Computational Physics - Department of Physics

50615 Many-body approaches to studies of electronic systems: Hartree-Fock theory and Density Functional Theory

respectively. The functiong(ri)is an arbitrary function, and by the substitutiong(ri) =ψν(ri)
we get

Vμex(ri)ψν(ri) =

(∫

ψ∗μ(rj)

1

ri j

ψν(rj)drj

)

ψμ(ri). (15.29)

We may then rewrite the Hartree-Fock equations as

HiHFψν(ri) =ενψν(ri), (15.30)

with

HiHF=hi+

N

∑

μ= 1

Vμd(ri)−

N

∑

μ= 1

Vμex(ri), (15.31)

and wherehiis defined by equation (15.7).

15.4.3Detailed solution of the Hartree-Fock equations

We show here the explicit form of the Hartree-Fock for heliumand beryllium
Let us introduce
ψnlmlsms=φnlml(r)ξms(s)

withsis the spin ( 1 / 2 for electrons),msis the spin projectionms=± 1 / 2 , and the spatial part
is
φnlml(r) =Rnl(r)Ylml(rˆ)

withYthe spherical harmonics andunl=rRnl. We have for helium

Φ(r 1 ,r 2 ,α,β) =√^1

2

φ 100 (r 1 )φ 100 (r 2 )

[

ξ↑( 1 )ξ↓( 2 )−ξ↑( 2 )ξ↓( 1 )

]

,

The direct term acts on
1
√
2

φ 100 (r 1 )φ 100 (r 2 )ξ↑( 1 )ξ↓( 2 )

while the exchange term acts on

−

1

√

2

φ 100 (r 1 )φ 100 (r 2 )ξ↑( 2 )ξ↓( 1 ).

How do these terms get translated into the Hartree and the Fock terms?
The Hartree term
Vμd(ri) =

∫

ψμ∗(rj)

1

ri j

ψμ(rj)drj,

acts onψλ(ri) =φnlml(ri)ξms(si), that is it results in

Vμd(ri)ψλ(ri) =

(∫

ψμ∗(rj)^1

ri j

ψμ(rj)drj

)

ψλ(ri),

and accounting for spins we have

Vnlmd l↑(ri)ψλ(ri) =

(∫

ψ∗nlml↑(rj)^1

ri j

ψnlml↑(rj)drj

)

ψλ(ri),

and
Vnlmd l↓(ri)ψλ(ri) =

(∫

ψ∗nlml↓(rj)^1

ri j

ψnlml↓(rj)drj

)

ψλ(ri),