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Variational Determination of the Two-Particle Density Matrix As a Quantum Many-Body Technique

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Variational Determination of the Two-Particle Density Matrix As a Quantum Many-Body Technique Variational determination of the two-particle density matrix as a quantum many-body technique Brecht Verstichel promotor: Prof. Dr. Dimitri Van Neck co-promotor: Prof. Dr. Patrick Bultinck Proefschrift ingediend tot het behalen van de academische graad van Doctor in de Wetenschappen: Fysica Universiteit Gent Faculteit Wetenschappen Vakgroep Fysica en Sterrenkunde Academiejaar 2011-2012 FACULTEIT WETENSCHAPPEN Variational determination of the two-particle density matrix as a quantum many-body technique Brecht Verstichel Promotor: Prof. dr. Dimitri Van Neck Co-promotor: Prof. dr. Patrick Bultinck Proefschrift ingediend tot het behalen van de academische graad van Doctor in de Wetenschappen: Fysica Universiteit Gent Faculteit Wetenschappen Vakgroep Fysica en Sterrenkunde Academiejaar 2011-2012 Dankwoord Ik zou graag enkele mensen bedanken die direct of indirect hebben bijgedragen aan deze thesis. Eerst mijn promotor Dimitri, voor de goede begeleiding en om mij de kans te geven dit interessante onderzoek te verrichten. Also a big word of thanks to Paul, Patrick and Helen. It was really fun working on this subject with you. Ook bedankt aan iedereen in het CMM, en in het bijzonder mijn theorie collega's Matthias, Sebastian, Ward en Diederik. Marc mag ook zeker niet vergeten worden. Matthias en ik hebben ook een mooie tijd beleefd op het INW, alwaar Pieter, Lesley, Wim, Arne, Tom, Piet en vele anderen zorgden voor een zeer leuke werkomgeving. Ook bedankt aan de vrienden waar ik regelmatig mee afspreek. Zonder mijn ouders had ik nooit de kansen en de nodige stimulatie gekregen om verder te studeren, dus ook bedankt. Mijn zus, Greet, stond ook altijd klaar om te helpen. En ik kan ook mijn petekindje Woutje niet genoeg bedanken voor zijn constructieve bijdrage aan het schrijfproces. En natuurlijk een heel grote knuffel voor Christine. i Contents Dankwoord i Contents iii 1 Introduction1 2 N-representability5 2.1 Definitions of N-representability.........................5 2.1.1 Dual definition of N-representability...................6 2.2 Standard N-representability conditions......................7 2.2.1 N-representability of the 1DM......................9 2.2.2 Necessary conditions for the 2DM.................... 11 2.2.2.1 Two-index conditions...................... 12 2.2.2.2 Three-index conditions..................... 13 2.2.2.3 The primed conditions...................... 15 2.3 Non-standard N-representability conditions................... 18 2.3.1 Subsystem constraints........................... 18 2.3.1.1 Fractional N-representability.................. 18 2.3.1.2 Subsystem 2DM's are fractional-N representable....... 19 2.3.1.3 Applicability of subsystem constraints............. 21 2.3.2 The sharp conditions............................ 21 2.3.2.1 Sharp bounds on I(Γ)...................... 22 2.3.2.2 Sharp bounds on Q(Γ)..................... 25 2.3.2.3 Sharp bounds on G(Γ)...................... 26 2.3.3 Diagonal constraints............................ 27 3 Semidefinite programming 31 3.1 Hermitian adjoint maps.............................. 31 3.2 Primal and dual semidefinite programs...................... 33 3.3 Formulation of v2DM as a semidefinite program................. 33 3.3.1 Primal formulation............................. 34 3.3.2 Dual formulation.............................. 35 iii iv CONTENTS 3.4 Interior point methods............................... 35 3.4.1 The central path.............................. 36 3.4.2 Dual-only potential reduction method.................. 37 3.4.2.1 Solution for fixed penalty.................... 38 3.4.2.2 Outline of the algorithm.................... 40 3.4.2.3 Adding linear inequality constraints.............. 41 3.4.2.4 Adding linear equality constraints............... 42 3.4.3 Primal-dual predictor-corrector method................. 43 3.4.3.1 Equations of motion....................... 43 3.4.3.2 The overlap matrix....................... 45 3.4.3.3 Solution to the equations of motion.............. 48 3.4.3.4 Outline of the algorithm.................... 50 3.4.3.5 Adding linear inequality constraints.............. 51 3.4.3.6 Adding linear equality constraints............... 53 3.5 Boundary point method.............................. 54 3.5.1 The augmented Lagrangian........................ 54 3.5.2 Solution to the inner problem....................... 55 3.5.3 Outline of the algorithm.......................... 56 3.5.4 Adding linear equality and inequality constraints............ 58 3.6 Computational performance of the methods................... 58 3.6.1 Interior point methods........................... 58 3.6.2 Boundary point method.......................... 