Aurora School 2010
Few Body Methods in Nuclear Physics - Lecture IV
Nir Barnea
The Hebrew University, Jerusalem, Israel
Sept. 2010
Aurora School 2010
Outline
1 Hyperspherical Harmonics - Short History
2 Construction
3 Summary
4 Anti Symmetrization
5 Convergence
6 The EIHH method
7 Examples
Aurora School 2010
Hyperspherical Harmonics - Short History
The Hyperspherical Harmonics
1
The HH were introduced in 1935 by Zernike
and Brinkman.
x1
x2
x3
x4
x5
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Hyperspherical Harmonics - Short History
The Hyperspherical Harmonics
1
The HH were introduced in 1935 by Zernike
and Brinkman.
2
They were reintroduced 25 years later by
Delves and Smith.
x1
x2
x3
x4
x5
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Hyperspherical Harmonics - Short History
The Hyperspherical Harmonics
1
The HH were introduced in 1935 by Zernike
and Brinkman.
2
They were reintroduced 25 years later by
Delves and Smith.
3
In the 1970 Reynal and Revai derived the HH
transformation coefficients.
x1
x2
x3
x4
x5
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Hyperspherical Harmonics - Short History
The Hyperspherical Harmonics
1
The HH were introduced in 1935 by Zernike
and Brinkman.
2
They were reintroduced 25 years later by
Delves and Smith.
3
In the 1970 Reynal and Revai derived the HH
transformation coefficients.
4
and in 1972 Kil’dushov derives the HH
recoupling coefficients.
x1
x2
x3
x4
x5
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Coordinates
Hyperspherical (HS )coordinates are
D-dimensional generalization of the spherical
coordinates
qX
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
Using the tree structure one can easily construct
the hyperspherical coordinates
1
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
Each coordinate corresponds to a leaf
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Coordinates
Hyperspherical (HS )coordinates are
D-dimensional generalization of the spherical
coordinates
qX
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
Using the tree structure one can easily construct
the hyperspherical coordinates
1
Each coordinate corresponds to a leaf
2
The root of the tree is associated with ρ
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Coordinates
Hyperspherical (HS )coordinates are
D-dimensional generalization of the spherical
coordinates
qX
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
Using the tree structure one can easily construct
the hyperspherical coordinates
1
Each coordinate corresponds to a leaf
2
The root of the tree is associated with ρ
3
Each junction, including the root, is
associated with an angle
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Coordinates
Hyperspherical (HS )coordinates are
D-dimensional generalization of the spherical
coordinates
qX
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
Using the tree structure one can easily construct
the hyperspherical coordinates
1
Each coordinate corresponds to a leaf
2
The root of the tree is associated with ρ
3
Each junction, including the root, is
associated with an angle
4
Left branch multiply by sin(θ)
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Coordinates
Hyperspherical (HS )coordinates are
D-dimensional generalization of the spherical
coordinates
qX
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
Using the tree structure one can easily construct
the hyperspherical coordinates
1
Each coordinate corresponds to a leaf
2
The root of the tree is associated with ρ
3
Each junction, including the root, is
associated with an angle
4
Left branch multiply by sin(θ)
5
Right branch multiply by cos(θ)
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Coordinates
Hyperspherical (HS )coordinates are
D-dimensional generalization of the spherical
coordinates
qX
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
Using the tree structure one can easily construct
the hyperspherical coordinates
1
Each coordinate corresponds to a leaf
2
The root of the tree is associated with ρ
3
Each junction, including the root, is
associated with an angle
4
Left branch multiply by sin(θ)
5
Right branch multiply by cos(θ)
6
The HS representation of each coordinates is
fixed by the path from root to leaf.
