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

Ψ

0

(

r

1

,

r

2

)

is the optimal product of one-electron orbitals

ϕ

1s

(

r

1

)

ϕ

1s

(

r

2

)

approximating the

1

2

s

ground-state configuration. An improved representation of the ground state can be obtained by a superposition containing excited

1

S

electronic configurations, including

2

s

,

2

p

,

3

2

d

, …, with relative contributions determined by the variational principle.

Shull and Löwdin [3] represented the total wavefunction in the form

Ψ(

r

1

,

r

2

)=

∞

∑

L=0

f

L

(

r

1

,

r

2

)

2L+1

2

P

L

(cosΘ)

,

where

Θ

is the angle between

r

1

and

r

2

, while

P

L

(cosΘ)

is the Legendre polynomial of degree

L

. Note that the functions

P

L

(cosΘ)

contain internal angular dependence but can still represent atomic

S

states, such as

2

p

1

S

and

3

2

d

1

S

, with

L=0

. In the computations presented in this Demonstration, we consider only the

1

2

s

,

2

s

,

2

p

and

3

2

d

contributions to CI. The

1

2

s

contribution is taken as the HF function

Ψ

0

(

r

1

,

r

2

)

, which can be very closely approximated using double-zeta orbitals

ϕ

1s

(r)=A(

-αr

e

+a

-βr

e

)

.

The

2

s

contribution is represented by the orthogonalized Slater-type function

ϕ

2s

(r)=B(1-kr)

-ζr

e

.

Together, these two contributions can closely approximate the

S

-limit to the CI function, as defined by Shull and Löwdin. The

P

and

D

contributions are represented using simple Slater-type orbitals:

ϕ

2p

(r)=

2

3

5/2

ξ

r

-ξr

e

and

ϕ

3d

(r)=

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

(cosΘ)

for

2

p

and

Ψ

D

=

ϕ

3d

(

r

1

)

ϕ

3d

(

r

2

)

5

2

P

2

(cosΘ)

for

3

2

d

.

All the relevant matrix elements of the Hamiltonian are then computed; for example,

H

1s,1s

=21s-

1

2

∇

-

Z

r

1s+1s1s

1

r

12

1s1s

,

H

1s,2s

=1s2s

1

r

12

1s2s

,

and so forth. All energies are expressed in Hartree atomic units:

1hartree=27.211eV

.

You can select the level of configuration interaction:

Ψ

0

,

S

-limit,

S

+

P

or

S

+

P

+

D

, and the values of the exponential parameters

ζ

,

ξ

and

η

. 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.

Plots of the radial distribution function for each component configuration are also shown on the left. The relative magnitudes are not to scale.