5 Application of the Uniformly Charged Sphere Stabilization
103
angular and spin moments, L z , S z —its projections on the z-axis and α—all quantum numbers, which were not introduced explicitly. For the two-electron system the
wave function can be represented by [23, 36–39]
Ψ
LS
=
μ
a μ D μ + ˆ
A SL
ν
f ν (1)Γ ν (2).
(5.3)
The first term on the right-hand side of (5.3) refers to the electron correlation and it
is defined as the linear combination of Slater determinants D μ with coefficients a μ .
The second term refers to the asymptotic part of the wave function at r i ≥ R and it
can be written as a set of antisymmetrized products of the single-particle functions
f ν (1) and Γ ν (2). The operator ˆ
A SL is taken in the form ˆ
A SL = (1 − P 12 )/
√
2, where
P 12 is the permutation operator.
Let’s define the finite set of single-particle functions {ϕ i }, i = 1, 2, . . . , N, where
each function is a product of independently normalized spatial and spin parts:
ϕ i = χ i (r)τ (σ ), ϕ i |ϕ j = δ ij . To define both terms of (5.3) the {ϕ i } set is split
in the subsets {ϕ Γ
i }, {ϕ
f
i } and {ϕ D
i }. The first subset {ϕ Γ
i } determines the orthonormal target states {Γ ν }, Γ ν =
i c νi ϕ Γ
i , where c νi are some coefficients and index
ν corresponds to the full set of quantum numbers {n ν , l Γ
ν , s Γ
ν , l Γ
zν , s Γ
zν }, characterizing uniquely the state of target. The target functions should satisfy the condition
Γ μ (i)| ˆ
H s (i) − E νs |Γ ν (i) = 0, i.e. these functions may be exact or obtained by
the linear variational method. To describe the scattered electron state let’s introduce
the single-particle functions f ν =
j b νj ϕ
f
j expanded over the subset {ϕ
f
j } of the
initial set {ϕ i } with coefficients b νj . The last subset is used to define the functions
D μ = ˆ
A SL {ϕ D
i (1)ϕ D
j (2)}.
In the general case the basic formulation of the variational method depends on the
choice of {ϕ Γ
i }, {ϕ
f
i } and {ϕ D
i } sets. For example, the case when the initial choice
does not provide the orthogonality conditions A SL (f ν (1)Γ ν (2))|D μ (1, 2) = 0 was
discussed earlier [23, 37]. Further it will be assumed that the wave function variation
preserves the orthogonality of both terms of (5.3) due to the corresponding choice of
the single-particle subsets. In this case the basic equations of the variational method
are
Γ ν (2)
ˆ
H s − E
Ψ
LS = 0,
(5.4)
D μ | ˆ
H s − E
Ψ
LS = 0,
(5.5)
where the integration in (5.4) is realized over all the variables of the second particle and in (5.5) over all the variables of both particles. As has been shown
earlier, if the wavefunction of the scattered electron f ν is written in the form
f ν (1) = r 1
−1 F ν (r 1 )Y l ν m ν (ˆ r 1 )τ ν (σ 1 ), where ˆ
r 1 represents the polar (θ ) and azimuthal
(φ) angles, m ν + l Γ
zν = L z and Y lm (ˆ r) is a spherical harmonic, (5.4) can be transformed to a system of the radial integro-differential equations. In our case this system of equations will differ slightly from its original form [23, 36–39] and have the
form
−
d 2
dr 2 +
l ν (l ν + 1)
r 2
+ ˜
V s (r) − k
2
νs
F ν (r) =
ν
A νν (r) +
μ
B νμ (r), (5.6)
103
angular and spin moments, L z , S z —its projections on the z-axis and α—all quantum numbers, which were not introduced explicitly. For the two-electron system the
wave function can be represented by [23, 36–39]
Ψ
LS
=
μ
a μ D μ + ˆ
A SL
ν
f ν (1)Γ ν (2).
(5.3)
The first term on the right-hand side of (5.3) refers to the electron correlation and it
is defined as the linear combination of Slater determinants D μ with coefficients a μ .
The second term refers to the asymptotic part of the wave function at r i ≥ R and it
can be written as a set of antisymmetrized products of the single-particle functions
f ν (1) and Γ ν (2). The operator ˆ
A SL is taken in the form ˆ
A SL = (1 − P 12 )/
√
2, where
P 12 is the permutation operator.
Let’s define the finite set of single-particle functions {ϕ i }, i = 1, 2, . . . , N, where
each function is a product of independently normalized spatial and spin parts:
ϕ i = χ i (r)τ (σ ), ϕ i |ϕ j = δ ij . To define both terms of (5.3) the {ϕ i } set is split
in the subsets {ϕ Γ
i }, {ϕ
f
i } and {ϕ D
i }. The first subset {ϕ Γ
i } determines the orthonormal target states {Γ ν }, Γ ν =
i c νi ϕ Γ
i , where c νi are some coefficients and index
ν corresponds to the full set of quantum numbers {n ν , l Γ
ν , s Γ
ν , l Γ
zν , s Γ
zν }, characterizing uniquely the state of target. The target functions should satisfy the condition
Γ μ (i)| ˆ
H s (i) − E νs |Γ ν (i) = 0, i.e. these functions may be exact or obtained by
the linear variational method. To describe the scattered electron state let’s introduce
the single-particle functions f ν =
j b νj ϕ
f
j expanded over the subset {ϕ
f
j } of the
initial set {ϕ i } with coefficients b νj . The last subset is used to define the functions
D μ = ˆ
A SL {ϕ D
i (1)ϕ D
j (2)}.
In the general case the basic formulation of the variational method depends on the
choice of {ϕ Γ
i }, {ϕ
f
i } and {ϕ D
i } sets. For example, the case when the initial choice
does not provide the orthogonality conditions A SL (f ν (1)Γ ν (2))|D μ (1, 2) = 0 was
discussed earlier [23, 37]. Further it will be assumed that the wave function variation
preserves the orthogonality of both terms of (5.3) due to the corresponding choice of
the single-particle subsets. In this case the basic equations of the variational method
are
Γ ν (2)
ˆ
H s − E
Ψ
LS = 0,
(5.4)
D μ | ˆ
H s − E
Ψ
LS = 0,
(5.5)
where the integration in (5.4) is realized over all the variables of the second particle and in (5.5) over all the variables of both particles. As has been shown
earlier, if the wavefunction of the scattered electron f ν is written in the form
f ν (1) = r 1
−1 F ν (r 1 )Y l ν m ν (ˆ r 1 )τ ν (σ 1 ), where ˆ
r 1 represents the polar (θ ) and azimuthal
(φ) angles, m ν + l Γ
zν = L z and Y lm (ˆ r) is a spherical harmonic, (5.4) can be transformed to a system of the radial integro-differential equations. In our case this system of equations will differ slightly from its original form [23, 36–39] and have the
form
−
d 2
dr 2 +
l ν (l ν + 1)
r 2
+ ˜
V s (r) − k
2
νs
F ν (r) =
ν
A νν (r) +
μ
B νμ (r), (5.6)