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Harmonics
1
Hyperspherical coordinates
qP
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Harmonics
1
2
Hyperspherical coordinates
qP
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
In hyperspherical coordinates
∆=
∂2
K̂ 2
D−1 ∂
+
−
∂ρ2
ρ ∂ρ ρ2
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Harmonics
1
2
Hyperspherical coordinates
qP
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
In hyperspherical coordinates
∆=
3
x1
x2
x3
x4
x5
∂2
K̂ 2
D−1 ∂
+
−
∂ρ2
ρ ∂ρ ρ2
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
ρK Y[K] (Ω) is a Harmonic polynomial.
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Harmonics
1
2
Hyperspherical coordinates
qP
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
In hyperspherical coordinates
∆=
∂2
K̂ 2
D−1 ∂
+
−
∂ρ2
ρ ∂ρ ρ2
3
ρK Y[K] (Ω) is a Harmonic polynomial.
4
The HH are eigenstates of K̂ 2
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
K̂ 2 Y(Ω) = K(K + D − 2)Y(Ω)
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Harmonics
1
2
Hyperspherical coordinates
qP
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
In hyperspherical coordinates
∆=
∂2
K̂ 2
D−1 ∂
+
−
∂ρ2
ρ ∂ρ ρ2
3
ρK Y[K] (Ω) is a Harmonic polynomial.
4
The HH are eigenstates of K̂ 2
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
K̂ 2 Y(Ω) = K(K + D − 2)Y(Ω)
5
Using the tree structure one can easily
construct HH starting from the leafs and
uniting branches.
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Harmonics
1
2
Hyperspherical coordinates
qP
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
In hyperspherical coordinates
∆=
∂2
K̂ 2
D−1 ∂
+
−
∂ρ2
ρ ∂ρ ρ2
3
ρK Y[K] (Ω) is a Harmonic polynomial.
4
The HH are eigenstates of K̂ 2
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
K̂ 2 Y(Ω) = K(K + D − 2)Y(Ω)
5
6
Using the tree structure one can easily
construct HH starting from the leafs and
uniting branches.
Each junction is associated with a quantum
number.
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The Hyperspherical Harmonics
1
2
Hyperspherical coordinates
qP
x1 , x2 , x3 , ...xD −→ ρ =
xi2 , ΩD−1
x1
x2
x3
x4
x5
In hyperspherical coordinates
∆=
∂2
K̂ 2
D−1 ∂
+
−
∂ρ2
ρ ∂ρ ρ2
3
ρK Y[K] (Ω) is a Harmonic polynomial.
4
The HH are eigenstates of K̂ 2
=
=
=
=
=
ρ cos(α) cos(β) cos(δ)
ρ cos(α) cos(β) sin(δ)
ρ cos(α) sin(β)
ρ sin(α) cos(γ)
ρ sin(α) sin(γ)
The “Tree” diagram
K̂ 2 Y(Ω) = K(K + D − 2)Y(Ω)
5
Using the tree structure one can easily
construct HH starting from the leafs and
uniting branches.
6
Each junction is associated with a quantum
number.
