Computational Physics - Department of Physics

49815 Many-body approaches to studies of electronic systems: Hartree-Fock theory and Density Functional Theory

of 2 ( 2 l+ 1 ) 2 ( 2 · 1 + 1 ) = 6. This leads to ten possible values forα,β,...,ν. Fig. 15.1 shows the
possible quantum numbers which the ten first elements can have.

s p

K

L

H

s p s p

K

L

He Li

s p s p s p s p

n= 1

n= 2

Be B C N

s p s p s p

n= 1

n= 2

O F Ne

Fig. 15.1The electronic configurations for the ten first elements. We let an arrow which points upward to
represent a state withms= 1 / 2 while an arrow which points downwards hasms=− 1 / 2.

If we consider the helium atom with two electrons in the 1 sstate, we can write the total
Slater determinant as

Φ(r 1 ,r 2 ,α,β) =

√^1

2

∣∣

∣∣ψα(r^1 )ψα(r^2 )

ψβ(r 1 )ψβ(r 2 )

∣∣

∣∣, (15.2)

withα=nlmlsms= ( 1001 / 21 / 2 )andβ=nlmlsms= ( 1001 / 2 − 1 / 2 )or usingms= 1 / 2 =↑and
ms=− 1 / 2 =↓asα=nlmlsms= ( 1001 / 2 ↑)andβ=nlmlsms= ( 1001 / 2 ↓). It is normal to skip
the quantum number of the one-electron spin. We introduce therefore the shorthandnlml↑or
nlml↓)for a particular state. Writing out the Slater determinant

Φ(r 1 ,r 2 ,α,β) =

√^1

2

[

ψα(r 1 )ψβ(r 2 )−ψβ(r 1 )ψγ(r 2 )

]

, (15.3)

we see that the Slater determinant is antisymmetric with respect to the permutation of two
particles, that is
Φ(r 1 ,r 2 ,α,β) =−Φ(r 2 ,r 1 ,α,β),

For three electrons we have the general expression

Φ(r 1 ,r 2 ,r 3 ,α,β,γ) =

√^1

3!

∣∣

∣∣

∣∣

ψα(r 1 )ψα(r 2 )ψα(r 3 )

ψβ(r 1 )ψβ(r 2 )ψβ(r 3 )

ψγ(r 1 )ψγ(r 2 )ψγ(r 3 )

∣∣

∣∣

∣∣. (15.4)

Computing the determinant gives

Φ(r 1 ,r 2 ,r 3 ,α,β,γ) =√^1 3!

[

ψα(r 1 )ψβ(r 2 )ψγ(r 3 )+ψβ(r 1 )ψγ(r 2 )ψα(r 3 )+ψγ(r 1 )ψα(r 2 )ψβ(r 3 )−

ψγ(r 1 )ψβ(r 2 )ψα(r 3 )−ψβ(r 1 )ψα(r 2 )ψγ(r 3 )−ψα(r 1 )ψγ(r 2 )ψβ(r 3 )

]

.(15.5)

We note again that the wave-function is antisymmetric with respect to an interchange of any
two electrons, as required by the Pauli principle. For anN-body Slater determinant we have
thus (omitting the quantum numbersα,β,...,ν)