478
9 The Unification of Structure and Reaction Frameworks
+
n>nmax
u n (r i ) u n (r f ) [[Ψ
J T f
T f
| ⊗ ⊗u n f j f τ f |]
J A
M A
| ˆ
H T + ˆ
H P |[|Ψ
J T i
T i
⊗ |u n i j i τ i
J A
M A
+
n i =n f
n i >nmax
n f ≤nmax
u n i (r i ) u n f (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
H T + ˆ
H P |[Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
+
n i =n f
n i ≤nmax
n f >nmax
u n i (r i ) u n f (r f )
J T f
T f
⊗ ⊗u n f f j f τ f |]
J A
M A
| ˆ
H T + ˆ
H P | ˆ
A {|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
.
(9.177)
B. The sum in which n i = n f and n i > n max will be shown to vanish. Indeed, if n i >
n max and n f ≥ 0, the state |u n i i j i τ i is orthogonal both to all states occupied in
|Ψ
J T f ,M T f
T f
and to |u n f f j f τ f . Moreover, ( ˆ
V res − ˆ
U 0 ) = ( ˆ
V res − ˆ
U 0 ) A−1 in this
case, as all the matrix elements of ( ˆ
V res − ˆ
U 0 ) involving |u n i i j i τ i are negligible.
Hence, one has
ˆ
A {{Ψ
J T f
T f
| ⊗ u n f f j f τ f |}
J A
M A
| ˆ
H |[|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= = ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
T + ˆ
U basis + ( ˆ
V res − ˆ
U 0 )|{|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= = ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
|( ˆ
V res − ˆ
U 0 )|{|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= = ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
|( ˆ
V res − ˆ
U 0 ) A−1 |{|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= 0 ,
(9.178)
where one has used the fact that [|Ψ
J T i
T i
⊗ |u n i f j f τ f
J A
M A
and ˆ
A {|Ψ
J T f
T f
⊗
|u n f i j i τ i
J A
M A
do not have a common Slater determinant in their expansions,
implying that the ˆ
T + ˆ
U basis matrix element vanishes, and that (V res − U 0 ) A−1
does not act on |u n i i j i τ i . Evidently, this argument also applies if n i and n f are
exchanged, so that the sum in which n i = n f and n f > n max vanish as well.
Consequently, only the sum involving n i = n f and n i , n f ≤ n max remains.
If | i j i τ i = | f j f τ f , the terms of the sum involving n > n max in
Eq. (9.177) all vanish. Indeed,
[[Ψ
J T f
T f
| ⊗ ⊗u n f j f τ f |]
J A
M A
| ˆ
H T + ˆ
H P |[|Ψ
J T i
T i
⊗ |u n i j i τ i
J A
M A
= =Ψ
J T f
T f
| ˆ
H T |Ψ
J T i
T i
n f j f τ f |u n i j i τ i + +Ψ
J T f
T f
|Ψ
J T i
T i
n f j f τ f | ˆ
H P |u n i j i τ i
= 0 ,
(9.179)
as |u n f j f τ f and |u n i j i τ i are mutually orthogonal eigenvectors of ˆ
H P .
9 The Unification of Structure and Reaction Frameworks
+
n>nmax
u n (r i ) u n (r f ) [[Ψ
J T f
T f
| ⊗ ⊗u n f j f τ f |]
J A
M A
| ˆ
H T + ˆ
H P |[|Ψ
J T i
T i
⊗ |u n i j i τ i
J A
M A
+
n i =n f
n i >nmax
n f ≤nmax
u n i (r i ) u n f (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
H T + ˆ
H P |[Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
+
n i =n f
n i ≤nmax
n f >nmax
u n i (r i ) u n f (r f )
J T f
T f
⊗ ⊗u n f f j f τ f |]
J A
M A
| ˆ
H T + ˆ
H P | ˆ
A {|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
.
(9.177)
B. The sum in which n i = n f and n i > n max will be shown to vanish. Indeed, if n i >
n max and n f ≥ 0, the state |u n i i j i τ i is orthogonal both to all states occupied in
|Ψ
J T f ,M T f
T f
and to |u n f f j f τ f . Moreover, ( ˆ
V res − ˆ
U 0 ) = ( ˆ
V res − ˆ
U 0 ) A−1 in this
case, as all the matrix elements of ( ˆ
V res − ˆ
U 0 ) involving |u n i i j i τ i are negligible.
Hence, one has
ˆ
A {{Ψ
J T f
T f
| ⊗ u n f f j f τ f |}
J A
M A
| ˆ
H |[|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= = ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
T + ˆ
U basis + ( ˆ
V res − ˆ
U 0 )|{|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= = ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
|( ˆ
V res − ˆ
U 0 )|{|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= = ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
|( ˆ
V res − ˆ
U 0 ) A−1 |{|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
= 0 ,
(9.178)
where one has used the fact that [|Ψ
J T i
T i
⊗ |u n i f j f τ f
J A
M A
and ˆ
A {|Ψ
J T f
T f
⊗
|u n f i j i τ i
J A
M A
do not have a common Slater determinant in their expansions,
implying that the ˆ
T + ˆ
U basis matrix element vanishes, and that (V res − U 0 ) A−1
does not act on |u n i i j i τ i . Evidently, this argument also applies if n i and n f are
exchanged, so that the sum in which n i = n f and n f > n max vanish as well.
Consequently, only the sum involving n i = n f and n i , n f ≤ n max remains.
If | i j i τ i = | f j f τ f , the terms of the sum involving n > n max in
Eq. (9.177) all vanish. Indeed,
[[Ψ
J T f
T f
| ⊗ ⊗u n f j f τ f |]
J A
M A
| ˆ
H T + ˆ
H P |[|Ψ
J T i
T i
⊗ |u n i j i τ i
J A
M A
= =Ψ
J T f
T f
| ˆ
H T |Ψ
J T i
T i
n f j f τ f |u n i j i τ i + +Ψ
J T f
T f
|Ψ
J T i
T i
n f j f τ f | ˆ
H P |u n i j i τ i
= 0 ,
(9.179)
as |u n f j f τ f and |u n i j i τ i are mutually orthogonal eigenvectors of ˆ
H P .