Configuration Interaction for the Helium Isoelectronic Series
Configuration Interaction for the Helium Isoelectronic Series
Configuration interaction (CI) provides a systematic method for improving on single-configuration Hartree–Fock (HF) computations [1, 2]. This Demonstration considers the two-electron atoms in the helium isoelectronic series. The HF wavefunction (,) is the optimal product of one-electron orbitals ()() approximating the ground-state configuration. An improved representation of the ground state can be obtained by a superposition containing excited S electronic configurations, including , , , …, with relative contributions determined by the variational principle.
Ψ
0
r
1
r
2
ϕ
1s
r
1
ϕ
1s
r
2
1
2
s
1
2
2
s
2
2
p
3
2
d
Shull and Löwdin [3] represented the total wavefunction in the form
Ψ(,)=(,)(cosΘ)
r
1
r
2
∞
∑
L=0
f
L
r
1
r
2
2L+1
2
P
L
where is the angle between and , while (cosΘ) is the Legendre polynomial of degree . Note that the functions (cosΘ) contain internal angular dependence but can still represent atomic states, such as and , with . In the computations presented in this Demonstration, we consider only the , , and contributions to CI. The contribution is taken as the HF function (,), which can be very closely approximated using double-zeta orbitals
Θ
r
1
r
2
P
L
L
P
L
S
2S
2
p
1
3S
2
d
1
L=0
1
2
s
2
2
s
2
2
p
3
2
d
1
2
s
Ψ
0
r
1
r
2
ϕ
1s
-αr
e
-βr
e
The contribution is represented by the orthogonalized Slater-type function
2
2
s
ϕ
2s
-ζr
e
Together, these two contributions can closely approximate the -limit to the CI function, as defined by Shull and Löwdin. The and contributions are represented using simple Slater-type orbitals:
S
P
D
ϕ
2p
2
3
5/2
ξ
-ξr
e
and
ϕ
3d
2
3
2
5
7/2
η
2
r
-ηr
e
with
Ψ
P
ϕ
2p
r
1
ϕ
2p
r
2
3
2
P
1
for and
2
2
p
Ψ
D
ϕ
3d
r
1
ϕ
3d
r
2
5
2
P
2
for .
3
2
d
All the relevant matrix elements of the Hamiltonian are then computed; for example,
H
1s,1s
1
2
2
∇
Z
r
1
r
12
H
1s,2s
1
r
12
and so forth. All energies are expressed in Hartree atomic units: .
1hartree=27.211eV
You can select the level of configuration interaction: , -limit, + or ++, and the values of the exponential parameters , and . The built-in Mathematica function Eigenvalues then finds the lowest eigenvalue for the corresponding Hamiltonian matrix. The results are represented graphically on a barometer display, comparing them to the exact values of the energy.
Ψ
0
S
S
P
S
P
D
ζ
ξ
η
Plots of the radial distribution function for each component configuration are also shown on the left. The relative magnitudes are not to scale.