Methodecalvulsdefinitive .pdf



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´
Hele

Heff

Heff =

n
X

(hµ + Veff (µ))

µ

Veff µ

µ

Heff Ψ = EΨ

Ψ

Φi

| Ψ(1, 2, ......, n) >=| Φ1 Φ2 ...Φi ......Φn >

H(1)Φi (1) = εi Φi (1)

H(1) = h(1) +

m
X

Ji (1)

i

εi

Φi

χp

| Φi >=

m
X

Cin | χn >

n=1

h

N

h(1) =

−4µ X ZI

2
rµI
I=1

Ji (1)

Ji (1) =

Z

Φ∗i (2)

1
Φi (2)dτ
r12

HSCF Φi (1) = εSCF
Φi (1)
i

HSCF

HSCF (1) = h(1) +

m
X

(Ji (1) − Ki (1))

i

εi

Φi

h

N

h(1) =

−4µ X ZI

2
rµI
I=1

Φi

Ji (1)

@ A

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Φi

1
Φi (2)dτ
r12

Z

Φ∗i (2)

Z

Φ∗i (2)

1
Φj (2)dτ
r12

Ji (1)

Ki (1)

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Ji (1) =

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·
Ki (1)

Ki (1) =

Φi

Φi

HSCF

χp

Φi

Φi =

m
X

Cpi χp

p=1

HSCF | Φi >= εSCF
| Φi >
i

HSCF

m
X

Cpi | χp >= εSCF
i

p=1

m
X

m
X

Cpi | χp >

p=1

SCF
Cpi [HSCF
Spq ] = 0
pq − εi

p=1

SCF
det | HSCF
Spq |= 0
pq − εi

HSCF
pq = Hpq +

m XX
X
i

r

Cir Cis [2(pq | rs) − (pr | qs)]

s

N

Hpq =< χp |

−4µ X ZI

| χq >
2
rµI
I=1

Spq =< χp | χq >
χp

χq

Φj

@ A Ÿ

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1
| χr χs >
r12

(pq | rs) =< χp χq |
|

α

β

α Φα
i
β Φβi
(





SCF (α)

(α)

HSCF (α) Φi (1) = εi
Φi (1
SCF(β) (β)
HSCF(β) Φi (1) = εi
Φi (1)



X


γ

Cγpi χp
 Φi =
p







X

avec
γ = α ou β

SCF(α)

Spq ] = 0

SCF(β)

Spq ] = 0

SCF(α)

− εi
pi [Hpq

p=1


X

SCF(β)
Cβpi [Hpq
− εi

p=1

SCF(α)

Hpq

SCF(α)
Hpq
= Hpq +

m X
X
i

SCF(β)
Hpq
= Hpq +

α

ir Cis [(pq | rs) − (pr | qs)] +

r,s

m X
X
i

SCF(β)

Hpq

m X
X
j

Cβir Cβis [(pq | rs) − (pr | qs)] +

r,s



X


γrs

P
=
Cγir Cγis
 r,s
i






avec
γ = α ou β

β
Prs = Pα
rs + Prs

»

r,s

m X
X
j

α

ir Cis (pq | rs)

r,s

Cβir Cβis (pq | rs)

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SCF(α)

Hpq

@ A

SCF(β)

j Ë N S CDILG[@ d4ABŸ

Hpq
XX
= Hpq +
[Prs (pq | rs) − Pγrs (pr | qs)]

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½

SCF(γ)


 Hpq




r

s

avec
γ = α ou β

χ(n,l,m) = Nn,l,m rn−1 exp(−ζr)Ylm (θ, ϕ)

|

exp(−αr2

exp(−ζr

χp (1)χq (1)

6=

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l r[œ ¤3· p
l r3œ ¤_» p
E-G3@ S`Ÿ
(pq | rs) = (pq | rs)δpq δrs

χp

R

χq

6=

χp χq τ

Spq =< χp | χq >= δpq

X

Cpi [HSCF
pq − δpq εi ] = 0

p

HSCF

X
1
Pqq (pp | qq)
HSCF
pp = Hpp − Ppp (pp | pp) +
2
q
 SCF
 Hpq = Hpq − 21 Ppq (pp | qq)
avec

p 6= q

occ
X



Ppp = 2 Cjp Cjp




j=1
et

occ

X



P
=
2
Cjp Cjq

 pq

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l r3œ ´3¡ p
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l r3œ ´[  p
Å
j=1

|

|

|

|

HSCF

HSCF
pp = Upp +

X
B

HSCF
pq =

X
B

B
Vpp
+

X

q∈A

X
1
Ppp [(pp | qq) − (pq | pq)] +
Prs (pp | rs)
2
r,s∈B

X
1
B
Vpq
+ Ppq [(pq | pq) − (pp | qq)] +
Prs (pp | rs)
2
r,s∈B

HSCF
= Hpr −
pr

1 XX
Pqs (pp | rs)
2
q∈A s∈B

B
Vpq
=< χp |

ZB
| χq >
rB

ZB

1
Upp =< χp | − 4 −VA | χp >
2

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l r3œ ´[Å p
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= −ZB (pq | sB sB )

