484
9 The Unification of Structure and Reaction Frameworks
B. The first term in Eq. (9.188) is a finite sum and can be calculated numerically
using Gamow shell model formulas as the many-body states therein can be
expanded using Slater determinants. The second sum with N CM ≤ N CMmax and
N
CM > N CMmax will be shown to be negligible:
H
J A M A
cc
N CM , N
CM
=
J T
T | ⊗ ⊗N CM L CM J int J P M P |
J A
M A
ˆ
A ˆ
H ˆ
A
|Ψ
J T
T
⊗ |N
CM L
CM J
int J
P M
P
J A
M A
=
J T
T | ⊗ ⊗N CM L CM J int J P M P |
J A
M A
ˆ
H T + ˆ
H P + ˆ
A ˆ
H TP ˆ
A
|Ψ
J T
T
⊗ |N
CM L
CM J
int J
P M
P
J A
M A
=
J T
T | ⊗ ⊗N CM L CM J int J P M P |
J A
M A
ECM + E T + E int
|Ψ
J T
T
⊗ |N
CM L
CM J
int J
P M
P
J A
M A
+
n,n
c n c
n SD n | ˆ
H TP |SD n
= (E CM + E T + E int ) δ cc δ N CM N
CM
+
n,n
c n c
n SD n | ˆ
H TP |SD n
=
n,n
c n c
n SD n | ˆ
H TP |SD n
=
n,n
α,β,γ ,δ∈T ,p
c n c
n αβ| ˆ
V res − ˆ
U 0 |γ δ SD n | a
+
α a
+
β a δ a γ |SD n ,
(9.189)
where |α , |β , |γ , and |δ stand for single-particle states that are occupied
in target and projectile and where c n and c
n are the coefficients of the Slater
determinants expansion of many-body composites (see Eq. (9.111)):
|Ψ
J T
T ⊗ |N CM L CM J int J P M P
J A
M A
=
n
c n |SD n
|Ψ
J T
T ⊗ |N
CM L
CM J
int J
P M
P
J A
M A
=
n
c
n |SD n .