1. Università degli Studi and INFN, MIlano
Density Functional Theory for stable and exotic nuclei (plus extensions) LECTURE I
Gianluca Colò
Università degli Studi and INFN, MIlano
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2. Università degli Studi and INFN, MIlano
Outline (I)
Introduction (basically, motivations for DFT).
DFT: the original Hohenberg-Kohn theorem; transferring the theory from electronic systems to nuclei; basics on Skyrme, Gogny (and relativistic) functionals.
Properties of finite nuclei within DFT: ground-state properties like masses …
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

3. Università degli Studi and INFN, MIlano
Advantages of DFT (I)
The principle is relatively simple: write the energy of a system as a functional of the density and minimize !
Aims at being a theory that reconciles all observables: total energy of the nucleus, collective rotations and vibrations, single-particle states (to some extent) …
To be confronted with more accurate theories having more limited goals !
11Li
2n halo nucleus
S2n= 300 keV
9Li n
From: T. Nakamura
154Sm vs. 40Ca
From: P. Flip
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

4. Università degli Studi and INFN, MIlano
Advantages of DFT (II)
From: D. Vretenar
Can access a very large part of the nuclear chart, if not all
Excited states
Link with nuclear matter and compact objects like neutron stars
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

5. The many-body fermion problem
We are concerned with a quantum system governed by:
In the case of electrons and ions, one has clearly:
The N-electrons Schrödinger equation cannot be solved.
DFT is the most popular framework in the physics of electronic systems and in chemistry, due to its conceptual elegance and capability to be accurate at relatively low computational cost.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

6. The Hohenberg-Kohn theorem
The original theorem and its proof can be found in P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)1. We have in mind a system of interacting fermions (H = T + V) in some external potential Vext.
There exist a functional of the fermion density
and the part denoted by F is universal (for nuclei, it would be the only part).
b) It holds:
or, in other words, at least in principle this functional has a minimum at the exact ground-state density where it assumes the exact energy as a value.
1 More than 20,000 citations (2017)
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Università degli Studi and INFN, MIlano

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Proof
We wish to show the one-to-one correspondence between the external potential and the ground-state density. For every potential we have a given ground-state density, because the Schrödinger equation defines a unique ground-state wave function (we assume no degeneracy).
We need to show now that the above two mappings can be inverted in a unique way. For the first one, we proceed with a reductio ad absurdum. Let us assume that two different external potentials lead to the same w.f.
By subtracting we find:
Therefore the two potentials are not different and we have come to a contradiction.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

8. Università degli Studi and INFN, MIlano
We need to show that the mapping between w.f. and density can also be inverted. Once more, with a reductio ad absurdum, let us start by assuming that two different wavefunctions lead to the same density. For what said above, it means that two different potentials lead to the same density.
The second terms at r.h.s. are equal:
By subtracting we come to
which is in contradiction with the variational principle because Ψ1 (Ψ2) is the g.s. of H1 (H2), so that the l.h.s is negative, the r.h.s. is positive !
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9. Università degli Studi and INFN, MIlano
The Kohn-Sham scheme
One considers an “auxiliary” systems of independent particles in “orbitals”.
Reminder: their single-particle w.f.’s form the Slater determinant
We can assume that the associated densities span all possible forms, including the exact density. We build a KS functional in the following way:
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Università degli Studi and INFN, MIlano

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The good, the bad and …
The HK theorem has been generalised in (almost all) possible ways: degenerate ground-state, magnetic systems (m or spin densities), finite T, relativistic case … (the list might be not exhaustive).
In all cases one then build the energy functional E and solves:
Then, all quantities are in principle given (!).
The REAL weak point is that the various proofs of HK theorems do not give any clue on HOW to build the functional F.
Since there are many approaches to build it, the real disadvantage of DFT is that there are many realizations of such theory !
One must use the functional derivative:
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11. Università degli Studi and INFN, MIlano
… the ugly
In the physics of matter, DFT has still problems: (i) gaps of solids are underestimated, (ii) Van Der Waals systems are also not well described, (iii) there are other issues with strongly correlated systems.
It is domain for specialists … and even in nuclear physics we have too many E[ϱ]’s !
From: K. Burke
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12. Università degli Studi and INFN, MIlano
Nuclear saturation
Described by the Bethe-Weiszäcker mass formula:
B/A is ~ 8 MeV
The volume part is ~16 MeV
M. Waroquier et al., Phys. Rep. 148, 249 (1987).
Corresponding density:
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13. Saturation as a minimum in E/A
The empirical saturation point is a stable point for matter with N=Z .
E/A OF SYMMETRIC MATTER HAS A MINIMUM AT ϱ0 = 0.16 fm-3; THE VALUE AT THE MINIMUM IS -16 MeV (E/A without mass = -B/A).
We will see that naïve mean-field does not provide saturation while DFT does.
G.C., Phys. Part. Nucl. 39, 557 (2008).
M. Waroquier et al., Phys. Rep. 148, 249 (1987).
Università degli Studi and INFN, MIlano
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14. Nuclear saturation with constant V: no way !
Let us assume that we wish to obtain saturation in a naïve mean-field picture with a constant V.
A two-body potential with constant matrix elements (a 𝛿-force which is a constant in k-space) would give at the Fermi energy:
The same potential would give in mean-field (Hartree-Fock):
The two equations are incompatible !
T. Nakatsukasa et al., Rev. Mod. Phys. 88, (2016)
Università degli Studi and INFN, MIlano
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams

