210
5 Formulation and Implementation of the Gamow Shell Model
determinant coupled Hartree-Fock potential can be defined from the variational
principle, demanding that these excitations vanish, similarly to the Hartree-Fock
procedure:
GS | ˆ
H | Ψ k = 0 ,
(5.62)
where
|Ψ k = [a
†
kkj a nnj ]
0
0 |Ψ GS .
(5.63)
In this expression, nnj is the last occupied shell of the multi-Slater determinant coupled Hartree-Fock configuration of , j quantum numbers, and kkj is a scattering
shell. a
†
kkj and a nnj are coupled to zero in order to have |Ψ k coupled to J . Using
Eqs. (5.61) and (5.62), one obtains:
2j + 1 nnj |h|kkj GS | [a
†
nnj a kkj ]
0
0 | Ψ k
−
λ occ J
√
2J + 1 occ nnj |V |λ occ kkj J
× ×Ψ GS | [[a
†
λ occ
a
†
nnj ]
J [ a λ occ a kkj ]
J ]
0
0 | Ψ k = 0 ,
(5.64)
where
• α = nnj and β = kkj in the one-body part of Eqs. (5.61) and (5.64) as |Ψ k is a
1p-1h excitation of |Ψ GS
• β = nnj , δ = kkj and α = γ = λ occ in the two-body part of Eqs. (5.61) and
(5.64) because, on the one hand, |Ψ k is a 1p-1h excitation of |Ψ GS and, on the
other hand, the shell indices associated to α = nnj resonant states can be chosen
so that they are always smaller than the shell indices associated to β = kkj
scattering states.
One can then define the multi-Slater determinant coupled Hartree-Fock potential
from Eq. (5.64):
MSDHF |β = =α|h|β
−
λ occ J
√
2J + 1 occ α|V |λ occ β J GS | [[a
†
λ occ
a
†
nnj ]
J [ a λ occ a kkj ]
J ]
0
0 | Ψ k
√
2j + 1 GS | [a
†
nnj a kkj ]
0
0 | Ψ k
(5.65)
U MSDHF is spherical and independent of k as the latter only induces angular
momentum coupling and rearrangement phases (see Eq. (5.65)). It is solved selfconsistently, diagonalizing ˆ
H in the configuration of lowest energy to obtain