 Λn, (5.47)withΛn = 1(n!)2 ∑ij...ab...λij...ab...N [a„ia„j ...abaa] , (5.48)where once again λij...ab... are unknown coefficients to be determined. The variation of the func-tional of Eq. (5.46) is given byδε(Λ, T ) = PδΛQHP +P(1 +ΛQ) [H,δT ]P. (5.49)From the first term of Eq. (5.49), imposing stationary conditions on the functional by requiringδε = 0, gives QHP = 0, (5.50)which is the decoupling condition of Eq. (5.27) and thus equivalent to the CC amplitudesequations in Eq. (5.36). Making use of the decoupling condition and ∆E0 = PHP, the secondterm of Eq. (5.49) can be rewritten asP(1 +ΛQ) [H,δT ]P = P(H +ΛQH −∆E0Λ)QδTP, (5.51)51Chapter 5. Coupled-cluster theorywhere to obtain this result we have expanded the commutator and used the resolution of theidentity P +Q = 1ˆ. Requiring stationarity of Eq. (5.51) givesP(H +ΛQH −∆E0Λ)Q = 0. (5.52)While the decoupling condition can be used to solve for the unknown tab...ij... amplitudes, Eq. (5.52)can be used to solve for the unknown de-excitation operator amplitudes λij...ab.... Because Λ andT are de-excitation and excitation operators, respectively, they must satisfyΛP = 0, ΛQ = Λ, PT = 0, QT = T, (5.53)and so we can writePΛQHQ = P [Λ,H]Q +PHΛQ= P (ΛH)CQ +∆E0PΛQ +PHQΛQ, (5.54)where the commutator, as shown in Eqs. (5.39) and (5.40), gives only fully connected terms.With the above result, Eq. (5.52) then becomesP(H + (ΛH)C+PHQΛ)Q = 0, (5.55)which is the final equations we need to solve in order to obtain the Λ amplitudes. Notice thatthe first two terms of Eq. (5.55) are connected, while the last one also involves disconnectedterms. Using the reference and excited states, the above equation can be rewritten as0 = ⟨Φ0∣HNeT ∣Φnpnh⟩C + ⟨Φ0∣Λ (HNeT )C ∣Φnpnh⟩C+ ∑mh,mp⟨Φ0∣HNeT ∣Φmpmh⟩C⟨Φmpmh∣Λ∣Φnpnh⟩, (5.56)where ∣Φnpnh⟩ denotes an np−nh excited state. If ∣Φnpnh⟩ is a singly excited state ∣Φai ⟩, then thelast term in Eq. (5.56) vanishes since ⟨Φmpmh∣Λ∣Φai ⟩ = 0 ∀m = 1, ...,A, and we have⟨Φ0∣HNeT ∣Φai ⟩C + ⟨Φ0∣Λ (HNeT )C ∣Φai ⟩C = 0, (5.57)which are also known as Λ1 equations. When projecting onto doubly excited states we thenobtain the Λ2 equations0 = ⟨Φ0∣HNeT ∣Φabij ⟩C + ⟨Φ0∣Λ (HNeT )C ∣Φabij ⟩C+ ⟨Φ0∣HNeT ∣Φai ⟩C⟨Φai ∣Λ∣Φabij ⟩ + ⟨Φ0∣HNeT ∣Φaj ⟩C⟨Φaj ∣Λ∣Φabij ⟩+ ⟨Φ0∣HNeT ∣Φbi⟩C⟨Φbi ∣Λ∣Φabij ⟩ + ⟨Φ0∣HNeT ∣Φbj⟩C⟨Φbj ∣Λ∣Φabij ⟩.(5.58)Projecting onto triple excited states one would obtain equivalent equations for Λ3 and so onfor higher excitations.52Chapter 5. Coupled-cluster theoryThe de-excitation operator Λ, as we will see in later sections, is fundamental for the cal-culation of expectation values of operators, which require the knowledge of the left- and right-ground states. In fact, the coupled-cluster Hamiltonian H is non-Hermitian and the left- andright- ground states have to be found separately by solving the CC equations and the Λ equa-tions, respectively.5.4 Coupled-cluster equation of motionThe CC exponential ansatz in Eq. (5.28) defines the structure of the ground state. In orderto study the full spectrum of the Hamiltonian HN we need to be able to solve the Schro¨dingerequation for an arbitrary excited state ∣Ψf ⟩, i.e.,HN ∣Ψf ⟩ = ∆Ef ∣Ψf ⟩. (5.59)A reasonable ansatz for the excited state is to assume we can obtain it by acting on the groundstate ∣Ψ0⟩ with an excitation operator Rf ,∣Ψf ⟩ =Rf ∣Ψ0⟩ =RfeT ∣Φ0⟩, (5.60)where we used the exponential ansatz of Eq. (5.28) and we define Rf asRf = r0 +∑iaraiN [a„aai] + 14 ∑ijab rabijN [a„aa„bajai] + ... . (5.61)Using the ansatz of Eq. (5.60), the Schro¨dinger equation for the excited states becomesHRf ∣Φ0⟩ = ∆EfRf ∣Φ0⟩, (5.62)where we used [Rf , exp(T )] = 0 since excitation operators commute (as discussed beforeEq. (5.40) in Section 5.2). Equation (5.62) shows that Rf ∣Φ0⟩ is an eigenstate of the simi-larity transformed Hamiltonian H. However, since H is non-Hermitian, we also have a lefteigenvalue equation ⟨Φ0∣LfH = ⟨Φ0∣Lf∆Ef , (5.63)where now Lf is a de-excitation operator defined asLf = l0 +∑ialiaN [a„iaa] + 14 ∑ijab lijabN [a„ia„jabaa] + ... (5.64)Left and right eigenfunctions correspond to the same eigenvalues, are bi-orthogonal and can benormalized such that ⟨Φ0∣LfRl∣Φ0⟩ = δfl. (5.65)53Chapter 5. Coupled-cluster theoryTherefore, the completeness relation is rewritten as1ˆ =∑fRf ∣Φ0⟩⟨Φ0∣Lf . (5.66)By taking f = 0 in Eqs. (5.62) and (5.63) we find the left and right ground state Schro¨dingerequations for the similarity transformed HamiltonianHR0∣Φ0⟩ = ∆E0R0∣Φ0⟩,⟨Φ0∣L0H = ⟨Φ0∣L0∆E0, (5.67)which, when compared with the energy functional of Eq. (5.46) give R0 = 1 and L0 = 1+Λ. Wecan introduce the new notation⟨ΦL0 ∣ = ⟨Φ0∣(1 +Λ) and ∣ΦR0 ⟩ = ∣Φ0⟩, (5.68)and rewrite Eq. (5.67) asH ∣ΦR0 ⟩ = ∆E0∣ΦR0 ⟩,⟨ΦL0 ∣H = ⟨ΦL0 ∣∆E0. (5.69)Like the T and Λ operators, also the Lf and Rf operators have unknown amplitudes rab...ij... andlij...ab.... The spectrum of the Hamiltonian can be obtained by solving either the right- or left-eigenstate problem independently. If we focus on the right-eigenstate problem, by multiplyingEq. (5.34) on the left by Rf and subtracting it from Eq. (5.62) we haveHRf ∣Φ0⟩ −RfH ∣Φ0⟩ = ∆EfRf ∣Φ0⟩ −∆E0Rf ∣Φ0⟩, (5.70)and calling ωf = ∆Ef −∆E0 we then obtain[H,Rf ] ∣Φ0⟩ = ωfRf ∣Φ0⟩. (5.71)The above equation can then be used to find the amplitudes of the Rf operator. The commuta-tor can be replaced by a connected product as we did for the similarity transformed Hamiltonianin Eq. (5.38), thus obtaining the so-called coupled-cluster equations of motion (EOM)HRf ∣Φ0⟩C = ωfRf ∣Φ0⟩. (5.72)Using the projection operators P and Q we can rewrite Eq. (5.72) as(HRfP)C = ωfRfP, (5.73)54Chapter 5. Coupled-cluster theoryand inserting P + Q = 1ˆ on the left-hand side in between H and Rf , and before Rf on theright-hand side, we obtain(HPRfP)C + (HQRfP)C = ωfPRfP + ωfQRfP. (5.74)Projecting the above equation first on the Q space and then on the P space we find(QHPRfP)C + (QHQRfP)C = ωfQRfP,(PHPRfP)C + (PHQRfP)C = ωfPRfP. (5.75)For the ground state, i.e., when f = 0, by definition we have ω0 = 0 and R0 = 1. In this case theabove equations give the following decoupling conditions(QHP)C= 0,(PHP)C= 0. (5.76)By making use of the decoupling conditions, Eq. (5.75) simplifies to(QHQRfP)C = ωfQRfP,(PHQRfP)C = ωfr0P, (5.77)where we used PRfP = r0P. Equation (5.77) represents the set of coupled equations whichhave to be solved to find the amplitudes rab...ij... and the excited states energies ∆Ef = ωf +∆E0.5.5 The integral transforms coupled-cluster methodIn the past decades, the LIT method has been successfully applied in conjunction with differentfew-body ab initio approaches for a variety of break-up reactions, from photon scattering toelectron scattering (see Refs. [21, 77] for a review and references therein). However, calculationshave always been limited to light nuclei not exceeding mass number A ≈ 7, the reason being theprohibitive complexity required to extend the computations to heavier systems. Only recently,the LIT method has been successfully extended to medium-mass nuclei [1, 105], an achieve-ment made possible by providing a coupled-cluster theory formulation of the Lorentz integraltransform. In the following sections we introduce the Lorentz integral transform formulation ofcoupled-cluster theory (LIT-CC), which will be used later on to extend the calculations of thedipole response function and the electric dipole polarizability to medium-mass nuclei.5.5.1 The similarity transformed LITRecall the form of the dipole response function from Eq. (3.55),R(ω) = ⨋f∣⟨Ψf ∣Θ∣Ψ0⟩∣2 δ(Ef −E0 − ω), (5.78)55Chapter 5. Coupled-cluster theorywhich measures the total electric dipole transition strength between the nuclear ground state∣Ψ0⟩ and all the final states ∣Ψf ⟩. Introducing the reference Slater determinant ∣Φ0⟩ and normalordering Θ with respect to ∣Φ0⟩, the response function can be rewritten asR(ω) = ⨋f∣⟨Ψf ∣ΘN ∣Ψ0⟩∣2 δ(∆Ef −∆E0 − ω)= ⨋f⟨Ψ0∣Θ„N ∣Ψf ⟩⟨Ψf ∣ΘN ∣Ψ0⟩δ(∆Ef −∆E0 − ω), (5.79)where we used Θ = ΘN − ⟨Φ0∣Θ∣Φ0⟩, E0 = Eref + ∆E0, Ef = Eref + ∆Ef and ⟨Ψf ∣Ψ0⟩ = 0.Making use of the exponential ansatz from Eq. (5.29) for the ground state and Eq. (5.60) withRf ∣ΦR0 ⟩ = ∣ΦRf ⟩ and ⟨ΦL0 ∣Lf = ⟨ΦLf ∣, we obtainR(ω) = ⨋f⟨ΦL0 ∣e−TΘ„NeT ∣ΦRf ⟩⟨ΦLf ∣e−TΘNeT ∣ΦR0 ⟩δ(∆Ef −∆E0 − ω)= ⨋f⟨ΦL0 ∣Θ„∣ΦRf ⟩⟨ΦLf ∣Θ∣ΦR0 ⟩δ(∆Ef −∆E0 − ω), (5.80)where we introduced the similarity transformed operator Θ = e−TΘNeT . Following the sameprocedure we used in Eq. (4.10) we can now calculate the Lorentz integral transform IL(σ)which becomesIL(σ) =Γpi⟨ΦL0 ∣Θ„ (H −∆E0 − σ + iΓ)−1 (H −∆E0 − σ − iΓ)−1 Θ∣ΦR0 ⟩, (5.81)where to obtain the above equation we followed the steps used in Eq. (4.10). Note that, becausewe used Eq. (5.80), the operators HN and ΘN have now been replaced by their similaritytransformed counterparts and the denominator depends on the correlation energy ∆E0 = E0 −Eref , rather than the ground-state energy E0. By introducing zL = σ + iΓ we then have in amore compact formIL(σ) = Γpi⟨ΦL0 ∣Θ„ (H −∆E0 − z∗L)−1 (H −∆E0 − zL)−1 Θ∣ΦR0 ⟩. (5.82)5.5.2 The LIT-CC equationsThe product of the denominators in Eq. (5.82) can be rewritten as1(H −∆E0 − z∗L) (H −∆E0 − zL) = − i2Γ [ 1(H −∆E0 − zL) − 1(H −∆E0 − z∗L)] . (5.83)Similarly to Eq. (4.11), if we introduce∣Ψ˜R(z∗L)⟩ = (H −∆E0 − z∗L)−1 Θ∣ΦR0 ⟩,∣Ψ˜R(zL)⟩ = (H −∆E0 − zL)−1 Θ∣ΦR0 ⟩, (5.84)56Chapter 5. Coupled-cluster theoryand we make use of Eq. (5.83), we obtainIL(σ) = − i2pi[⟨ΦL0 ∣Θ„∣Ψ˜R(zL)⟩ − ⟨ΦL0 ∣Θ„∣Ψ˜R(z∗L)⟩] . (5.85)The states defined in Eq. (5.84) are auxiliary states, i.e., they are not eigenstates of the Hamil-tonian and can be viewed as mathematical tools. However, we can assume these states are builtby applying some correlation operator to the reference state. Here we make the ansatz∣Ψ˜R(z∗L)⟩ =R(z∗L)∣ΦR0 ⟩ and ∣Ψ˜R(zL)⟩ =R(zL)∣ΦR0 ⟩, (5.86)which is similar to the coupled-cluster equations of motion ansatz of Eq. (5.60) for the excitedstates of the similarity transformed Hamiltonian. The two excitation operators R(z∗L) andR(zL) are expanded as in Eq. (5.61) and depend on some unknown amplitudes which can bedetermined by solving(H −∆E0 − z∗L)R(z∗L)∣ΦR0 ⟩ = Θ∣ΦR0 ⟩,(H −∆E0 − zL)R(zL)∣ΦR0 ⟩ = Θ∣ΦR0 ⟩. (5.87)We can use Eq. (5.34) to eliminate the ∆E0 dependence of the above equation, thus obtaining(HR(z∗L))C ∣ΦR0 ⟩ = z∗LR(z∗L)∣ΦR0 ⟩ +Θ∣ΦR0 ⟩,(HR(zL))C ∣ΦR0 ⟩ = zLR(zL)∣ΦR0 ⟩ +Θ∣ΦR0 ⟩. (5.88)From now on we will refer to Eq. (5.88) as the LIT-CC equations. For later purposes it is usefulto rewrite Eq. (5.85) in matrix notation. To do so, we note that the operator R(zL) can berewritten asR(zL) =r0(zL) +∑iaraiN [a„aai] + 14 ∑ijab rabijN [a„aa„bajai] + ...= ∑αAαrα(zL) = A ⋅ r(zL), (5.89)where the index α labels np-nh excitations, r(zL) is a column vector with elements rα(zL) =r0(zL), rai (zL), rabij (zL), ... and A is a row vector whose elements are strings of normal-orderedcreation and annihilation operators. With Eq. (5.89) and by projecting the second of Eqs. (5.86)on a ∣β⟩ ≡ ∣Φβpβh⟩ excited state, we have0 = ⟨β∣ (HR(zL))C ∣ΦR0 ⟩ − zL⟨β∣R(zL)∣ΦR0 ⟩ − ⟨β∣Θ∣ΦR0 ⟩= ∑α[⟨β∣ (HAα)C ∣ΦR0 ⟩ − zL⟨β∣Aα∣ΦR0 ⟩] rα(zL) − ⟨β∣Θ∣ΦR0 ⟩= ∑αMβα(zL)rα(zL) − SRβ .(5.90)57Chapter 5. Coupled-cluster theoryIn the above expression we made use of ⟨β∣Aα∣ΦR0 ⟩ = ⟨β∣α⟩ = δβα, and we introduced the matrixMβα(zL) = ⟨β∣ (HAα)C ∣ΦR0 ⟩ − zLδβα, (5.91)and the vectorSRβ = ⟨β∣Θ∣ΦR0 ⟩. (5.92)From Eq. (5.90) we see that the amplitudes vector r(zL) can be easily found by solvingr(zL) = [M(zL)]−1 SR. (5.93)Equivalent expressions to Eqs. (5.89), (5.91) and (5.93) hold for R(z∗L).5.5.3 Non-symmetric Lanczos algorithmWith the results of the previous sections, the LIT in Eq. (5.85) can be rewritten asIL(σ) = − i2pi⟨ΦL0 ∣Θ„ [R(zL) −R(z∗L)] ∣ΦR0 ⟩. (5.94)Making use of Eqs. (5.89) and (5.93), the LIT becomesIL(σ) = − i2pi⟨ΦL0 ∣Θ„∑α[rα(zL) − rα(z∗L)]Aα∣ΦR0 ⟩= − i2piSLα∑αβ{[Mαβ(zL)]−1 − [Mαβ(z∗L)]−1}SRβ= − i2piSL {[M(zL)]−1 − [M(z∗L)]−1}SR,(5.95)whereSLα = ⟨ΦL0 ∣Θ„∣α⟩. (5.96)To numerically evaluate the LIT of Eq. (5.95) we rely again on the Lanczos algorithm. InCC theory, however, the matrix we would have to tri-diagonalize is non Hermitian and so wecannot use the Lanczos algorithm described in Section 4.3, which works only for symmetricmatrices. There exist generalized variants of the standard Lanczos algorithm, e.g., the Arnoldimethod or the non-symmetric Lanczos algorithm [106], which we will use here. The algorithmis similar to the one in Section 4.3. Given an n × n matrix A, one defines two vectors w0 andv0 such that w0v0 = 1 and set w−1 = v−1 = 0, and b0 = 0. Then, for each i = 1, ...,m ≤ n, thematrix elements of the tridiagonal representation Atr of A can be obtained viaai =wiAvi, (5.97)pi = Avi − aivi − bivi−1, (5.98)si =wiA − aiwi − biwi−1, (5.99)58Chapter 5. Coupled-cluster theorybi+1 = √sipi, (5.100)vi+1 = pibi+1 , wi+1 = sibi+1 . (5.101)Note that the algorithm described above already includes a Gram-Schmidt orthogonalizationstep to prevent numerical instabilities due to loss of orthogonality in the Lanczos vectors wiand vi. To apply the Lanczos algorithm to Eq. (5.95) we choose the two initial pivots to bev0 = SR√SLSR,w0 = SL√SLSR.(5.102)Following a procedure similar to Subsection 4.3, using the definition of the identity we have1ˆ = [M(zL)]tr [M(zL)]−1tr → ∑k{[M(zL)]tr}ik {[M(zL)]−1tr }kj = δij , (5.103)1ˆ = [M(z∗L)]tr [M(z∗L)]−1tr → ∑k{[M(z∗L)]tr}ik {[M(z∗L)]−1tr }kj = δij . (5.104)By setting j = 0 and defining Xk(zL) = {[M(zL)]−1tr }k0 and Xk(z∗L) = {[M(z∗L)]−1tr }k0 the aboveequations become∑k{[M(zL)]tr}ikXk(zL) = δi0, (5.105)∑k{[M(z∗L)]tr}ikXk(z∗L) = δi0. (5.106)Equations (5.105) and (5.106) are systems of equations equivalent to Eq. (4.29) and by takingk = 0 we can use Cramer’s rule from Eq. (4.32) to solve for X0(zL) and X0(z∗L), getting thecontinued fractionsX0(zL) = 1(a0 − zL) − b21(a1−zL)− b21(a2−zL)− b23(a3−zL)−..., (5.107)X0(z∗L) = 1(a0 − z∗L) − b21(a1−z∗L)− b21(a2−z∗L)− b23(a3−z∗L)−.... (5.108)To obtain the two continued fractions shown above, we need the coefficients ai and bi which, bylooking at Eqs. (5.105) and (5.106), we see are the elements of the tridiagonal representationof the matrices M(zL) and M(z∗L) obtained using the Lanczos pivots of Eq. (5.102). Once the59Chapter 5. Coupled-cluster theorycontinued fractions have been calculated, the LIT can be easily evaluated viaIL(σ) = − i2pi(SL ⋅ SR) [X0(zL) −X0(z∗L)] . (5.109)The calculation of the LIT using continued fractions is extremely efficient: to obtain theLIT at different values of σ and Γ, one needs to tridiagonalize the M matrices only once. Then,once the Lanczos coefficients have been stored, it is trivial and fast to vary σ and Γ. On theother hand, if we were to get the LIT using Eq. (5.94), we would have to solve the LIT-CCequations as many times as all the possible combinations of σ and Γ. That is why Eq. (5.94)has been used only as a check of the method, but never for production calculations.5.5.4 The similarity transformed Stieltjes for the polarizabilityIn Eq. (4.18) we have shown the electric dipole polarizability can be calculated via an integraltransform with a Stieltjes kernel. Due to the similarity of the Stieltjes transform with the LIT,we can follow a procedure similar to the one in the previous sections to rewrite the Stieltjestransform as IS(σ) = ⟨ΦL0 ∣Θ„R(σ)∣ΦR0 ⟩. (5.110)Using CC theory, the electric dipole polarizability in Eq. (4.21) can be calculated fromαD = 2α limσ→0+ IS(σ) = 2α limσ→0+⟨ΦL0 ∣Θ„R(σ)∣ΦR0 ⟩. (5.111)Making use of the matrix notation for the excitation operator R(σ) we haveIS(σ) = SL [M(σ)]−1 SR, (5.112)and by defining the initial Lanczos pivots as in Eq. (5.102), we obtainIS(σ) = (SL ⋅ SR)X0(σ). (5.113)The polarizability can then be calculated asαD = 2α limσ→0+ (SL ⋅ SR) 1(a0 − σ) − b21(a1−σ)− b21(a2−σ)− b23(a3−σ)−..., (5.114)where, again, the coefficients ai and bi belong to the tridiagonal representation of M(σ) andare obtained by repeated application of the non-symmetric Lanczos steps.60Chapter 5. Coupled-cluster theory5.6 Diagrammatic notationIn previous sections we often talked about connected or fully contracted terms. Here we intro-duce diagrammatic representations for the states and operators of CC theory, where the choiceof the term “connected” will finally become clear. Diagrammatic techniques turn out to bevery useful when solving CC equations as they lead to more clear and compact results. A fulland complete treatment of CC diagrammatic techniques can be found in Ref. [91], while herewe just give a practical overview on the subject. We represent states by∣Φai ⟩ = a i and ⟨Φai ∣ = a i , (5.115)where upward lines denote particle states and downward lines hole states. Operators can alsobe represented graphically. Using a dashed line to denote the action of the operator, a one-bodyoperator takes the formFN =∑ab ba + ∑ij ij + ∑aia i + ∑aia i , (5.116)which corresponds to the normal-ordered part of Eq. (5.16) where the sum is performed explic-itly, i.e.,FN =∑ab⟨a∣f ∣b⟩N [a„aab] +∑ij⟨i∣f ∣j⟩N [a„iaj]+∑ai⟨a∣f ∣i⟩N [a„aai] +∑ai⟨i∣f ∣a⟩N [a„iaa] . (5.117)A normal-ordered two-body operator would then be represented byVN = 14∑abcda bc d+ 14∑ijklk li j+ ∑aibjab ij+ 12∑abciaci b + 12∑ijkajik a + 12∑abciab i c+ 12∑ijkaki j a+ 14∑abiji aj b+ 14∑ijabi aj b,(5.118)61Chapter 5. Coupled-cluster theoryThe diagrams in the above equation correspond to the algebraic expressionVN = 14∑abcd⟨ab∣∣cd⟩N [a„aa„badac] + 14 ∑ijkl⟨ij∣∣kl⟩N [a„ia„jakal]+ ∑ijab⟨ai∣∣bj⟩N [a„aa„iajab] + 12 ∑abci⟨ab∣∣ci⟩N [a„aa„baiac]+ 12∑ijka⟨ia∣∣jk⟩N [a„ia„aakaj] + 12 ∑aibc⟨ai∣∣bc⟩N [a„aa„iacab]+ 12∑ijka⟨ij∣∣ka⟩N [a„ia„jaaak] + 14 ∑abij⟨ab∣∣ij⟩N [a„aa„bajai]+ 14∑ijab⟨ij∣∣ab⟩N [a„ia„jabaa] .(5.119)The specific numerical factors in the above equation arise when, from Eq. (5.18), we explicitlylabel the p, q, r, s indexes with particle a, b and hole i, j labels. Comparing the algebraic expres-sions of the one- and two-body operators – Eqs. (5.117) and (5.119) – with their diagrammaticcounterparts – Eqs. (5.116) and (5.118) – it is clear that each outward and inward line withrespect to the operator (dashed line) is equivalent to creation and annihilation operators, re-spectively. Contractions between creation and annihilation operators can then be representedby connected diagrams, e.g.,∑il⟨a∣f ∣i⟩⟨l∣f ∣j⟩N [a„aaia„l aj] = ailj. (5.120)To simplify the diagrammatic notation even more, one introduces the convention that anyunlabelled – without an explicit letter label – line is summed over all possible hole or par-ticle indexes7. The numerical factors will be neglected in this new notation and they willbe reintroduced when evaluating full diagrams by following the rules of Appendix B.1 whichare summarized from Ref. [91]. Using the simplified notation we can represent the one- and7To further simplify the notation, we use arrows sparingly, only where it is necessary. The character (parti-cle/hole) of the lines without arrows can be inferred from the lines with arrows or can be chosen as one likes aslong as lines connected to the same vertex appear in particle/hole pairs. For example, the diagramcan be interpreted as or as .62Chapter 5. Coupled-cluster theorytwo-body operators asFN = + + + , (5.121)VN = + + + ++ + + + .