LLNL-PRES-777097This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC
Lecture 3: Ab Initio Theory of Nuclear Forces andNuclear Structure (continued)
TALENT Course 6: Theory of exploring nuclear reaction experiments
Sofia Quaglioni
LLNL-PRES-7770972
§How to describe the basic interactions of protons and neutrons?
§Can we describe nuclei as many-body systems of interacting protons and neutrons?
Content
Option 2: Solution of Schroedinger equationfor intrinsic + C.M. Hamiltonian
Work with A single particle coordinates and separate intrinsic and center of mass motion
LLNL-PRES-7770974
A-nucleon problem
H(A) = Ti + VNN(|ri-rj|) + V3NA
i=1
A
i<j=1
A
i<j<k=1ijk
s1 s2
r1r2
O
sA…
rA
LLNL-PRES-7770975
1) Solve single-particle problem:
We know how to solve the independent-particle problem
H(A) = Ti + U(ri) = hi
A
i=1
A
i=1
Single-particleHamiltonian
h ϕn(r) = εn ϕn(r)
~
e.g.: U(r) = mΩ2r2−12
LLNL-PRES-7770976
2) The antisymmetric A-nucleon solutions can be build as
We have that:
We know how to solve the independent-particle problem
1ϕk1(r1) ϕk1(r2) … ϕk1(rA)
ϕk2(r1) ϕk2(r2) … ϕk2(rA)
ϕkA(r1) ϕkA(r2) … ϕkA(rA)
… … …A!− det𝛟k =√−
H(A) 𝛟k = Ek 𝛟k with A
i=1Ek = εki
LLNL-PRES-7770977
3) Use the independent-particle model solutions as `basis states’ to build an ansatz for the A-nucleon wave function
What about our original A-nucleon problem?
Ψ (r1,r2,…,rA) = ck 𝛟k(r1,r2,…,rA) k
(H(A)-E) Ψ = 0 è ck (H(A)-E) 𝛟k= 0k
N
N
Maximum number of excitations
LLNL-PRES-7770978
4) Project the equation on the basis states (from the left)
What about our original A-nucleon problem?
ck ∫𝛟m(r1,…,rA) H(A) 𝛟k(r1,…,rA) dr1…drA*
= E ck ∫𝛟m(r1,…,rA)𝛟k(r1,…,rA) dr1…drAk
*
𝛿mk
k
N
N
Hmk ck = E cmk
N
LLNL-PRES-7770979
The A-nucleon Schrödinger equation becomes a linear algebra eigenvalue problem
§ The elements of the N×N Hamiltonian matrix are
§ And the unknown expansion coefficients ck are the elements of the eigenvector c
What about our original A-nucleon problem?
H mk = ∫𝛟m(r1,…,rA) H(A) 𝛟k(r1,…,rA) dr1…drA*
H c = E c uknown
LLNL-PRES-77709710
The A-nucleon Schrödinger equation becomes a linear algebra eigenvalue problem
§ The elements of the N×N Hamiltonian matrix are
§ And the unknown expansion coefficients ck are the elements of the eigenvector c
What about our original A-nucleon problem?
H c = E c uknown
H mk = ⟨𝛟m | H(A) | 𝛟k⟩ Short-hand notation
LLNL-PRES-77709711
§This is an ‘expansion’ technique: uses large (but finite!) expansion on A-body basis states
§Convergence to the exact result is approached (variationally) by increasing N (i.e., basis size)
§Antisymmetrization is trivial
§Did we forget about anything?
Some notes
LLNL-PRES-77709712
§ In the independent-particle problem, in general the c.m. motion is mixed with intrinsic motion, giving rise to spurious effects
§Exception: harmonic oscillator (HO) potential is exactly separable
What about the center of mass motion?
H(HO) = Ti + mΩ2r2A
i=1
~ −12
LLNL-PRES-77709713
§ In the independent-particle problem, in general the c.m. motion is mixed with intrinsic motion, giving rise to spurious effects
§Exception: harmonic oscillator (HO) potential is exactly separable
What about the center of mass motion?
