5 Application of the Uniformly Charged Sphere Stabilization
105
Assuming that ΔΨ is small, we obtain the condition Ψ
LS |V s − V
s |Ψ
LS = 0 which
leads to the approximate equality V s ≈ V
s only if one of the external potential parameters (V 0 or R) has the same value for both potentials. It has been shown earlier [19, 20], that the exact solution F (r) for the external interval (r ≥ R) is
F (r) = U(l + 1 − a, 2l + 2, y)y
l+1 e
−y/2 ,
(5.13)
where (using the notations introduced above) r = λy, λ 2 = 2 /8m e (V 0 − E + E s ),
E s = E(H, 2 S, n = 1), a = 2m e e 2 Z 0 λ/ 2 and U(l + 1 − a, 2l + 2, y) is the Tricomi
function [40, 41]. Taking into account the approximate equality between the external
potential V s used in variational calculation and V
s required for exact calculations,
the function F (r) from (5.8) and (5.9) can be calculated using the relation (5.13)
with the parameters of the external potential and E. Because the parameters of the
external potential V 0 and R were chosen as constants in each energy calculation, the
stationarity condition δE| E=E = 0 leads to the stationarity of the parameter λ(E)
which is included in the equation for the exact radial function (5.13). Thus, if (5.13)
is used, the coefficients (5.8) and (5.9) and their ratio tan δ l (k) = B/A are stationary
with respect to small variations of λ.
Because this method to estimate the scattering parameters doesn’t require the
calculation of the coefficients b νi , one can solve the eigenvalue problem in the
basis of Slater determinants (CI method) instead of the system of equations (5.5)
and (5.10). The asymptotic part of the wave function (5.3) can be always written as ˆ
A SL
ν f ν (1)Γ ν (2) =
i
j d ij ˆ
A SL ϕ
f
j (1)ϕ Γ
i (2), where coefficients d ij =
ν b νj c νi . Accordingly, the calculation of the coefficients b νj with the known d ij
and c νi is possible only if the set of functions {Γ ν } has the same dimension as the
set {ϕ Γ
i }. At least two variants of the wave function construction in the Slater determinant basis satisfying this requirement are possible. In the first variant the initial
set {ϕ i } is separated onto the non-overlapping subsets {ϕ Γ
i } and {ϕ
f
i }, which are
used to define the asymptotic part. The correlation part includes the complete set of
the antisymmetrized products constructed from the single-particle functions {ϕ Γ
i }.
This variant corresponds to the restricted CI (RCI). In the second one wave function
contains the complete set of the antisymmetrized products of ϕ i without the explicit
fragmentation in the asymptotic and correlation parts (full CI method, or FCI).
5.3 The Calculation Method
The trial wave function was taken as the linear combination of configuration-state
functions (CSF) which are proper for the operators ˆ
S z (1, 2) and ˆ
S 2 (1, 2)
Ψ
LS
=
i,j
c ij Φ ij (1, 2),
(5.14)
105
Assuming that ΔΨ is small, we obtain the condition Ψ
LS |V s − V
s |Ψ
LS = 0 which
leads to the approximate equality V s ≈ V
s only if one of the external potential parameters (V 0 or R) has the same value for both potentials. It has been shown earlier [19, 20], that the exact solution F (r) for the external interval (r ≥ R) is
F (r) = U(l + 1 − a, 2l + 2, y)y
l+1 e
−y/2 ,
(5.13)
where (using the notations introduced above) r = λy, λ 2 = 2 /8m e (V 0 − E + E s ),
E s = E(H, 2 S, n = 1), a = 2m e e 2 Z 0 λ/ 2 and U(l + 1 − a, 2l + 2, y) is the Tricomi
function [40, 41]. Taking into account the approximate equality between the external
potential V s used in variational calculation and V
s required for exact calculations,
the function F (r) from (5.8) and (5.9) can be calculated using the relation (5.13)
with the parameters of the external potential and E. Because the parameters of the
external potential V 0 and R were chosen as constants in each energy calculation, the
stationarity condition δE| E=E = 0 leads to the stationarity of the parameter λ(E)
which is included in the equation for the exact radial function (5.13). Thus, if (5.13)
is used, the coefficients (5.8) and (5.9) and their ratio tan δ l (k) = B/A are stationary
with respect to small variations of λ.
Because this method to estimate the scattering parameters doesn’t require the
calculation of the coefficients b νi , one can solve the eigenvalue problem in the
basis of Slater determinants (CI method) instead of the system of equations (5.5)
and (5.10). The asymptotic part of the wave function (5.3) can be always written as ˆ
A SL
ν f ν (1)Γ ν (2) =
i
j d ij ˆ
A SL ϕ
f
j (1)ϕ Γ
i (2), where coefficients d ij =
ν b νj c νi . Accordingly, the calculation of the coefficients b νj with the known d ij
and c νi is possible only if the set of functions {Γ ν } has the same dimension as the
set {ϕ Γ
i }. At least two variants of the wave function construction in the Slater determinant basis satisfying this requirement are possible. In the first variant the initial
set {ϕ i } is separated onto the non-overlapping subsets {ϕ Γ
i } and {ϕ
f
i }, which are
used to define the asymptotic part. The correlation part includes the complete set of
the antisymmetrized products constructed from the single-particle functions {ϕ Γ
i }.
This variant corresponds to the restricted CI (RCI). In the second one wave function
contains the complete set of the antisymmetrized products of ϕ i without the explicit
fragmentation in the asymptotic and correlation parts (full CI method, or FCI).
5.3 The Calculation Method
The trial wave function was taken as the linear combination of configuration-state
functions (CSF) which are proper for the operators ˆ
S z (1, 2) and ˆ
S 2 (1, 2)
Ψ
LS
=
i,j
c ij Φ ij (1, 2),
(5.14)