(5.122)The cluster operators introduced in Eqs. (5.30) and (5.31) also have a diagrammatic represen-tation, for example, T1, T2 and T3 are denoted byT1 = , T2 = , T3 = , (5.123)while the de-excitation operators Λ1, Λ2 and Λ3 are represented byΛ1 = , Λ2 = , Λ3 = . (5.124)To each diagram we associate an excitation level by counting the number of top (+) andbottom (−) pairs of outgoing and ingoing open lines. For example, the T1 operator has excitationlevel +1 while the T2 has excitation level +2 and T3 has excitation level +3. The Λ operatorsin Eq. (5.124) have excitation levels −1,−2 and −3, respectively. The normal-ordered one-bodyoperator terms in Eq. (5.121) in order from left to right have excitation levels 0,0,+1 and −1respectively, while the terms of the normal-ordered two-body operator VN of Eq. (5.122) haveexcitation levels 0,0,0,+1,+1,−1,−1,+2 and −2.5.7 Coupled-cluster singles and doubles approximationFor a nuclear system of mass number A, the cluster operator T introduced in Eq. (5.30) is asum of A terms. However, from a computational point of view, a full expansion calculation isnot feasible and one usually has to truncate the expansion of the cluster operator. The sametruncation is usually adopted to Λ when solving the Λ–equations and it is, for consistency,extended to the excitation operators Rk and Lk as well if solving for the excitation spectrum.The most common truncation is the coupled-cluster singles and doubles (CCSD) approximation63Chapter 5. Coupled-cluster theorywhere one takesT = T1 + T2, (5.125)Λ = Λ1 +Λ2, (5.126)Rk =R(0)k +R(1)k +R(2)k , (5.127)Lk = L(0)k +L(1)k +L(2)k . (5.128)In this approximation, at the ground-state level we have to solve one equation for the correlationenergy ∆E0 and two equations for the unknown amplitudes tai and tabij , i.e.,∆E0 =⟨Φ0∣HNeT1+T2 ∣Φ0⟩C ,0 =⟨Φai ∣HNeT1+T2 ∣Φ0⟩C ,0 =⟨Φabij ∣HNeT1+T2 ∣Φ0⟩C ,(5.129)together with the Λ–equations (Eqs. (5.57), (5.58)) to determine λia and λijab.This is the first step to be performed before one proceeds with the calculation of the Lorentzand Stieltjes transforms.5.7.1 The m0 sum ruleBoth IL(σ) and IS(σ) from Eqs. (5.109) and (5.113) depend on a common factor SL ⋅SR. Usingthe similarity transformed response function of Eq. (5.80), it is easy to show that the productSL ⋅ SR is the strength of the dipole response function, also defined as the m0 sum rule,m0 = ∫ R(ω)dω = ⟨ΦL0 ∣Θ„Θ∣ΦR0 ⟩ = SL ⋅ SR. (5.130)Recalling the definition of ⟨ΦL0 ∣ and ∣ΦR0 ⟩ from Eq. (5.68), in the CCSD approximation we canwrite the m0 explicitly asm0 = ⟨Φ0∣(1 +Λ1 +Λ2) (Θ„NeT1+T2)C (ΘNeT1+T2)C ∣Φ0⟩. (5.131)In practice, we can separately evaluate SL and SR and then perform a simple vector productto obtain m0. To simplify the calculation we use diagrammatic techniques withSL = + and SR = + , (5.132)where SL and SR have been expanded only up to 1p-1h and 2p-2h excitations to be consistentwith the CCSD truncation. The bare dipole operator Θ is a hermitian one-body operator, i.e.,64Chapter 5. Coupled-cluster theoryΘ„ = Θ, and its normal-ordered form is represented asΘN = + + + . (5.133)The CCSD similarity transformed dipole operator is given byΘ = (ΘNeT1+T2)C = [ΘN (1 + T1 + T2 + 12T 21 + T1T2)]C , (5.134)where higher powers in T give zero contributions since the dipole operator, being a one-bodyoperator, can at most be connected with 2 excitation operators. The term T 22 /2 is also droppedsince, when contracted with ΘN , it would give a diagram with excitation level of +3 whichwould exceed the current truncation of Eq. (5.132). Using the diagrammatic representations ofΘN and T we can rewrite Eq. (5.134) asΘ = + + + ++ + + ++ + ++ + ++ .(5.135)To reduce the number of diagrams and simplify the calculations we introduce effective diagramsfor the similarity transformed operator Θ. To this aim we define= + = ⟨a∣Θ∣b⟩ −∑i⟨a∣t1∣i⟩⟨i∣Θ∣b⟩, (5.136)65Chapter 5. Coupled-cluster theory= + = ⟨j∣Θ∣i⟩ +∑a⟨a∣t1∣i⟩⟨j∣Θ∣a⟩, (5.137)= = ⟨i∣Θ∣a⟩, (5.138)= + ++ += ⟨a∣Θ∣i⟩ +∑b⟨b∣t1∣i⟩⟨a∣Θ∣b⟩ −∑j⟨a∣t1∣j⟩⟨j∣Θ∣i⟩+ ∑jb⟨ab∣t2∣ij⟩⟨j∣Θ∣b⟩ −∑jb⟨a∣t1∣j⟩⟨b∣t1∣i⟩⟨j∣Θ∣b⟩, (5.139)= ++ += − P (ij)∑kc⟨c∣t1∣i⟩⟨ab∣t2∣kj⟩⟨k∣Θ∣c⟩ + P (ab)∑c⟨ac∣t2∣ij⟩⟨b∣Θ∣c⟩− P (ab)∑kc⟨a∣t1∣k⟩⟨cb∣t2∣ij⟩⟨k∣Θ∣c⟩ − P (ij)∑k⟨ab∣t2∣ik⟩⟨k∣Θ∣j⟩, (5.140)= = −∑j⟨ab∣t2∣ij⟩⟨j∣Θ∣c⟩, (5.141)= = ∑b⟨ab∣t2∣ij⟩⟨k∣Θ∣b⟩. (5.142)The algebraic expressions are obtained using the rules in Appendix B.1. Furthermore, forthe T amplitudes we used the notation tai = ⟨a∣t1∣i⟩ and tabij = ⟨ab∣t2∣ij⟩. Note that, while the66Chapter 5. Coupled-cluster theorybare operator Θ is a one-body operator only, the similarity transformed Θ is now a two-bodyoperator as the terms in Eqs. (5.140), (5.141) and (5.142) have two-body vertices similar tosome of the diagrams of VN in Eq. (5.122). Using this new notation, it is straightforward to seethatΘ∣Φ0⟩ = + , (5.143)and the equivalent algebraic expression can be easily found using Eqs. (5.139) and (5.140). Thecalculation of the left pivot SL involves a product with the Λ operator as well, but it is quiteeasy if we combine the effective diagrams of Θ with the Λ diagrams defined Eq. (5.124). Bykeeping only the terms which have total de-excitation levels −1 and −2 (consistent with theCCSD approximation), one has⟨Φ0∣(1+Λ1 +Λ2)Θ„ = + ++ + ++ + + .(5.144)Defining λijab ≡ ⟨ij∣λ2∣ab⟩ and λia ≡ ⟨i∣λ1∣a⟩, and using the rules in Appendix B.1 together withthe expressions of the effective diagrams in Eqs. (5.136)–(5.142), one has= −∑jb⟨j∣λ1∣a⟩⟨b∣t1∣j⟩⟨i∣Θ„∣b⟩ −∑j⟨j∣λ1∣a⟩⟨i∣Θ„∣j⟩, (5.145)= ∑b⟨i∣λ1∣b⟩⟨b∣Θ„∣a⟩ −∑jb⟨i∣λ1∣b⟩⟨b∣t1∣j⟩⟨j∣Θ„∣a⟩, (5.146)= − 12∑jkbc⟨bc∣t2∣jk⟩⟨ji∣λ2∣bc⟩⟨k∣Θ„∣a⟩, (5.147)= − 12∑jkbc⟨bc∣t2∣jk⟩⟨jk∣λ2∣ba⟩⟨i∣Θ„∣c⟩, (5.148)67Chapter 5. Coupled-cluster theory= ∑jb⟨ij∣λ2∣ab⟩ [∑c⟨c∣t1∣j⟩⟨b∣Θ„∣c⟩−∑kc⟨b∣t1∣k⟩⟨c∣t1∣j⟩⟨k∣Θ„∣c⟩ + ⟨b∣Θ„∣j⟩+∑kc⟨bc∣t2∣jk⟩⟨k∣Θ„∣c⟩ −∑k⟨b∣t1∣k⟩⟨k∣Θ„∣j⟩] , (5.149)= P (ab)P (ij)⟨i∣λ1∣a⟩⟨j∣Θ„∣b⟩, (5.150)= P (ab)∑c⟨ij∣λ2∣ac⟩⟨c∣Θ„∣b⟩− P (ab)∑kc⟨ij∣λ2∣ac⟩⟨c∣t1∣k⟩⟨k∣Θ„∣b⟩, (5.151)= − P (ij)∑k⟨ik∣λ2∣ab⟩⟨j∣Θ„∣k⟩− P (ij)∑kc⟨ik∣λ2∣ab⟩⟨c∣t1∣k⟩⟨j∣Θ„∣c⟩, (5.152)where P (pq) = 1 − Ppq and Ppq is the permutation operator between p and q. The operatorP (pq) is needed in diagrams with pairs of inequivalent lines which have to be anti-symmetrized.The m0 sum rule, defined in Eq. (5.130), is obtained by contracting the diagrams of SR inEq. (5.143) with the diagrams of SL from Eq. (5.144) and keeping only the closed diagrams,i.e.,m0 = SL ⋅ SR = + ++ + ++ +68Chapter 5. Coupled-cluster theory= + , (5.153)where in the last equality we used the definition in Eq. (5.132) and we grouped the contribu-tions from the one- and two-body parts of SL and SR. In particular, the first five diagramsof Eq. (5.153) are the one-body contributions, while the last three represent the two-bodycontributions.5.7.2 The continued fraction and the Hamiltonian tridiagonalizationTo calculate the LIT and the Stieltjes transforms, one needs to calculate continued fractions inwhich the coefficients are found by tridiagonalizing the matrix of Eq. (5.91) or, in other words,the similarity transformed Hamiltonian since H and M differ only by a diagonal term. Thisis done by repeatedly applying the non-symmetric Lanczos steps of Eqs. (5.97)−(5.101) to H,and thus one needs to iteratively evaluate products between H and the Lanczos vectors. Thisprocedure is similar to what one usually does to solve the coupled-cluster EOM (left-hand sideterm in Eq. (5.72)). To be consistent with the CCSD approximation, we expand the Lanczosvectors wi and vi as8wi = + ,(5.154)vi = + .By definition, the two pivots w0 and v0 are equivalent to SL and SR except for a normalizationfactor m0 (see Eq. (5.102)).Similarly to what we did to find the effective diagrams of the Θ operator, one can alsodefine effective diagrams for the similarity transformed Hamiltonian. A full derivation of thesediagrams goes beyond the purpose of this work and the interested reader can find a completeand detailed derivation in Ref. [91]. The diagrammatic representation of H is9H = + + + ++ + + + +8We introduce a new symbol, the three horizontal lines superimposed to a wavy line, to denote the Lanczosvectors.9We represent the action of the similarity transformed Hamiltonian operator by a wavy line and a star.69Chapter 5. Coupled-cluster theory+ + + . (5.155)Equation (5.155) has been truncated at the 2p–2h level, but the full untruncated H includesalso three- and four-body vertices. Also note that the truncation of the similarity transformedHamiltonian is unrelated to the truncation of the original Hamiltonian H at the two- or three-body level. In fact, some of the diagrams in Eq. (5.155) contain contributions from three-bodyforces (if the latters are included in H). With Eqs. (5.154) and (5.155) we can then calculatethe product between H and vi. The one-body contribution is(Hvi)(1) = + ++ + + , (5.156)and the two-body contribution reads(Hvi)(2) = + ++ ++ . (5.157)The diagrams in Eqs. (5.156) and (5.157) are the same appearing in the solution of the EOMand their algebraic expression can be found in Ref. [91]. Equivalent equations are found for theproduct with wi, with the one-body contribution being equal to(wiH)(1) = + ++ + . (5.158)70Chapter 5. Coupled-cluster theoryThe above diagrams are the upside-down equivalents of the diagrams in Eq. (5.156), except forthe third term. In fact, the upside-down version of the third term in Eq. (5.156) is identicallyzero, i.e., = 0 , (5.159)since the second line of Eq. (5.129) can be shown to be equivalent to the similarity transformedHamiltonian diagram0 = ⟨Φai ∣HeT1+T2 ∣Φ0⟩C = , (5.160)which corresponds to the CC equation for the singles amplitudes. The two-body contributionsare, once again, the upside-down counterparts of the diagrams in Eq. (5.157),(wiH)(2) = + ++ ++ . (5.161)So far we derived diagrams in the CCSD approximation. The next natural step is to extend thecalculations by including the next power in the T expansion, namely triples excitations. Wewill discuss how to achieve this in the next section.5.8 Linear triples correctionsWhile the calculation of observables in CCSD approximation is relatively cheap from a com-putational point of view, the full inclusion of triples excitations adds considerable complexityto the CC equations. The number of the 3p–3h configurations increases factorially with themass number A, thus requiring larger and larger computer storage capabilities. Various ap-proximations, either iterative [107, 108] or non-iterative [109] have been developed to includehigher-order corrections to the singles and doubles truncation. Non-iterative, perturbativetriples methods, rely on perturbation theory arguments and consider only some higher-orderrelevant diagrams. Several implementations of these approximate methods exist, see for exam-ple Refs. [8, 91, 110, 111]. Here, we adopt an iterative approximation, namely the CCSDT-1approximation [107], where at the ground-state level one expands the CC exponential aseT ≈ eT1+T2 + T3. (5.162)71Chapter 5. Coupled-cluster theoryWithin this approximation, the T1 equation will have a new contribution VNT3, while theequation for T2 will have both FNT3 and VNT3 contributions. Furthermore, the CCSD equationsin Eq. (5.129) must be supplemented by an equation for the tabcijk amplitudes of T3 which onlyhave three terms,0 = FNT3 + VNT3 + VNT2. (5.163)From Eq. (5.163) we see that tabcijk can be written as a function of tabij and thus does not needto be stored. For our case we also require a 3p–3h equivalent approximation for the left-states,i.e.,Λ = Λ1 +Λ2 +Λ3, (5.164)and one has to solve the Λ3 equations to find the amplitudes λijkabc in a similar approximationto the one used in Eq. (5.163). Once the Λ3 and T3 amplitudes are found by solving for theground state, one can study the corrections to the one-body similarity transformed operator Θof Eq. (5.134). In the evaluation of Θ, we now use the full expansion of the exponential whichgivesΘ = (ΘNeT1+T2+T3)C= [ΘN (1 + T1 + T2 + 12T 21 + T1T2 + T3 + 12T 22 + T1T3)]C= Θ(CCSD) + [ΘN (T3 + 12T 22 + T1T3)]C.(5.165)From the above equations we see that in the CCSDT-1 approximation we have three morecontributions to the similarity transformed Θ operator. These three terms correspond to thenew diagrams(ΘNT3)C = ++ ++ , (5.166)(12ΘNT22 )C= , (5.167)(ΘNT1T3)C = + . (5.168)72Chapter 5. Coupled-cluster theoryThe last term of Eq. (5.165) is a new contribution to the effective diagram defined in Eq. (5.140),so that we can redefine the effective diagram as=(CCSD)+ . (5.169)All the other terms in Eqs. (5.166), (5.167) and (5.169) are collected together into new three-body effective diagrams for Θ as= = −P (c/ab)∑k⟨abc∣t3∣ijk⟩⟨k∣Θ∣d⟩, (5.170)= = P (k/ij)∑c⟨abc∣t3∣ijk⟩⟨l∣Θ∣c⟩, (5.171)= + ++ += −P (i/jk)∑ld⟨l∣Θ∣d⟩⟨d∣t1∣i⟩⟨abc∣t3∣ljk⟩ + P (c/ab)∑d⟨abd∣t3∣ijk⟩⟨c∣Θ∣d⟩− P (j/ik)P (b/ac)∑ld⟨ad∣t2∣ij⟩⟨l∣Θ∣d⟩⟨bc∣t2∣lk⟩ − P (a/bc)∑ld⟨l∣Θ∣d⟩⟨a∣t1∣l⟩⟨dbc∣t3∣ijk⟩− P (k/ij)∑l⟨abc∣t3∣ijl⟩⟨l∣Θ∣k⟩,(5.172)where P (i/jk) = 1 − Pij − Pik. The m0 sum rule of Eq. (5.131) with triples now readsm0 = ⟨Φ0∣(1 +Λ1 +Λ2 +Λ3) (Θ„NeT1+T2+T3)C (ΘNeT1+T2+T3)C ∣Φ0⟩. (5.173)Accordingly, we will now have to re-define Eq. (5.132) for SL and SR up to 3p–3h excitations,i.e.,SR = + + , (5.174)73Chapter 5. Coupled-cluster theorySL = + + . (5.175)Using the CCSD expressions for SL and SR – Eqs. (5.143) and (5.144) – together with thenew effective diagrams for the Θ operator in Eqs. (5.169)−(5.172), we can find the new triplescontributions to SL and SR. The right vector SR is straightforward to find and it isSR = + += SR(CCSD) + + ,(5.176)where in the last equality we explicitly separated the CCSD and triples contributions. A largernumber of new terms emerges for the left vector SL. Making use of Eq. (5.144) one hasSL = SL(CCSD) + (CCSD) ++ + ++ + , (5.177)where in the first diagram the two-body vertex is the one defined in CCSD approximation inEq. (5.140). For the calculation of the polarizability, triples corrections enter not only in them0, but also in the continued fraction. A full inclusion of triples would require us to re-evaluatethe products in Eqs. (5.156)−(5.158) and (5.161) as the similarity transformed Hamiltonianin Eq. (5.155) contains three-body vertices. Moreover, we would need a new equation for the3p–3h contributions to the Lanczos step. Since we are using the CCSDT-1 approximation atthe ground-state level, for the Lanczos steps we use a consistent approximation (see Ref. [112]for a details), called the EOM-CCSDT-1 approximation. Within this framework, we have newtwo- and three-body contributions to wiH, i.e.,(wiH)(1) = (wiH)(1)(CCSD) , (5.178)(wiH)(2) = (wiH)(2)(CCSD) + + , (5.179)74Chapter 5. Coupled-cluster theory(wiH)(3) = ++ ++ + . (5.180)For the right product with vi, together with new two- and three-body contributions, one alsohas new one-body contributions, i.e.,(Hvi)(1) = (Hvi)(1)(CCSD) + , (5.181)(Hvi)(2) = (Hvi)(2)(CCSD) + + ,+ (5.182)(Hvi)(3) = ++ + . (5.183)Here, the dotted lines represent the action of the bare Hamiltonian H as it is clear from thecontext. In conclusion, within the CCSDT-1 and EOM-CCSDT-1 approximations the calcula-tion of the m0 sum rule requires the evaluation of 5 new diagrams for SR and 16 new diagramsfor SL, while the polarizability needs 16 more diagrams which would contribute to the diago-nalization of the similarity transformed Hamiltonian and so to the continued fraction.In Section 7.4 we will employ this approximation and the above diagrams and compareit with the CCSD truncation. This will help to estimate the uncertainty of the many-bodymethod.75Chapter 6Spherical coupled-clusterThe treatment of CC theory in Chapter 5 has been in terms of single-particle orbitals denoted bya, b, ..., i, j, ... which are intended to be collective indexes for all the quantum numbers associatedwith a given orbit 10. Our calculations are performed by employing the single-particle states ofthe harmonic-oscillator (HO) basis which are characterized by the oscillator frequency h̵Ω andwhere the basis is truncated in units of major oscillators shellsN . We denote the truncation levelby a parameter Nmax, i.e., the maximum value of oscillator shells considered. We supplementthe HO oscillator basis by spin and isospin quantum numbers so that a single-particle state ∣p⟩is written as ∣p⟩ ≡ ∣np, (lp, sp)jpmp⟩⊗ ∣τp, τ zp ⟩ ≡ ∣αp, jpmp⟩. (6.1)Here np is the principal quantum number of the orbit p, and lp, sp = 1/2 are its orbital angularmomentum and spin, respectively; jp is the total angular momentum of the state and mp itsprojection. Following the notation introduced in Eq. (3.40), the isospin is denoted by τp = 1 andits projection τ zp equals +1 for proton states and −1 for neutron orbits. To simplify the notation,in the last equality of Eq. (6.1) we introduced the index αp = {np, (lp, sp), τp, τ zp } to denote theset of all quantum numbers except for the total angular momentum jp and its projection mp.Using this notation, two- and three-body states can be represented as∣pq⟩ ≡∣αp, jpmp;αq, jqmq⟩, ∣pqr⟩ ≡ ∣αp, jpmp;αq, jqmq;αr, jrmr⟩. (6.2)So far, we presented coupled-cluster theory in an m-scheme formulation, as for each orbit weexplicitly take into account the projections mp of the total angular momentum jp. While the m-scheme representation allows for an easy-to-follow formulation of CC theory, in this scheme wehave an ever increasing number of 2p−2h and 3p−3h configurations for the states in Eq. (6.2) asone increases the model space (Nmax, the size of the single-particle basis) and the mass numberof the nucleus under study. Calculations in m-scheme with 3p−3h configurations are limited tovery small model-space sizes, and in Ref. [97], Hagen et al. show that even in 4He it is hard togo beyond Nmax = 6.The only way to reduce the computational cost without sacrificing the size of the HO basisexpansion is to take advantage of possible symmetries of the system. The nuclear Hamiltonian isinvariant under rotations and if we limit ourselves to the study of closed-shell and closed-subshellnuclei, both H and the T,Λ operators are scalars under rotations. This formalism enables us to10In this context, by orbits we mean single-particle states.76Chapter 6. Spherical coupled-clusterdo calculations in significantly larger model spaces. In fact, the matrix element storage needsare greatly reduced since the reduced matrix elements for the operators do not depend onthe m projection anymore. This approach, also known as coupled-angular momentum scheme,J−scheme or spherical coupled-cluster, has already been successfully applied to a variety ofmedium- and heavy-mass nuclei [8, 38, 113], thus extending the reach of coupled-cluster theory.In the following, we briefly introduce the formalism and notation of spherical coupled-cluster.The J−coupled expressions for the diagrams in Eq. (5.178)−(5.183) are equivalent to the ones forthe coupled-cluster EOM method and are presented in Ref. [93], while the J−scheme expressionsfor the m0 diagrams of Sections 5.7.1 and 5.8 are reported in Appendix B.3 and B.4. Thediagrams for the triples contributions of the Lanczos steps have been derived by G. Hagen andwill be published soon [114].6.1 Two- and three-body statesTo suppress the m−dependence of the matrix elements of our operators, we define the equivalentof the states in Eq. (6.2) in a coupled-angular momentum framework,{∣p⟩⊗ ∣q⟩}JpqMpq = ∣αp, αq; jpjq;JpqMpq⟩ ≡ ∣pq⟩, (6.3)where we coupled the single orbits angular momenta jp and jq to total momentum Jpq andprojection Mpq. In the last equality we introduced ∣pq⟩ as a compact notation to denote thecoupled state. Using the completeness of the coupled states1ˆ = ∑JpqMpq∣αp, αq; jpjq;JpqMpq⟩⟨αp, αq; jpjq;JpqMpq ∣ (6.4)we find the relation between coupled and uncoupled states to be∣αp, jpmp;αq, jqmq⟩ = ∑JpqMpq⟨αp, αq; jpjq;JpqMpq ∣αp, jpmp;αq, jqmq⟩×× ∣αp, αq; jpjq;JpqMpq⟩. (6.5)Equation (6.5) shows that the coupled state can be separated in two contributions where one ofthe two is independent of the single orbits angular momentum projections m. By introducingthe notation ⟨αp, αq; jpjq;JpqMpq ∣αp, jpmp;αq, jqmq⟩ ≡ CJpqMpqjpmpjqmq (6.6)to represent Clebsch-Gordan coefficients, we can rewrite Eq. (6.5) in a more compact notationas ∣pq⟩ = ∑JpqMpqCJpqMpqjpmpjqmq∣pq⟩. (6.7)77Chapter 6. Spherical coupled-clusterThe inverse relation can be found by using properties of the Clebsch-Gordan coefficients andwe obtain ∣pq⟩ = ∑mpmqCJpqMpqjpmpjqmq∣pq⟩. (6.8)Three-particle states can be treated in a similar way. However, given a state ∣pqr⟩, one hasdifferent choices for the coupling order. Here, we choose a coupling order where we first couplep→ q and then (pq)→ r, meaning{{∣p⟩⊗ ∣q⟩}JpqMpq ⊗ ∣r⟩}JpqrMpqr = ∣αp, αq, αr; jpjq;Jpqjr;JpqrMpqr⟩ ≡ ∣pqr⟩. (6.9)Repeated use of Eqs. (6.6) and (6.7), and with the notation of Eq. (6.9), we obtain∣pqr⟩ = ∑JpqrMpqrJpqMpqCJpqMpqjpmpjqmqCJpqrMpqrJpqMpqjrmr∣pqr⟩, (6.10)and ∣pqr⟩ = ∑mpmqMpqmrCJpqMpqjpmpjqmqCJpqrMpqrJpqMpqjrmr∣pqr⟩. (6.11)6.2 Matrix elements and the Wigner-Eckart theoremGiven a two-body spherical tensor operator T Jµ(2) of rank J , we can make use of Eq. (6.6) torelate its matrix elements from m-scheme to J−scheme,⟨pq∣T Jµ(2) ∣rs⟩ = ∑JpqMpqJrsMrsCJpqMpqjpmpjqmqCJrsMrsjrmrjsms⟨pq∣T Jµ(2) ∣rs⟩. (6.12)For a three-body operator T Jµ(3) , making use of Eq. (6.9), we find⟨pqr∣T Jµ(3) ∣stu⟩ = ∑JpqMpqJpqrMpqr∑JstMstJstuMstuCJpqMpqjpmpjqmqCJpqrMpqrJpqMpqjrmr××CJstMstjsmsjtmtCJstuMstuJstMstjumu⟨pqr∣T Jµ(3) ∣stu⟩.