H(HO)~A
i<j=1−12
= Tint + + Tcm + AmΩ2R2mΩ2 (ri-rj)22A
HintHcm
LLNL-PRES-77709714
§ Solutions of independent-particle problem for the HO potential factorize as
§ For each eigenvalue E there will be a set of degenerate eigenstates all sharing the same intrinsic structure but different c.m. excitations
§ This degeneracy can be broken by adding to the Hamiltonian a Lawson term
§ New eigenvalues are 𝐸& = 𝐸 + 𝜆𝑁+,ℏΩ, i.e. 𝑁+, >0 states are pushed to high energy
Elimination of spurious motion of the center of mass
𝐻& →𝐻 + ℒ ℒ = 𝜆 𝐻+,3 − 56ℏ3
LLNL-PRES-77709715
Second Quantization§ One of the most useful representations in many-body theory
• : the state with no particles (the vacuum)
• : creation operator, creates a fermion in the state i
• : annihilation operator, annihilates a fermion in the state i
• Anticommutation relations:
• So that the Slater determinant can be written as:
0
ai
+
ai
+, a
j
+{ }= ai, a
j{ }= 0, ai
+, a
j{ }= ai, a
j
+{ }= δij
Pauli principle in second quantization
ai
ai
+0 = i , a
i
+i = 0
aii = 0 , a
i0 = 0
ai
+aj
+= −a
j
+ai
+
φn
(A)=1
A!
ϕi
r1( ) ϕ
i
r2( ) … ϕ
i
rA( )
ϕj
r1( ) ϕ
j
r2( ) ϕ
j
rA( )
ϕl
r1( ) ϕ
l
r2( ) … ϕ
l
rA( )
= al
+…a
j
+ai
+0 ,
l >> j > i
implicitly assumes we have already chosen the form of the single-particle states, (i = 1,2,3, ... A) as dictated by some mean-field potential
LLNL-PRES-77709716
§ How are Slater determinants (SD) actually represented in a computer program?§ We are dealing with fermions, so a single-particle state is either occupied or empty, which in
computer language translates to either 1’s or 0’s
§ A very useful approach is a bit representation known as M-scheme§ If the mean-field is spherically symmetric, the single-particle states will have good j, mj
§ A single integer represents a complicated slater determinant
§ While the many-body SD states will have goodM, they do not have good J. States of good Jmust be projected and will be a combination of Slater determinants. Same for T and MT
§ Eigenstates obtained by the diagonalization of the Hamiltonian in SD basis will have a good J and (approximately) good T
= 21+ 2
3+ 2
5+ 2
6=106
0p3/2 0s1/20p1/2
1 010 0 1 10
1 113 1 3 11 ----
a1, 32,−12
+a1, 32, 32
+a1, 12, 12
+a0, 12,−12
+0 =
, j, jz
2mj
LLNL-PRES-77709717
§ Many-body HO Slater determinants
— Antisymmetrization is trivial— Good M, MT and parity quantum numbers, but not J and T
• Huge number of basis states
Multi-particle states in the Slater Determinant basis – Summary
r1
σ1
τ1,r2
σ2
τ2,,rA
σ
A
τAa+
la
+
ja+
i 0
=1
A!
ϕi
r1( ) ϕ
i
r2( ) … ϕ
i
rA( )
ϕj
r1( ) ϕ
j
r2( ) ϕ
j
rA( )
ϕl
r1( ) ϕ
l
r2( ) … ϕ
l
rA( )
ϕnljmj
12mt(r,σ ,τ )
= Rnl(r) Y
l(r̂)⊗ χ S
12
(σ )
mj
j
χ T12mt(τ )
LLNL-PRES-77709718
Hard core of nuclear interaction scatters nucleons to high momenta
V(r)Vkk’ = ∫ eik’r V(r)e-ikrdr
= ⟨k| V |k’⟩
Fourier Transform
J = 0 S = 1ℓ = 0 T = 0
LLNL-PRES-77709719
Hard core of nuclear interaction scatters nucleons to high momenta
V(r)Vkk’ = ∫ eik’r V(r)e-ikrdr
= ⟨k| V |k’⟩
Fourier Transform
J = 0 S = 1ℓ = 0 T = 0
Very large N values (basis sizes) are required to reach convergent solution!
LLNL-PRES-77709720
Basis dimension grows rapidly with A!
Convergence can be a challenge!
