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.

COUPLED-CLUSTER CALCULATIONS OF GROUND AND EXCITED STATES OF

NUCLEI

Marta Włoch,a

Jeffrey R. Gour,a and Piotr Piecucha,b

a Department of Chemistry,Michigan State University,East Lansing, MI 48824

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 PHYSICSCHEMISTRY 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 PHYSICSNUCLEAR 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.

pro

tons

neutrons

82

50

28

28

50

82

2082

28

20

126

Abinitiofew-body

calculations

Shell Model

The landscapeand the models

LargeLarge--scalescalecomputingcomputing

pro

tons

neutrons

82

50

28

28

50

82

2082

28

20

126

Abinitiofew-body

calculations

Shell Model

The landscapeand the models

pro

tons

neutrons

82

50

28

28

50

82

2082

28

20

126

Abinitiofew-body

calculations

Shell ModelShell Model

The landscapeand the models

LargeLarge--scalescalecomputingcomputingLargeLarge--scalescalecomputingcomputing

MANY-BODY TECHNIQUES DEVELOPED IN ONE AREA SHOULD BE

APPLICABLE TO ALL AREAS

MOLECULAR ELECTRONIC STRUCTURE: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.

NUCLEAR STRUCTURE: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:ELECTRONIC STRUCTURE:Bond breaking in F2

NUCLEAR STRUCTURE:NUCLEAR STRUCTURE:Binding energy of 4He

(4 shells)

Method Energy (MeV)

osc|H’|osc -7.211

HF|H’|HF -10.520

CCSD -21.978

CR-CCSD(T) -23.524

Full Shell Model (Full CI)

-23.484

Nucleus 4 shells 7 shells4He 4E4 9E68B 4E8 5E1312C 6E11 4E1916O 3E14 9E24

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 AFTER THE INTRODUCTION OF DIAGRAMMATIC METHODS AND COUPLED-CLUSTER THEORY TO CHEMISTRY BY COUPLED-CLUSTER THEORY TO CHEMISTRY BY JJÍRI ČÍŽEKÍRI ČÍŽEK AND AND JOE JOE PALDUSPALDUS AND AFTER THE DEVELOPMENT OF DIAGRAM FACTORIZATION AND AFTER THE DEVELOPMENT OF DIAGRAM FACTORIZATION TECHNIQUES BY TECHNIQUES BY ROD BARTLETTROD BARTLETT, QUANTUM CHEMISTS HAVE LEARNT , QUANTUM CHEMISTS HAVE LEARNT HOW TO GENERATE EFFICIENT COMPUTER CODES FOR ALL KINDS OF HOW TO GENERATE EFFICIENT COMPUTER CODES FOR ALL KINDS OF COUPLED-CLUSTER METHODSCOUPLED-CLUSTER METHODS

EXAMPLE: IMPLEMENTATION OF THE CCSD METHODEXAMPLE: IMPLEMENTATION OF THE CCSD METHOD

FACTORIZED CCSD EQUATIONSFACTORIZED CCSD EQUATIONS

(WITH A MINIMUM no2nu

4 OPERATION COUNT AND nonu3 MEMORY

REQUIREMENTS)

RECURSIVELY

GENERATED

INTERMEDIATES

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

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

See K. Kowalski, D.J. Dean, M. Hjorth-Jensen, T. Papenbrock, and P. Piecuch, Phys. Rev. Lett., 2004.

=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 Extension of the active-space EOMCC methods to excited states of radicals via the electron-attached and ionized EOMCC formalismsradicals 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’+cmHcm)

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)

Propertiesl- 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 A-1, A+1) A-1, A+1)

Ground and Excited States of 16O (Idaho-A)

Ground State

Idaho-A Binding Energy, No Coulomb: -7.46 MeV/nucleon (CCSD)

-7.53 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 StateIdaho-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 4He and 16O (Argonne V8′)

The coupled-cluster approach accurately reproduces the very expensive full shell model results at a fraction of a cost.

Ground-state properties of 16O, Idaho-A

Form factor

Exp.: 2.73±0.025 fm

CCSD: 2.51 fm

Ground and Excited States of Open-Shell Systems Around 16O (N3LO)

Total Binding Energies in MeV

18.85

3.064.14

15.66

15O

17O

16O

Zero Order Estimate of 3- State of 16O

Coupled Cluster: 18.85-3.06 = 15.79 MeV

Experiment: 15.66-4.14 = 11.52 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 16O with Various Potentials

• The non-local N3LO and CD-Bonn interactions give much stronger binding than the local Argonne V18 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)

SummaryWe 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.

AcknowlegementsThe 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