dft

1. 1
Nuclear DFT: Questions and Challenges
Witold Nazarewicz (Tennessee)
First FIDIPRO-JSPS Workshop on Energy Density Functionals in Nuclei
October 25-27, 2007, Keurusselka, (Jyväskylä) FINLAND
• Introduction
• Recent/relevant examples
• Questions and challenges
• Perspectives

2. Introduction

4. number of nuclei ~ number of processors!

5. Density Functional Theory
(introduced for many-electron systems)
The Hohenberg-Kohn theorem states that the ground state electron density minimizes
the energy functional:
P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas”, Phys. Rev. 136, B864 (1964)
M. Levy, Proc. Natl. Acad. Sci (USA) 76, 6062 (1979)
an universal
functional
The original HK proof applies to systems with nondegenerate ground states. It proceeds
by reductio ad absurdum, using the variational principle. A more general proof was given by
Levy.
The minimum value of E is the ground state electronic energy
Since F is a unique functional of the charge density, the energy
is uniquely defined by 
Electron density is the fundamental variable
proof of the Hohenberg-Kohn theorem is not constructive,
hence the form of the universal functional F is not known
Since the density can unambiguously specify the potential, then contained within the
charge density is the total information about the ground state of the system. Thus what
was a 4N(3N)-variable problem (where N is the number of electrons, each one having
three Cartesian variables and electron spin) is reduced to the four (three) variables
needed to define the charge density at a point.

6. W. Kohn and L.J. Sham, "Self-Consistent Equations Including Exchange and Correlation
Effects,” Phys. Rev. 140, A1133 (1965)
Density Functional Theory
Kohn-Sham equations
• Takes into account shell effects
• The link between T and  is indirect, via the orbitals f
• The occupations n determine the electronic configuration
Orbitals f form a complete set. The occupations n are given by the Pauli principle (e.g.,
n=2 or 0). The variation of the functional can be done through variations of individual s.p.
trial functions with a constraint on their norms. It looks like HF, but is
replaced by E[].
Kohn-Sham equation
Kohn-Sham potential (local!)
has to be
evaluated
approximately

7. Local Density Approximation (LDA)
for the exchange+correlation potential:
• One performs many-body calculations for an infinite system with a constant density 
• The resulting energy per particle is used to extract the xc-part of the energy, exc(),
which is a function of 
• The LDA of a finite system with variable density (r) consists in assuming the local
xc-density to be that of the corresponding infinite system with density  =(r):
• The formalism can be extended to take spin degrees of freedom, by introducing spin-
up and spin-down densities (‘Local Spin Density’ formalism, LSD or LSDA)
The LDA is often used in nuclear physics (Brueckner et al., Negele). It states that the
G-matrix at any place in a finite nucleus is the same as that for nuclear matter at the
same density, so that locally one can calculate G-matrix as in nuclear matter calculation.
Pairing extension: DFT for semiconductors; also: the Hartree-Fock-
Bogoliubov (HFB) or Bogoliubov-de Gennes (BdG) formalism.

8. Mean-Field Theory ⇒ Density Functional Theory
• mean-field ⇒ one-body densities
• zero-range ⇒ local densities
• finite-range ⇒ gradient terms
• particle-hole and pairing
channels
• Has been extremely successful.
A broken-symmetry generalized
product state does surprisingly
good job for nuclei.
Nuclear DFT
• two fermi liquids
• self-bound
• superfluid

9. Nuclear Local s.p. Densities and Currents
isoscalar (T=0) density
isovector (T=1) density
isovector spin density
isoscalar spin density
current density
spin-current tensor density
kinetic density
kinetic spin density
+ analogous p-p densities and currents

10. Justification of the standard Skyrme functional: DME
In practice, the one-body density matrix is strongly peaked around
r=r’. Therefore, one can expand it around the mid-point:
The Skyrme functional was justified in such a way in, e.g.,
•Negele and Vautherin, Phys. Rev. C5, 1472 (1972); Phys. Rev. C11, 1031 (1975)
•Campi and Bouyssy, Phys. Lett. 73B, 263 (1978)
… but nuclear EDF does not have to be related to any given
effective two-body force!
Actually, many currently used nuclear energy functionals are not
related to a force
DME and EFT+RG