LLNL-PRES-77709721
§ Introduce unitary transformation: 𝒰 (𝒰⨥𝒰 = 𝟙)
Effective interactions from unitary transformations of bare Hamiltonian
E = ⟨Ψ| H(A) |Ψ⟩
E = ⟨Ψ| 𝒰⨥𝒰 H(A) 𝒰⨥𝒰 |Ψ⟩
= (⟨Ψ|𝒰⨥) 𝒰H(A)𝒰⨥ (𝒰|Ψ⟩)
= ⟨Ψ| H(A) |Ψ⟩~ ~ ~
Bare Hamiltonian,
wave function
EffectiveHamiltonian,
wave function
LLNL-PRES-77709722
Example: Similarity renormalization group (SRG) transformation
⟨H𝛌 = 𝒰𝛌H 𝒰⨥𝛌 k|
H(2) |k’⟩~ dH𝛌 = -4 [𝜼(𝛌),H𝛌 ]
d𝛌 𝛌5~ ~
Flow parameter
ʹk
k⟨k| H(2) |k’⟩𝛌0 > 𝛌1> 𝛌2 …
Plane wave
𝛌~
Two-body Hamiltonian in momentum space
LLNL-PRES-77709723
§ Start from:
§ Compute derivative with respect to 𝛼 = =>
§ Setting with hermitean
§ Customary choice in nuclear physics …kinetic energy operator
Similarity renormalization group (SRG) transformation
Hα =Uα HUα+
UαUα+ =Uα
+Uα =1
dHα
dα=dUα
dαHUα
+ +UαHdUα
+
dα=dUα
dαUα
+UαHUα+ +UαHUα
+Uα
dUα+
dα
=dUα
dαUα
+Hα +HαUα
dUα+
dα= ηα,Hα[ ]
anti-Hermitiangenerator
ηα ≡dUα
dαUα
+ = −ηα+
ηα = Gα,Hα[ ] Gα
Gα = T
LLNL-PRES-77709724
Example: Similarity renormalization group (SRG) transformation
dH𝛌 = -4 [𝜼(𝛌),H𝛌 ]d𝛌 𝛌5~ ~
𝛌 = 20 fm-1
⟨k| H(2) |k’⟩𝛌~
Low and high momentum components coupled
⟨H𝛌 = 𝒰𝛌H 𝒰⨥𝛌 k|
H(2) |k’⟩~
Initial condition
Hα=0 = Hλ=∞ = H λ 2 =1/ α
LLNL-PRES-77709725
Example: Similarity renormalization group (SRG) transformation
dH𝛌 = -4 [𝜼(𝛌),H𝛌 ]d𝛌 𝛌5~ ~
𝛌 = 2 fm-1
⟨k| H(2) |k’⟩𝛌~
Low and high momentum components de-coupled
⟨H𝛌 = 𝒰𝛌H 𝒰⨥𝛌 k|
H(2) |k’⟩~
Hα=0 = Hλ=∞ = H λ 2 =1/ α
Initial condition
LLNL-PRES-77709726
Example: Similarity renormalization group (SRG) transformation
dH𝛌 = -4 [𝜼(𝛌),H𝛌 ]d𝛌 𝛌5~ ~
𝛌 = 2 fm-1
⟨k| H(2) |k’⟩𝛌~
Can work with smaller N values (basis sizes)!
⟨H𝛌 = 𝒰𝛌H 𝒰⨥𝛌 k|
H(2) |k’⟩~
Hα=0 = Hλ=∞ = H λ 2 =1/ α
Initial condition
LLNL-PRES-77709727
Example: Similarity renormalization group (SRG) transformation
dH𝛌 = -4 [𝜼(𝛌),H𝛌 ]d𝛌 𝛌5~ ~
𝛌 = 2 fm-1
⟨k| H(2) |k’⟩𝛌~
⟨H𝛌 = 𝒰𝛌H 𝒰⨥𝛌 k|
H(2) |k’⟩~
See: Bogner, Furnstahl, Schwenk, Prog. Part. Nucl. Phys. 65 (2010)
Hα=0 = Hλ=∞ = H λ 2 =1/ α
Initial condition
LLNL-PRES-77709728
§This sounds to good to be true …
§What’s the catch?
Question
LLNL-PRES-77709729
§The transformation (e.g., SRG) generates a ‘new’, softer NN interaction
§Unitarily equivalent to the bare NN potential in the two-nucleon sector only!
§ Induces 3-body and, in general, up to A-body forces even starting from an NN potential
Notes on effective interactions
LLNL-PRES-77709730
§ Evolution induces many-nucleon terms (up to A)
§ determined in A=2 system; determined in A=3 system, etc.