(6.13)Although the coupled matrix elements on the right-hand side of Eqs. (6.12) and (6.13) do notdepend on the single orbits projection quantum numbers anymore, we still need to keep trackof the projections of the total angular momenta of the many-body state and the projection ofthe angular momentum J of the operator T Jµ. To suppress this residual dependence on theangular momentum projections, the Wigner-Eckart theorem turns out to be extremely useful.The theorem relates the matrix elements of a spherical tensor operator T Jµ to its reduced78Chapter 6. Spherical coupled-clustermatrix elements, i.e., for a one-body operator11⟨p∣T Jµ∣q⟩ = ⟨αp, jpmp∣T Jµ∣αq, jqmq⟩= CjpmpjqmqJµ⟨αp, jp∣∣T J ∣∣αq, jq⟩ = CjpmpjqmqJµ⟨p∣∣T J ∣∣q⟩, (6.14)where we use ⟨p∣∣T J ∣∣q⟩ to denote the reduced matrix elements and it is implicit that whendealing with reduced matrix elements the states ⟨p∣ and ∣q⟩ do not depend on any projectionquantum number. Equation (6.14) is easily generalized to two- and three-body operators,⟨pq∣T Jµ(2) ∣rs⟩ = CJpqMpqJrsMrsJµ⟨pq∣∣T J(2)∣∣rs⟩, (6.15)and⟨pqr∣T Jµ(3) ∣stu⟩ = CJpqrMpqrJstuMstuJµ⟨pqr∣∣T J(3)∣∣stu⟩, (6.16)where we had to use the coupled two- and three-body states since the Wigner-Eckart theoremin Eq. (6.13) couples the angular momentum of the operator with the total angular momenta ofthe left- and right-states. To obtain a relation between the uncoupled matrix elements and thereduced matrix elements we can substitute Eqs. (6.15) and (6.16) into Eqs. (6.12) and (6.13),thus obtaining⟨pq∣T Jµ(2) ∣rs⟩ = ∑JpqMpqJrsMrsCJpqMpqjpmpjqmqCJrsMrsjrmrjsmsCJpqMpqJrsMrsJµ⟨pq∣∣T J(2)∣∣rs⟩, (6.17)and⟨pqr∣T Jµ(3) ∣stu⟩ = ∑JpqMpqJpqrMpqr∑JstMstJstuMstuCJpqMpqjpmpjqmqCJpqrMpqrJpqMpqjrmr××CJstMstjsmsjtmtCJstuMstuJstMstjumuCJpqrMpqrJstuMstuJµ⟨pqr∣∣T J(3)∣∣stu⟩.(6.18)Equations (6.17) and (6.18) are an important result: we can factorize the matrix elementsof our operators so that the dependence on the projection quantum numbers is condensed ina geometric factor given by Clebsch-Gordan coefficients. This means that, given a certainoperator, we only need to store its reduced matrix elements, thus tremendously lowering thememory storage requirements of the calculations.6.3 Permutation operatorsThe action of the permutation operator Ppq introduced in Section 5.7.1 in the m-scheme ex-pressions for the m0 diagrams also takes a different form in the J−coupled formalism. We recall11Here we use a slightly different convention than the one used by Varshalovich [115] and Edmonds [71].79Chapter 6. Spherical coupled-clusterthat the action of P (pq) on a two-body uncoupled state isP (pq)∣pq⟩ = (1 − Ppq)∣pq⟩ = ∣pq⟩ − ∣qp⟩. (6.19)Using Eq. (6.7) we haveP (pq)∣pq⟩ = (1 − Ppq) ∑JpqMpqCJpqMpqjpmpjqmq∣pq⟩= ∑JpqMpqCJpqMpqjpmpjqmq∣pq⟩ − ∑JpqMpqCJpqMpqjqmqjpmp∣qp⟩= ∑JpqMpqCJpqMpqjpmpjqmq(∣pq⟩ − (−1)jp+jq−Jpq ∣qp⟩)= ∑JpqMpqCJpqMpqjpmpjqmqP˜ (pq)∣pq⟩,(6.20)where we used Eq. (B.8) and in the last equality we definedP˜ (pq) = 1 − (−1)jp+jq−JpqPpq, (6.21)with Ppqthe permutation operator acting on the coupled states. For the three-body states thepermutation operator action isP (r/pq)∣pqr⟩ = P (rp)P (rq)∣pqr⟩ = (1 − Prp − Prq)∣pqr⟩= ∣pqr⟩ − ∣rqp⟩ − ∣prq⟩. (6.22)Proceeding as in Eq. (6.21), we want to rewrite the states ∣rqp⟩ and ∣prq⟩ so that we can collectall the angular factors and define a permutation operator like in Eq. (6.21) which acts on thecoupled states only. Thus, we use Eq. (6.10) to get∣rqp⟩ = ∑JrqMrqJrqpMrqpCJrqMrqjrmrjqmqCJrqpMrqpJrqMrqjpmp∣rqp⟩, (6.23)and ∣prq⟩ = ∑JprMprJprqMprqCJprMprjpmpjrmrCJprqMprqJprMprjqmq∣prq⟩. (6.24)To obtain the same Clebsch-Gordan coefficient of Eq. (6.10), we need to decouple the coefficientsin Eqs. (6.23) and (6.24), and re-couple them so that p is coupled to q and (pq) to r. We canmake use of Eq. (B.10) to re-couple the angular momenta and the property in Eq. (B.8) to80Chapter 6. Spherical coupled-clusterrearrange the Clebsch-Gordan coefficients so that one obtains∣rqp⟩ = − ∑JpqrMpqrJpqMpqCJpqMpqjpmpjqmqCJpqrMpqrJpqMpqjrmr∑JrqJˆpqJˆrq⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jp jq Jpqjr Jpqr Jrq⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ ∣rqp⟩, (6.25)∣prq⟩ = ∑JpqrMpqrJpqMpqCJpqMpqjpmpjqmqCJpqrMpqrJpqMpqjrmr∑Jpr(−1)jq+jr+Jpq+Jpr JˆpqJˆpr××⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jp jq JpqJpqr jr Jpr⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ ∣prq⟩.(6.26)With Eqs. (6.10), (6.25) and (6.26) we can then rewrite Eq. (6.22) asP (r/pq)∣rpq⟩ = ∑JpqrMpqrJpqMpqCJpqMpqjpmpjqmqCJpqrMpqrJpqMpqjrmrP˜ (r/pq)∣pqr⟩,(6.27)where P˜ (r/pq) is now the anti-symmetrization operator acting on the coupled states defined asP˜ (r/pq) = 1 +∑JrqJˆpqJˆrq⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jp jq Jpqjr Jpqr Jrq⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭Prp−∑Jpr(−1)jq+jr+Jpq+Jpr JˆpqJˆpr⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jp jq JpqJpqr jr Jpr⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭Prq .(6.28)6.4 J-coupled coupled-cluster diagramsWith the definitions in the previous sections, all CC diagrams, equations and algebraic expres-sions can be translated in the J−coupled scheme. The advantage is that we can work withreduced matrix elements of operators, thus lowering the memory requirements of our calcu-lations. As an example, here we show how we can translate the calculation of the effectivediagram in Eq. (5.141) from m-scheme to J−coupling scheme. The excitation operator T is atensor operator of rank 0, i.e., T ≡ T 00, and we denote its amplitudes as t001 for the one-bodypart, t002 for the two-body part and so on. The dipole operator Θ ≡ ΘJµ has rank J = 1, butfor generalization purposes we will not assign a fixed value to J . Formally, Eq. (5.141) is thenrewritten as ⟨ab∣ΘJµ∣ic⟩ = − ∑αj ,jjmj⟨ab∣t002 ∣ij⟩⟨j∣ΘJµ∣c⟩, (6.29)81Chapter 6. Spherical coupled-clusterwhere the similarity transformed dipole operator is also of rank J since it is a product of ascalar operator and another tensor of rank J . The sum over the orbit index j is intended to beover all the quantum numbers of the orbit, and so in Eq. (6.29) we have explicitly separated thetotal angular quantum numbers from the others. We make now use of Eq. (6.17) and rewritethe left-hand side of Eq. (6.29) as⟨ab∣ΘJµ∣ic⟩ = ∑JabMabJicMicCJabMabjamajbmbCJicMicjimijcmcCJabMabJicMicJµ⟨ab∣∣ΘJ ∣∣ic⟩. (6.30)For the right-hand side, as we have a one-body matrix element, we also need to use Eq. (6.14),and we get (neglecting the minus sign and the sum over the αj quantum numbers),∑jjmj⟨ab∣t002 ∣ij⟩⟨j∣ΘJµ∣c⟩ = ∑jjmj∑JabMabJijMijCJabMabjamajbmbCJijMijjimijjmjCJabMabJijMij00⟨ab∣∣t02∣∣ij⟩××CjjmjjcmcJµ⟨j∣∣ΘJ ∣∣c⟩= ∑jjmj∑JabMabCJabMabjamajbmbCJabMabjimijjmj⟨ab∣∣t02∣∣ij⟩××CjjmjjcmcJµ⟨j∣∣ΘJ ∣∣c⟩,(6.31)where in the last equality we usedCJabMabJijMij00 = δJabJijδMabMij . (6.32)To obtain expressions only in terms of reduced matrix elements, we want to simplify the angularcoefficients appearing in Eq. (6.30). To do so, the angular momenta in Eq. (6.31) must be re-coupled. While a ad b are already coupled in the right order, we need to decouple i and j andre-couple i with c. Making use of Eq. (B.10) to re-couple the second and third Clebsch-Gordancoefficients in the last equality of Eq. (6.31), and then using Eq. (B.8), we obtain∑jjmj⟨ab∣t002 ∣ij⟩⟨j∣ΘJµ∣c⟩ = ∑jj∑JabMabCJabMabjamajbmb ∑JicMicCJicMicjcmcjimiCJabMabJµJicMic×× (−1)2Jab Jˆicjˆj⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc ji JicJab J jj⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ab∣∣t02∣∣ij⟩⟨j∣∣ΘJ ∣∣c⟩= ∑JabMabJicMicCJabMabjamajbmbCJicMicjimijcmcCJabMabJicMicJµ∑jjJˆicjˆj×× (−1)ji+jc+Jab+J⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc ji JicJab J jj⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ab∣∣t02∣∣ij⟩⟨j∣∣ΘJ ∣∣c⟩. (6.33)82Chapter 6. Spherical coupled-clusterThe Clebsch-Gordan coefficients and the sums over the coupled angular momenta in the lastequality of Eq. (6.33) are exactly the same appearing in Eq. (6.30). If we equate the two termsas in Eq. (6.29), we can simplify these factors and finally obtain⟨ab∣∣ΘJ ∣∣ic⟩ = −∑jJˆicjˆj(−1)ji+jc+Jab+J⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc ji JicJab J jj⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ab∣∣t02∣∣ij⟩⟨j∣∣ΘJ ∣∣c⟩, (6.34)where here the sum over j is intended to be over all the quantum numbers but the projectionmj . The final result of Eq (6.34) is independent of all the angular projection quantum numbersand only requires the storage of the reduced matrix elements of the relevant operators. Theprocedure followed in this section can be used to rewrite and solve Eq. (5.178)−(5.183) ina J−coupled scheme and has been successfully used to rewrite the CC and EOM equations(see Ref. [93] and references therein for a detailed summary), and extended to calculationsof selected open-shell nuclei in Ref. [116]. Here, we use the J−coupled scheme to performcalculations of the dipole response function and electric dipole polarizability of selected medium-mass nuclei [1, 2, 5–7, 105].83Chapter 7ResultsIn this Chapter we use the coupled-cluster method to calculate electromagnetic observables forselected nuclei. We start in Section 7.1 by benchmarking the CCSD method with an exact cal-culation based on the effective interaction hyperspherical harmonics (EIHH) method [117, 118].For this purpose we will use the Entem and Machleidt chiral NN force derived at next-to-next-to-next-to-leading order (N3LO) [39]. In Section 7.2 we study 16,22O and 40,48Ca with thesame NN force and we calculate αD and the response function. In the subsequent sections westudy the effects of three-nucleon forces (3NFs) by employing the chiral NN+3NF interactionNNLOsat [18]. As we will see, observables strongly depend on the Hamiltonian employed in thecalculations and so we will investigate how to estimate uncertainties by using different interac-tions and exploiting correlations (Section 7.3.2). In Section 7.3.3 we use these correlations topredict αD and the neutron-skin radius, i.e., the difference between proton and neutron radii,in the double-magic 48Ca nucleus. Finally, we add triples corrections at the CCSDT-1 level,benchmark the corrections in 4He again with exact EIHH calculations, and present new resultsfor 16O.Throughout all this Chapter, the calculations are performed employing a Hartree-Fock (seeRefs. [95, 96] for details) basis built on the spherical HO eigenfunctions with characteristicfrequency h̵Ω, radial quantum number n and angular momentum l. For the calculations inSections 7.1 and 7.2 with NN forces we vary the model-space size from 7 up to 19 majoroscillator shells, corresponding to Nmax = 2n+l from 6 to 18, limiting l ≤ 10. When adding 3NFsin Sections 7.3, we use a maximum Hartree-Fock basis built on 15 major harmonic-oscillatorshells. We thus vary the model-space size up to Nmax = 14 and we truncate the 3NFs matrixelements at E3max = Nmax = 14 for 4He and 16,22O, with E3max = εa + εb + εc being the energy(in units of h̵Ω) of the three-body state ∣abc⟩. For our purposes, this truncation provides well-enough converged results. In fact, for the more challenging neutron-rich 22O nucleus, increasingE3max to 16 leads to a variation in energy of only 400 keV, corresponding to a 0.3% variation.7.1 Benchmarking the new methodIn Chapter 5, by reformulating the LIT approach within the CC theory we have obtained anew method to tackle break-up observables in light- and medium-mass nuclei. Moreover, thespherical formulation of CC theory presented in Chapter 6 allows us to reach model-space sizes84Chapter 7. Resultslarge enough to obtain converged results for the LIT and the polarizability. For the benchmarkspresented in this section, we used the chiral NN N3LO interaction from Ref. [39].7.1.1 Convergence and Lanczos coefficientsThe Lanczos approach outlined in Section 5.5.3 has few important advantages for the LITmethod. First of all, the result in Eqs. (5.109) and (5.114) suggests the tridiagonalization ofM has to be done only once regardless of the value of σ and Γ, and so different LITs can beeasily obtained with one calculation only. Moreover, one can usually converge with reasonablyfew Lanczos vectors (depending on the nucleus and the excitation operator under study). InFig. 7.1 we show the convergence of the LIT with respect to the number of Lanczos coefficientsm for 4He in panels (a) and (b) and 16O in panels (c) and (d).−20−100102030r m[%]4He− Γ = 10 MeV(a) 4He− Γ = 20 MeV(b)m = 20m = 40m = 600 20 40 60 80 100σ[MeV]−20−100102030r m[%]16O− Γ = 10 MeV(c)0 20 40 60 80 100σ[MeV]16O− Γ = 20 MeV(d)Figure 7.1: Convergence of the LIT with respect to the number of Lanczos coefficients m in 4He –panels (a) and (b) – and in 16O – panels (c) and (d). All panels show the ratio in Eq. (7.1) as a functionof σ. Panels (a) and (c) show results for Γ=10 MeV, while panels (b) and (d) for Γ = 20 MeV.85Chapter 7. ResultsFor all the four panels we show the ratiorm = 100 × IL(σ) − I(m)L (σ)IL(σ) , (7.1)where ImL (σ) is the LIT calculated with m < 100 Lanczos coefficients and IL(σ) is the con-verged result obtained with 100 Lanczos coefficients. Panels (a) and (b) in Fig. 7.1 show resultsfor 4He using two different values of Γ, namely 10 MeV and 20 MeV. Panels (c) and (d) showthe same for 16O. As expected, we see the convergence depends on the value of Γ and on thenucleus as well. For larger values of Γ, one observes faster convergence. In fact, for large Γ,the LIT is less sensitive to the fine structure of the response function, which depends on thenumber of Lanczos coefficients. The low-energy part (small σ) is affected less by the numberof Lanczos coefficients as it is dominated by the lowest eigenvalues of M and not by the higherones. Finally, the convergence is slower in 16O compared to 4He, thus suggesting that LITcalculations for heavier nuclei will require either larger values of Γ or a larger number m ofLanczos coefficients to obtain the same level of convergence observed in 4He.As the polarizability is mostly sensitive to the low-energy region of the spectrum, we expectit to converge much faster with respect to the number of Lanczos coefficients. Indeed, this isthe case as shown in Fig. 7.2, where the polarizability in 4He is presented as a function of thenumber of Lanczos coefficients m used in Eq. (5.114).0 2 5 7 10 12 15 17 20m0.0650.070.0750.080.0850.09αD[fm3 ](a)4He0 2 5 7 10 12 15 17 20m0.30.3250.350.3750.40.4250.450.4750.5αD[fm3 ](b)16OFigure 7.2: Convergence of the electric dipole polarizability αD in4He with respect to the number ofLanczos coefficients m employed in the continued fraction of Eq. (5.114).The polarizability shows excellent convergence and already at around 10-15 Lanczos coef-ficients it is converged. With 15 Lanczos coefficients the error with respect to the convergedvalue is 0.3%, which shows the advantage of using directly Eq. (5.114) instead of an integrationof the response function (which involves all the complexities of the inversion procedure, seeSection 4.4 for details). The behaviour is similar in heavier nuclei and thus the polarizability86Chapter 7. Resultsrequires less computational effort to calculate than the full LIT.7.1.2 Model-space convergenceIn Fig. 7.3, from Ref. [1], the LIT of the 4He dipole response function is shown as a function ofσ at fixed Γ = 10 MeV. In panel (a), the EIHH results are presented for different model-spacesizes, represented by different values of the grand-angular momentum Kmax [1, 117, 118].0.00.0050.010.0150.020.025I L(σ)[fm2 MeV−1]Γ = 10 MeV4He(a) Kmax = 8Kmax = 10Kmax = 12Kmax = 14Kmax = 16Kmax = 180.0 20.0 40.0 60.0 80.0 100.0 120.0σ [MeV]0.00.0050.010.0150.02I L(σ)[fm2 MeV−1]Γ = 10 MeV4He(b) Nmax = 8Nmax = 10Nmax = 12Nmax = 14Nmax = 16Nmax = 18Figure 7.3: Panel (a) – Convergence of IL(σ) in 4He with Γ = 10 MeV as a function of Kmax in theEIHH expansion. Panel (b) – Convergence of IL(σ) calculated with coupled-cluster with Γ = 10 MeV asa function of Nmax for an HO frequency of h̵Ω = 20 MeV.The convergence is fast and excellent. In panel (b), we show the results computed with CC atthe singles and doubles approximation for model spaces of Nmax = 8,10,12,14,16,18 and for avalue of the underlying HO frequency h̵Ω = 20 MeV. Compared to the EIHH calculations, thecoupled-cluster method shows a larger difference between the smallest and largest model-spaceresults. However, the LIT is well converged when Nmax = 18 is used and does not change whenvarying the underlying HO frequency.87Chapter 7. ResultsFor the mass number A = 4, extensive studies have been performed with the accurate EIHHmethod [117, 118]. By comparing EIHH and CC results for 4He, where the same interactionand excitation operator are used, we can study the convergence pattern and assess the accuracyof the approximations introduced in the CCSD scheme.7.1.3 4He test caseAt this point it is interesting to compare both the EIHH and CCSD converged results. InFig. 