Hpq

Hpq = Spq

βp

βp + β q
2

βq

AM1

(pq | rs) = γpq γrs (pp | rr)
|


 (pp | qq) = γpq = γAB
avec

et q ∈ B
p∈A

γAB

γAB = (s2A | s2B )


SCF


 Hpp




¯ G[@TS

PBB

HSCF
X
PBB γAB
= Hpp − 21 Ppp γAA +
B6=A

avec
p∈A

 SCF
 Hpq = Hpq − 21 Ppq γAB
avec

p ∈ A et q ∈ B

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À V1@ d<S \[CW@%Ÿ

PBB =

X

p∈B

rTÉ

Ppp

l r3œ ·3É p
l r3œ ·+r p


X
1
SCF

(PBB γBB − VAB )

 Hpp = Hpp + (PAA − 2 Ppp )γAA +
B6=A

avec
p∈A





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PAA
PBB
VAB ≈ χp |VB |χp
βpq βAB Spq

 SCF
 Hpq = βpq − 21 Ppq γAB
avec

p 6= q

VB

βAB

HSCF

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X
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 HSCF
Prr [(pp | rr) − 21 (pr | pr)] +
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pp = Upp +
r

A6=B

Si (p, r) ∈ A







1
HSCF
pp = βAB Spq − 2 Ppq γAB
Si p ∈ A et q ∈ B

3
1
HSCF
pq = 2 Ppq (pq | pq) − 2 Ppq (pp | qq)
Si (p, q) ∈ A

HSCF
pq

|
Hpp

Ppq

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r ¤
|

Φi

Φi

Φ∗i

|Ψ0

|

|Ψ0

| J >=| Ψki >=| Φ1 , Φ1 , ......, Φi , Φi , Φ∗k , Φi , ......., Φn , Φn >

Φi

Φ∗k

| J >=| Ψki

Φj



Φ1 Φ∗k




i∗
>=| Φ1 , Φ1 , ......, Φ∗k , Φi , Φj , Φi , ......., Φn , Φn >
j

Φi

|



|Ψ0


ρ(→
r)


ρ(→
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r-´





E(ρ(→
r )) = T(ρ(→
r )) + Exc (ρ(→
r )) + Vexterieur(ρ(→
r )) + J(ρ(→
r ))


T(ρ(→
r ))


J(ρ(→
r ))

H=

n
X
i=1


Vchange (ρ(→
r ))

n





ρ(→
r ) Exc (ρ(→
r ))


Vexterieur(ρ(→
r ))

N

n

4i X X ZiI X



+
Jj + Vchange (ρ(→
r )) + Vcorrlation (ρ(→
r ))
2
riI
i=1 I=1

j=1


Vcorrlation (ρ(→
r ))


ρ(→
r)


E(ρ(→
r )) = 2

occ
X

< Φi | −

i=1

4i

| Φi > +U(ρ(→
r ))
2

Eion−ion

Vexterieur

Exc


U(ρ(→
r )) = Eion−ion(R) +

Z




d→
r Vexterieur(→
r )ρ(→
r)+





4i
∂U

+

2
∂ρ(→
r)

∂U
= Veff = 2

∂ρ(→
r)

Veff

Z



Z Z



ρ(→
r )ρ(→
r 0)



+ Exc [ρ(→
r )]
d→
r d→
r0 →



r − r0

E(ρ(→
r ))



Φi (→
r ) = εi Φi (→
r)


ρ(→
r 0)
d



d→
r0 →
Exc [ρ(→
r )] + Vexterieur(→
r)
0 +



| r − r | dρ

Φi


ρ(→
r)

Veff


ρ(→
r)=

n
X


| Φi (→
r ) |2

i=1

R



ρ(→
r )d→
r =





Exc

Exc

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l r3œ Å4´ p
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rT·

Exc = [ρ(→
r )] =

Z




εxc [ρ(→
r )]ρ(→
r )d→
r



rn N1/3

X
n

X
4(→
r )i
1



+ V( ri ) +
H=
2
r
i<j ij
i=1



ri


V(→
ri ) =

Heff =



N
(3 − ( rRi )2
− 2R
−N
ri

r≤R
r≥R


n

X
4(→
ri )

+ Veff (→
ri )

2
i=1

Veff

Veff =

−U0
1 + exp[(r − R)/ε]

U0

ε

Ω⊥ =

ΩZ =





2+δ
2−δ

2+δ
2−δ

1/3

−2/3



¯ ω0 (l2 − < l2 >n )
hi = − 42i + 12 mω02 (Ω2⊥ l2 + Ω2z z2 ) − U h
2
2
2
l =x +y

l r3œ Å3· p

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