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 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.
126
The landscape
and the models
protons
•
28
20
82
ro
r-p
50 cess
o
-pr
rp
82
50
8
28
neutrons
2
20
28
12
A=
60
A~
ry
heo
l T Field
a
n
io ean
nct
Fu nt M
ity siste
s
n
De lf-con
se
Shell Model
Ab initio
few-body
calculations
LargeLarge-scale
computing
s
ces
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.
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 F2
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
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
Nucleus
4 shells
7 shells
4He
4E4
9E6
8B
4E8
5E13
12C
6E11
4E19
16O
3E14
9E24
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 no2nu4 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
HIGH ACCURACY AND RELATIVELY LOW COMPUTER COST …
=exact (full CI)
See K. Kowalski, D.J. Dean, M. Hjorth-Jensen, T. Papenbrock,
and P. Piecuch, Phys. Rev. Lett., 2004.
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
4s
px ,py
Creating
CH
CH+
4s
px ,py
Active Space
3s
3s
2s
2s
1s
1s
Creating
OH
OH-
Bare Hamiltonian (N3LO, Idaho-A, etc.)
Effective Hamiltonian (e.g., G-matrix, Lee-Suzuki)
PR-EOMCCSD (A-1)
1h & 2h-1p r-amplitude eqs.
Center of mass corrections (H = H’+cmHcm)
PA-EOMCCSD (A+1)
1p & 2p-1h r-amplitude eqs.
Sorting 1- and 2-body integrals of H
CCSD (ground state)
t-amplitude equations
Properties
 equations
“Triples”
energy
corrections
CR-CCSD(T)
(A A-1, A+1)
EOMCCSD (excited states)
r-amplitude equations
Properties
l- and ramplitude
equations
“Triples”
energy
corrections
CR-EOMCCSD(T)
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 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 CoupledCluster Results for
the Total Binding
Energies of 4He and
16O (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 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
Excitation Energies in MeV
15O
Zero Order Estimate of 3- State of 16O
Coupled Cluster: 18.85-3.06 = 15.79 MeV
18.85
15.66
16O
17O
3.06
4.14
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
Ground and Excited States of Open-Shell Systems
Around 16O with Various Potentials
Binding Energy per Nucleon (MeV)
Excitation Energies (MeV)
• 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.
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