7.4, we compare the LITs for the values of Γ = 10 and 20 MeV in panel (a) and (b),respectively. The coupled-cluster results are shown to overlap for two different values of the0.00.0020.0040.0060.0080.010.012I L(σ)[fm2 MeV−1]Γ = 20 MeV4He(a)0.0 20.0 40.0 60.0 80.0 100.0 120.0σ [MeV]0.00.0050.010.0150.02I L(σ)[fm2 MeV−1]Γ = 10 MeV4He(b) EIHH (Kmax = 18)CCSD (h¯Ω = 26MeV)CCSD (h¯Ω = 20MeV)Figure 7.4: Comparison of IL(σ) in 4He at Γ = 20 MeV – panel (a) – and 10 MeV – panel (b) –calculated with the CCSD scheme with Nmax = 18 and two values of h̵Ω = 20 and 26 MeV against theLIT from EIHH.harmonic-oscillator frequency. They also agree very well with the EIHH result, especially forΓ = 20 MeV. For the finer resolution scale of Γ = 10 MeV, minor differences are observed. It isknown that calculations of the LIT with smaller Γ tend to be more cumbersome. In fact, as Γdecreases, the Lorentz kernel approaches the δ−function, facing again the continuum problem88Chapter 7. Results(for σ above the break-up threshold). Consequently, the Lorentz states of Eq. (5.84) approachthe vanishing boundary condition at farther and farther distances. However, since the conver-gence of the LIT is very good, as also demonstrated by the h̵Ω−independence, we attributethe small differences with respect to the EIHH result to the truncations inherent in the CCSDapproximation.To further quantify the role of coupled-cluster truncations, it is interesting to compare thedipole response functions obtained by the inversion of both the calculated IL(σ). For theinversions we use the method outlined in Refs. [75, 78] and in Section 4.4.3, which looks for theregularized solution of the integral transform equation. As in Refs. [1, 105], we regularize thesolution using the nonlinear ansatz of Eqs. (4.56) and (4.57), i.e.,fn(ω,β) = ω3/2e−αpi(Z−1)√ µω e− ωnβ , (7.2)where β is a nonlinear parameter. Since the first channel is p+3 H and it involves the Coulombforce, a Gamow prefactor is included; the reduced mass µ isµ = mpm3Hmp +m3H . (7.3)The coefficients cn in Eq. (4.51) and the parameter β are obtained by a least square fitof the calculated IL(σ) with the integral transform of the regularized ansatz in Eq. (4.52),requiring that the resulting response function is zero below the threshold energy ωth, whereparticle emission starts. Since the first break-up channel is the proton-triton, ωth is obtainedby the difference of the ground-state energies of 4He and 3H, i.e., ωth = E0(4He)−E0(3H). TheCCSD approximation and the particle-removed equation-of-motion method [113, 119], lead tobinding energies of 23.97 and 7.37 MeV for 4He and 3H, respectively, and thus to a thresholdenergy of ωth = 16.60 MeV. With the employed chiral NN N3LO interaction [39], the correctbinding energies obtained from the EIHH method are 25.39 (7.85) MeV for 4He (3H), leading toa slightly different ωth = 17.54 MeV. Because for 4He we know the precise threshold results withthe N3LO potential, we require the response function to be zero below 17.54 MeV also whenwe invert the CCSD calculations. Figure 7.5 shows the comparison of the response functionsobtained by inverting IL(σ) from the coupled-cluster and EIHH calculations shown in Fig. 7.4.89Chapter 7. Results0.0 20.0 40.0 60.0 80.0 100.0 120.0ω [MeV]ω)[fm2 MeV−1]4HeEIHHCCSDFigure 7.5: Comparison of the 4He dipole response function calculated with CCSD (h̵Ω = 20,26 MeVand Nmax = 18) with the EIHH result. The bands are obtained by inverting LITs with both Γ = 10 and20 MeV.For the coupled-cluster calculations, we found that the inversions are insensitive to h̵Ω. Inprinciple, the inversion should also not depend on the parameter Γ. We use Γ = 10 MeV andΓ = 20 MeV to gauge the quality of the inversions. For the EIHH, the inversions obtained fromthe LITs at Γ = 10 and 20 MeV overlap perfectly, proving the precision of these calculations. Incase of the coupled-cluster, the two values of Γ lead to slightly different inversions, as shown inFig. 7.5, where the blue band accounts for these differences. The band is very narrow and can beviewed as a numerical uncertainty associated with the inversion procedure. Overall, the CCSDresponse function is very close to the virtually exact EIHH result. This suggests that the ef-fect of high order excitations in coupled-cluster – triples (T3) and quadruples (T4) – is very small.The electric dipole polarizability defined as an inverse energy weighted sum rule of the dipoleresponse function is expected to be sensitive to the low-energy region of the spectrum. FromFig. 7.5 we see negligible differences of the response functions calculated with EIHH and CCSDmethods below 30 MeV and so we also expect the calculated polarizability to agree betweenthe two methods.In Fig. 7.6, we show the electric dipole polarizability of 4He as a function of the model-space size Nmax. The solid horizontal line (black) is the polarizability obtained with the EIHHmethod for Kmax = 18. The lines with scatter-symbols are the results from the coupled-clustermethod obtained for different values of the frequency h̵Ω of the harmonic-oscillator basis usedin the calculation.90Chapter 7. Results6 8 10 12 14 16 18Nmax0.αD[fm3 ]4He h¯Ω = 20 MeVh¯Ω = 24 MeVh¯Ω = 26 MeVEIHHFigure 7.6: The electric dipole polarizability obtained in the CCSD approach in 4He as a functionof the model-space size Nmax for different values of the harmonic-oscillator frequency h̵Ω (dashed red,dotted blue, solid green). The converged value at Nmax = 18 is compared with the result obtained fromthe EIHH method (solid black).As expected, the polarizability is independent of the oscillator frequency as one increases Nmax,and converges rapidly to a value αD = 0.0815 fm3 [5], which compares fairly well with the EIHHresult of αD = 0.0831 fm3 (calculated as in Ref. [120]) and αD = 0.0822(5) fm3 obtained usingthe no-core-shell-model with the same interaction [121]. The small difference between the CCSDand EIHH results are attributed to the truncation to singles and doubles excitations only incoupled-cluster theory.7.2 Oxygen and Calcium isotopes with chiral NN forcesThe benchmarks on 4He suggest that the Lorentz integral transform formulation of the CCSDmethod can be employed for the computation of the dipole response and the electric dipolepolarizability. Theoretical uncertainties with respect to the model space and the inversion ofthe LIT are well under control. Because one of the advantages of CC is its mild computationalscaling with increasing mass number, we now turn our attention to medium-mass nuclei withclose-shell or close-subshell nature. In the following we will study 16,22O [1, 5, 105] and 40,48Ca [1,3, 6].7.2.1 Oxygen isotopesWe first investigate the convergence of the LIT as a function of the model-space size Nmax. InFig. 7.7, we present the LITs for Γ = 10 MeV (panels (a) and (b)) and Γ = 20 MeV (panels (c)and (d)) for both 16O and 22O.91Chapter 7. Results−20 0 20 40 60 80 1000.050.10.15I L(σ)[fm2 MeV−1](a)16OΓ = 10 MeV0 20 40 60 80 100(b)22OΓ = 10 MeV−20 0 20 40 60 80 100σ [MeV] L(σ)[fm2 MeV−1](c)16OΓ = 20 MeV0 20 40 60 80 100σ [MeV](d)22OΓ = 20 MeVNmax = 8Nmax = 10Nmax = 12Nmax = 14Nmax = 16Nmax = 18Figure 7.7: Panels (a) and (c) – Convergence of IL(σ) in 16O at Γ = 10 and 20 MeV as a function ofNmax for an harmonic-oscillator frequency of h̵Ω = 26 MeV. Panels (b) and (d) – Convergence of IL(σ)in 22O at Γ = 10 and 20 MeV as a function of Nmax for an harmonic-oscillator frequency of h̵Ω = 24MeV.The results are presented for a range of Nmax between 8 and 18, the convergence is rathergood and, as expected, it is better for the larger value of Γ. For Γ = 10 MeV, the maximumdifference between the LITs at Nmax = 16 and Nmax = 18 is found to be about 2% in 16O and4% in 22O in the region of interest 0 − 100 MeV. Next, in Fig. 7.8 we compare the convergedLITs for 22O versus 16O for the width Γ = 10 MeV and different values of the underlying HOfrequency parameter h̵Ω = 20 MeV (dotted line), 24 MeV (dashed line) and 26 MeV (solid line).For the best converged frequencies there is a residual h̵Ω dependence which is of roughly 2%and 5% for 16O and 22O respectively. Considering both model-space convergence and residualh̵Ω dependence, and by adding the latters in quadrature, the overall theoretical error associatedto the LITs for Γ = 10 MeV in the CCSD scheme amounts to 3% in 16O and 6% in 22O. Thesecan be considered as the uncertainty σb from the model-space truncations and approximationsof our ab initio method.92Chapter 7. ResultsFrom Fig. 7.8 we see that the 22O total strength is larger than that of 16O. The total dipoleresponse strength m0, also known as the bremsstrahlung sum rule and defined in Eq. (5.130),m0 = ∫ R(ω)dω = ⟨ΦL0 ∣Θ„Θ∣ΦR0 ⟩, (7.4)is also the strength of the LIT,m0 = ∫ IL(σ)dσ. (7.5)0 20 40 60 80 100σ [MeV] L(σ)[fm2 MeV−1] 22O16OΓ = 10 MeVFigure 7.8: Comparison of IL(σ) at Γ = 10 MeV for 22O (black lines) and 16O (blue lines). Differentharmonic-oscillator frequencies have been used: h̵Ω = 20 MeV (dotted lines), h̵Ω = 24 MeV (dashed lines)and h̵Ω = 26 MeV (solid lines).Integrating the curves in Fig. 7.8 using Eq. (7.5), and also calculating m0 with Eq. (7.4),we obtain the values m0(16O) = 4.64(1) fm2 and m0(22O) = 6.73(4) fm2. On the other hand,the bremsstrahlung sum rule can also be written as [122]m0 ∝ (NZA)2R2NZ , (7.6)where RNZ is the difference between the proton and the neutron centers of mass and Z andN the proton and neutron numbers, respectively. If one assumes that the two centers of massdo not differ much in 16O and 22O, the ratio between the m0 of16O and 22O using Eq. (7.6)is m0(22O)/m0(16O) = 1.27, which is within 13% of the ratio we predict from our calculations.Thus, the difference in strength can be qualitatively explained by the difference in neutronnumbers of the two isotopes.Another interesting feature of Fig. 7.8 is that the peak of the 16O LIT is slightly shifted tothe right with respect to the one of the 22O LIT; moreover, for the heavier isotope, we observe93Chapter 7. Resultsa large contribution to the strength at low energies (small σ values). The latter observationsuggests the possibility of a pygmy dipole resonance.To the purpose of comparing with experimental data, we perform the inversion of thecomputed LIT using the ansatz of Eq. (7.2). We first test the inversion on 16O. Within theCCSD scheme, the binding energy of 16O is 107.24 MeV and with the more precise perturbative-triples approach, Λ-CCSD(T) [8, 110], it becomes 121.47 MeV. The threshold energy in thiscase is the difference between the binding energy of 16O and 15N, and is computed using theparticle-removed equation-of-motion theory [8]. For the 16O photo disintegration reaction ωthbecomes then 14.25 MeV and in the inversion we require the response function to be zero belowthis threshold. For a fixed value of Γ, several choices of the number of basis states ν lead tobasically the same inversion. For 16O the inversions obtained from the LIT at Γ = 10 MeV areslightly different than those obtained from the LIT at Γ = 20 MeV [1]. This is due to the factthat the corresponding LITs (see Fig. 7.8) are converged only at a few-percent level and not tothe sub-percent level. Because such a difference is very small, we will interpret it as a numericalerror of the inversion and consider a band made by all of these inversions together as our finalresult in the CCSD scheme. The latter is presented in panel (a) of Fig. 7.9 in comparison to thedata by Ahrens et al. [64] and also to the more recent evaluation by Ishkhanov et al. [123, 124].10 20 30 40 50 60 70ω [MeV]ω)[fm2 MeV−1](a)16OCCSDIshkhanov et al.Ahrens et al.6 8 10 12 14 16 18 20ω [MeV](b) 22OCCSDLeistenschneider et al.Figure 7.9: Panel (a) – Comparison of the 16O dipole response function calculated with coupled-cluster(blue band) against experimental data by Ahrens et al. [64] (black triangles), and Ishkhanov et al. [123](red circles). Panel (b) – Comparison of the 22O dipole response function calculated in the LIT-CCSDscheme (blue band) against experimental data by Leistenschneider et al. [67] (orange squares). Forboth the panels the theoretical result are shifted on the experimental threshold energies to allow for acomparison of the GDR and PDR shapes.The blue band represents the CCSD result shifted on the experimental threshold ωexpth = 12.127MeV [125] to allow for a direct comparison of the shape of the GDR peak. The position of the94Chapter 7. ResultsGDR in 16O is rather well reproduced by our calculations. We find that the theoretical widthof the GDR is larger than the experimental one, while the tail region above 40 MeV is well de-scribed within uncertainties. The total dipole strength from our calculations, mth0 (16O) = 4.6(1)fm2, compares fairly well with mexp0 (16O) = 4.8(3) fm2 obtained by integrating Ahrens et inversion in 22O is obtained by imposing the strength to be zero below the theoreticalthreshold energy of 5.6 MeV, which differs from the experimental value of 6.85(6) MeV. Again,to compare the shape of our inversion with the data, in panel (b) of Fig. 7.9 we display theresponse function from the inversion shifted to the experimental threshold. Note that such ashift does not affect the shape of the inversion. Since in 22O the first break-up channel corre-sponds to the emission of a neutron, we did not include the Gamow prefactor in Eq. (7.2) in theinversion. In Fig. 7.9(b) we find a smaller resonant peak very close to the emission threshold.The existence of such a peak is a stable feature, independent on the inversion uncertainties.The latter are represented by the width of the bands, obtained by inverting LITs with Γ = 5,10and 20 MeV and varying the number of basis functions ν in Eq. (4.51). The low-lying peakfrom our inversion well reproduces the experimental one. While presently we cannot undoubt-edly confirm it, such low-energy peaks in the dipole response are often interpreted as dipolemodes of the excess neutrons against an 16O core (see, e.g. Ref. [126]). This is not the firsttime that the LIT approach suggests the existence of a low-energy dipole mode. In fact, EIHHcalculations [127] predict a similar, but much more pronounced peak in 6He for semi-realisticinteractions.In Fig. 7.10 we show the convergence of the polarizability of 16O as a function of the model-space size Nmax and different values of the underlying HO frequency h̵Ω.6 8 10 12 14 16 18Nmax0.αD[fm3 ]16O h¯Ω = 20 MeVh¯Ω = 24 MeVh¯Ω = 26 MeVAhrens et al.Figure 7.10: The electric dipole polarizability in 16O as a function of the model-space size Nmax fordifferent values of the harmonic-oscillator frequency h̵Ω (dashed red, dotted blue, solid green). Theconverged value at Nmax = 18 is compared to the experimental value (light green band) from Ref. [64].95Chapter 7. ResultsAgain, the convergence is very fast and the final value αthD (16O) = 0.461 fm3 is indepen-dent of h̵Ω. However, the converged value underestimates by 20% the experimental one ofαexpD (16O) = 0.568(9) fm3, obtained by integrating the weighted photoabsorption data fromAhrens et al. [64]. Using the same NN N3LO interaction, Hagen et al. obtained a chargeradius for 16O of rthch(16O) = 2.24 fm [35], which also underestimates the experimental value ofrexpch (16O) = 2.699(5) fm [128] by 17%, thus hinting to a possible deficiency of the Hamiltonian.For 22O the convergence of the polarizability with the current interaction is extremely slowand even at a model space of Nmax = 18 the result is not fully converged. We will return on the22O polarizability in Section 7.4, where the use of softer interactions will help to speed up theconvergence.7.2.2 Calcium isotopesData from photoabsorption experiments exist in 40Ca [64]. Experiments were performed onnatural Calcium targets, where 40Ca makes up for about 97% of the total abundance. 48Cais also an extremely interesting nucleus since it is neutron rich, has doubly-magic structure,and can now be reached by nuclear ab initio methods and other approaches such as densityfunctional theory (DFT), thus representing a bridging point between the two methods. Fur-thermore, 48Ca is almost stable and when it decays, it does so by double-beta decay. It isthe lightest nuclide known to undergo double-beta decay. This medium-mass nucleus has alsoattracted a lot of experimental interest lately, with the planned Calcium Radius Experiment(CREX) [129] at Jefferson Lab, and the Mainz Radius Experiment (MREX) [130] in Mainz,aiming at a measurement of the radius of its weak charge distribution. It is a candidate forsearches in neutrinoless double-beta decays [131] and measurements of its electric dipole polar-izability have been recently performed by the Osaka-Darmstadt collaboration [7].In Fig. 7.11 we show the dipole response function obtained from the inversion of the LIT inthe CCSD approximation for 40Ca. The inversion has been obtained making use of the ansatzin Eq. (7.2) and with a threshold energy ωth = 12.8 MeV obtained with the particle-removedequation-of-motion method. The blue band has been obtained by inverting IL(σ) calculatedat model space Nmax = 18 and three different values of the underlying HO parameter, namelyh̵Ω = 20,24 and 26 MeV. The width of the band is calculated by taking different widths ofthe LIT to invert (Γ = 5,10 and 20 MeV), and by varying the number ν of basis functions.The coupled-cluster result is compared with photoabsorption data on natural calcium fromAhrens et al. [64]. For the comparison we shifted the curve in Fig. 7.11 to the experimentalthreshold value, ωexpth = 8.38 MeV [125]. The location of the GDR predicted using the NN-N3LO interaction is found at slightly lower excitation energy with respect to the experiment.The result is quite encouraging: although broader and lower in strength than the experimentalone, a giant resonance is clearly observed.96Chapter 7. Results0 20 40 60 80 100ω [MeV]ω)[fm2 MeV−1]40CaCCSDAhrens et al.Figure 7.11: Comparison of the CCSD dipole response function obtained from the inversion of the LITwith Nmax = 18 and three values of h̵Ω = 20,24 and 26 MeV (blue band) with the photoabsorption dataof Ref. [64] (black squares).In Fig. 7.12 we show the convergence of the polarizability in (a) 40Ca and (b) 48Ca fordifferent values of h̵Ω. The convergence of 40Ca with respect to Nmax is excellent and almostno residual h̵Ω-dependence is seen at the maximum model-space size. For 48Ca the convergenceis slower and because of the increasing computational cost we could only reach Nmax = 16. Theslow αD convergence in48Ca also reflects itself in a slow convergence of the LIT, preventing usto obtain a stable inversion.With the present N3LO nucleon-nucleon interaction we predict a polarizability αD(40Ca) =1.47(2) fm3, both from the continued fraction and integration of the response function. Thisvalue is rather low in comparison to the experimental value αexpD (40Ca) = 1.87(3) fm3, rep-resented in panel (a) of Fig. 7.12 by the green band. The experimental value is obtained bycombining the data from Ref. [64] with a refined set of data in the giant resonance region mea-sured by the same group [132], as discussed by Birkhan et al. [7], and is much smaller thanthe one quoted in Ref. [64]. The preference of the data set from Ref. [132] is motivated by apreliminary comparison with 40Ca(p, p′) results taken at Osaka [7].For 48Ca we obtain αD(48Ca) = 1.83(6) fm3, where the uncertainty is calculated by lookingat the residual h̵Ω dependence, shown in panel (b) of Fig. 7.12. Our value is slightly lower, butcompatible with the experimental one of αexpD (48Ca) = 2.07(22) fm3 [7].97Chapter 7. Results6 8 10 12 14 16αD[fm3 ](a)40Cah¯Ω = 20 MeVh¯Ω = 24 MeVh¯Ω = 26 MeVAhrens et al.6 8 10 12 14 16 18Nmax1.