COUPLED-CLUSTER CALCULATIONS OF GROUND AND EXCITED STATES OF NUCLEI Marta Włoch, a Jeffrey R. Gour, a and Piotr Piecuch a,b a Department of Chemistry,Michigan State University, East Lansing, MI b Department of Physics and Astronomy and NSCL Theory Group, Michigan State University, East Lansing, MI 48824

QUANTUM MANY-BODY METHODS CROSS THE BOUNDARIES OF MANY DISCIPLINES … CHEMISTRY AND MOLECULAR PHYSICS Accurate ab intio electronic structure calculations, followed by various types of molecular modeling, provide a quantitative and in-depth understanding of chemical structure, properties, and reactivity, even in the absence of experiment. By performing these calculations, one can solve important chemical problems relevant to combustion, catalysis, materials science, environmental studies, photochemistry, and photobiology on a computer. NUCLEAR PHYSICS Physical properties, such as masses and life-times, of short-lived nuclei are important ingredients that determine element production mechanisms in the universe. Given that present nuclear structure facilities and the proposed Rare Isotope Accelerator will open significant territory into regions of medium-mass and heavier nuclei, it becomes imperative to develop predictive (i.e. ab initio) many-body methods that will allow for an accurate description of medium-mass systems that are involved in such element production. A highly accurate ab initio description of many-body correlations from elementary NN interactions provides a deep insight into (only partially understood) interactions themselves.

MANY-BODY TECHNIQUES DEVELOPED IN ONE AREA SHOULD BE APPLICABLE TO ALL AREAS

MOLECULAR ELECTRONIC STRUCTURE: Molecular orbital (MO) basis set (usually, linear combination of atomic orbitals (LCAO) obtained with Hartree-Fock or MCSCF). Examples of AO basis sets: 6-311G++(2df,2pd), cc-pVDZ, MIDI, aug-cc-pVTZ.basis setscc-pVDZ NUCLEAR STRUCTURE: Example: Harmonic-oscillator (HO) basis set.

The key to successful description of nuclei and atomic and molecular systems is an accurate determination of the MANY-PARTICLE CORRELATION EFFECTS. INDEPENDENT-PARTICLE-MODEL APPROXIMATIONS, such as the popular Hartree-Fock method, are inadequate and DO NOT WORK !!! ELECTRONIC STRUCTURE: Bond breaking in F 2 NUCLEAR STRUCTURE: Binding energy of 4 He (4 shells) MethodEnergy (MeV)  osc |H’|  osc   HF |H’|  HF  CCSD CR-CCSD(T) Full Shell Model (Full CI)

Nucleus4 shells7 shells 4 He4E49E6 8B8B4E85E13 12 C6E114E19 16 O3E149E24 Many-particle correlation problem in atoms, molecules, nuclei, and other many-body systems is extremely complex … Dimensions of the full CI spaces for many-electron systems Dimensions of the full shell model spaces for nuclei Full CI = Full Shell Model (=exact solution of the Schrödinger equation in a finite basis set) has a FACTORIAL scaling with the system size (“N! catastrophe”) Highly accurate yet low cost methods for including many-particle correlation effects are needed to study medium-mass nuclei.

SINGLE-REFERENCE COUPLED-CLUSTER (CC) THEORY

Standard Iterative Coupled Cluster Methods

AFTER THE INTRODUCTION OF DIAGRAMMATIC METHODS AND COUPLED-CLUSTER THEORY TO CHEMISTRY BY JÍRI ČÍŽEK AND JOE PALDUS AND AFTER THE DEVELOPMENT OF DIAGRAM FACTORIZATION TECHNIQUES BY ROD BARTLETT, QUANTUM CHEMISTS HAVE LEARNT HOW TO GENERATE EFFICIENT COMPUTER CODES FOR ALL KINDS OF COUPLED-CLUSTER METHODS EXAMPLE: IMPLEMENTATION OF THE CCSD METHOD

FACTORIZED CCSD EQUATIONS (WITH A MINIMUM n o 2 n u 4 OPERATION COUNT AND n o n u 3 MEMORY REQUIREMENTS)

RECURSIVELY GENERATED INTERMEDIATES Matrix elements of the similarity transformed Hamiltonian serve as natural intermediates …