7
Each junction adds a factor
R ,αL )
N cosKR (θ) sinKL (θ)P(α
(K−KR −KL )/2 (cos(2θ))
x5
x4
@ γq
@
x3
x2
@
@@δq
@
@
@
@
@αq
@βq
x1
Aurora School 2010
Construction
The recursive step
Assume that we have constructed HS coordinates
for two subsystems L, R of dimensions νL , νR and
coordinates ρL , ΩL , ρR , ΩR
The HS Transformation
ρL
ρR
=
=
ρ sin(α)
ρ cos(α)
ρq L , νL
ρqR , νR
@
@
@
@
@αq
Aurora School 2010
Construction
The recursive step
Assume that we have constructed HS coordinates
for two subsystems L, R of dimensions νL , νR and
coordinates ρL , ΩL , ρR , ΩR
The HS Transformation
ρL
ρR
Leads to
dV
=
=
ρ sin(α)
ρ cos(α)
=
=
ρq L , νL
ρqR , νR
@
@
@
@
@αq
ρLνL −1 dρL dΩL ρνRR −1 dρR dΩR
ρνL +νR −1 dρ sinνL −1 (α) cosνR −1 (α)dαdΩL dΩR
Aurora School 2010
Construction
The recursive step
Assume that we have constructed HS coordinates
for two subsystems L, R of dimensions νL , νR and
coordinates ρL , ΩL , ρR , ΩR
The HS Transformation
ρL
ρR
Leads to
dV
K̂ 2 = −
=
=
ρ sin(α)
ρ cos(α)
=
=
ρq L , νL
ρqR , νR
@
@
@
@
@αq
ρLνL −1 dρL dΩL ρνRR −1 dρR dΩR
ρνL +νR −1 dρ sinνL −1 (α) cosνR −1 (α)dαdΩL dΩR
K̂L2
K̂R2
(νR − νL ) − (νL + νR − 2) cos(2α) ∂
∂2
+
+
+
2
2
∂α
sin(2α)
∂α sin (α) cos2 (α)
Aurora School 2010
Construction
The recursive step
Assume that we have constructed HS coordinates
for two subsystems L, R of dimensions νL , νR and
coordinates ρL , ΩL , ρR , ΩR
The HS Transformation
ρL
ρR
Leads to
dV
K̂ 2 = −
=
=
ρ sin(α)
ρ cos(α)
=
=
ρq L , νL
ρqR , νR
@
@
@
@
@αq
ρLνL −1 dρL dΩL ρνRR −1 dρR dΩR
ρνL +νR −1 dρ sinνL −1 (α) cosνR −1 (α)dαdΩL dΩR
K̂L2
K̂R2
(νR − νL ) − (νL + νR − 2) cos(2α) ∂
∂2
+
+
+
2
2
∂α
sin(2α)
∂α sin (α) cos2 (α)
The eigenstates of K̂ 2 are of the form Φ(α, ΩL , ΩR ) = ϕ(α)ΦL (ΩL )ΦR (ΩR )
K̂L2 ΦL (ΩL )
K̂R2 ΦR (ΩR )
K̂ 2 ΦR (Ω)
=
=
=
Here Ω = (α, ΩL , ΩR ) and ν = νL + νR
KL (KL + νL − 2)ΦL (ΩL )
KR (KR + νR − 2)ΦR (ΩR )
K(K + ν − 2)Φ(Ω)
Aurora School 2010
Construction
Setting x = cos(2α) and substituting ϕ(x) = (1 − x)KL /2 (1 + x)KR /2 P(x) we get
h
(1 − x2 )
i
d2
d
+ (b − a − (a + b + 2)x) + n(n + a + b + 1) P = 0
dx2
dx
where
a
b
n
=
=
=
KL + νL /2 − 1
KR + νR /2 − 1
(K − KL − KR )/2
The Jacobi polinomial, P(a,b)
n (x), is the solution of this equation, therfore
L +νL /2−1,KR +νR /2−1)
ϕ(x) = N(1 − x)KL /2 (1 + x)KR /2 P(K
(x)
(K−KL −KR )/2
Aurora School 2010
Construction
An important example - The Jacobi coordinates
The normalized equal mass Jacobi coordinates
r
1
r2 − r1
η1 =
2
r
2
1
η2 =
r3 − (r2 + r3 )
3
2
...
r
A − 2
1
ηA−2 =
rA−2 −
(r1 + r2 + · · · + rA−3 )
A−1
A−2
r
A − 1
1
ηA−1 =
rA−1 −
(r1 + r2 + · · · + rA−1 )
A
A−1
Note that the hyperadial coordinate
ρ2 =
X
η2i =
i
is symmetric under particle permutations
A
1X
(ri − rj )2
A i<j
Aurora School 2010
Construction
The common “Tree”
ηN
ηN−1
ηN−2
y x z
y x z
y x z
φ
φ
φ
q
qθ
qθ
m@q
mN−1
mN−2
N θ
@q
`N
`N−1
`@q
N−2
@
@
@
@
@
η1
y x z
φ
q
m@
1 θ
`1 q
@ α2
@q K2
@
@
@
@
@
η2
y x z
φ
q
m@q
2 θ
`2
@
@
@
@
@
@
@
@
@ αN−2
@q KN−2
@ αN−1
@q KN−1
@ αN
@q KN
N
hY
i
Y[K] =
Y`j , mj (η̂j )
j=1
×
N
hY
j=2
` ,K
3j−5
(` + 21 ,Kj−1 + 2 )
Nj,Kj j j−1 (sin αj )`j (cos αj )Kj−1 Pµjj
(cos(2αj ))
i
Aurora School 2010
Construction
The Merits of the HH expansion
A complete set of basis functions.
