# Presentation: 666115

## Building Nuclei from the Ground Up: Nuclear Coupled-cluster Theory

- David J. Dean
**Oak Ridge National Laboratory**
**Nuclear Coupled-cluster Collaboration:****T. Papenbrock, K. Roche, Oak Ridge National Laboratory****P. Piecuch, M. Wloch, J. Gour, Michigan State University ****M. Hjorth-Jensen, Oslo**
- Funding: DOE-NP, SciDAC, DOE-ASCR

## “Given a lump of nuclear material, what are its properties, and how does it interact?”

- How do we describe nuclei we cannot measure?
**Robust, predictive nuclear theory exists for structure and reactions**
**Nuclear data needed to constrain theory**
**Goal is the Hamiltonian****	****– Bare intra-nucleon Hamiltonian****	****– Energy density functional **
**Mission relevant to NP, NNSA**
**Half of all elements heavier than iron produced in r-process where limited (or no) experimental information exits**

- “Given a lump of nuclear material, what are its properties, and how does it interact?”

## Pushing the nuclear boundaries

- The Leadership Computing Facility effort will:
**Enlarge ab-initio square to mass 100 **
**Enable initial global DFT calculations with restored symmetries **

- All Regions**:**** Nuclear cross-section efforts ****(NNSA, RIA, Nuclear Energy)**

**Nuclear DFT effort**

**Thermal Properties regions**

**Nuclear Coupled Cluster effort**

## Nuclear interactions: Cornerstone of the entire theoretical edifice

**Solved up to mass 12 with GFMC, converged mass 8 with diagonalization.  We want to go much further!**

- Nuclear interactions: Cornerstone of the entire theoretical edifice

**Real three-body interactions****derived from QCD-based****effective theories**

- Method of Solution:
**Nuclear Coupled-Cluster Theory**

**Depends on spin, angular momentum, and nucleon****(proton and neutron) quantum numbers. Complicated interactions.**

-   

-      ***i***** **  V*r****i***, *r****j***V***NNN***

***2***

***i *****= 1, ******

***-h******-2***

***2M******i***

***i****** < j***

## Coupled-cluster theory: Ab initio in medium mass nuclei

**It boils down to a set of coupled, nonlinear algebraic equations**
**Storage of both amplitudes and interactions is an issue as problems scale up**
**Largest problem so far: 40 Ca with 2 million unknowns, 7 peta-ops to solve once****(up to 10 runs per publishable result)**
**Breakthrough science: Inclusion of 3-body force into CC formalism (6d tensor)****(work in progress); weakly bound systems (see next)**

- Correlated ground-statewave function

- Correlationoperator

- Reference Slaterdeterminant

- Energy

- Amplitude equations

**  **1  **2  **3  *L*

- +

*E*  exp*** *** exp*** *

- *ab*Lexp*** *** exp*** *  *ab*L_x0008_**  

*ij*L

*ij*L

- –

## Coupled-cluster theory: Ab initio in heavier nuclei

- Effect of 3-body clusters on states
**Collaborations with P. Piecuch (MSU)**
**Implemented CR-CCSD(T) for the nuclear problem. **
**Calculated excited states with EOMCCSD and CR-EOMCCSD(T)**
**Calculated A+/-1 systems. **

- Coupled-cluster theory: Ab initio in heavier nuclei

- Results:
**Excited state is high with the two-body force alone (lack of good “l-splitting”) **
**Triple amplitudes are small and have little effect on states **
**Different forces yield different spin-orbit splittings **
**Some further oscillator dependence in A+/-1 systems compared with A **

**Dean and Hjorth-Jensen, **PRC69, 054320 (2004)**; Kowalski, Dean, Hjorth-Jensen, Papenbrock, Piecuch,**PRL92, 132501 (2004)**; Wloch, Dean, Gour, Hjorth-Jensen, Papenbrock, Piecuch, **PRL94, 21501 (2005)**;****Gour, Piecuch, Wloc, Hjorth-Jensen, Dean, **PRC (2006)**; Hagen, Dean, Hjorth-Jensen, Papenbrock, **PRL (2006)

**4**

**6**

**8**

**infty**

**Number of Oscillator Shells**

**Wolch et al., PRL (2005)**

**0**

**5**

**0**

**5**

**0**

**5**

**0**

**E = (MeV)**

**E* (0****+****) = 19.8 MeV**

**E* (3****-****) = 12.0 MeV**

***E****g.s.**** = –120.5 MeV**

## Solution of coupled-cluster equation 

**System of non-linear coupled algebraic equations ******** solve by iteration**
**n = number of neutrons and protons**
**N = number of basis states**
**Solution tensor memory**
** ****(N-n)**********2*********n**********2**
**Interaction tensor memory**
** ****N**********4**
**Operations count scaling**
** ****O(n**********2*********N**********4)**
** ****O(n**********4*********N**********4) with 3-body**
** ****O(n**********3*********N**********5) at CCSDT **

- Solution of coupled-cluster equation

- Many such terms exist
- Cast into a matrix-matrix multiply algorithm
- Parallel issue: block sizes of V and t

- Basic numerical operation:

*t**new *(*ab*, *ij*) = *V(kl, cd)t**old **(cd, ij)t**old **(ab, kl)*

*k, l = *1, *n**c, d = n+*1*, N*

## Weakly bound and unbound nuclei – requires continuum states: Complex coupled-clusters [Hagen et al. PRL (2006)]

- Weakly bound and unbound nuclei – requires continuum states: Complex coupled-clusters [Hagen et al. PRL (2006)]

**Single-particle basis includes bound, resonant, non-resonant continuum, and scattering states (5s5p5d4f4g protons; 20s20p5d4f4g neutrons)**

**He chain results**

- N3LO Vlowk (****= 1.9 fm-1)

## Code parallelism

***V(ab, c1, d1)***
***t2***** partial sum**

***V(ab, c1, d2)***
***t2***** partial sum**

****

***V(ab, c2, d2)***
***t2***** partial sum**

***V(ab, c2, d1)***
***t2***** partial sum**

****

***V(ab, c1, d1)***
***t2 *****partial sum**

***V(ab, c1, d2)***
***t2 *****partial sum**

***V(a, b, c2, d2)***
***t2***** partial sum**

***V(ab, c2, d1)***
***t2***** partial sum**

- Global reduce (sum) *t2*, distribute

## Future direction

**Current algorithm scales to 1K processors with about 20% efficiency. Attacking problems in mass 16 region is doable with current code **
**Develop algorithm that spreads both the two-body matrix elements and the CC amplitudes (in collaboration with Ken Roche) ******** Enables nuclei in the mass 100 region and should scale to 100K processors**
**Design further parallel algorithms that calculate nuclear properties to calculate densities and electromagnetic transition amplitudes **
**Eventual time-dependent CC for fission dynamics**

## Contacts

- Contact name
- David J. Dean
- Physical Sciences Directorate
- Nuclear Theory
- (865) 576-5229
- deanj@ornl.gov

-    Dean_NCCT_0611