3.7 Numerical Implementation of the Berggren Completeness Relation
119
where
s(k, r) =
2
π
sin(kr) ,
(3.94)
so that one has just added and subtracted the = 0 Fourier-Bessel transform of the
Coulomb potential (C Coul ΔZ)/r. The converging character of Eq. (3.93) will be
demonstrated in Exercise XI.
Exercise XI
One will demonstrate that the subtraction method allows to integrate out the
singularities occurring due to the infinite range of the Coulomb potential.
A. Show that the first radial integral of Eq. (3.93) can always be calculated with
the complex rotation.
B. Calculate the second integral of Eq. (3.93) analytically.
C. Based on the results obtained in A and B, conclude that Eq. (3.93) can always
be calculated precisely, noticing that singularities occur only in the second
integral of Eq. (3.93).
In the off-diagonal method, diagonal infinite matrix elements are replaced by offdiagonal matrix elements situated in the vicinity of the diagonal of the matrix. This
is motivated by the analytical approximation of the integral of ln |k − k | between
k = 0 and k = k max acquired from the trapezoidal rule (see Exercise XII). The
discretization scheme is defined in the following way. First one calculates all offdiagonal matrix elements, where complex scaling can always be applied. Diagonal
matrix elements involving resonances are also straightforward to calculate with
complex scaling, as divergences can only occur when integrating two scattering
states of close k values. Afterwards one replaces the initially diverging diagonal
matrix elements, that is, those involving scattering states, by off-diagonal matrix
elements lying close to the diagonal:
Reg
+∞
0
u(k, r)V Coul (ΔZ, r)u(k, r)dr
→ Reg
+∞
0
u(k
+ , r)V Coul (ΔZ, r)u(k
− , r)dr,
(3.95)
where
k
±
= k ±
Δk
4π
,
(3.96)
as suggested by the Exercise XII. However, Δk in (3.96) remains to be fixed.
119
where
s(k, r) =
2
π
sin(kr) ,
(3.94)
so that one has just added and subtracted the = 0 Fourier-Bessel transform of the
Coulomb potential (C Coul ΔZ)/r. The converging character of Eq. (3.93) will be
demonstrated in Exercise XI.
Exercise XI
One will demonstrate that the subtraction method allows to integrate out the
singularities occurring due to the infinite range of the Coulomb potential.
A. Show that the first radial integral of Eq. (3.93) can always be calculated with
the complex rotation.
B. Calculate the second integral of Eq. (3.93) analytically.
C. Based on the results obtained in A and B, conclude that Eq. (3.93) can always
be calculated precisely, noticing that singularities occur only in the second
integral of Eq. (3.93).
In the off-diagonal method, diagonal infinite matrix elements are replaced by offdiagonal matrix elements situated in the vicinity of the diagonal of the matrix. This
is motivated by the analytical approximation of the integral of ln |k − k | between
k = 0 and k = k max acquired from the trapezoidal rule (see Exercise XII). The
discretization scheme is defined in the following way. First one calculates all offdiagonal matrix elements, where complex scaling can always be applied. Diagonal
matrix elements involving resonances are also straightforward to calculate with
complex scaling, as divergences can only occur when integrating two scattering
states of close k values. Afterwards one replaces the initially diverging diagonal
matrix elements, that is, those involving scattering states, by off-diagonal matrix
elements lying close to the diagonal:
Reg
+∞
0
u(k, r)V Coul (ΔZ, r)u(k, r)dr
→ Reg
+∞
0
u(k
+ , r)V Coul (ΔZ, r)u(k
− , r)dr,
(3.95)
where
k
±
= k ±
Δk
4π
,
(3.96)
as suggested by the Exercise XII. However, Δk in (3.96) remains to be fixed.