Chap 9 Electron-electron interactions
• e-e interaction and Pauli exclusion principle (chap 17)
• Hartree approximation
• Hartree-Fock approximation
• Exchange-correlation hole
• Density functional theory
Dept of Phys
M.C. Chang
What’s missing with the non-interacting electrons?
• exchange effect
• screening effect
• normalization of band gap (and band structure, FS)
• quasiparticle, collective excitation
• superconductivity
•…
Beyond non-interacting electron
 pi2
 1
H  
 U ion (ri )    Vee (ri  rj ),
i 1  2m
 2 ii ,j j
N
e2
Vee (ri  rj ) 
,
| ri  rj |
H  (r1 , r2 , )  E  (r1 , r2 , )
This is a differential eq. with N=1023 degrees of freedom.
We need approximations.
Hartree approximation (1928):
 



assume (r1 , r2 , )   1 (r1 ) 2 (r2 ) N (rN )
No quantum correlation
～ classical particles
H  (r1 , r2 , )  E  (r1 , r2 , )

|  j (r ') |2 





j
 
 2  U ion (r )  i (r )  e 2  d 3r '
 i (r )   i i ( r )
2
m
|
r

r
'
|




2
Hartree (or direct) potential VeeH (r)
Each electron moves in the potential from all the other electrons,
• Need to be solved self-consistently (by iteration).
• Self-consistency doesn’t mean the result is correct.
What’s wrong with the HA?
• The manybody wave function violates the Pauli principle
• The calculated total energy is positive (means the electron gas is unstable)
Self-consistent Hartree approximation
1. choose initial  i 

2. construct n(r )   |  i (r ) |  VeeH (r )  e 2  d 3r '
2
i
n(r ')
| r r '|

2


3. solve  
 2  U ion (r )  VeeH ( r )  'i ( r )   i 'i ( r )
 2m


4. construct n '(r )   |  'i (r ) |
2
i

5. if n '(r )  n(r )   , then STOP.
else let  i (r )   'i ( r ), GOTO 2
E. Kaxiras, Atomic and Electronic Structure of Solids, p.46
Hartree-Fock approximation (1930):
• A brief review of variational principle (single particle version)
H (r )   (r )
the ground state can be obtained by minimizing  H  ,
under the constraint    1
i.e.

 H        1   0 (   )

*
 (r )
• Hartree-Fock approximation
assume  (r1 , s1 , r2 , s2

 1 (r1 , s1 )  1 (r2 , s2 )
1  2 (r1 , s1 )  2 (r2 , s2 )
)
N!
 N (r1 , s1 )  N (r2 , s2 )
 1 (rN , sN )
 2 (rN , sN )
 N (rN , sN )
p2
1 N
e2
 H    i
 U (r )  i 
i  j
i  j

2
m
2
|
r

r
'
|
i 1
i , j 1 ( i  j )
Hartree
energy
1 N
e2

 i  j | r  r '|  j i
2 i , j 1 (i  j )
Fock
energy
N
Variational principle

N



H






1



i
i
i
0
 * (r , s) 
i 1

See handwritten note
 2 2

H
    Uion (r )  Vee (r )  (r , s)    d 3r ' veeF (r , s; r ' s ') (r ', s ')    (r , s)
s'
 2m

VeeH (r )  e2  d 3r '
  (r ' s ')
i
i,s '
veeF (r , s; r ' s ')   e 2
| r r '|

*
i
2
Sum over filled
states only
 e 2  d 3r '
n(r ')
Hartree (or direct) potential
| r r '|
(r ' s ') i (r , s)
i
| r r '|
 s ,s '
Fock (or exchange) potential
• The exchange potential exists only between electrons with parallel spins.
• The exchange potential is non-local! This makes the HFA much harder to calculate!
• Still need self-consistency.
• Again, no guarantee on the correctness (even qualitatively) of the self-consistent result!
Hartree-Fock theory of uniform electron gas
• The jellium approximation
Positive Ion charges
Uniform background
Smooth off
• Below we show that plane waves are sol’ns of HFA
eik r
assume  k (r , s) 
s
V
This cancels with
 V
H
ee
(en0 )
 e  d r '
| r r '|
3
U ion (r )  e  d 3r '
2
e
3
F
d
r
'
v
(
r
,
s
;
r
'
s
')

(
r
',
s
')


s ' 
ee
k
V
d r'
4 e 2
=
V
3
e
(en0 )
| r r '|
 s,s '
ik '( r  r ')
k 's'
| r r '|
eik r '
s'
V
1
 k (r , s)

2
k' | k  k ' |
2

4 e 2
2
 
 
V
 2m

1
 k (r )   k k (r )

2 
k ' k F | k  k ' | 
ε(k)
∴ HF energy
2
k2
d 3k '
1
2
 (k ) 
 4 e 
2m
(2 )3 | k  k ' |2
k  kF
k 2 2e 2  1 1  x 2 1  x


k

ln
2m
 F  2
4x
1 x
2

k
,
x


kF

k/kF
“DOS” D(k(ε)) (arb. unit)
Lindhard function F(x)
|F’(x)|
Slope at kF has
logarithmic div
k/kF
• If, when you remove one electron from an N-electron system, the other N-1
wave functions do not change, then |ε(k)| is the ionization energy (Koopman’s
theo. 1933).
• In reality, the other N-1 electrons would relax to screen the hole created by
ionization. (called “final state effect”, could be large.)
• Total energy of the electron
gas (in the HFA)
EHF 
Substrat double counting
between k and k’
1