N
X
∗
Y[K]
(Ω0 )Y[K] (Ω)
[K]
δ(ρ − ρ0 ) Y
=
δ(ηi − η0i )
ρD−1
i=1
Easy transformation between configuration and momentum space
ei
P
ηj q j
=
(2π)D/2 X K ∗
i Y[K] (Ωq )Y[K] (Ω)JK+D/2−1 (Qρ)
(Qρ)D/2−1 [K]
Good asymptotics.
With appropriate choice of Jacobi coordinates and states clusterization can be
”easily” treated.
Aurora School 2010
Construction
Close relatives, the HH and the HO basis
In view of the definition of the hyperspherical coordinates it is evident that the
HO Hamiltonian, written in the form
!
!
N
X
∆j
1 ∂2
3N + 4 ∂
K̂ 2
2 2
,
+
−
+
ω
ρ
− + ω2 η2j =
2
2 ∂ρ2
ρ ∂ρ ρ2
j=1
has eigenvectors of the form
ΨHO = Rnρ (ρ) Y[K]
with eigenvalues
EN = ~ω N +
!
!
3(A − 1)
3(A − 1)
= ~ω 2nρ + K +
.
2
2
Therefore the HH grand angular quantum number K can be associated with the
quanta of excitations of the HO wave function.
Given PHO (Nmax ), The space spanned by the HO states with N ≤ Nmax , and
PHH (Kmax ), The space spanned by the HH states with K ≤ Kmax = Nmax and a
complete hyperadial basis, then PHO (Nmax ) ⊂ PHH (Kmax ).
Aurora School 2010
Summary
The HH expansion in 4 steps
1. Remove the center of mass
~ c.m. , ~η1~η2 . . . ~ηA−1
~r1 ,~r2 , . . .~rA −→ R
2. Introduce hyperspherical coordinates
~η1~η2 . . . ~ηA−1 −→ ρ =
q
η21 + η22 + . . . + η2A−1 , Ω
3. Expand the wave function using hyperspherical harmonics
X
Ψ(ρ, Ω) =
R[K] (ρ)Y[K] (Ω)
K≤Km ax
4. Solve the Schrödinger equation
H=−
! X
X
3A − 4 ∂
K̂ 2
1 ∂2
+
Vij +
+
−
Vijk
2
2
2 ∂ρ
ρ ∂ρ ρ
i<j
i<j<k
Aurora School 2010
Summary
Not so fast !!!
There are two Major obstacles
1
The HH basis has no good
permutational symmetry.
(Anti)Symmetrization must be
enforced.
2
For some nuclear forces the
convergence of the HH expansion is
notoriously slow and must be
accelerated.
Aurora School 2010
Anti Symmetrization
Strategies
1
Apply the anti-symmetrization to the
HH basis
X
 =
sign(g)ĝ
g∈SA
Aurora School 2010
Anti Symmetrization
Strategies
1
Apply the anti-symmetrization to the
HH basis
X
 =
sign(g)ĝ
g∈SA
At a cost of A! operations.
Aurora School 2010
Anti Symmetrization
Strategies
1
Apply the anti-symmetrization to the
HH basis
X
 =
sign(g)ĝ
g∈SA
2
Do it recursively in steps
 = 1 − (1, A) − (2, A) − . . . (A − 1, A)
... × 1 − (1, 3) − (2, 3) × 1 − (1, 2)
At a cost of A! operations.