15. Finite-range is not enough
Due to its short-range, the Fourier transform of the nucleon-nucleon force does not have a strong k-dependence
One must add explicit momentum dependence or exchange terms
It has been shown that Gaussians or Yukawas with exchange terms can produce saturation:
(Very) wrong effective mass !
G. Accorto, R. Romano, A. Rancati, X. Roca-Maza, G.C. (to be published).
Università degli Studi and INFN, MIlano
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams

16. A note on exchange operators
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17. Reminder on effective mass
Here, E is the single-particle energy, NOT the total energy.
First term: E-mass
Second term: k-mass
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18. Effective mass vs. levels in finite nuclei
C. Mahaux et al., Phys. Rep. 120, 1 (1985)
Experimental main peaks of transfer reactions like (d,p) can be reproduced by different Woods-Saxon potentials. Sometimes this is equivalent to a single potential if m* is introduced.
The empirical value of m*/m is between 0.7 and 1.
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19. Università degli Studi and INFN, MIlano
Nuclear DFT (I)
Slater determinant
1-body density matrix
Strategies:
Start from a two-body (three-body) effective force and define H = T + Veff [ϱ]. Write the expectation value on a general Slater determinant “DFT is an exactification of Hartree-Fock” (W. Kohn)
Write directly the Energy Density Functional (EDF).
Università degli Studi and INFN, MIlano
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams

20. Università degli Studi and INFN, MIlano
Nuclear DFT (II)
If the only way to reconcile to some extent different properties of nuclei are density-dependent interactions, ultimately these are just “EDF generators” !
Energy density =
E is a functional
Mechanism for saturation
Flexibility (i.e., enough terms) to account for the many features of finite nuclei and of the EoS
Suitable for extensive computational efforts
Linkable to more fundamental theories (?)
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21. Università degli Studi and INFN, MIlano
The Gogny force
There are two Gaussians with different ranges and spin/isospin dependence
The introduction of a density-dependent term seemed unavoidable. This must be zero-range.
The last term (spin-orbit) is zero-range for simplicity.
The great advantage of the Gogny force is that it seems adequate not only to enter the HF equations but to describe also nuclear superfluidity.
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

22. Università degli Studi and INFN, MIlano
The Skyrme force
attraction
short-range repulsion
There are velocity-dependent terms which mimick the finite-range. They are related to m*.
The last term is a zero-range spin-orbit.
In total: 10 free parameters to be fitted (typically).
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

23. A note on the functional derivative
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24. Università degli Studi and INFN, MIlano
Let us see how it works in practice in a simplified case. A simple two-body force generates E ~ ϱ2. NOTE: In this slide we call E only the potential energy part.
This is the potential energy. The KS equation reads
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
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25. Realistic functionals
One has to consider several possible densities (including spin densities, isospin densities, gradients …).
Non-locality well mocked up by gradients, kinetic energy density etc. - although there are non-local functionals, there does not seem to be strong need.
48Ca
r [fm]
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General EDFs
Question: at which order should one stop the gradient expansion ? The answer is likely to be related to the high momentum cutoff of the theory (EFT philosophy).
All possible terms for EDFs (those compatible with symmetries) are too many.
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27. Analogies with electronic DFT
J. Perdew, K. Schmidt, AIP Conf. Proc. 577, 1 (2001).
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28. Summary of modern functionals and applications
Skyrme (SEDF)
Gogny (GEDF)
Relativistic MF or HF
(CEDF)
local functionals (evolved from Veff ÷ δ(r1-r2))
non-local from Veff having Gaussian shape
covariant functionals
(Dirac nucleons exchanging effective mesons)
Typical number of parameters ~ 10.
Error on masses of the order of 1 MeV (can go further down).
Trends of charge radii and deformations fairly well reproduced.
Can be extended to the case in which the density is not a single Slater determinant (shape coexistence)
Good description of excitations such as giant resonances, rotational bands.
They can also be applied to β and ββ transitions.
There is current interest in large amplitude motion, reactions etc.
(…)
SIF Summer Course Nuclear Physics with Stable and Radioactive Ion Beams
Università degli Studi and INFN, MIlano

29. Masses/binding energies of nuclei
Typical energy density functionals have errors ≈ MeV for the masses. It means about 0.1 % to 1 % (!).
However, we are often interested in differences of masses, for instance for nuclear reactions or nuclear decays.
Interesting scaling: compare with dissociation energy in molecules (eV).
The Bruxelles-Montréal collaboration designed specific Skyrme forces for mass systematics. Quite successful at the expense of ad hoc terms.
Phenomenological part: so-called “Wigner” term, collective correlations - 9 parameters.
Standard Skyrme force
with 10 parameters
Volume pairing force (cf. later):
p-p, n-n, either even or odd
plus cutoff - 5 parameters
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No clear difference between nonrelativistic and covariant functionals.
Problem of the “arches” !
Issue with weakly bound nuclei.
HFB17
HFB16
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