480
9 The Unification of Structure and Reaction Frameworks
where E T is the energy of |Ψ
J T
T , and where the following one-body completeness relations have been used
n
u n (r i )u n (r f ) = δ(r i − r f )
n
e n u n (r i )u n (r f ) =
¯
h
2
2m
−
d 2
dr 2
i
+
+ 1)
r 2
i
δ(r i − r f ) + U basis (r i , r f ) .
(9.183)
D. One can now calculate H if from the Berggren basis expansion involving arbitrary
targets |Ψ
J T i
T i
and |Ψ
J T f
T f
and projectile channels |r i i j i τ i and |r f f j f τ f
The incoming and outgoing channels will be denoted by c i and c f , respectively.
The expansion coefficients of many-body composite states in a basis of Slater
determinants will be denoted as c
(i)
α i and c
(f)
α f for initial and final A-body wave
functions, respectively (see Eq. (9.7)):
[|Ψ
J T i
T i
⊗ |u n i j τ
J A
M A
=
α i
c
(i)
α i
|SD α i
[|Ψ
J T f
T f
⊗ |u n f j τ
J A
M A
=
α f
c
(f)
α f
|SD α f .
Let us evaluate Eq. (9.177) using the results of A, B, C:
H if =
¯
h 2
2m
−
d 2
dr 2
i
+
+ 1)
r 2
i
+ E T
δ(r i − r f ) δ c i c f + U basis (r i , r f ) δ c i c f
+
nmax
n i n f
u n (r i ) u n (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f j τ |}
J A
M A
| ˆ
H | ˆ
A {|Ψ
J T f
T f
⊗ |u n i j τ
J A
M A
− δ c i c f
n≤nmax
u n (r i ) u n (r f ) [[Ψ
J T f
T f
| ⊗ ⊗u n j τ |]
J A
M A
| ˆ
H T + ˆ
H P |[|Ψ
J T f
T f
⊗ |u n j τ
J A
M A
=
¯
h 2
2m
−
d 2
dr 2
i
+
+ 1)
r 2
i
+ E T
δ(r i − r f ) δ c i c f + U basis (r i , r f ) δ c i c f
+
nmax
n i n f α i α f
u n i (r i )u n f (r f )c
(i)
α i c
(f)
α f SD α f | ˆ
H |SD α i − δ c i c f
n≤nmax
(E T + e n ) u n (r i )u n (r f )
≡
¯
h 2
2m
−
d 2
dr 2
i
+
+ 1)
r 2
i
+ E T
δ(r i − r f ) δ c i c f + U basis (r i , r f ) δ c i c f
+ V c i c f (r i , r f ) ,
(9.184)