§ In actual calculations so far only terms up to
§ Three types of SRG-evolved Hamiltonians used — NN only: Start with initial T+VNN and keep — NN+3N-induced: Start with initial T+VNN and keep— NN+3N-full: Start with initial T+VNN+VNNN and keep
SRG evolution for A-nucleon system
Hα = Hα[1] + Hα
[2] + Hα[3] + Hα
[4] +...+ Hα[A]
Hα[3]
Hα[1] + Hα
[2]
Hα[1] + Hα
[2] + Hα[3]
Hα[1] + Hα
[2] + Hα[3]
Varying 𝜆 provides diagnostic tool to asses contribution of omitted many-body terms, tests unitarity
Hα[3]
LLNL-PRES-77709731
Example: convergence of 4He ground-state energy with chiral NN+3N forces
Jurgenson et al., PRL 103, 082501
SRG NN+3N
Bare NN+3N
LLNL-PRES-77709732
Example: convergence of 4He ground-state energy with chiral NN+3N forces
1 2 3 4 5 10 20λ [fm
−1]
−29
−28
−27
−26
−25
−24
Gro
und-S
tate
Ener
gy [
MeV
]
NN-onlyNN+NNN-induced+NNN-initial
4He N
3LO (500 MeV)
Expt.
Figure 29: Alpha particle binding energy during SRG evolution. Thecurves correspond to those in Fig. 28.
7.6 7.8 8 8.2 8.4 8.6 8.8
Eb(3H) [MeV]
24
25
26
27
28
29
30
Eb(4
He)
[M
eV]
Tjon line for NN-only potentialsSRG NN-only
SRG NN+NNN (λ >1.7 fm−1
)
8.45 8.528.2
28.3
28.4
N3LO
λ=3.0λ=1.2
λ=2.5
λ=1.5
λ=2.0
λ=1.8
Expt.
(500 MeV)
Figure 30: Correlation plot of the binding energies of the alpha parti-cle and the triton. The dotted line connects (approximately) the locusof points found for phenomenological potentials, and is known as theTjon line [40].
potential. The change in energy with λ reflects the vi-olation of unitarity by omission of the induced three-body force. When this induced 3NF is included (“NN+ NNN–induced”), the energy is independent of λ forA = 3. If we now turn to the alpha particle (4He) inFig. 29, we see similar behavior, except now the inclu-sion of the induced 3NF does not lead to a completelyflat curve at the lowest λ values. If there is sufficientconvergence, this is a signal of missing induced 4NF.
In both cases, it is evident that starting with an initialNN-only interaction (in this case, an N3LO(500 MeV)
1 2 3 4 5 10λ
−40
−30
−20
−10
0
10
20
30
40
g.s
. E
xp
ecta
tio
n V
alu
e
<Trel><V
NN>
<V3N
>
1 2 3 4 5 10λ
−1−0.5
0
<Trel><V
NN>
<V3N
>
3H
h- ω = 28
Nmax
= 18
Nmax
-A3 = 32 NN+NNN
Figure 31: Contributions to the triton binding energy during SRG evo-lution. Plotted are the expectation values of the kinetic energy, thetwo-body potential, and the three-body potential [17].
interaction [29]), does not reproduce experiment. Thethird line in each plot of Figs. 28 and 29 shows that aninitial 3NF (labeled NNN) contribution leads to a goodreproduction of experiment. The triton energy is part ofthe fit of this initial force, but the alpha particle energyis a prediction. Note that the magnitude of the NN-onlyvariation is comparable to the initial 3NF needed. Thisis an example of the natural size of the 3NF being mani-fested by the running of the potential (which is, in effect,the beta function).
The nature of the evolution is illustrated in Fig. 30,which is a correlation plot of the binding energies ineach nucleus. The dotted line is known as the Tjonline for NN-only phenomenological potentials. It wasfound that different potentials that fit NN scattering datagave different binding energies, but that they clusteredaround this line. With the SRG evolution starting withjust an NN potential, the path follows the line, passingfairly close to the experimental point. With an initialNNN force and keeping the induced 3-body part, thetrajectory is greatly reduced (see inset), at least until λis small.