11. Not all terms are equally important. Some probe specific observables
pairing
functional
Construction of the functional
Perlinska et al., Phys. Rev. C 69, 014316 (2004)
Most general second order expansion in densities and their derivatives
p-h density p-p density

12. A remark: physics of exotic nuclei is demanding
Interactions
• Poorly-known spin-isospin
components come into play
• Long isotopic chains crucial
Interactions
Many-body
Correlations
Open
Channels
Open channels
• Nuclei are open quantum systems
• Exotic nuclei have low-energy decay
thresholds
• Coupling to the continuum important
•Virtual scattering
•Unbound states
•Impact on in-medium Interactions
Configuration interaction
• Mean-field concept often questionable
• Asymmetry of proton and neutron
Fermi surfaces gives rise to new
couplings (Intruders and the islands of
inversion)
• New collective modes; polarization
effects

13. Nuclear DFT
Global properties, global calculations
* Global DFT mass calculations: HFB mass formula: m~700keV
• Taking advantage of high-performance computers
M. Stoitsov et al.
S. Goriely et al., ENAM’04

14. S. Cwiok, P.H. Heenen, W. Nazarewicz
Nature, 433, 705 (2005)
Mass differences, global calculations
Stoitsov et al., PRL 98, 132502 (2007)

15. Global calculations of ground-state spins and parities
for odd-mass nuclei
L. Bonneau, P. Quentin, and P. Möller, Phys. Rev. C 76, 024320 (2007)

16. Global calculations of the lowest 2+ states
Terasaki et al.
QRPA
Bertsch et al.
Sabbey et al.

17. Isospin dynamics
important!
High-spin terminating states
Zdunczuk et al.,Phys.Rev. C71, 024305(2005)
Stoitcheva et al., Phys. Rev. C 73, 061304(R) (2006)
• Excellent examples of single-
particle configurations
• Weak configuration mixing
• Spin polarization; probing time-
odd terms!
• Experimental data available

18. Prog. Part. Nucl. Phys. 59, 432 (2007)

19. Different
deformabilities!
Shell effects in metastable minima
seem to be under control
P.H. Heenen et al.,
Phys. Rev. C57, 1719 (1998)
Important data needed to fix
the deformability of the NEDF:
• absolute energies of SD states
• absolute energies of HD states
Advantages:
• large elongations
• weak mixing with ND structures
Example: Surface Symmetry Energy
Microscopic LDM and Droplet Model Coefficients: P.G. Reinhard et al. PRC 73, 014309 (2006)

20. Beyond Mean Field
nuclear collective dynamics, correlation energy
Large-scale GCM+proj correlation
energy calculations
M. Bender et al.: PRC 73, 034322
(2006); PRL 94, 102503 (2005)
PN Projected HFB equations
M. Stoitsov et al.: PRC75,
014308 (2007)

21. Questions and challenges

22. Intrinsic-Density Functionals
J. Engel, Phys. Rev. C75, 014306 (2007)
Generalized Kohn-Sham Density-Functional Theory via Effective
Action Formalism
M. Valiev, G.W. Fernando, cond-mat/9702247
B.G. Giraud, B.K. Jennings, and B.R. Barrett, arXiv:0707.3099 (2007);
B.G. Giraud, arXiv:0707.3901 (2007)
How to extend DFT to finite, self-bound systems?

23. What are the missing pieces?
What is density dependence?
(ph and pp channels)
Spin-isospin sector (e.g., tensor)
Momentum dependence of the effective mass?
• Induced interaction
• Isovector and isoscalar

24. How to root nuclear DFT in a microscopic theory?
ab-initio - DFT connection
NN+NNN - EDF connection (via EFT+RG)

25. Ab-initio - DFT Connection
• One-body density matrix is the key quantity to study
• “local DFT densities” can be expressed through (x,x’)
• Testing the Density Matrix Expansion and beyond
UNEDF Homework
• Introduce external potential
• HO for spherical nuclei (amplitude of
zero-point motion=1 fm)
• 2D HO for deformed nuclei
• Density expressed in COM
coordinates
• Calculate x,x’) for 12C, 16O and
40,48,60Ca (CC)
• Perform Wigner transform to relative
and c-o-m coordinates q and s
• Extract , J, 
• Analyze data by comparing with
results of DFT calculations and low-
momentum expansion studies.
• Go beyond I=0 to study remaining
densities (for overachievers)
Negele and Vautherin:
PRC 5, 1472 (1972)
isospin
UNEDF
Pack Forest meeting