Aurora School 2010
Anti Symmetrization
Strategies
1
Apply the anti-symmetrization to the
HH basis
X
 =
sign(g)ĝ
At a cost of A! operations.
g∈SA
2
Do it recursively in steps
 = 1 − (1, A) − (2, A) − . . . (A − 1, A)
... × 1 − (1, 3) − (2, 3) × 1 − (1, 2)
At a cost of (A − 1) operations.
Aurora School 2010
Anti Symmetrization
Strategies
1
Apply the anti-symmetrization to the
HH basis
X
 =
sign(g)ĝ
At a cost of A! operations.
g∈SA
2
3
Do it recursively in steps
 = 1 − (1, A) − (2, A) − . . . (A − 1, A)
... × 1 − (1, 3) − (2, 3) × 1 − (1, 2)
Generate HH states from HO Slater
determinant.
1 2
Det{HO} = e− 2 ρ ρK Â Y(Ω)X(si , ti )
At a cost of (A − 1) operations.
Aurora School 2010
Anti Symmetrization
Strategies
1
Apply the anti-symmetrization to the
HH basis
X
 =
sign(g)ĝ
At a cost of A! operations.
g∈SA
2
3
Do it recursively in steps
 = 1 − (1, A) − (2, A) − . . . (A − 1, A)
... × 1 − (1, 3) − (2, 3) × 1 − (1, 2)
Generate HH states from HO Slater
determinant.
1 2
Det{HO} = e− 2 ρ ρK Â Y(Ω)X(si , ti )
At a cost of (A − 1) operations.
Probably the only viable way to
extend HH calculations to large
A. There are CM issues.
Aurora School 2010
Anti Symmetrization
Strategies
1
Apply the anti-symmetrization to the
HH basis
X
 =
sign(g)ĝ
At a cost of A! operations.
g∈SA
2
Do it recursively in steps
 = 1 − (1, A) − (2, A) − . . . (A − 1, A)
... × 1 − (1, 3) − (2, 3) × 1 − (1, 2)
3
Generate HH states from HO Slater
determinant.
1 2
Det{HO} = e− 2 ρ ρK Â Y(Ω)X(si , ti )
4
Use the group of kinematic rotations
ηi −→ η0i = ĝηi
At a cost of (A − 1) operations.
Probably the only viable way to
extend HH calculations to large
A. There are CM issues.
Aurora School 2010
Anti Symmetrization
Recurcive (Anti) Symmetrization
Few comments on the permutation group SA
Each state in a irrep of SA is uniquely defined by irreos of the group-subgroup
chain
S1 ⊂ S2 ⊂ S3 · · · ⊂ SA−2 ⊂ SA−1 ⊂ SA
This basis is called the Yamanouchi basis
|YA i = |ΓA YA−1 i = |ΓA ΓA−1 ΓA−2 . . . Γ2 Γ1 i
The sum of all 2-body transpositions is a class sum operator
X
Ĉ2 (n) =
(i, j)
i<j
It commutes with the permutation group
[Ĉ2 (n), Sn ] = 0
Given Γn−1 the eigenvalues of Ĉ2 (n) uniquely identify the irreps of Sn
Aurora School 2010
Anti Symmetrization
Recurcive Construction
Assume that after n − 1-steps our invariant subspace has the basis |Γn−1 Yn−2 βn−1 i
[Ĉ2 (n), Sn ] = 0 =⇒ [Ĉ2 (n), Ĉ2 (k)] = 0 for every k ≤ n
Consequently (thanks to Shur’s lemmas)