Figures 31 and 32 show individual contributions tothe energy in the form of ground-state matrix elementsof the kinetic energy, two-body, three-body, and (im-plied) four-body potentials. The hierarchy of contri-butions is quite clear but the graphs also manifest thestrong cancellations between the NN and kinetic energycontributions. These cancellations magnify the impactof higher-body forces. Even so, it appears that a trunca-tion including the NNN but omitting higher-body forces
R.J. Furnstahl / Nuclear Physics B (Proc. Suppl.) 228 (2012) 139–175154
π π π
c1, c3, c4 cD cE
Figure 27: Leading three-body forces from chiral EFT. These contri-butions represent three different ranges: long-range 2-pion exchange,short-range contact with one-pion exchange, and pure contact interac-tion.
energy correct. In fact, low-energy effective theory tellus generalized diagrams such as those in Fig. 26 withfour or more legs imply that there are A-body forces(and operators) initially!
However, there is a natural hierarchy predicted fromchiral EFT, whose leading contributions are given inFig. 27 (we’ll return to this in Section 4.1 and supplyadditional details). If we stop the flow equations be-fore induced A-body forces are unnaturally large or ifwe can tailor the SRG Gs to suppress their growth, wewill be ok. (Another option is to choose a non-vacuumreference state, which is what is done with in-mediumSRG, to be discussed later.) Note that analytic boundson A-body growth have not been derived, so we need toexplicitly monitor the contribution in different systems.But the bottom line that makes the SRG attractive asa method to soften nuclear Hamlitonians is that it is atractable method to evolve many-body operators.
To include the 3NF using SRG with normal-orderingin the vacuum, we start with the SRG flow equationdHs/ds = [[Gs,Hs],Hs] (e.g., with Gs = Trel). Theright side is evaluated without solving bound-state orscattering equations, unlike the situation with Vlow k, sothe SRG can be applied directly in the three-particlespace. The key observation is that for normal-orderingin the vacuum, A-body operators are completely fixed inthe A-particle subspace. Thus we can first solve for theevolution of the two-body potential in the A = 2 space,with no mention of the 3NF (either initial or induced),and then use this NN potential in the equations appliedto A = 3.
What about spectator nucleons? There is a decou-pling of the 3NF part. We can see this from the first-quantized version of the SRG flow equation,
dVs
ds=
dV12
ds+
dV13
ds+
dV23
ds+
dV123
ds= [[Trel,Vs],Hs] , (42)
where we isolate the contributions from each pair andthe 3NF. Using each SRG equation for the two-bodyderivatives, we can cancel them against terms on the
1 2 3 4 5 6 7 10 20λ [fm−1 ]
−8.6
−8.4
−8.2
−8.0
−7.8
−7.6
−7.4G
roun
d-St
ate
Ene
rgy
[MeV
]
NN-onlyNN + NNN-inducedNN + NNN
3 H
Expt.
Figure 28: Triton binding energy during SRG evolution. The threecurves are for an initial potential with only NN components wherethe induced 3NF is not kept (“NN-only”), for the same initial NNpotential but keeping the induced 3NF (“NN + NNN-induced”), andwith an initial NNN included as well (“NN + NNN”).
right side. The result is [12]:
dV123
ds= [[T12,V12], (T3 + V13 + V23 + V123)]
+ {123→ 132} + {123→ 231}+[[Trel,V123],Hs] . (43)
The key is that there are no “multi-valued” two-bodyinteractions remaining (i.e., dependence on the excita-tion energy of unlinked spectators); all the terms areconnected. An implementation of these equations in amomentum basis would be very useful and has very re-cently been achieved by Hebeler [37]. But an alternativeapproach has also succeeded: a direct solution in a dis-crete basis [38, 16, 17].
The idea is that the SRG flow equation is an opera-tor equation, and thus we can choose to evolve in anybasis. If one chooses a discrete basis, than a separateevolution of the three-body part is not needed. This wasfirst done for nuclei by Jurgenson and collaborators in2009 using an anti-symmetrized Jacobi harmonic oscil-lator (HO) basis [16]. The technology for working withsuch a basis had already been well established for appli-cations to the no-core shell model (NCSM) [39]. Thisapproach leads to SRG-evolved matrix elements of thepotential directly in the HO basis, which is just whatis needed for many-body applications such as NCFC orcoupled cluster.