αD[fm3 ](b)48Cah¯Ω = 20 MeVh¯Ω = 24 MeVh¯Ω = 26 MeVBirkhan et al.Figure 7.12: Panel (a) – The electric dipole polarizability in 40Ca as a function of the model-space sizeNmax for different values of the harmonic-oscillator frequency h̵Ω (dashed red, dotted blue, solid green).The converged values at Nmax = 18 is compared to the re-evaluated experimental value [7] of Ref. [64](green band). Panel (b) – The electric dipole polarizability in 48Ca as a function of the model-space sizeNmax for different values of the harmonic-oscillator frequency h̵Ω (dashed red, dotted blue, solid green).The best converged values at Nmax = 16 are compared to the recent experimental value obtained byBirkhan et al. [7] (orange band).Aside from 48Ca, where convergence is slower, we observe a systematic trend in the resultsfor the dipole polarizability when increasing the mass number A: our calculations underesti-mate αD in16O and 40Ca. On the other hand, the benchmarks performed on 4He showed anexcellent agreement with exact calculations, with very small discrepancies due to truncationsto singles and doubles in the CC scheme. Size-extensivity of CC theory [99–101] ensures thatthe accuracy of the method is not affected by the size of the system [102, 103], and so weexpect the truncations effect to be very small in heavier systems as well. Discrepancies betweenthe calculated values and the experimental data can then be mainly attributed to deficienciesof the employed Hamiltonian. It is worth to mention that with the present NN interaction40Ca is about 20 MeV overbound [133] and with a charge radius rch = 3.05 fm [1], which is98Chapter 7. Resultsconsiderably smaller than the experimental value of 3.4776(19) fm [128]. This points towards ageneral problem of the present Hamiltonian, which does not provide good saturation propertiesof nuclei, leading to too small radii and consequently too small polarizabilities.7.3 Chiral three-body interactionsIn Section 7.2 we have been using a NN force at N3LO to perform calculations for medium-massnuclei. However, the importance of 3NF to correctly describe nuclear properties has been high-lighted in various works [32, 134–137]. Even when supplementing chiral NN Hamiltonians with3NF, often calculations still yield too large separation energies [138], overbind medium-mass andheavy nuclei by about 1 MeV per nucleon and strongly underestimate charge radii [1, 31–38].For this reason, in the last years, substantial efforts have been devoted to the development ofmore accurate chiral Hamiltonians that can overcome the aforementioned shortcomings [16–18].In this section we augment the nuclear Hamiltonian to include 3NFs from χEFT. We willfirst use a new chiral NNLOsat interaction [18], which has been shown to accurately reproduceradii and binding energies up to the Calcium isotopes region which are consistent with theempirical saturation point of symmetric nuclear matter [18]. We then investigate correlationsbetween observables (e.g., between the polarizability and the radii) using a family of NN+3NFchiral interactions [16]. The three-nucleon part of these Hamiltonians is adjusted to the bindingenergy of 3H and the charge radius of 4He, and yield a realistic saturation point of nuclear mat-ter [16] as well as reproduce two-neutron separation energies of the calcium isotopic chain [139].In summary, we restrict ourselves to the use of the most modern three-body Hamiltonians,which reasonably well reproduce known experimental data for the radii in the mass region ofinterest.In this section we also want to investigate different methods to calculate the polarizability.In many cases the experimental results are limited to a narrow energy range and the calculationof the polarizability as performed in the previous section only gives the value of the full αDsum rule. As shown in Chapter 4, the electric dipole polarizability can be calculated in threedifferent ways:ˆ compute the LIT for the dipole response, obtain R(ω) from its inversion and computethe dipole polarizability from Eq. (4.8), i.e., asαD = 2α∫ R(ω)ωdω; (i)ˆ obtain the polarizability directly from the LIT using Eq. (4.15) for Γ→ 0, i.e.,αD = 2α limΓ→0∫ IL(σ)σ dσ; (ii)99Chapter 7. Resultsˆ use the continued fraction as in Eq. (5.114) or, in other words, obtain the polarizabilityfrom the Stieltjes integral transform viaαD = 2α limσ→0+ IS(σ). (iii)Method (ii) is in principle a discretization of the continuum and it will be interesting to compareit with the other two methods. Method (iii) is the less computational expensive one since,as shown in the benchmark of Fig. 7.2, we do not need many Lanczos coefficients to obtainconverged results for the polarizability. Method (ii) requires, in principle, converged LITs atleast at low energy and method (i) adds another layer of complexity since to obtain R(ω)the LITs must be converged also at medium energies, thus requiring the calculation of moreLanczos coefficients. Depending on the problem at hand, the above methods give differentways to approach the calculation of the polarizability and can also be used as a check forself-consistency. In fact, all the methods should return the same results.7.3.1 4He revisitedPanel (a) of Fig. 7.13 shows the electric dipole polarizability of 4He obtained from the continuedfraction (method (iii)) with the NNLOsat interaction, as a function of the model-space sizeNmax. The four curves represent calculations with different values of the oscillator frequencyh̵Ω = 18,20,22 and 24 MeV. The convergence in Nmax is excellent and independence on h̵Ω isreached with Nmax = 14, with an uncertainty from the different values of h̵Ω of about 0.1%. Thecalculation is compared to the combined experimental values from Refs. [140, 141] which havebeen extracted by fitting to experimental photoabsorption data from Arkatov et al. [142, 143](green band). In panel (b) of Fig. 7.13 we compare the three different ways to calculate thedipole polarizability for 4He. Methods (i) and (ii) require an integration in energy and wepresent αD(ε), where ε is the upper limit of the integration. The blue band shows method (i),i.e., αD is obtained by integrating the weighted response function where R(ω) stems from aninversion of the LIT. The width of the blue band is an estimate of the uncertainty involved inthe inversion procedure. The red solid line shows method (ii), i.e., αD obtained from the LITat small Γ. The black dashed line shows the result from method (iii), i.e., using the continuedfraction from the Stieltjes integral transform which corresponds to the converged result ofpanel (a) in Fig. 7.13. The green band represents, again, the experimental data from Arkatovet al. [140–143]. We note that the different methods yield the same dipole polarizability. Theintegration methods (i) and (ii) exhibit a similar dependence on the integration range ε, thedifference being that the former is smooth while the latter increases in steps. The value obtainedwith the NNLOsat interaction, αD = 0.0735 fm3 [2] is lower than the one found in Section 7.1.3,αNND = 0.0815 fm3 [5], with NN forces, and is in much better agreement with the experimentalvalue of 0.074(9) fm3 [140–143].100Chapter 7. Results6 8 10 12 14Nmax0.αD[fm3 ](a)4Heh¯Ω = 22 MeVh¯Ω = 24 MeVh¯Ω = 20 MeVh¯Ω = 18 MeVArkatov et al.0 20 40 60 80 100 120 140 160 180ε [MeV]αD(ε)[fm3 ](b) 4He(i)(ii)(iii)Arkatov et al.Figure 7.13: Panel (a) – The electric dipole polarizability in 4He as a function of the model-spacesize Nmax. Curves for different values of h̵Ω, the underlying harmonic-oscillator frequency, are shown.The green band is the experimental data from Arkatov et al. [140–143]. Panel (b) – The electric dipolepolarizability αD(ε) in 4He as a function of the integration energy ε: using the LIT and method (i)in blue (band), using method (ii) in red (solid), and using the continued fraction with the Stieltjestransform (method (iii)) in black (dashed). Calculations are performed for h̵Ω = 22 MeV and Nmax = 14.The green band represents the experimental data from Arkatov et al. [140–143].In Figure 7.14 we show the response function of 4He for different theoretical calculations withNN-only and NN+3NF interactions. For both calculations, the response function is obtainedfrom the inversion of the LIT as described in Refs. [1, 2, 75, 78, 105] and Section 4.4.3 of thiswork, with the width of the band being an estimate of the inversion uncertainty. The dark bandis the result obtained with the coupled-cluster method from Fig. 7.5 using the chiral N3LO NNinteraction [39]. The light blue band represents the calculation with NNLOsat [18] and has beenobtained by inverting the LIT with Γ = 10 and 20 MeV calculated at Nmax = 14 and h̵Ω = 22MeV. This is also the curve that has been integrated with method (i) in Figure 7.13(b). Toemphasize the shape of the response function in the continuum, in Fig. 7.14 we shifted the N3LO101Chapter 7. Resultscurve to the experimental threshold, i.e., from 17.54 MeV (theory) to 19.8 MeV (experiment),while the NNLOsat threshold well reproduces the experimental one since the potential has beentuned to reproduce the binding energies of 4He and 3H.20 30 40 50 60 70ω [MeV]ω)[fm2 MeV−1]4He Nakayama et al.Arkatov et al.Nilsson et al.Shima et al.Tornow et al.NN(N3LO) (CCSD)NNLOsat (CCSD)Figure 7.14: 4He photoabsorption response function calculated with interactions (see text for details)compared with experimental data from Nakayama et al. [144] (blue circles), Arkatov et al. [142, 143](white squares), Nilsson et al. [145] (yellow squares), Shima et al. [146, 147] (magenta circles) and Tornowet al. [148] (green squares).We find that the NNLOsat response function, which includes three-nucleon forces, presents alarger peak with respect to NN -only calculations. Finally, the theoretical results are comparedwith the experimental data from Nakayama et al. [144] (blue circles), Arkatov et al. [142, 143](white squares), Nilsson et al. [145] (yellow squares), Shima et al. [146, 147] (magenta circles)and Tornow et al. [148] (green squares).7.3.2 Medium-mass nuclei: Oxygen and CalciumGiven the discrepancies in the description of the dipole polarizability with the NN-N3LO inter-action only, it is now interesting to study 16,22O with NN+3NF using the NNLOsat Hamiltonian.With respect to the NN interaction used in previous calculations, NNLOsat converges faster.Indeed, with NNLOsat we are able to obtain reasonably converged results for the polarizabilityof 22O.In Fig. 7.15 we show the electric dipole polarizability for 16O (a) and 22O (b) as a functionof the model-space size calculated with the NNLOsat interaction. We observe that the curvesfor different h̵Ω values converge very nicely and only a small residual dependence remains at102Chapter 7. Resultsthe largest model-space size Nmax = 14. The neutron-rich 22O seems to converge slower thanits double-magic neighbour 16O.αD[fm3 ](a)16Oh¯Ω = 18 MeVh¯Ω = 20 MeVh¯Ω = 22 MeVAhrens et al.6 8 10 12 14Nmax0.αD[fm3 ](b)22Oh¯Ω = 18 MeVh¯Ω = 20 MeVh¯Ω = 22 MeVFigure 7.15: Electric dipole polarizability in 16,22O with the NNLOsat interaction. Panel (a) – Electricdipole polarizability in 16O as a function of the model-space size Nmax for different values of the h̵Ωparameter. The theoretical result is compared with the experimental data from Ahrens et al. [64] (greenband). Panel (b) – Electric dipole polarizability in 22O as a function of the model-space size Nmax fordifferent values of the HO parameter h̵Ω.This is because the excess neutrons in 22O are loosely bound, making the wave function moreextended and thus the convergence slower. We also note that αD of22O is larger than for 16O,thus suggesting an enhancement of the dipole strength at low energy due to the neutron excess.This enhancement is a possible signature of a PDR in 22O as observed experimentally by Leis-tenschneider et al. [67]. The values of Fig. 7.15(b) represent the polarizability integrated up toinfinity. Since the experimental data are limited to energies below 25 MeV, it is not possible tocompare the results of Fig. 7.15(b) with the experimental one obtained by integrating the data103Chapter 7. Resultsfrom Leistenschneider et al. [67]. A possible comparison method will be discussed later on inthis section.In panel (a) of Fig. 7.15 we also compare our result with the weighted integral of the experi-mental data from Ahrens et al. [64] (green band) and we see a very nice agreement as opposedto the calculations with NN forces only in Fig. 7.10 which largely underestimated αD in16O.In Fig. 7.16 we calculated the charge radii of 16O (a) and 22O (b), which have been obtainedfrom the point-proton radius taking into account contributions from nucleonic charge radii (seeRef. [6] and Appendix C for details).16 18 20 22h¯Ω [MeV]2.62.652.72.752.82.85r ch[fm](a)16ONmax = 10Nmax = 12Nmax = 14Exp.14 16 18 20 22h¯Ω [MeV]2.62.652.72.752.82.85r ch[fm](b)22ONmax = 10Nmax = 12Nmax = 14Figure 7.16: Charge radii in 16O (a) and 22O (b) as a function of the HO oscillator frequency h̵Ωand for different values of model-space size Nmax calculated with the NNLOsat interaction. In16O thetheoretical result is compared with the experimental data [128] (green band).In both panels we show the convergence behaviour, where each curve represents a model-space104Chapter 7. Resultssize Nmax as a function of the HO frequency h̵Ω. We see that for the largest model-space sizeNmax = 14 the curves tend to flatten, showing independence with respect to h̵Ω. In both cases,h̵Ω = 20 MeV is the best converging frequency as the charge radius value is almost unaffectedwhen going from Nmax = 10 to Nmax = 14. Selecting the best converging three values of h̵Ωwe obtain rthch(22O) = 2.739(2) fm and rthch(16O) = 2.712(2) fm, the latter overestimating byonly 0.5% the experimental value of rexpch (16O) = 2.699(5) fm [128] and shown in panel (a) ofFig. 7.16 with a green band. No experimental data exist for the charge radius of 22O.Panel (a) of Fig. 7.17 compares the results from the three methods (i), (ii) and (iii) to obtainthe polarizability for 16O.105Chapter 7. Results0.αD(ε)[fm3 ](a) 16OAhrens et al.(i)(ii)(iii)20 40 60 80 100 120 140 160ε [MeV]αD(ε)[fm3 ](b) 22O(ii)(iii)Figure 7.17: The electric dipole polarizability αD(ε) in 16O (a) and 22O (b) as a function of theintegration limit ε obtained with the NNLOsat interaction. The blue band is obtained integrating theweighted response function as in method (i); the red solid curve is calculated integrating the weightedLIT at small Γ as in method (ii); the black dashed line is obtained from the continued fraction andthe Stieltjes integral transform (method (iii)). Calculations are performed with Nmax = 14 and h̵Ω = 22MeV in 16O and h̵Ω = 18 MeV in 22O. The green band in panel (a) represents the experimental data for16O [64].The blue band shows the integration of the weighted response function (see Fig. 7.18) as inmethod (i), and the width of the band takes into account the uncertainty of the inversion. Thered solid line refers to method (ii) and the black dashed line is the reference value calculatedwith the continued fraction using method (iii). Again, we find good agreement of the resultsfor the dipole polarizability, as well as agreement with the experimental data [64] (green band).In panel (b) we show the running sum of αD for22O. We used the largest model space and thefastest converging frequency of h̵Ω = 18 MeV. We find good agreement between method (i), (ii)and (iii). However, because the LIT convergence is not good enough to obtain a stable inversion,106Chapter 7. Resultswe do not include method (i) in the figure for 22O. Nevertheless, by looking at the ladderedcurve we learn about the convergence of αD as a function of the energy. This can be used tomake a comparison with the experimental data from Leistenschneider et al. [67]. To compareour calculations with experimental data we integrate the experimental strength of Ref. [67] up tothe available energy range of about 18 MeV above threshold, obtaining αexpD (18MeV) = 0.43(4)fm3. This value is much lower than our calculated αthD = 0.86(4) fm3 shown in panel (b) ofFig. 