26. Density Matrix Expansion for RG-Evolved Interactions
S.K. Bogner, R.J. Furnstahl et al.
see also:
EFT for DFT
R.J. Furnstahl
nucl-th/070204

27. DFT Mass Formula
(can we go below 500 keV?)
P.G. Reinhard 2004
•need for error and covariance analysis
(theoretical error bars in unknown regions)
•a number of observables need to be considered
(masses, radii, collective modes)
•different terms sensitive to particular data
•only data for selected nuclei should be used
How to optimize the search? SVD can help…
Fitting theories of nuclear binding energies
G. F. Bertsch, B. Sabbey and M. Uusnakki,
Phys. Rev. C71, 054311(2005)

28. http://orph02.phy.ornl.gov/workshops/lacm08/unedf.html
Nicolas Schunck <schuncknf@ornl.gov>
http://www.phy.ornl.gov/theory/papenbro/jan08.htm
Thomas Papenbrock <papenbro@mail.phy.ornl.gov>

29. J. Dobaczewski and J. Dudek, Phys. Rev. C52, 1827 (1995)
M. Bender et al., Phys. Rev. C65, 054322 (2002)
H. Zdunczuk, W. Satula and R. Wyss, Phys. Rev. C71, 024305 (2005)
very poorly determined Can be adjusted to the Landau parameters
•Important for all I>0 states (including low-spin states in odd-A
and odd-odd nuclei)
•Impact beta decay
•Influence mass filters (including odd-even mass difference)
•Limited experimental data available
How to parameterize time-odd pieces?

30. DFT for odd-A and odd-odd nuclei
•For J>0 odd-T fields come into play
•For J>0 nuclear density is usually triaxial
•Many quasi-particle states need to be considered
•Pairing has to be treated carefully as blocking can result in a phase transition
•In odd-odd nuclei, residual p-n interaction appears
•Good agreement with even-even masses can be misleading
N. Schunck et al.
K. Rutz , M. Bender, P. -G. Reinhard and J. A. Maruhn
Phys. Lett. B468, 1 (1999)
HFB+SLy4, full blocking, alpha chains in SHE
S. Ćwiok, W. Nazarewicz, and P. H. Heenen
Phys. Rev. Lett. 83, 1108 (1999)
Recent work by M. Zalewski et al. on SO, tensor,
odd-T fields, and self-consistency
odd-T≠0

31. How to restore broken symmetry in DFT?
J. Dobaczewski et al., Phys. Rev. C (2007); arXiv:0708.0441
• The transition density matrices contains complex poles. Some
cancellation appears if the ph and pp Hamiltonians are the same
• The projection operator cannot be defined uniquely
• Problems with fractional density dependence
• Projected DFT yields questionable results when the pole appears close to
the integration contour. This often happens when dealing with PESs
see also: M. Bender, T. Duguet, D. Lacroix, in preparation.
S. Krewald et al.,Phys. Rev. C 74, 064310 (2006).

32. Can dynamics be incorporated directly into the functional?
Example: Local Density Functional Theory for Superfluid Fermionic Systems: The
Unitary Gas, Aurel Bulgac, Phys. Rev. A 76, 040502 (2007)
See also:
Density-functional theory for fermions
in the unitary regime
T. Papenbrock
Phys. Rev. A72, 041603 (2005)
Density functional theory for fermions
close to the unitary regime
A. Bhattacharyya and T. Papenbrock
Phys. Rev. A 74, 041602(R) (2006)

33. Perspectives

34. • Young talent
• Focused effort
• Large theoretical collaborations
•UNEDF (US+collaborators)
•EFES (JP)
•in works: ARTHENS, EURONS-2 networks (EU)
• Data from terra incognita
• High-performance computing
• Interaction with computer scientists
What is needed/essential?