X
Ĉ2 (n)|Γn−1 Yn−2 βn−1 i =
MβΓn−1β0 |Γn−1 Yn−2 β0n−1 i
β0n−2
n−1 n−1
Diagonalizing Ĉ2 (n) we will get states that belong to well define irrep Γn of Sn
P
Note that Ĉ2 (n) = Ĉ2 (n − 1) + n−1
i=1 (i, n)
In our basis
hΓn−1 Yn−2 βn−1 |Ĉ2 (n − 1)|Γn−1 Yn−2 β0n−1 i = δβn−1 β0n−1 C2 (Γn−1 )
Now lets focus on the second part
n−1
n−1
X
X
(i, n) =
(i, n − 1)(n − 1, n)(i, n − 1)
i=1
i=1
The operation of a permutation on a basis state can be expressed through
X
0
(i, n − 1)|Γn−1 Yn−2 βn−1 i =
DΓYn−1
g(i,n−1) |Γn−1 Yn−2
βn−1 i
0 Y
0
Yn−2
n−2 n−2
Aurora School 2010
Anti Symmetrization
Summing up these results
hΓn−1 Yn−2 βn−1 |Ĉ2 (n)|Γn−1 Yn−2 β0n−1 i = δβn−1 β0n−1 C2 (Γn−1 )
n−1
X
X
∗
0
00
g(i,n−1) DΓYn−1
g(i,n−1) hΓn−1 Yn−2
βn−1 |(n − 1, n)|Γn−1 Yn−2
β0n−1 i
+
DΓYn−1
0 Y
00 Y
i=1 Y 0 ,Y 00
n−2 n−2
n−2 n−2
n−2 n−2
These matrix elements do not depend on Yn−2 therefore we can sum on Yn−2 and
devide by |Γn−1 |
Realising that
X
Yn−2
∗
DΓYn−1
0 Y
n−2 n−2
g(i,n−1) DΓYn−1
00 Y
n−2 n−2
X −1 Γn−1
g(i,n−1) =
(D )Y Y 0
n−2 n−2
Yn−2
g(i,n−1) DΓYn−1
00 Y
n−2 n−2
0 Y 00
g(i,n−1) = δYn−2
n−2
Finally
hΓn−1 Yn−2 βn−1 |Ĉ2 (n)|Γn−1 Yn−2 β0n−1 i = δβn−1 β0n−1 C2 (Γn−1 )
n−1 X
+
hΓn−1 Yn−2 βn−1 |(n − 1, n)|Γn−1 Yn−2 β0n−1 i
|Γn−1 | Y
n−2
Aurora School 2010
Anti Symmetrization
Recurcive Construction - Summary
Through the recurcive method the O(A!) problem is reduced into (A − 1) steps
At each step only the ME of the transposition (n − 1, n) should be calculated
The resulting construction is a unitary transformation from the original basis to
the symmetrized basis
This unitary transformation is most naturally expressed through coefficients of
fractional parentage (CFPs)
In the construction presented above there is a problem of consistent phase
between Γn−1 ∈ Γn and Γ0n−1 ∈ Γn . Can be easily solved.
A. Novoselsky, J. Kateriel and R. Gilmore, J. Math. Phys. 29 1368 (1988)
Aurora School 2010
Anti Symmetrization
The group of kinematic rotations
The scalar operators
Xk,l = i(ηk · ∇l − ηl · ∇k )
are the generators of the orthoonal group, O(A − 1), of kinematical rotations. It is
evident that
[K̂ 2 , Xk,l ] = [Lj , Xk,l ] = 0
HH functions that belong to well defined irrep of O(A − 1) can be constructed
recurcivly trough diagonalization of the 2nd Casimir operator
C2 (A − 1) =
A−1
X
Xi,j2
i<j
Using the property
C2 (A − 1) = C2 (A − 2) +
A−1
X
2
Xi,A−1
i
The algebra and the irreps of O(n) are outside the scope of these lectures.
