# 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.