35. Jaguar Cray XT4 at ORNL
No. 2 on Top500
• 11,706 processor nodes
• Each compute/service node
contains 2.6 GHz dual-core AMD
Opteron processor and 4 GB/8 GB
of memory
• Peak performance of over 119
Teraflops
• 250 Teraflops after Dec.'07 upgrade
• 600 TB of scratch disk space
1Teraflop=1012 flops
1peta=1000 tera

36. 36
Example: Large Scale Mass Table Calculations
Science scales with processors
 The SkM* mass table contains 2525 even-even nuclei
 A single processor calculates each nucleus 3 times (prolate, oblate, spherical) and
records all nuclear characteristics and candidates for blocked calculations in the
neighbors
 Using 2,525 processors - about 4 CPU hours (1 CPU hour/configuration)
 The even-even calculations define 250,754 configurations in odd-A and odd-odd nuclei
assuming 0.5 MeV threshold for the blocking candidates
 Using 10,000 processors - about 25 CPU hours
Even-Even Nuclei
Jaguar@
Odd and odd-odd Nuclei
M. Stoitsov, HFB+LN mass table, HFBTHO

37. Summary
A. Staszczak, J. Dobaczewski, W. Nazarewicz, in preparation
Bimodal fission in nuclear DFT
nucl-th/0612017
Guided by data on short-lived nuclei, we are embarking on a comprehensive
study of all nuclei based on the most accurate knowledge of the strong inter-
nucleon interaction, the most reliable theoretical approaches, and the massive
use of the computer power available at this moment in time. The prospects
look good.

38. Backup

39.  Comparison between adaptive 3-D multi-wavelet
(iterative) and direct 2D and 3D harmonic oscillator (HO)
bases (direct)
 One-body test potential (currently no spin-orbit)
 Pöschl-Teller-Ginocchio (PTG): analytical solutions
 2-cosh : simulates fission process; HO expansion
expected to work poorly at large separations
Example: 3-D Adaptive Multiresolution Method for Atomic Nuclei
(taking advantage of jpeg200 technology)
PTG Exact 2D-HO 3D-HO Wavelets (Prec.: 10-3)
-39.740 042 -39.740 041 -39.740 021 -39.740
-18.897 703 -18.897 675 -18.897 059 -18.898
-0.320 521 -0.314 751 -0.256 288 -0.320
N
State
(2-cosh)
Wavelet HO basis 1-cosh
1 1/2 (+) -22.1542 -22.0963 -22.2396
2 1/2 (-) -22.1540 -22.0961 -
3 1/2 (+) -9.1011 -9.0200 -9.2115
4 1/2 (-) -9.0927 -9.0121 -
5 3/2 (-) -9.0926 -9.0121 -9.2115
6 1/2 (+) -9.0913 -9.0107 -
7 3/2 (+) -9.0912 -9.0107 -9.2115
8 1/2 (-) -9.0857 -9.0051 -
9 1/2 (+) -1.6541 -1.6008 -1.6225
10 1/2 (-) -1.4472 -1.3944 -
G. Fann, N. Schunck, et al.

40. The origin of SO splitting can be attributed to 2-body SO and tensor forces,
and 3-body force
R.R. Scheerbaum, Phys. Lett. B61, 151 (1976); B63, 381 (1976); Nucl. Phys. A257, 77
(1976); D.W.L. Sprung, Nucl. Phys. A182, 97 (1972); C.W. Wong, Nucl. Phys. A108,
481 (1968); K. Ando and H. Bando, Prog. Theor. Phys. 66, 227 (1981); R. Wiringa and
S. Pieper, Phys. Rev. Lett. 89, 182501 (2002)
The maximum effect is in spin-unsaturated systems
Discussed in the context of mean field models:
Fl. Stancu, et al., Phys. Lett. 68B, 108 (1977); M. Ploszajczak and M.E. Faber, Z.
Phys. A299, 119 (1981); J. Dudek, WN, and T. Werner, Nucl. Phys. A341, 253 (1980);
J. Dobaczewski, nucl-th/0604043; Otsuka et al. Phys. Rev. Lett. 97, 162501 (2006);
Lesinski et al., arXiv:0704.0731,…
…and the nuclear shell model:
T. Otsuka et al., Phys. Rev. Lett. 87, 082502 (2001); Phys. Rev. Lett. 95, 232502
(2005)
Example: Spin-Orbit and Tensor Force
(among many possibilities)
j<
j>
F
2, 8, 20
Spin-saturated systems
j<
j>
F
28, 50, 82, 126
Spin-unsaturated systems

41. Importance of the tensor interaction far from stability
[523]7/2
[411]1/2
Proton emission
from 141Ho