Aurora School 2010
Anti Symmetrization
(Anti) Symmetrization via Kinematic rotations
Using the group of kinematic rotations η0i = ĝηi , the HH symmetrization can be
carried out in two steps
HH −→ O(A−1)
and
O(A−1) −→ SA
In short we use the following group-subgroup chain
O3(A−1) ⊂ O3 ⊗ O(A−1) ⊂ O3 ⊗ SA
K
LM ΛA−1
LM YA
Aurora School 2010
Anti Symmetrization
(Anti) Symmetrization via Kinematic rotations
Using the group of kinematic rotations η0i = ĝηi , the HH symmetrization can be
carried out in two steps
HH −→ O(A−1)
and
O(A−1) −→ SA
In short we use the following group-subgroup chain
O3(A−1) ⊂ O3 ⊗ O(A−1) ⊂ O3 ⊗ SA
K
LM ΛA−1
LM YA
6-body system: A comparison between direct symmetrization and symmetrization
through the kinematical group O(A-1)
K
HH → SA
HH → O(A−1)
O(A−1) → SA
Ratio
2
4
6
8
198
11308
516647
107
56
728
8771
84700
108
1528
16511
127544
1.21
5.01
20.44
47.12
Aurora School 2010
Convergence
Convergence - Statement of the problem
For potentials with Coulomb type singularities the HH expansion of Ψ converge
−2
as Kmax
For Gaussian potentials Ψ converge as e−cKmax .
Actually. The problem is not the slow convergence rate but rather the fast
growth in the number of HH states.
Aurora School 2010
Convergence
Convergence
Strategies
1
Correlations: CHH, PHH, CFHHM ...
Ψ(ρ, Ω) = F(rij )
X
R[K] (ρ)Y[K] (Ω)
K≤Km ax
2
Basis Reduction: The potential basis (Fabre de-la Ripple), The Pisa Group,
Efros.
For a Bose-system this expansion may take the following form
XX
XX
R(2)
R(2,2)
Ψ(ρ, Ω) =
[K] (ρ)Y[K] (Ωij ) +
[K] (ρ)Y[K] (Ωij,kl ) + . . .
ij
3
[K]
ij,kl [K]
Effective interaction for the HH expansion. Replace the bare potential by an
effective one:
(2)
V (2) −→ Veff
(3)
V (3) −→ Veff
Aurora School 2010
The EIHH method
The Effective Interaction Hyperspherical Harmonics (EIHH)
(2)
The Veff
is derived from a ”2-body” Hamiltonian
H2 (ρ) =
√
1 K̂ 2
+ V(~r = 2ρ sin αN · η̂N ) ,
2m ρ2
Aurora School 2010
The EIHH method
The Effective Interaction Hyperspherical Harmonics (EIHH)
(2)
The Veff
is derived from a ”2-body” Hamiltonian
H2 (ρ) =
√
1 K̂ 2
+ V(~r = 2ρ sin αN · η̂N ) ,
2m ρ2
The effective Hamiltonian is constructed through the Lee-Suzuki similarity
transformation
H2 eff (ρ) = U † (ρ)H2 (ρ)U(ρ) ;
1+ω
U= p
P(1 + ω† ω)P
Aurora School 2010
The EIHH method
The Effective Interaction Hyperspherical Harmonics (EIHH)
(2)
The Veff
is derived from a ”2-body” Hamiltonian
H2 (ρ) =
√
1 K̂ 2
+ V(~r = 2ρ sin αN · η̂N ) ,
2m ρ2
The effective Hamiltonian is constructed through the Lee-Suzuki similarity
transformation
H2 eff (ρ) = U † (ρ)H2 (ρ)U(ρ) ;
The operator ω = QωP is given by
X
hq|ii =
hq|ω|pihp|ii ;
α
1+ω
U= p
P(1 + ω† ω)P
H2 |ii = Ei |ii
Aurora School 2010
The EIHH method
The Effective Interaction Hyperspherical Harmonics (EIHH)
(2)
The Veff
is derived from a ”2-body” Hamiltonian
H2 (ρ) =
√
1 K̂ 2
+ V(~r = 2ρ sin αN · η̂N ) ,
2m ρ2
The effective Hamiltonian is constructed through the Lee-Suzuki similarity
transformation
H2 eff (ρ) = U † (ρ)H2 (ρ)U(ρ) ;
1+ω
U= p
P(1 + ω† ω)P
The operator ω = QωP is given by
X
hq|ii =
hq|ω|pihp|ii ;
H2 |ii = Ei |ii
α
Finally the effective interaction is given by
V2 eff (ρ) = H2 eff (ρ) −
1 K̂ 2
2m ρ2
Aurora School 2010
Examples
4-body ground-state - Bare vs Effective
Aurora School 2010
Examples
4-body ground-state
Convergence of the EIHH method for 4 He binding energy Eb [MeV] and root mean
1
square matter radius hr2 i 2 [fm] with AV18 and AV18+UIX potentials.