COUPLED-CLUSTER METHODS PROVIDE THE BEST COMPROMISE BETWEEN HIGH ACCURACY AND RELATIVELY LOW COMPUTER COST …

See K. Kowalski, D.J. Dean, M. Hjorth-Jensen, T. Papenbrock, and P. Piecuch, Phys. Rev. Lett., =exact (full CI)

Beyond the Standard CC Methods Approximate Higher-Order Methods Excited States Properties Open-Shell and Other Multi-Reference Problems

ACTIVE-SPACE CC AND EOMCC APROACHES (CCSDt, CCSDtq, EOMCCSDt, etc.) [Piecuch, Oliphant, and Adamowicz, 1993, Piecuch, Kucharski, and Bartlett, 1998, Kowalski and Piecuch, 2001, Gour, Piecuch, and Włoch, 2005]

Particle Attached and Particle Removed EOMCC Theory Particle Attaching Particle Removing Solve the Eigenvalue Problem

Extension of the active-space EOMCC methods to excited states of radicals via the electron-attached and ionized EOMCC formalisms  x  y     CH + Creating CH  x  y     OH - Creating OH Active Space

Bare Hamiltonian (N3LO, Idaho-A, etc.) Effective Hamiltonian (e.g., G-matrix, Lee-Suzuki) Center of mass corrections (H = H’+  cm H cm ) Sorting 1- and 2-body integrals of H CCSD (ground state) t-amplitude equations Properties  equations “Triples” energy corrections EOMCCSD (excited states) r-amplitude equations CR-CCSD(T) Properties l- and r- amplitude equations “Triples” energy corrections CR-EOMCCSD(T) PR-EOMCCSD (A-1) 1h & 2h-1p r-amplitude eqs. PA-EOMCCSD (A+1) 1p & 2p-1h r-amplitude eqs. (A  A-1, A+1)

Ground and Excited States of 16 O (Idaho-A) Ground State Idaho-A Binding Energy, No Coulomb: MeV/nucleon (CCSD) MeV/nucleon (CR-CCSD(T)) Approx. Coulomb: +0.7 MeV/nucleon Idaho-A + Approx. Coulomb: -6.8 MeV/nucleon N3LO (with Coulomb): -7.0 MeV/nucleon Experiment: -8.0 MeV/nucleon (approx. -1 MeV due to three-body interations) J=3- Excited State Idaho-A Excitation Energy: 11.3 MeV (EOMCCSD) 12.0 MeV (CR-EOMCCSD(T)) Experiment: 6.12 MeV (5-6 MeV difference)

Comparison of Shell Model and Coupled- Cluster Results for the Total Binding Energies of 4 He and 16 O (Argonne V 8′ ) The coupled-cluster approach accurately reproduces the very expensive full shell model results at a fraction of a cost.

Ground-state properties of 16 O, Idaho-A Form factor Exp.: 2.73±0.025 fm CCSD: 2.51 fm

Ground and Excited States of Open-Shell Systems Around 16 O (N 3 LO) Total Binding Energies in MeV O 17 O 16 O Zero Order Estimate of 3 - State of 16 O Coupled Cluster: = MeV Experiment: = MeV 4.27 MeV difference, which accounts for most of the 5-6 MeV discrepancy between the previously shown EOMCC result and experiment Excitation Energies in MeV

Ground and Excited States of Open-Shell Systems Around 16 O with Various Potentials The non-local N 3 LO and CD-Bonn interactions give much stronger binding than the local Argonne V 18 interaction. The different binding energies and spin-orbit splittings indicate that different potentials require different 3-body interactions. The relative binding energies of these nuclei for the various potentials are in good agreement with each other and with experiment. Binding Energy per Nucleon (MeV) Excitation Energies (MeV)

Summary We have shown that the coupled-cluster theory is capable of providing accurate results for the ground and excited state energies and properties of atomic nuclei at the relatively low computer cost compared to shell model calculations, making this an ideal method for performing accurate ab initio calculations of medium-mass systems. Acknowlegements The research was supported by: The National Science Foundation (through a grant to Dr. Piecuch and a Graduate Research Fellowship to Jeffrey Gour) The Department of Energy Alfred P. Sloan Foundation MSU Dissertation Completion Fellowship (Jeffrey Gour) And a special thanks to Morten Hjorth-Jensen and David Dean for providing us with the effective interactions and integrals that made these calculations possible