60 3.6.3 Comparison of scaling properties..................... 61 4 Symmetry adaptation of the 2DM 63 4.1 Spin symmetry................................... 63 4.1.1 Singlet ground state............................ 65 4.1.2 Higher-spin ground state.......................... 66 4.1.3 Decomposition of the two-index constraints............... 68 4.1.4 Decomposition of the three-index constraints.............. 69 4.2 Symmetry in atomic systems........................... 73 4.2.1 Symmetry under spatial rotations.................... 73 4.2.2 Reflection symmetry and parity...................... 74 4.2.3 The 2DM for atomic systems....................... 74 4.2.4 Two-index constraints for atomic systems................ 75 4.2.5 The three-index constraints for atomic systems............. 77 4.3 Symmetry in the one-dimensional Hubbard model............... 82 4.3.1 Translational invariance.......................... 85 4.3.1.1 The 2DM for translationally invariant systems........ 86 4.3.1.2 Two-index constraints for translationally invariant systems. 86 4.3.1.3 Three-index constraints for translationally invariant systems 87 4.3.2 Parity.................................... 88 4.3.2.1 Translationally invariant 1DM with parity.......... 89 4.3.2.2 Translationally invariant 2DM with parity.......... 90 4.3.2.3 Parity symmetric form of the two-index constraints..... 91 4.3.2.4 Parity symmetric form of the three-index constraints.... 93 CONTENTS v 5 Applications 99 5.1 The isoelectronic series of Beryllium, Neon and Silicon............. 99 5.1.1 Spin and angular momentum constraints................. 100 5.1.1.1 Imposing the spin constraints for S = 0............ 101 5.1.1.2 Imposing the spin constraints for S= 6 0............ 102 5.1.1.3 Spin and angular momentum projection............ 102 5.1.2 Results and discussion........................... 103 5.1.2.1 Ground-state energy....................... 104 5.1.2.2 Correlation energy........................ 110 5.1.2.3 Ionization energies........................ 115 5.1.2.4 Correlated Hartree-Fock-like single-particle energies..... 115 5.1.2.5 Natural occupations....................... 116 5.2 Dissociation of diatomic molecules........................ 117 5.2.1 v2DM dissociates molecules into fractionally charged atoms...... 117 5.2.2 Subsystem constraints cure the dissociation limit............ 118 5.2.2.1 Implementation......................... 119 5.2.2.2 Numerical verification...................... 121 5.2.2.3 Extension to polyatomic systems................ 122 5.2.2.4 Generalization of the atomic subspace constraints...... 123 5.3 The one-dimensional Hubbard model....................... 123 5.3.1 Results................................... 123 5.3.1.1 Ground-state energy....................... 123 5.3.1.2 Momentum distribution..................... 126 5.3.1.3 Correlation functions...................... 129 5.3.2 Failure in the strong-correlation limit.................. 133 5.3.2.1 The non-linear hopping constraint............... 137 5.3.2.2 The Gutzwiller projection constraint.............. 141 6 Restoring the strong-correlation limit in the Hubbard model 145 6.1 An intermediary object: the 2.5DM........................ 145 6.1.1 The consistency conditions........................ 146 6.1.1.1 Case of one equality....................... 147 6.1.1.2 Case of two equalities...................... 147 6.1.1.3 Case of three equalities..................... 148 6.2 The spin-adapted lifting conditions........................ 149 6.3 Formulation as a semidefinite program...................... 153 6.3.1 The overlap matrix............................. 154 6.4 Results........................................ 154 7 Conclusions and outlook 157 A Mathematical concepts and notation 161 B Angular momentum algebra 165 B.1 Spin coupling.................................... 165 B.2 Spherical tensor operators............................. 167 vi CONTENTS C Hermitian adjoint maps 169 C.1 v2DM formalism.................................. 169 C.1.1 Spin symmetry............................... 169 C.1.2 Spin and angular momentum symmetry................. 172 C.1.3 Translational invariance.......................... 173 C.1.4 Translational invariance with parity................... 175 C.2 v2.5DM formalism................................. 177 Bibliography 183 Nederlandstalige samenvatting 191 List of published papers 197 CHAPTER 1 Introduction In the introductory Section, one is supposed to set the stage for the coming Chapters. This stage is the quantum many-body problem. Its importance lies in the fact that it provides the fundamental description of processes in fields as varied as atomic, molecular, solid state and nuclear physics. Apart from the thrill of exploring physical phenomena at the quantum level there is also the realization that predicting and manipulating such microscopic processes has powered much of the 20th century technology, and
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