AV18
1
Kmax
Eb
hr2 i 2
6
8
10
12
14
16
18
20
25.312
25.000
24.443
24.492
24.350
24.315
24.273
24.268
1.506
1.509
1.520
1.518
1.518
1.518
1.518
1.518
AV18+UIX
Eb
26.23
27.63
27.861
28.261
28.324
28.397
28.396
28.418
1
hr2 i 2
1.456
1.428
1.428
1.427
1.428
1.430
1.431
1.432
Aurora School 2010
Examples
4-body ground-state
Convergence of the EIHH method for 4 He binding energy Eb [MeV] and root mean
1
square matter radius hr2 i 2 [fm] with AV18 and AV18+UIX potentials.
AV18
1
Kmax
Eb
hr2 i 2
6
8
10
12
14
16
18
20
25.312
25.000
24.443
24.492
24.350
24.315
24.273
24.268
1.506
1.509
1.520
1.518
1.518
1.518
1.518
1.518
FY [Nogga]
FY [Lazauskas]
HH [Viviani]
GFMC [Wiringa]
24.25
24.22
24.21
1.516
1.512
AV18+UIX
Eb
26.23
27.63
27.861
28.261
28.324
28.397
28.396
28.418
1
hr2 i 2
1.456
1.428
1.428
1.427
1.428
1.430
1.431
1.432
28.50
28.46
28.34
1.428
1.44
Aurora School 2010
Examples
The effective 3-body force
Convergence of 6 Li ground state with the AV8’ NN potential
Kmax
B.E V (2) eff
B.E. (KQ(3) = 16)
B.E. (KQ(3) = 20)
2
4
6
8
10
39.80
33.43
31.02
31.13
30.23
38.83
30.37
31.11
31.17
31.22
30.27
30.94
30.94
30.88
Aurora School 2010
Examples
The effective 3-body force
Convergence of 6 Li ground state with the AV8’ NN potential
Kmax
B.E V (2) eff
B.E. (KQ(3) = 16)
B.E. (KQ(3) = 20)
2
4
6
8
10
39.80
33.43
31.02
31.13
30.23
38.83
30.37
31.11
31.17
31.22
30.27
30.94
30.94
30.88
NCSM [Navratil]
GFMC [Pieper]
30.30
29.70(5)
Aurora School 2010
Examples
EIHH for Non-local interactions
(2)
Local potential: Veff
(ρ) was derived using the fact that V is diagonal in configuration
space.
Non-local potential: (or in general) we can construct EI from a ”2-body”
Hamiltonian of the form:
1 2 1
hn|H2 |ni =
K̂ hn| 2 |ni + hn|VA,A−1 |ni
2m
ρ
In this case
(2) 0
(2)
hn|Veff
|n i = δn,n0 hn|Heff
−
1 K̂ 2
|ni + (1 − δn,n0 )hn|V (2) |n0 i
2m ρ2
Aurora School 2010
Examples
The HH and EIHH methods
summing up
HH basis - not so easy to manipulate good asymptotics
Using sophisticated methods it is possible to perform A ≥ 4 calculations
Convergence is achieved either through the Lee-Suzuki effective interaction or
by selecting states.
Parallel computing
The Lanczos steps |bi = H|ai can be naturally devided between different nodes.
Each node calculates its part of |bi
the vector is reconstructed after each step