In Fig. 28, the comparison of two-body-only to fulltwo-plus-three-body evolution is shown for the triton(3H). The NN-only curve uses the evolved two-body
R.J. Furnstahl / Nuclear Physics B (Proc. Suppl.) 228 (2012) 139–175 153 omitted induced 4N
omitted induced 3N
Jurgenson et al., PRL 103, 082501
LLNL-PRES-77709733
Ab initio no-core shell model (NCSM)
§Bare/effective (e.g., SRG-evolved) NN+3N forces
§ Superposition of HO wave functions
§ Jacobi/single-particle coordinates
§Preserves translational invariance of intrinsic wave function
§ ‘Diagonalizes’ Hamiltonian matrix
§A ≲ 16
1max += NN
Nmax … maximal allowed HO excitation above the lowest
possible A-nucleon configuration
Full Nmax space: All basis states with N ≤ Nmax
LLNL-PRES-77709734
Ab initio no-core shell model (NCSM)
ΨSDA = cSDNjΦSDNj
HO (!r 1,!r 2 , ... ,
!r A )j∑
N=0
Nmax
∑ =ΨA ϕ000 (!RCM )
ΨSDA = cSDNjΦSDNj
HO (!r 1,!r 2 , ... ,
!r A )j∑
N=0
Nmax
∑ =ΨA ϕ000 (!RCM )
ΨA = cNiΦNiHO ( !η 1,
!η 2 ,...,
!η A−1)
i∑
N=0
Nmax
∑
§Bare/effective (e.g., SRG-evolved) NN+3N forces
§ Superposition of HO wave functions
§ Jacobi/single-particle coordinates
§Preserves translational invariance of intrinsic wave function
§ ‘Diagonalizes’ Hamiltonian matrix
§A ≲ 16
LLNL-PRES-77709735
§ Jacobi-coordinate HO basis
Small dimensions (no center of mass d.o.f; Coupled to J, T and parity)
NN (and NNN) interaction depends on Jacobi (relative) coordinates and/or momenta
Complicated antisymmetrization (involved, specialized algebra)
−Formalism depends on number of nucleons A
—Most efficient for A = 3,4 systems
Jacobi-coordinate vs. Slater Determinant HO basis
LLNL-PRES-77709736
§ Slater Determinant HO basis
Antisymmetrization is trivial; powerful second quantization representation
Center of mass d.o.f included, but can be removed exactly
−Physical eigenstates contain 0hW center of mass motion
Huge dimensions (M-scheme basis, no good J, T, only MJ, MT , and parity)
3N interaction needs to be transformed to Slater determinant basis
−Huge number of input matrix elements
—Most efficient for A>4 systems
Jacobi-coordinate vs. Slater Determinant HO basis
LLNL-PRES-77709737
Ab initio no-core shell model (NCSM)
§Bare/effective (e.g., SRG-evolved) NN+3N forces
§ Superposition of HO wave functions
§ Jacobi/single-particle coordinates
§Preserves translational invariance of intrinsic wave function
§ ‘Diagonalizes’ Hamiltonian matrix
§A ≲ 16
Works well if wave function is localized(well-bound states)
LLNL-PRES-77709738
Ab initio no-core shell model (NCSM)
Example: energy spectrum of nuclear states of the 10B nucleus
Helped to point out the fundamental importance of 3N forces in structure calculations
PRL 99, 042501 (2007)
Ener
gy sp
ectr
um o
f nuc
lear
stat
es (M
eV)
NN+3N
LLNL-PRES-77709739
Ab initio no-core shell model (NCSM)
§Bare/effective (e.g., SRG-evolved) NN+3N forces
§ Superposition of HO wave functions
§ Jacobi/single-particle coordinates
§Preserves translational invariance of intrinsic wave function
§ ‘Diagonalizes’ Hamiltonian matrix
§A ≲ 16
Does not works as well for nuclei with exoticdensities (halo nuclei)
LLNL-PRES-77709740
Ab initio no-core shell model (NCSM)
§Bare/effective (e.g., SRG-evolved) NN+3N forces
§ Superposition of HO wave functions
§ Jacobi/single-particle coordinates
§Preserves translational invariance of intrinsic wave function
§ ‘Diagonalizes’ Hamiltonian matrix
§A ≲ 16
Definitively not adapted to the description of
scattering wave functions!
LLNL-PRES-77709741
§Green’s function Monte Carlo
§Nuclear Lattice Effective Field Theory
§Coupled Cluster theory
§ In-Medium SRG
§Gorkov-Green function theory
§Many-Body Perturbation Theory
§Ab initio valence-space shell model
Ab initio community extremely successful in describing the static properties of nuclei
Explosion of ab initio methods pushing to medium-mass nuclei
What about the dynamics between nuclei (scattering States with E>0)?