7.15, which corresponds to the integration of the strength up to infinity. The theoreticalresult exceeds the experimental value by a factor of two and we also find that the integration ofthe theoretical strength over the first 18 MeV exhausts the 87% of the polarizability sum rule.On the other hand, Leistenschneider et al. observed a PDR extending for about 3 MeV abovethe neutron emission threshold of Sn = 6.85 MeV. Integrating the data over this interval yieldsa dipole polarizability αexpD (PDR) = 0.07(2) fm3. While our calculations in Fig. 7.17 do notreproduce the experimental threshold, integration over the first 3 MeV of the curve obtainedwith method (ii), and considering the different h̵Ω frequencies, yields αthD (PDR) = 0.05(1) fm3.This is consistent with the experimental result.Finally, in Fig. 7.18 we show the response function for 16O calculated with the NN-N3LOinteraction (from panel (a) of Fig. 7.9) with the dark band, and then with NNLOsat in light.The calculations are compared with the experimental data from Ahrens et al. [64] (triangleswith error bars) and Ishkhanov et al. [123, 124] (red circles). The response function withNNLOsat has been obtained again by inverting the LIT with both Γ = 10 and 20 MeV andat frequency h̵Ω = 22 MeV. The large error band for the NNLOsat results from the fact thatthe largest available model-space size in our calculation, namely Nmax = 14 is smaller than theNmax = 18 used for the N3LO potential. Similarly to what is done for 4He, also in this casewe plot the theoretical curves to start from the experimental threshold. We shift the N3LO(NNLOsat) curve from the theoretical threshold of 14.25 (10.69 [18]) MeV to the experimentalvalue of 12.1 MeV. In particular, for the NNLOsat, we have found that by using the thresholdenergy as a fit parameter in the inversion, we find that αD is correctly reproduced only witha threshold energy varied around 5% of the experimental value. Overall, it is interesting tosee that three-nucleon forces enhance the strength in the GDR region, slightly improving thecomparison with the experimental data.107Chapter 7. Results10 20 30 40 50 60ω [MeV]ω)[fm2 MeV−1]16O Ahrens et al.Ishkhanov et al.NN(N3LO) (CCSD)NNLOsat (CCSD)Figure 7.18: 16O photoabsorption response function calculated with the LIT-CCSD method with theN3LO NN interaction (dark band) and NNLOsat (light band). The red circles are the experimental datafrom Ishkhanov et al. [123, 124] while the white triangles with error bars are the experimental resultsby Ahrens et al. [64].Moving to the Calcium isotopes, calculations with NNLOsat predict a value of 3.48 fm [2,6] for the charge radius of 40Ca which compares fairly well with the experimental value of3.4776(19) fm [128]. On the other hand, the polarizability with NNLOsat is found to be 2.08fm3 which, if compared to the re-evaluated experimental value of 1.87(3) fm3 [7, 64], overes-timates αD by 11%. The uncertainty from basis truncations and from the approximation ofthe cluster operator T at single and double excitations is not expected to be the source ofthe disagreement. Furthermore, using NN and NN+3NF interactions we found that the de-pendence on the Hamiltonian used is much larger than the truncations of CCSD, estimatedat a few percent based on size extensivity. We can attempt to probe systematic theoreticaluncertainties due to the employed interaction by considering results from different families ofHamiltonians. To do so we study correlations between the polarizability and the charge radius,which is expected in medium- and heavy-mass nuclei from the nuclear droplet models [149, 150].This approach has been widely used both in DFT and ab initio calculations to investigate avariety of observables [2, 6, 40, 151–155]. To study such correlations, one needs a consider-able number of different interactions, so that results spanning a wide range of values for theobservables under investigation can be obtained. To this purpose, we choose to use similarityrenormalization group (SRG) [156] and Vlow−k [157] evolutions as a tool to generate a set ofphase-shift equivalent two-body interactions. We then use NNLOsat for calculation with 3NFand a set of NN+3NF from Hebeler et al. where 3NF are added at N2LO to the chiral N3LOnucleon-nucleon potential [16, 39] we used in Sections 7.1 and 7.2. Furthermore, when adding108Chapter 7. Results3NF forces – without considering the induced three-body forces – the low-energy constants werecalibrated on light nuclei observables [16].Figure 7.19 shows αD as a function of rch in16O – panel (a) – and 40Ca – panel (b) – forvarious interactions. Empty symbols correspond to calculations with NN potentials only. In2.0 2.2 2.4 2.6 2.8rch [fm]αD[fm3 ]16O2.4 2.6 2.8 3.0 3.2 3.4 3.6rch [fm]αD[fm3 ]40Ca(exp)(a)(b)(c)(d)(e)(f)Figure 7.19: αD versus rch in16O and 40Ca. Empty symbols refer to calculations with NN potentialsonly: (a) SRG evolved Entem and Machleidt [39] interaction with Λ = 500 MeV and λ =∞,3.5,3.0,2.5,and 2.0 fm−1; (b) SRG evolved Entem and Machleidt [39] interaction with Λ = 600 MeV and λ = 3.5,3.0,and 2.5 fm−1; (c) are SRG evolved CD-Bonn [52] potential with λ = 4.0 and 3.5 fm−1; (d) Vlow−k [157]evolved CD-Bonn interaction with λ = 3.0,2.5 and 2.0 fm−1; (e) Vlow−k-evolved AV18 [55] interactionwith λ = 3.0 and 2.5 fm−1. The red diamonds (f) refer to calculations that include 3NF: the large one isobtained with NNLOsat, and the others from chiral interactions as in Ref. [16]. The green bands (exp)show the experimental data for the radius [64, 128], and for the polarizability we use the re-evaluatedexperimental values obtained by integrating the photoabsorption data from Ahrens et al. [64].109Chapter 7. Resultsparticular, in Fig. 7.19, the circles (a) are obtained from SRG evolved Entem and Machleidt [39]interaction with cutoff Λ = 500 MeV and, in order of decreasing rch values, λ =∞,3.5,3.0,2.5,and 2.0 fm−1, while for the squares (b) we used the same interaction with cutoff λ = 600 MeVand, in order of decreasing rch values, λ = 3.5,3.0, and 2.5 fm−1. The points represented with tri-angles pointing up (c) are calculations with SRG evolved CD-Bonn [52] potential with, in orderof decreasing rch value, λ = 4.0 and 3.5 fm−1, while triangles pointing down (d) are calculationswith the Vlow−k [157] evolved CD-Bonn interaction and λ = 3.0,2.5 and 2.0 fm−1. The hexagons(e) are calculations with Vlow−k-evolved AV18 [55] interaction and λ = 3.0 and 2.5 fm−1, in orderof decreasing radius. The red diamonds (f) are calculations including 3NFs. The larger reddiamond is the value obtained with NNLOsat, while the smaller ones are the potentials fromHebeler et al. from Ref. [16]. The error bars for the calculations represent uncertainties aris-ing both from the coupled-cluster truncation scheme and the model-space truncations and areestimated to be of the order of 1% for the charge radius [18, 93] and 2% for the polarizability(see Sections 7.1.3 and 7.3.3 for more details). Finally, the green bands are the experimentalvalues for the polarizability [64] and the charge radius [128].From Fig. 7.19 it is clear that αD and rch are strongly correlated. We also note that NNinteractions alone systematically underestimate both αD and rch, while the inclusion of 3NFsimproves the agreement with data. It is evident that one cannot blindly use a correlation be-tween theoretical data points to extrapolate to experimental results. The data based on NNinteractions, even when extrapolated with a simple linear or quadratic curve, do not meet theexperimental values. In contrast, the results from NN+3NFs can be interpolated in the vicinityof the experimental data (when, e.g., the charge radius is known) to yield a sensible predictionfor the dipole polarizability.We can also use the different three-body interactions to gauge the uncertainty on the po-larizability running sum. In Fig. 7.20 we show the running sum of the polarizability αD(ε) as afunction of the upper integration threshold ε of method (ii). Panel (a) shows the running sumin 16O and panel (b) in 40Ca. The grey band is the result obtained using the family of chiralNN+3NF interactions from Hebeler et al. [16] and NNLOsat and it is compared to the weightedintegral of the experimental cross sections from Ahrens et al. [64] (green bands). For 16O wealso plot in blue the running sum from Fig. 7.17 obtained using method (i) and the NNLOsatinteraction. The theoretical calculations qualitatively reproduce the experimental data. Thefull polarizability sum rule, i.e., the integral with limit ε → ∞ – also equivalent to the valueobtained from the continued fraction of method (iii) – is in good agreement with the experi-mental value. By looking at the steepness of the running sums around the GDR peak region wecan see that both theoretical and experimental data show a narrow GDR in 40Ca and a widerone in 16O. In 16O the theory predicts less strength below 25 MeV than the experimental data,which is in line with what we previously observed in the inverted response function of Fig. 7.18.110Chapter 7. Results0.αD(ε)[fm3 ](a) 16OχEFT(i)− NNLOsatAhrens et al.0 20 40 60 80 100ε [MeV]αD(ε)[fm3 ](b) 40CaχEFTAhrens et al.Figure 7.20: Running sum of the electric dipole polarizability in 16O (panel (a)) and 40Ca (panel (b))as a function of the upper integration limit ε. The gray bands are calculations based on a set of chiraltwo- plus three-nucleon interactions [16, 18] and are compared with the integrated experimental data ingreen [64]. For 16O we also present the running sum obtained using the response function obtained withthe NNLOsat interaction (blue band).In the low-energy region, experimental data and theoretical results are not in perfect agree-ment. The gray bands are calculated by integrating the LIT at Γ = 0.01 MeV, for which theconvergence is quite slow compared to Γ = 10 and 20 MeV. Furthermore, as we have seen inSection 7.1, the LIT-CCSD approximation slightly underestimates the polarizability in 4He,thus suggesting that triples contributions would probably enhance αD and possibly shift thecurves of the running sums to lower energies.Finally, to conclude, in Table 7.1 we summarize the results obtained with the NNLOsatinteraction for 4He and 22O, while in Table 7.2 we present the results obtained with the differentNN+3NF in 16O and 40Ca.111Chapter 7. ResultsTable 7.1: Theoretical values of αD for different nuclei calculated with the NNLOsat interaction incomparison to experimental data from [141–143] and other calculations from Refs. [121] (a), [118] (b)and [158] (c) for 4He. For 22O we compare to the value obtained integrating the data from Ref. [67] firstover the whole energy range (d) and then only the first 3 MeV of the strength (e), corresponding to thelow-lying dipole strength. Values are expressed in fm3. The theoretical uncertainties of our calculationsstem from the h̵Ω dependence in the model space with Nmax = 14.Nucleus Theory Exp4He 0.0735(1) 0.074(9)0.0673(5)a0.0655b0.0651c0.0694c22O 0.86(4) 0.43(4)d0.05(1) 0.07(2)eTable 7.2: List of results in 16O and 40Ca with three-body Hamiltonians [16, 18] for ground-state energyper nucleon, charge radius and electric dipole polarizability, plotted in Fig. 7.19. For the notation ofthe potentials we follow Ref. [16]. Energies have been obtained with a Λ −CCSD(T) approximation [8,110, 111]. Experimental values are taken from Ref. [125] (energy), Ref. [128] (radius). The experimentalvalues of the polarizabilities have been obtained integrating the data from Ref. [64].16OInteraction E0/A [MeV] rch [fm] αD [fm3] -7.70 2.62 0.462.0/2.0(PWA) -7.14 2.74 0.541.8/2.0(EM) -7.98 2.60 0.442.2/2.0(EM) -7.50 2.63 0.482.8/2.0(EM) -7.16 2.67 0.52NNLOsat -7.68 2.71 0.58Experiment -7.98 2.6991(52) 0.57(1)40Ca2.0.2.0(EM) -8.22 3.35 1.672.0/2.0(PWA) -7.24 3.55 2.031.8/2.0(EM) -8.69 3.31 1.572.2/2.0(EM) -7.89 3.38 1.752.8/2.0(EM) -7.35 3.44 1.94NNLOsat -8.15 3.48 2.08Experiment -8.55 3.4776(19) 1.87(3)112Chapter 7. ResultsIn this Section we showed that 3NFs are fundamental to correctly describe radii and po-larizability in medium-mass nuclei. However, calculations show a strong dependence on theemployed Hamiltonian, from which correlations amongst observables emerge, e.g., αD and rch.Different families of interactions and correlations can be used to gauge the uncertainty due tothe input Hamiltonian. In the next Section we show an application of correlations to predictthe neutron-skin thickness, the difference of the neutron and proton root-mean-square radii,and electric dipole polarizability in 48Ca [2, 6, 7].7.3.3 Neutron skin and polarizability in 48CaThe nucleus 48Ca is of particular interest because it is neutron-rich, has doubly magic structureand the study of its dipole response and αD can offer insights into the properties of neutron-rich matter and its equation of state (EOS) (see Ref. [159] for a detailed review). The EOSof neutron-rich matter governs the properties of neutron-rich nuclei, the structure of neutronstars, and the dynamics of core-collapse supernovae [160, 161]. The largest uncertainty of theEOS at nuclear densities for neutron-rich conditions stems from the limited knowledge of thesymmetry energy J , which is the difference of the energies of neutron and nuclear matter atsaturation density, and the slope of the symmetry energy L, which is related to the pressure ofneutron matter [162, 163]. The symmetry energy also plays an important role in nuclei, where itcontributes to the formation of neutron skins in the presence of a neutron excess. Calculationsbased on DFTs pointed out that J and L can be correlated with isovector collective excitationsof the nucleus such as PDRs [164] and GDRs [165], thus suggesting that the neutron-skinthickness could be constrained by studying properties of collective isovector observables at lowenergy [166]. Thus, the electric dipole polarizability represents a viable tool to constrain theEOS of neutron matter and the physics of neutron stars, as well as to predict neutron skins inneutron-rich systems [167–172].Furthermore, 48Ca represents the first opportunity to compare αD and the neutron skinfrom ab initio calculations based on χEFT interactions with state-of-the-art DFT calculationsin the same nucleus, an important insight which will also impact the planned CREX and MREXexperiments [129, 130].Figure 7.21 shows the predicted values for the polarizability αD (a) and the neutron-skinradius rskin (b) calculated with coupled-cluster theory. The red diamonds represent the cal-culation with NNLOsat and the family of interactions from Ref. [16] also used in Fig. 7.19.Benchmark results for 4He show that CCSD calculations yield an intrinsic radius that is byabout 1% too large when compared to numerically exact calculations [18]. Using again sizeextensivity arguments, we assume that radii computed for heavier nuclei similarly exhibit anuncertainty of about 1%. Regarding the uncertainty due to the truncation of the model-space,we find that the point-nucleon radii in 48Ca increase by 0.02 fm when increasing the modelspace from E3max = 14 to E3max = 16. It is expected that increasing the model-space size113Chapter 7. Resultsbeyond the current limit will slightly increase the computed radii. Our CCSD computationsoverestimate the radii slightly, thus compensating for part of the model-space uncertainty. Wethereby arrive at a total method uncertainty of about 1% coming from both the CCSD ap-proximation and the model-space truncation. In Section 7.1 we also showed that the CCSDresult for the electric dipole polarizability αD for4He is within 1% of the numerically exactEIHH approach. Combining this uncertainty with the model-space truncation we arrive at anuncertainty estimate of 2% for αD in48Ca. These method uncertainties are shown as error barson the computed data in Fig. 3.2 3.4 3.6 3.8rch [fm]αD[fm3 ]48Ca(a)3.0 3.2 3.4 3.6 3.8rch [fm] skin[fm]48Ca(b)Figure 7.21: Electric dipole polarizability αD (a) and neutron skin rskin (b) plotted versus the chargeradius rch. The ab initio predictions with NNLOsat and the set of chiral interactions of Hebeler et al. [16]are represented with red diamonds. The theoretical error bars estimate uncertainties from truncations ofthe employed method and model space. The red line represents a linear fit to the data and the red bandencompasses all error bars and estimates systematic uncertainties. The vertical green line marks theexperimental value of rch [128]. Its intersection with the red line and the red band yields the horizontalblue band, giving the predicted range for the abscissa.The red band is obtained via an orthogonal least square fit [173] and the width of the bandis chosen to be the maximum between the 95% confidence interval and the orthogonal distanceof the calculated points with respect to the fit line. The green band represents the very preciseexperimental value for the charge radius [128]. By intersecting the red band from the theoreticalfit and the green experimental band, we are able to predict values (blue band) for αD in panel(a) and rskin in panel (b). Following this procedure we obtain 2.19 ≲ αthD (48Ca) ≲ 2.60 fm3 [6]and 0.12 ≲ rthskin(48Ca) ≲ 0.15 fm [6]. The predicted range for rskin is appreciably lower thanthe combined DFT estimate of 0.176(18) fm [152] and is well below the relativistic DFT valueof 0.22(2) fm [152]. On the other hand, the predicted value for αD is consistent with the DFTvalue of 2.306(89) fm3 [152].While no experimental data exist for the neutron-skin radius – CREX [129] and MREX [130]will measured it – the Darmstadt-Osaka collaboration recently obtained experimental valuesfor αD in48Ca [7]. In panel (a) of Fig. 7.22 we show the experimental photoabsorption cross-114Chapter 7. Resultssection of 48Ca from Birkhan et al. [7]. The Darmstadt-Osaka collaboration only measured theregion of the GDR (blue circles).Figure 7.22: Figure from Birkhan et al. [7]. Panel (a) – Combined photoabsorption cross sections in48Ca from Ref. [7] (blue circles) for excitation energy ω ≤ 25 MeV and from Ref. [64] (red squares)for 25 ≤ ω ≤ 60 MeV. Panel (b) – Running sum of the electric dipole polarizability in comparison toχEFT predictions, where the gray band is based on calculations with NNLOsat and the set of two- plusthree-nucleon interactions from Ref. [16].Reference [7] notes that the cross-sections of 40,48Ca are very similar and they are basicallythe same but for an energy shift of a few MeV. Birkhan et al. then proceed in combining theirdata in 48Ca with Ahrens et al. [64] data in 40Ca to reproduce the tails of the cross sectionat large energies (red squares). Panel (b) of Fig. 7.22 shows a comparison of the theoreticalresults to experiment, where the smooth band (blue and red) shows the running sum of theexperimental dipole polarizability with error bars. The laddered (gray) band is based on differ-ent chiral Hamiltonians, using the same NN+3NF interactions employed in Figs. 7.20 and 7.21.Once again, we find that the agreement between the experimental and theoretical results inFig. 7.22(b) is better for higher excitation energies ω. However, as discussed previously for16O and 40Ca, the position of the GDR is more affected by truncations and suffers from theslow convergence of the LIT at small Γ, which could lead to a shift of a few MeV. In addition,contributions from triple corrections could be important at low energies. Both of these trunca-115Chapter 7. Resultstion errors are not included in the uncertainty shown in Fig. 7.22(b), because it is difficult toquantify them without explicit calculations. With these taken into account, the steep rise inthe theoretical band around 20 MeV indicates the position of the GDR peak is consistent withthe experimental centroid.In Fig. 7.23, we present a detailed comparison of the experimental αD value with predic-tions from χEFT and state-of-the-art DFT calculations. The χEFT predictions (green trian-gles) are based on the set of chiral two- plus three-nucleon interactions by Hebeler at al. [16]and NNLOsat, whereas the DFT results are based the functionals SkM∗, SkP, SLy4, SV-min,UNEDF0 and UNEDF1 (see Ref. [6] for details). In addition, we show the χEFT predictionselected to reproduce the 48Ca charge radius obtained with the correlation plot in Fig. 7.21 andthe range of αD predictions [153] from DFTs providing a consistent description of polarizabilitiesin 68Ni [68], 120Sn [66], and 208Pb [65].αD(fm3)48CaFigure 7.23: Figure from Birkhan et al. [7]. Experimental electric dipole polarizability in 48Ca (blueband) and predictions from χEFT [16, 18] (green triangles) and DFT calculations [6, 7] (red squares).The green and black bar indicate the χEFT prediction selected to reproduce the 48Ca charge radius [6](see Fig. 7.21) and the range of αD predictions [153] from DFTs providing a consistent description ofpolarizabilities in 68Ni [68], 120Sn [66], and 208Pb [65], respectively.Taking only the interactions and functionals in Fig. 7.23 consistent with the experimentalrange implies a neutron skin in 48Ca of 0.14-0.20 fm, where the lower neutron skin in this range(< 0.15 fm) is given by our ab initio calculations. For the latter, the small neutron skin isrelated to the strong N = 28 shell closure, which leads to practically the same charge radii for40Ca (3.4776(19) fm [128]) and 48Ca (3.4771(20) fm [128]).7.4 Inclusion of triples effectsThe results obtained with the coupled-cluster method using the singles and doubles approxi-mation and three-body chiral interactions show very good agreement with experimental dataand with exact calculations in 4He. The small discrepancies between coupled-cluster and EIHH116Chapter 7. Resultscalculations observed in the dipole strength and the polarizability in 4He amount to roughly1.5%. Size extensivity of coupled-cluster theory suggests the small error propagates to heaviernuclei with the same magnitude. In what follows, we show a first study of triples effects in theCCSDT-1 approximation for αD and the m0 sum rule. We first revisit4He using the Entem-Machleidt NN interaction at N3LO previously used to benchmark coupled-cluster in the CCSDapproximation with EIHH calculations. We then proceed to study the heavier 16O.The major challenge when including triples excitations in coupled-cluster theory is repre-sented by the large amount of three-body matrix elements which one needs to store. As themodel space Nmax and the mass number A increase, the number of 3p–3h configurations growsfactorially. The increase in the number of configurations is shown in Fig. 7.24, where the num-ber of 2p–2h (diamonds) and 3p–3h (circles) configurations are plotted as a function of themodel-space size Nmax in4He (solid blue line) and 16O (dashed red line).6 8 10 12 14 16 18 20Nmax1031041051061071081091010#confs.2p2h3p3h4He16OFigure 7.24: Number of 2p–2h (diamonds) and 3p–3h (circles) configurations as a function of themodel-space size Nmax in4He (solid blue line) and 16O (dashed red line).While spherical coupled-cluster allows us to extend calculations to medium- and heavy-massnuclei, it is not enough to reduce the 3p−3h configurations number to a manageable size. There-fore, due to the limited amount of computational resources currently available, one has to comeup with a smart truncation scheme for the 3p–3h configurations to reduce the total amount ofmatrix elements. As discussed at the beginning of this chapter, for the matrix element of thethree-body forces one usually truncates the 3p–3h configurations using an energy cut – E3max117Chapter 7. Results– on the energy of the three-body state ∣abc⟩. In panel (a) of Fig. 7.25 we show the effect ofthis energy cut on the electric dipole strength m0 of4He calculated within the CCSDT-1 ap-proximation at Nmax = 6 (blue line). The values obtained at different truncations are comparedwith the full calculation result represented by the black dashed horizontal line. Evidently, onecannot truncate the calculation at an arbitrarily chosen value E3max, as convergence is reachedonly at full E3max space.0 3 6 9 12 15 18E3max [h¯Ω]1.0051.00751.011.01251.0151.01751.021.02251.025m0[fm2 ](a)4He0 3 6 9 12 15 18L3max [h¯](b)4HeFigure 7.25: Convergence of the m0 sum rule in4He in the CCSDT-1 approximation with the NN-N3LO interaction [39] and Nmax = 6 as a function of E3max (a) in blue and L3max (b) in red. The dashedline (black) represents the converged result at full E3max and L3max spaces.A better alternative to the energy cut is achieved by truncating the different partial-wave contributions, i.e., by imposing a truncation on the total orbital angular momentumL3max = la + lb + lc of the 3p–3h configuration ∣abc⟩. Such truncation is shown in panel (b)of Fig. 7.25 by the red line. This truncation on the total angular momentum has previouslybeen used by Hagen et al. [174] for the calculation of ground-state energies with, e.g., triplescorrections. The convergence in panel (b) of Fig. 7.25 is much faster than the one in panel (a)and the calculation is already fully converged at L3max = 10 − 12. A converged calculationin the E3max = 18 space requires 30% more configurations with respect to a converged onein the L3max = 10 space. Depending on the available computational resources, we could alsotruncate L3max = 8, which returns an error with respect to the converged result comparable toE3max = 16, but with 45% less 3p–3h configurations.In Fig. 7.26 we show the convergence of the polarizability in 4He calculated with the NN-N3LO interaction for h̵Ω = 26 MeV and different Nmax as a function of the angular-momentumtruncation L3max of the 3p–3h configurations. Different colours represent different Nmax. The118Chapter 7. Resultshorizontal solid black line is the result from EIHH calculations using the same interaction. Thepolarizability is practically converged for all model spaces with L3max = 15.0 3 6 9 12 15 18 21L3max0.0810.0820.0830.0840.0850.0860.0870.0880.0890.09αD[fm3 ]4Heh¯ω = 26 MeVNmax = 8Nmax = 10Nmax = 12Nmax = 14EIHHFigure 7.26: Convergence of αD in4He with the chiral NN-N3LO [39] interaction with respect to theangular momentum cut L3max at h̵Ω = 26 MeV, and for different model-space size Nmax. The blackhorizontal line is the exact result from EIHH calculations with the same interaction.The convergence in Nmax is also very good and Fig. 7.27 shows the convergence in Nmax fortwo different values of h̵Ω = 24 and 26 MeV at a fixed angular-momentum cut of L3max = 21.Overall, the convergence in L3max and Nmax is very good and shows only a small dependenceon h̵Ω at the maximum model space.8 10 12 14 16Nmax0.080.0820.0840.0860.0880.09αD[fm3 ]4Heh¯Ω = 24 MeVh¯Ω = 26 MeVEIHHFigure 7.27: Convergence of αD in4He with the chiral NN-N3LO [39] interaction with respect to themodel-space size Nmax for different values of h̵Ω and at fixed L3max = 21. The black horizontal line isthe exact result from EIHH calculations with the same interaction.119Chapter 7. ResultsThe m0 sum rule convergence is of similar quality to the one for αD, and in Fig. 7.28 wecompare the results for m0 (a) and αD (b) obtained with the coupled-cluster method in theCCSDT-1 approximation (red band) and at singles and doubles level (blue band). The blackline is the exact calculation with the EIHH method.0.860.880.90.920.940.960.981.0m0[fm2 ](a)4HeEIHHCCSDCCSDT− 1 0.0780.080.0820.0840.086αD[fm3 ](b)4HeFigure 7.28: The dipole strength and electric dipole polarizability in 4He using the NN-N3LO [39]interaction. Panel (a) – The m0 sum rule in the CCSD (blue band) and CCSDT-1 (red band) ap-proximations compared with the exact EIHH calculation (black line). Panel (b) – The electric dipolepolarizability αD in the CCSD (blue band) and CCSDT-1 (red band) approximations compared withthe exact EIHH calculation (black line).The widths of the blue and red bands are obtained by looking at the residual h̵Ω dependenceat maximum model space Nmax = 16 and L3max = 21. The red bands are wider than the blueones, thus suggesting the convergence in the CCSDT-1 approximation is a bit slower than inthe CCSD scheme. However, the inclusion of triples greatly improves the CCSD calculationsboth for m0 and αD with respect to the exact result. For the m0 sum rule, the CCSD calcula-tion underestimates the EIHH result by 1.62%, while the CCSDT-1 overshoots the exact EIHHcalculation only by 0.54%. The polarizability goes from a difference of 1.70% in the CCSD ap-proximation, to an astonishing 0.06% difference from the EIHH value in the CCSDT-1 scheme.The small residual discrepancy between the coupled-cluster and the EIHH calculations is dueto missing quadruples (T4) and non-linear triples terms which are not included in the CCSDT-1truncation scheme.Next, we use the NNLOsat interaction and proceed to investigate the impact of tripleson 16O. Due to the large number of 3p–3h configurations when A = 16 (see Fig. 7.24), withthe TRIUMF theory computing cluster we are able to do calculations only up to a maximummodel space of Nmax = 12 and L3max = 12. Panel (a) of Fig. 7.29 shows the convergence of thepolarizability as a function of the L3max cut and for different model-space sizes Nmax. Panel (b)shows the same for the m0 sum rule. The convergence for both αD and m0 with respect to L3maxis good and the curves flatten around L3max = 11. By comparing the Nmax = 10 (blue) with theNmax = 12 (red) curves, we note that full convergence in Nmax is not fully reached yet. However,120Chapter 7. Resultsfor each Nmax, the relative difference between the CCSD and CCSDT-1 (at L3max = 12) resultsis almost constant and amounts roughly to 4% for both m0 and polarizability.0.450.4750.50.5250.550.5750.60.625αD[fm3 ](a) Nmax = 6Nmax = 8Nmax = 10Nmax = 120 3 6 9 12L3max4.[fm2 ](b)Figure 7.29: Convergence of αD (panel (a)) andm0 (panel (b)) in16O with the NNLOsat [18] interactionwith respect to the angular momentum cut L3max at h̵Ω = 22 MeV, and for different model-space sizesNmax.The CCSDT-1 Nmax-converged values can be obtained, e.g., by an exponential extrapola-tion, using the various values in Nmax of αD and m0 at L3max = 12 from Fig. 7.29. The resultsof the extrapolation are shown in Fig. 7.30 and deviate by a 3% from the CCSD result obtainedat Nmax = 14. In panel (a) we compare the dipole strength sum rule m0 obtained in the CCSD(blue band) scheme with the extrapolated value in the CCSDT-1 (red band) approximation.The CCSD value is obtained at Nmax = 14 and h̵Ω = 22 MeV. The theoretical results are com-pared with the experimental value obtained by integrating the data from Ahrens et al. [64](green band). Panel (b) shows the same for the electric dipole polarizability.From Fig. 7.30 we see that the effect of triples is very small. The result with triples forthe m0 sum rule slightly deviates from the experimental value (panel (a)). On the other hand,panel (b) shows that the extrapolated polarizability obtained in the CCSDT-1 approximationis still in agreement with the experimental data. The uncertainty from our calculation, i.e.,∆(α(CCSD)D − α(CCSDT−1)D ), is of the same order of the experimental one and amounts to a 3%of the CCSD value.121Chapter 7. Results2.[fm2 ](a)16OCCSDCCSDT− 1Ahrens et al.0.40.450.50.550.60.650.7αD[fm3 ](b)16OFigure 7.30: The m0 sum rule (panel (a)) and the electric dipole polarizability (panel (b)) in16Ocalculated with the NNLOsat interaction within the CCSD (blue band) and CCSDT-1 (red band) ap-proximations. The theoretical results are compared with integrated experimental data (green band)from Ahrens et al. [64].In this last section we showed that the inclusion of triples improves the comparison betweenthe coupled-cluster method and the EIHH method for the calculation of the dipole strength andpolarizability in 4He. On other hand, the effect of triples is quite small and it is of the order of1-2% in 4He and 3-4% in 16O. When comparing our calculations with experimental data in 16O,we find that the contribution to the polarizability when going from the CCSD scheme to theCCSDT-1 approximation is of the same order of the experimental error. While heavier nucleiare at the moment not in reach of triples calculations, these results indicate that the CCSDapproximation is quite good. We will soon be able to study the effect of triples on neutron-richnuclei such as 22O and 8He. Progress in this direction is underway [175].122Chapter 8ConclusionsIn this work we presented an approach that combines the Lorentz integral transform with thecoupled-cluster method, which allows to extend the calculations of electromagnetic break-upobservables to medium-mass nuclei from first principles.In Chatper 2 we briefly reviewed the ideas behind modern nuclear interactions derived fromchiral effective field theory, while in Chapter 3 we developed the formalism for the calculationof electromagnetic observables in nuclei making use of the multipole expansion. The focus is oninclusive reactions and we derived the expression for the nuclear photoabsorption cross sectionwhich can then be used to obtain the dipole response function and the electric dipole polariz-ability.In Chapter 4 we treated the problem of the continuum with integral transforms and showeda variety of methods for their inversion, which requires the solution of an ill-posed problem.In Chapter 5 we introduced coupled-cluster theory and combined it with the integral trans-forms formalism. We studied different coupled-cluster truncations, namely the CCSD andCCSDT-1 approximations. However, coupled-cluster as used in quantum chemistry is too com-putationally expensive for heavy nuclear systems, and so in Chapter 6 we gave a brief introduc-tion on the spherical coupled-cluster method, which allows for computations in medium- andheavy-mass nuclei.Finally, in Chapter 7 we showed results for calculations on a variety of nuclei. These, areall original results, which have been obtained thanks to the body work of this thesis. The mainfindings are summarized in the following.In Section 7.1, we benchmarked the coupled-cluster using the singles and doubles approxi-mation against exact calculations in 4He. The benchmark is of good quality and gives us thenecessary confidence to perform computations in heavier nuclei [1].In Section 7.2 we presented results for the electric dipole strength and polarizability in 16,22Oand 40,48Ca using chiral two-body interactions [1, 105]. Our calculations yielded results for theresponse function of oxygen and calcium isotopes that are in semi-quantitative agreement with123Chapter 8. Conclusionsexperimental data. In 22O we found a very interesting dipole response function exhibiting asoft dipole mode at low energies consistent with experimental data from Leistenschneider etal. [67]. On the other hand, we observe a systematic underestimation of the electric dipolepolarizabilities, radii and binding energies as we progressively move to heavier nuclei, pointingto the inadequacy of chiral interactions at the two-body level to describe nuclear dynamics [1, 5].In Section 7.3 we augmented the two-body interactions with three-body forces and per-formed calculations for 4He, 16,22O and 40,48Ca. We investigated the running sum of the po-larizability as a viable tool to compare our calculations with experimental data where onlylower-lying dipole strengths are measured, and as a self-consistency check for our theoreticalmethod. Using the NNLOsat interaction in4He, 16O and 40Ca, we were able to obtain valuesfor the polarizabilities and the charge radii in good agreement with the experimental data [2].The comparison with data reveals the important role of three-nucleon forces for the correctdescription of medium-mass nuclei.Next, we studied 16O and 40Ca with different two and three-body interactions and observeda strong correlation between the dipole polarizability and the charge radius [2]. Using a setof selected chiral three-body Hamiltonians, we exploited this correlation to predict the electricdipole polarizability and the neutron-skin radius in 48Ca [6]. The use of a different set ofthree-body Hamiltonians allowed us to gauge the uncertainty in the calculations from the useof different interactions. We compared our results with recent measurements of the electricdipole polarizability in 48Ca and observed good agreement between theory and experiment [7].Finally, in Section 7.4 we studied the effect of triples in coupled-cluster theory for the elec-tric dipole polarizability and the dipole strength. We observed that the error from neglectingtriples and higher-order excitations in coupled-cluster is comparable to the uncertainty orig-inating from the model-space convergence. We showed that the inclusion of triples in 4Heimproves the comparison with exact calculation results at the sub-percent level. We then usedthe NNLOsat interaction to assess triples contributions in16O, where their effect is of a fewpercents. 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Effect of spin-orbit nuclear charge densitycorrections due to the anomalous magnetic moment on halonuclei. Phys. Rev. C, 82:014320, Jul 2010. doi: 10.1103/PhysRevC.82.014320. URL[177] J. L. Friar, J. Martorell, and D. W. L. Sprung. Nuclear sizes and the isotope shift.Phys. Rev. A, 56:4579–4586, Dec 1997. doi: 10.1103/PhysRevA.56.4579. URL AVector spherical harmonicsRecall that spherical harmonics are eigenfunctions of the angular momentum operator lˆ2 andits third component mˆ,Y lm(Ω) = Y lm(Θ, φ) =¿ÁÁÀ2l + 14pi(l −m)!(l +m)!Pml (cosΘ)eimφ, (A.1)where Pml (cosΘ) are associated Legendre polynomials. In order to define vector sphericalharmonics one first introduces the following polar unit vectorse+1 = − 1√2(ex + iey) ,e0 = ez ,e−1 = 1√2(ex − iey) ,(A.2)which satisfy e∗µ ⋅ eν = δµν and e∗µ = (−1)µeµ. Vector spherical harmonics are defined both inmomentum and coordinate space via an expansion over spherical harmonicsYlJM(Ω) =∑mλCJMlm1λYlm(Ω)eλ, (A.3)where the expansion coefficients CJMlm1λ are Clebsch-Gordan coefficients. When built in this way,the vector spherical harmonics are eigenfunctions of the total angular momentum operator Jˆ2,its third component Mˆ , and the orbital angular momentum operator lˆ2. They satisfy theorthogonality relation∫ dΩYl′J ′M ′(Ω)Y∗lJM(Ω) = δl′lδJ ′JδM ′M , (A.4)143Appendix A. Vector spherical harmonicsand the following properties in momentum spaceqˆ ×YJ+1JM (Ωq) = i√ J2J + 1YJJM(Ωq),qˆ ×YJJM(Ωq) = i√ J + 12J + 1YJ−1JM (Ωq) + i√J2J + 1YJ+1JM (Ωq),qˆ ×YJ−1JM (Ωq) = i√ J + 12J + 1YJJM(Ωq),qˆY JM(Ωq) = √ J2J + 1YJ−1JM (Ωq) −√J + 12J + 1YJ+1JM (Ωq),(A.5)where qˆ = q/∣q∣. From the second and the fourth lines of Eq. (A.5) also followsYJ−1JM (Ωq) = √ J2J + 1 qˆY JM(Ωq) − i√J + 12J + 1 qˆ ×YJJM(Ωq),YJ+1JM (Ωq) = −√ J + 12J + 1 qˆY JM(Ωq) − i√J2J + 1 qˆ ×YJJM(Ωq)(A.6)andqˆ ×YJJM(Ωq) = i√J + 1J qˆY JM(Ωq) + i√2J + 1JYJ+1JM (Ωq). (A.7)Taking now qˆ along the same direction of ez, and so e0 defined as in Eq. (A.2), we have thatlz =m = 0 and Θ = 0. Since P 0l (1) = 1, Eq. (A.1) becomesY l0(Ωq) = √2l + 14pi , (A.8)and the conjugate of Eq. (A.3) isY∗JJM(Ωq) =∑λCJMJ01λY∗J0 (Ωq)e∗λ = √2J + 14pi CJMJ01Me∗M , (A.9)where in the last equality the Clebsch-Gordan coefficient forces λ =M . From this last equationwe finally findqˆ ×Y∗JJM(Ωq) = √2J + 14pi CJMJ01M(qˆ × e∗M) = iM√2J + 14piCJMJ01Me∗M , (A.10)since if qˆ = e0, then e0 × e∗M = qˆ × e∗M = iMe∗M from Eq. (A.2).144Appendix BCoupled-cluster rules and diagramsB.1 Coupled-cluster rulesIn the following we summarize the rules to convert CC diagrams to their algebraic representa-tion. This list is taken from Ref. [91], where the rules are derived in detail, and is presentedhere in the CC spherical formalism, where we assume T and Λ to be scalar operators. As anexample, suppose we have the following diagram⟨i∣ΘJµ∣a⟩ = , (B.1)corresponding to the matrix element of a one-body tensor operator ΘJµ of rank J . The dashedline represents the action of a two-body tensor operator of rank J , i.e., ΘJµ and the horizontalsolid lines represent T3 (lower line) and Λ2 (upper line). To obtain the equivalent algebraicexpression we proceed in the following way:1. Label external, open lines with hole and particle indexes corresponding to the ones occur-ring in bra and ket of Eq. (B.1). Label internal lines with different particle-hole indexes,i abj ck dl . (B.2)2. With any one-body operator O vertex associate a factor ⟨out∣O∣in⟩. Analogously, for atwo-body operator, associate a factor ⟨outleftoutright∣O∣inleftinright⟩. Here, out and inrefer to the indexes of outward and inward lines connected to it:⟨kl∣ΘJµ∣cd⟩ . (B.3)3. With every T (Λ) vertex associate the relative amplitude:⟨kl∣ΘJµ∣cd⟩⟨ij∣λ002 ∣ab⟩⟨bcd∣t003 ∣jkl⟩ . (B.4)145Appendix B. Coupled-cluster rules and diagrams4. Sum over all internal lines labels:∑jklbcd⟨kl∣ΘJµ∣cd⟩⟨ij∣λ002 ∣ab⟩⟨bcd∣t003 ∣jkl⟩ . (B.5)5. Associate a factor 1/2 with each pair of equivalent internal lines. Two internal lines areconsidered equivalent if they connect the same two vertices, going in the same direction.In our example diagram we have two pairs of equivalent lines, i.e., the lines connectingthe ΘJµ operator with the T3 operator:14∑jklbcd⟨kl∣ΘJµ∣cd⟩⟨ij∣λ002 ∣ab⟩⟨bcd∣t003 ∣jkl⟩ (B.6)6. Associate a sign (−1)h−l, where h is the number of hole lines and l is the number of loops;in our case we have h = 4 and l = 4, where we count as loops also virtual loops, i.e., loopsthat can be created by connected external open lines. So our expression does not changein this step.7. Sum over all distinct permutations Pˆ of labels of inequivalent external particle lines andof inequivalent external hole lines, including a parity factor (−1)σ(Pˆ ). Lines are said to beinequivalent if the graph changes structure when exchanging indexes. Our diagram hasno inequivalent lines.The final expression of our diagram is⟨i∣ΘJµ∣a⟩ = 14∑jklbcd⟨kl∣ΘJµ∣cd⟩⟨ij∣λ002 ∣ab⟩⟨bcd∣t003 ∣jkl⟩. (B.7)B.2 Clebsch-Gordan coefficients relationsHere we present a lot of formulas used to derive the J−coupled expressions for the diagramsin the following sections. The proof of these and many other angular momentum relations arepresented in Ref. [115]. Latin letters represent angular momenta, while greek letters denote theangular momenta projections,Ccγaαbβ = (−1)a+b−cCcγbβaα , (B.8)∑αβCcγaαbβCc′γ′aαbβ = δcc′δγγ′ , (B.9)∑δCeεbβdδCdδaαfϕ = (−1)2e∑cγcˆdˆ CcγaαbβCeεfϕcγ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩a b ce f d⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ , (B.10)146Appendix B. Coupled-cluster rules and diagrams∑αCcγaαbβCdδaαfϕ =∑eε(−1)2e cˆdˆ CeεbβdδCeεfϕcγ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩a b ce f d⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ , (B.11)∑αβδCcγaαbβCeεdδbβCdδaαfϕ = (−1)b+c+d+f cˆdˆ Ceεcγfϕ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩a b ce f d⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ , (B.12)∑αβδCaαbβcγCdδbβeεCdδaαfϕ = (−1)b+c+d+f aˆdˆ2eˆ Ceεcγfϕ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩a b ce f d⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ , (B.13)and we denoted aˆ = √2ja + 1.B.3 m0 diagrams in J−coupled scheme (CCSD)We first present the diagrams needed for the calculation of the m0 at singles and doubles level.These diagrams have been used to obtain the results in Refs. [1, 2, 5–7, 105], but never pre-sented before as published material. The expressions for the EOM in the CCSD approximationin J−coupled scheme used for the calculation of the continued fraction and thus the polariz-ability have been previously published by Hagen et. al. [93] and we do not present them here.For each diagram we first present its expression in m-scheme obtained using the rules inAppendix B.1 from Ref. [91]. We then obtain their equivalent expression in J−coupled schemeas shown in the example of Section 6.4. We label with R (L) the diagrams contributing toSR (SL). With the subscripts 1,2,3 we will denote the 1p–1h, 2p–2h and 3p–3h contributionsrespectively. We will further use roman capital and greek letters to denote the various termscontributing. For example, the diagram denoted with L1Bα corresponds to a term contributingto SL at the 1p–1h level and it is the α (first) diagram of the B contribution of Eq. (B.38),namelyL1Bα = iab(B.14)147Appendix B. Coupled-cluster rules and diagramsB.3.1 Diagram R1A= a i + iba + a ij+ i aj b+ ibja (B.15)a i = ⟨a∣RJµ1Aα ∣i⟩ = ⟨a∣ΘJµ∣i⟩ (B.16)iba = ⟨a∣RJµ1Aβ ∣i⟩ =∑b⟨b∣t001 ∣i⟩⟨a∣ΘJµ∣b⟩ (B.17)aij= ⟨a∣RJµ1Aγ ∣i⟩ = −∑j⟨a∣t001 ∣j⟩⟨j∣ΘJµ∣i⟩ (B.18)i aj b= ⟨a∣RJµ1Aδ ∣i⟩ =∑jb⟨ab∣t002 ∣ij⟩⟨j∣ΘJµ∣b⟩ (B.19)ibja = ⟨a∣RJµ1Aε ∣i⟩ = −∑jb⟨b∣t001 ∣i⟩⟨a∣t001 ∣j⟩⟨j∣ΘJµ∣b⟩ (B.20)J−coupled representation⟨a∣∣RJ1Aα ∣i⟩ = ⟨a∣∣ΘJ ∣∣i⟩ (B.21)⟨a∣∣RJ1Aβ ∣i⟩ = ∑b⟨b∣∣t01∣∣i⟩⟨a∣∣ΘJ ∣∣b⟩ (B.22)⟨a∣∣RJ1Aγ ∣i⟩ = −∑j⟨a∣∣t01∣∣j⟩⟨j∣∣ΘJ ∣∣i⟩ (B.23)⟨a∣∣RJ1Aδ ∣i⟩ = ∑jb∑Jab(−1)jj+ji+Jab jˆj Jˆ2abjˆa⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jj jb Jja ji Jab⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ × ⟨ab∣∣t02∣∣ij⟩⟨j∣∣Θj ∣∣b⟩ (B.24)⟨a∣∣RJ1Aε ∣i⟩ = −∑jb⟨b∣∣t01∣∣i⟩⟨a∣∣t01∣∣j⟩⟨j∣∣ΘJ ∣∣b⟩ (B.25)B.3.2 Diagram R2A= i a jcb + i a b jk148Appendix B. Coupled-cluster rules and diagrams+ bkcj a i + jc kb a i (B.26)i a jcb = ⟨ab∣RJµ2Aα ∣ij⟩ = P (ab)∑c⟨ac∣t002 ∣ij⟩⟨b∣ΘJµ∣c⟩ (B.27)i a bjk= ⟨ab∣RJµ2Aβ ∣ij⟩ = −P (ij)∑k⟨ab∣t002 ∣ik⟩⟨k∣ΘJµ∣j⟩ (B.28)bkcj a i = ⟨ab∣RJµ2Aγ ∣ij⟩= − P (ab)∑kc⟨b∣t001 ∣k⟩⟨ac∣t002 ∣ij⟩⟨k∣ΘJµ∣c⟩ (B.29)jc kb a i = ⟨ab∣RJµ2Aδ ∣ij⟩= − P (ij)∑kc⟨c∣t001 ∣j⟩⟨k∣ΘJµ∣c⟩⟨ab∣t002 ∣ik⟩ (B.30)J−coupled representation⟨ab∣∣RJ2Aα ∣∣ij⟩ = P (ab)∑c(−1)ja+jc+Jab+J Jˆij jˆb⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc ja JijJab J jb⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ac∣∣t02∣∣ij⟩⟨b∣∣ΘJ ∣∣c⟩ (B.31)⟨ab∣∣RJ2Aβ ∣∣ij⟩ = − P (ij)∑k(−1)ji+jj+Jab+J Jˆij jˆk⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jj ji JijJab J jk⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ab∣∣t02∣∣ik⟩⟨k∣∣ΘJ ∣∣j⟩ (B.32)⟨ab∣∣RJ2Aγ ∣∣ij⟩ = − P (ab)∑kc(−1)ja+jc+Jab+J Jˆij jˆb⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc ja JijJab J jb⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨b∣∣t01∣∣k⟨⟨k∣∣ΘJ ∣∣c⟩⟨ac∣∣t02∣∣ij⟩(B.33)149Appendix B. Coupled-cluster rules and diagrams⟨ab∣∣RJ2Aδ ∣∣ij⟩ = − P (ij)∑kc(−1)ji+jj+Jab+J Jˆij jˆk⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jj ji JijJab J jk⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨c∣∣t01∣∣j⟨⟨k∣∣ΘJ ∣∣c⟩⟨ab∣∣t02∣∣ik⟩(B.34)B.3.3 Diagram L1Aa i = a i (B.35)⟨i∣LJµ1A∣a⟩ = ⟨i∣ [Θ„]Jµ ∣a⟩ (B.36)J−coupled representation⟨i∣∣LJ1A∣∣a⟩ = ⟨i∣∣ [Θ„]J ∣∣a⟩ (B.37)B.3.4 Diagram L1B= iab + a ji+ a j bi+ i b ja(B.38)iab = ⟨i∣LJµ1Bα ∣a⟩ =∑b⟨i∣λ001 ∣b⟩⟨b∣ [Θ„]Jµ ∣a⟩ (B.39)aji= ⟨i∣LJµ1Bβ ∣a⟩ = −∑j⟨j∣λ001 ∣a⟩⟨i∣ [Θ„]Jµ ∣j⟩ (B.40)ajbi= ⟨i∣LJµ1Bγ ∣a⟩ = −∑jb⟨j∣λ001 ∣a⟩⟨b∣t001 ∣j⟩⟨i∣ [Θ„]Jµ ∣b⟩ (B.41)ibja= ⟨i∣LJµ1Bδ ∣a⟩ = −∑jb⟨i∣λ001 ∣b⟩⟨b∣t001 ∣j⟩⟨j∣ [Θ„]Jµ ∣a⟩ (B.42)J−coupled representation⟨i∣∣LJ1Bα ∣∣a⟩ =∑b⟨i∣∣λ01∣∣b⟩⟨b∣∣ [Θ„]J ∣∣a⟩ (B.43)150Appendix B. Coupled-cluster rules and diagrams⟨i∣∣LJ1Bβ ∣∣a⟩ = −∑j⟨j∣∣λ01∣∣a⟩⟨i∣∣ [Θ„]J ∣∣j⟩ (B.44)⟨i∣∣LJ1Bγ ∣∣a⟩ = −∑jb⟨j∣∣λ01∣∣a⟩⟨b∣∣t01∣∣j⟩⟨i∣∣ [Θ„]J ∣∣b⟩ (B.45)⟨i∣∣LJ1Bδ ∣∣a⟩ = −∑jb⟨i∣∣λ01∣∣b⟩⟨b∣∣t01∣∣j⟩⟨j∣∣ [Θ„]J ∣∣a⟩ (B.46)B.3.5 Diagram L1C= bj icka+ bj akci(B.47)bjicka= ⟨i∣LJµ1Cα ∣a⟩ = −12 ∑jkbc⟨bc∣t002 ∣jk⟩⟨ji∣λ002 ∣bc⟩⟨k∣ [Θ„]Jµ ∣a⟩ (B.48)bjakci= ⟨i∣LJµ1Cβ ∣a⟩ = −12 ∑jkbc⟨bc∣t002 ∣jk⟩⟨jk∣λ002 ∣ba⟩⟨i∣ [Θ„]Jµ ∣c⟩ (B.49)J−coupled representation⟨i∣∣LJ1Cα ∣∣a⟩ = − 12 ∑jkcb∑Jbc Jˆ2bcjˆ2k⟨bc∣∣t02∣∣jk⟩⟨ji∣∣λ02∣∣bc⟩⟨k∣∣ [Θ„]J ∣∣a⟩δjkji (B.50)⟨i∣∣LJ1Cβ ∣∣a⟩ = − 12 ∑jkcb∑Jbc Jˆ2bcjˆ2c⟨bc∣∣t02∣∣jk⟩⟨jk∣∣λ02∣∣ba⟩⟨a∣∣ [Θ„]J ∣∣c⟩δjajc (B.51)B.3.6 Diagram L1Dbji a (B.52)⟨i∣LJµ1D ∣a⟩ =∑jb⟨ij∣λ002 ∣ab⟩⟨b∣RJµ1A∣j⟩ (B.53)J−coupled representation⟨i∣∣LJ1D ∣∣a⟩ =∑jb∑Jab(−1)ja+jb−Jab jˆbJˆ2abjˆi⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jb jj Jji ja Jab⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ij∣∣λ02∣∣ab⟩⟨b∣∣RJ1A∣∣j⟩ (B.54)151Appendix B. Coupled-cluster rules and diagramsB.3.7 Diagram L2Aa i b j = a i b j (B.55)⟨ij∣LJµ2A∣ab⟩ = P (ab)P (ij)⟨i∣λ001 ∣a⟩⟨j∣ [Θ„]Jµ ∣b⟩ (B.56)J−coupled representation⟨ij∣∣LJ2A∣∣ab⟩ = P (ab)P (ij)(−1)jb+ja+Jij+J Jˆabjˆj⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jb ja JabJij J jj⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨i∣∣λ01∣∣a⟩⟨j∣∣ [Θ„]J ∣∣b⟩ (B.57)B.3.8 Diagram L2B= i a jbc + i a b kj+ i a b k cj+ i a j c kb(B.58)i a jbc = ⟨ij∣LJµ2Bα ∣ab⟩ = P (ab)∑c⟨ij∣λ002 ∣ac⟩⟨c∣ [Θ„]Jµ ∣b⟩ (B.59)i a bkj= ⟨ij∣LJµ2Bβ ∣ab⟩ = −P (ij)∑k⟨ik∣λ002 ∣ab⟩⟨j∣ [Θ„]Jµ ∣k⟩ (B.60)i a bkcj= ⟨ij∣LJµ2Bγ ∣ab⟩= − P (ij)∑kc⟨ik∣λ002 ∣ab⟩⟨c∣t001 ∣k⟩⟨j∣ [Θ„]Jµ ∣c⟩ (B.61)i a jckb= ⟨ij∣LJµ2Bδ ∣ab⟩= − P (ab)∑kc⟨ij∣λ002 ∣ac⟩⟨c∣t001 ∣k⟩⟨k∣ [Θ„]Jµ ∣b⟩ (B.62)152Appendix B. Coupled-cluster rules and diagramsJ−coupled representation⟨ij∣∣LJ2Bα ∣∣ab⟩ = P (ab)∑c(−1)jb+ja+Jij+J Jˆabjˆc⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jb ja JabJij J jc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ij∣∣λ02∣∣ac⟩⟨c∣∣ [Θ„]J ∣∣b⟩ (B.63)⟨ij∣∣LJ2Bβ ∣∣ab⟩ = − P (ij)∑k(−1)jk+ji+Jij+J Jˆabjˆj⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jk ji JabJij J jj⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ik∣∣λ02∣∣ab⟩⟨j∣∣ [Θ„]J ∣∣k⟩ (B.64)⟨ij∣∣LJ2Bγ ∣∣ab⟩ = − P (ij)∑kc(−1)jk+ji+Jij+J Jˆabjˆj⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jk ji JabJij J jj⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ik∣∣λ02∣∣ab⟩⟨c∣∣t01∣∣k⟩⟨j∣∣ [Θ„]J ∣∣c⟩ (B.65)⟨ij∣∣LJ2Bδ ∣∣ab⟩ = − P (ab)∑kc(−1)ja+jb+Jij+J Jˆabjˆc⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jk ji JabJij J jc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ij∣∣λ02∣∣ac⟩⟨c∣∣t01∣∣k⟩⟨k∣∣ [Θ„]J ∣∣b⟩ (B.66)B.4 m0 diagrams in J−coupled scheme (CCSDT-1)In this section we present the new contributions arising in the CCSDT-1 approximation enteringthem0 sum rule. Each diagram is first represented inm-scheme notation and then the equivalentJ−coupling representation is reported. We use the same labelling scheme we used in the previoussection.B.4.1 Diagram R2Ba i b jc k(B.67)⟨ab∣RJµ2B ∣ij⟩ =∑kc⟨abc∣t003 ∣ijk⟩⟨k∣ΘJµ∣c⟩ (B.68)153Appendix B. Coupled-cluster rules and diagramsJ−coupled representation⟨ab∣∣RJ2B ∣∣ij⟩ = ∑kc∑Jabc(−1)jk+Jij+Jabc+1 jˆkJˆ2abcJˆab⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jk jc JJab Jij Jabc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨abc∣∣t03∣∣ijk⟩⟨k∣∣ΘJ ∣∣c⟩ (B.69)B.4.2 Diagram R3Ai a j dlb k c (B.70)⟨abc∣RJµ3A∣ijk⟩ = −P (j/ik)P (b/ac)∑ld⟨ad∣t002 ∣ij⟩⟨l∣ΘJµ∣d⟩⟨bc∣t002 ∣lk⟩ (B.71)J−coupled representation⟨abc∣∣RJ3A∣∣ijk⟩ = − P (j/ik)P (b/ac)∑ld∑JkdJbc(−1)ja+jb+jc+jl+jd+Jijk+1jˆlJˆabJˆij JˆijkJˆ2kdJˆ2bc××⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jd jk JkdJbc J jl⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jd jk JkdJijk ja Jij⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩Jkd ja JijkJabc J Jbc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jb ja JabJabc jc Jbc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ad∣∣t02∣∣ij⟩⟨l∣∣ΘJ ∣∣d⟩⟨bc∣∣t02∣∣lk⟩B.4.3 Diagram R3BThis diagram is a contribution to the three-body part of the SR vector.i a j b kdc = i a j b kdc + i a j b k dlc . (B.72)While the above diagrams can be calculated separately, it is more convenient to define an154Appendix B. Coupled-cluster rules and diagramsintermediate χA,⟨b∣χJµA ∣a⟩ =ba =ba +aib, (B.73)so that only the left-hand side of Eq. (B.72) needs to be evaluated. The m– and J−schemerepresentations of the above intermediate are⟨b∣χJµA ∣a⟩ = ⟨b∣ΘJµ∣a⟩ −∑i⟨b∣t001 ∣i⟩⟨i∣ΘJµ∣a⟩, (B.74)⟨b∣∣χJA∣∣a⟩ = ⟨b∣∣ΘJ ∣∣a⟩ −∑i⟨b∣∣t01∣∣i⟩⟨i∣∣ΘJ ∣∣a⟩. (B.75)With the intermediate in Eq. (B.73), the m–scheme representation of the left-hand diagram ofEq. (B.72) is⟨abc∣RJµ3B ∣ijk⟩ = P (c/ab)∑d⟨abd∣t003 ∣ijk⟩⟨c∣χJµA ∣d⟩. (B.76)In J−scheme one has⟨abc∣∣RJ3B ∣∣ijk⟩ = P (c/ab)∑d(−1)jd+Jab+Jabc+J Jˆijk jˆc⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jd Jab JijkJabc J jc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨abd∣∣t03∣∣ijk⟩⟨c∣∣χJA∣∣d⟩.(B.77)B.4.4 Diagram R3CThis diagram is a contribution to the three-body part of the SR vector.i a j b clk = i a j b clk + i a j b cldk . (B.78)As in Eq. (B.72) we can define a new intermediate χB,⟨b∣χJµB ∣a⟩ = ij= ij+jai, (B.79)155Appendix B. Coupled-cluster rules and diagramswhich m– and J−scheme expressions are⟨j∣χJµB ∣i⟩ = ⟨j∣ΘJµ∣i⟩ +∑a⟨a∣t001 ∣i⟩⟨j∣ΘJµ∣a⟩, (B.80)⟨j∣∣χJB ∣∣i⟩ = ⟨j∣∣ΘJ ∣∣i⟩ +∑a⟨a∣∣t01∣∣i⟩⟨j∣∣ΘJ ∣∣a⟩. (B.81)With the intermediate in Eq. (B.79), the m–scheme representation of the left-hand diagram ofEq. (B.78) becomes⟨abc∣RJµ3C ∣ijk⟩ = −P (k/ij)∑l⟨abc∣t003 ∣ijl⟩⟨l∣χJµB ∣k⟩. (B.82)In J−scheme one has⟨abc∣∣RJ3C ∣∣ijk⟩ = − P (k/ij)∑l(−1)jk+Jij+Jabc+J Jˆijk jˆl ××⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jk Jij JijkJabc J jl⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨abc∣∣t03∣∣ijl⟩⟨l∣∣χJB ∣∣k⟩. (B.83)B.4.5 Diagram L1EThis is a new contribution to the one-body part of SL, i.e., the first and the second diagramsin Eq. (5.177),(CCSD)+ . (B.84)From Eq. (5.176) we find that the two-body part of SR in the CCSDT-1 approximation is givenR2A(T )=(CCSD)+ . (B.85)We can then combine the two diagrams of Eq. (B.84) in a unique diagram which depends onR2A(T ) = R2A +R2B and all we need to compute isR2A(T )a i b j c k , (B.86)156Appendix B. Coupled-cluster rules and diagrams⟨i∣LJµ1E ∣a⟩ = 14 ∑jkbc⟨ijk∣λ003 ∣abc⟩⟨bc∣RJµ2A(T )∣jk⟩. (B.87)J−coupled representation⟨i∣∣LJ1E ∣∣a⟩ = 14 ∑jkbc ∑JabJbcJijJjk∑Jabc(−1)ji+jj+jk+jb+jc+Jbc+Jabc Jˆ2bcJˆ2abcJˆabJˆjkJˆijjˆi××⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jb jc JbcJabc ja Jab⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jj jk JjkJabc ji Jij⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩Jbc Jjk Jji ja Jabc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ijk∣∣λ03∣∣abc⟩⟨bc∣∣RJ2A(T )∣∣jk⟩(B.88)B.4.6 Diagram L2CThis is a new contribution to the two-body part of SL and it can be written in terms of thediagram R1A in Eq. (B.15).i a j bk cD1(B.89)⟨ij∣LJµ2C ∣ab⟩ =∑kc⟨ijk∣λ003 ∣abc⟩⟨c∣RJµ1A∣k⟩ (B.90)J−coupled representation⟨ij∣∣LJ2C ∣∣ab⟩ = ∑kc∑Jabc(−1)jc+Jab+Jabc+1 jˆcJˆ2abcJˆij⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc jk JJij Jab Jabc⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ijk∣∣λ03∣∣abc⟩⟨c∣∣RJ1A∣∣k⟩(B.91)B.4.7 Diagram L2D= += a i c k ldbj+ a i c k dljb(B.92)157Appendix B. Coupled-cluster rules and diagramsa ic k ldbj= ⟨ij∣LJµ2Dα ∣ab⟩= − 12P (ij) ∑lkcd⟨kli∣λ003 ∣acb⟩⟨cd∣t002 ∣kl⟩⟨j∣ [Θ„]Jµ ∣d⟩ (B.93)a ic k dljb= ⟨ij∣LJµ2Dβ ∣ab⟩= − 12P (ab) ∑lkcd⟨ikj∣λ003 ∣acd⟩⟨cd∣t002 ∣kl⟩⟨l∣ [Θ„]Jµ ∣b⟩ (B.94)J−coupled representation⟨ij∣∣LJ2Dα ∣∣ab⟩ = − 12P (ij) ∑klcd ∑JcdJacbJacJik(−1)ji+jk+jl+jb+jc+jd+Jac+Jcd+Jij+Jab+J JˆikJˆ2cdJˆacJˆ2acbJˆabjˆj××⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jk jl JcdJacb ji Jjk⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ja jb JabJacb jc Jac⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩Jcd jc jdJab ji Jacb⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jd ji JabJij J jj⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ikl∣∣λ03∣∣acb⟩⟨cd∣∣t02∣∣kl⟩⟨j∣∣ [Θ„]J ∣∣d⟩ (B.95)⟨ij∣∣LJ2Dβ ∣∣ab⟩ = − 12P (ab) ∑klcd ∑JcdJacdJacJik(−1)jj+jk+ja+jb+jc+jd+Jcd+Jik+J JˆikJˆ2cdJˆacJˆ2acdJˆabjˆl××⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ji jj JijJacd jk Jik⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc jd JcdJacd ja Jac⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩Jcd jk jlJij ja Jacd⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jb ja JabJij J jl⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭×× ⟨ikj∣∣λ03∣∣acd⟩⟨cd∣∣t02∣∣kl⟩⟨l∣∣ [Θ„]J ∣∣b⟩ (B.96)158Appendix B. Coupled-cluster rules and diagramsB.4.8 Diagram L3Aa i b j c k = a i b j c k (B.97)⟨ijk∣LJµ3A∣abc⟩ = P (c/ab)P (k/ij)⟨ij∣λ002 ∣ab⟩⟨k∣ [Θ„]Jµ ∣c⟩ (B.98)J−coupled representation⟨ijk∣∣LJ3A∣∣abc⟩ = P (c/ab)P (k/ij)(−1)jc+Jab+Jijk+J Jˆabcjˆk⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc Jab JabcJijk J jk⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ij∣∣λ02∣∣ab⟩⟨k∣∣ [Θ„]J ∣∣c⟩(B.99)B.4.9 Diagram L3BThis diagram is a contribution to the three-body part of the SL vector.i a j b kcd = i a j b kcd + i a j b kd l c. (B.100)We can make use of the intermediate in Eq. (B.73) to calculate the left-hand diagram inEq. (B.100), which m– and J−scheme expressions are⟨ijk∣LJµ3B ∣abc⟩ = P (c/ab)∑d⟨ijk∣λ003 ∣abd⟩⟨d∣ [χ„A]Jµ ∣c⟩, (B.101)and⟨ijk∣∣LJ3B ∣∣abc⟩ = P (c/ab)∑d(−1)jc+Jab+Jijk+J Jˆabcjˆd⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jc Jab JabcJijk J jd⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ijk∣∣λ03∣∣abd⟩⟨d∣∣ [χ„A]J ∣∣c⟩.(B.102)159Appendix B. Coupled-cluster rules and diagramsB.4.10 Diagram L3CThis diagram is a contribution to the three-body part of the SL vector.i a j b clk= i a j b c lk+ i a j b c ldk(B.103)We can make use of the intermediate in Eq. (B.79) to calculate the left-hand diagram inEq. (B.103), which m– and J−scheme expressions are⟨ijk∣LJµ3C ∣abc⟩ = −P (k/ij)∑l⟨ijl∣λ003 ∣abc⟩⟨k∣ [χ„B]Jµ ∣l⟩, (B.104)and⟨ijk∣∣LJ3C ∣∣abc⟩ = − P (k/ij)∑l(−1)jl+Jij+Jijk+J Jˆabcjˆk⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩jl Jij JabcJijk J jk⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭⟨ijl∣∣λ03∣∣abc⟩⟨k∣∣ [χ„B]J ∣∣l⟩.(B.105)160Appendix CRadius operatorThe results for the radii shown in Chapter 7 have been calculated implementing the radiusoperator in the center of mass frame. As in Section 3.3, we define the z projection of theisospin operator to be +1 for protons and −1 for neutrons, i.e.,τ z ∣p⟩ = +∣p⟩ and τ z ∣n⟩ = −∣n⟩. (C.1)Then, the point-proton and point-neutron radii operators squared of a nucleus with mass num-ber A and Z protons and N neutrons are defined asr2p = 1Z A∑i=1(1 + τzi2) (ri −Rcm)2 ,r2n = 1N A∑i=1(1 − τzi2) (ri −Rcm)2 (C.2)where the center of mass radius is defined asRcm = 1AA∑i=1 ri. (C.3)We can manipulate Eq. (C.2) in order to isolate the one- and two-body contributions to theradii. Let’s focus on the proton radius for the moment and write explicitly the square on theright-hand side of the first line of Eq. (C.2) asr2p = 1Z A∑i=1(1 + τzi2)(r2i − 2ri ⋅Rcm +R2cm) . (C.4)Making use of Eq. (C.3) we can rewrite2ri ⋅Rcm = 2AriA∑j=1 rj , (C.5)and noting thatA∑i=11 + τ zi2= Z∑i=1 = Z andA∑i=11 − τ zi2= N∑i=1 = N, (C.6)161Appendix C. Radius operatortogether with Eq. (C.3), we can rewrite Eq. (C.2) asr2p = 1Z A∑i=1(1 + τzi2) r2i − 2ZA A∑i=1(1 + τzi2) ri ⋅ A∑j=1 rj + 1A2A∑i,j=1 ri ⋅ rj= 1ZA∑i=1(1 + τzi2) r2i + 2A A∑i,j=1 ri ⋅ rj [ 12A − 1 + τzi2Z] . (C.7)To isolate the one-body and two-body contributions we now split the sum over i, j into thediagonal (i = j) and off-diagonal (i ≠ j) terms, i.e,r2p = 1Z A∑i=1(1 + τzi2) r2i + 2A A∑i=1 r2i [ 12A − 1 + τzi2Z] + 4AA∑i<j ri ⋅ rj [ 12A − 1 + τzi2Z] . (C.8)The first two terms in Eq. (C.8) represent the one-body part of the squared point-proton radiusoperator, while the last term is the two-body term. By following the same procedure, ananalogous expression can be obtained for the point-neutron radius operator, which readsr2n = 1N A∑i=1(1 − τzi2) r2i + 2A A∑i=1 r2i [ 12A − 1 − τzi2N] + 4AA∑i<j ri ⋅ rj [ 12A − 1 − τzi2N] . (C.9)Once the point-proton radius of Eq. (C.8) has been calculated, the charge radius can beobtained making use of [176]r2ch = r2p +R2p + NZ R2n + 34M2 + r2so. (C.10)Here R2p = 0.769 fm is the mean squared charge radius of a single proton, R2n = 0.116 fm isthat of a single neutron, 3/(4M2) = 0.033 fm is the relativistic Darwin-Foldy correction [177],and r2so is a spin-orbit nuclear charge-density correction, which depends on the nucleus understudy. For the value of the spin-orbit correction we typically use a mean-field estimate, which issimpler to obtain than the exact one and has been shown to be a very good approximation [6].Note that Eq. (C.10) is valid when neglecting meson exchange corrections, which are expectedto be small since they enter at N3LO in the χEFT expansion.162


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