(
k
)

V
(
k
)

F

2
k  kF 
3 2

 3 2 k F2 3 e 2 k F 
3 e2
2
 N


N
k
a

k
a




F 0
F 0
2
5
2
m
4

5
2
ma
2

2
a


0
0


E
2.21 0.916
4
 HF  2 
, in which  (rs a0 )3  n
N
rs
rs
3
2
2


e2
or
 rs in a0 = 2 , E in Ry=

me
2
a
2ma02 
0

• More on the HF energy
1
1
2
EHF    d 3rVeeH (r )  (r , s)    d 3r  d 3r ' veeF (r , s; r ' s ') * (r , s) (r ', s ')
2 ,s
2 ,s,s '

=
1
i,
2
3
3 ns ' ( r ') ns ( r ) 1
2
3
3
e
d
rd
r
'

e
d
r
d
r
'


2 s,s ' 
| r r '|
2 s,s '  
*
i
(r ' s ') i (r , s) * (r , s) (r ', s ')
 ss '
| r r '|
n (r ')ns (r ) HF
1
= e2  d 3rd 3r '  s '
g ss ' (r , r ')
2
| r r '|
s,s '
• Pair correlation function
g ssHF' (r , r ')  1 

2
*
i
(r ' s ') i (r , s )
i
ns ' (r ')ns (r )
 ss '
The conditional probability to find a spin-s’ electron at r’,
when there is already a spin-s electron at r.
• In the jellium model (ns=N/2V),
g ssHF' ( r , r ')  1 
2
N
2
e
k  kF
ik ( r  r ')
 ss '
2
r | r  r ' |,
2
 sin k F r  k F r cos k F r 
 3

= 1 3
j1 ( k F r )   ss '
  ss ' or 1  
3 3
k
r
k
r

F

 F

• Fock (exchange) potential keeps electrons with the same spin apart
(This is purely a quantum statistical effect)
exchange
hole
interaction of the electron with the “hole”:
 X hole
HF
en0 1  g
(r )  1
3 e2 k F
 (  e)  d r
 
r
2
4 
3
• Beyond HFA
Now there is a hole even if the electrons have different spins!
exchange-correlation (xc) hole
(named by Wigner)
could be non-spherical
in real material
DOS for an electron gas
in HA and HFA
HFA
HA
What’s wrong with HFA?
• In the HFA, the DOS goes to zero at the Fermi energy.
HFA gets the specific heat and the conductivity seriously wrong.
• The band width is 2.4 times too wide (compared to free e)
• The manybody wave function is not necessarily a single Slater determinant.
Beyond HFA:
• Green function method: diagrammatic perturbation expansion
• Density functional theory: inhomogeneous electron gas, beyond jellium
• Quantum Monte Carlo
•…
Green function method (diagrammatic perturbation expansion)
Correction to
free particle
energy
RPA
Random phase approx.
(RPA)
• The energy correction beyond HFA is called correlation energy (or stupidity energy).
EC = EEXACT - EHF
• Gell-Mann+Bruckner’s result (1957, for high density electron gas)
E/N = 2.21/rS2 + 0 - 0.916/rS + 0.0622 ln(rS) - 0.096 + O(rS)
= E K + E H - EF + EC
(E in Ry, rS in a0)
• This is still under the jellium approximation.
• Good for rS<1, less accurate for electrons with low density (Usual metals, 2 < rS < 5)
• E. Wigner predicted that very low-density electron gas (rS > 10?) would
spontaneously form a non-uniform phase (Wigner crystal)
Luttinger, Landau, and quasiparticles
• Modification of the Fermi sea
due to e-e interaction (T=0)
Perturbation to all orders
Z
(if perturbation is valid)
• There is still a jump that defines the FS (Luttinger, 1960).
Its magnitude Z (<1) is related to the effective mass of a QP.
• A quasiparticle (QP) ＝ an electron “dressed” by other electrons.
A strongly interacting electron gas = a weakly interacting gas of QPs.
(Landau, 1956)
• It is a quasi-particle because, it has a finite life-time. Therefore,
its spectral function has a finite width:
Free
electron
QP
This peak sharpens as
we get closer to the FS
(longer lifetime)
Na: 3DEG with rs = 3.99
From Holzmann’s slide
Density functional theory
Cited 8578 times (by 11/26/2011)
Cited 18298 times (by 11/26/2011)
沈呂九
“density functional theory" appears in title or abstract
“impact”
1964
“availability of computing power”
N

2
  pi  U (r )   1 V (r  r )   (r , r , )  E  (r , r , )
 ee i j  1 2

ion i 
1 2

2 i, j
i 1  2m

i j


Particle density
n(r )  N  d 3r2  d 3r3
 d r  (r , r ,
3
*
N
2
) (r , r2 , )
Usually: Uion → Ψ → n
DFT:
n → Ψ → Uion
The 1st Hohenberg-Kohn theorem
The potential Uion is a unique functional of the ground state density n.
Pf:
suppose U, U’ give the same ground state density, n=n’
EG   G H  G ; EG '   G ' H '  G '
EG '   G H '  G   G H  U ' U  G  EG   G U ' U  G
 EG '  EG   d 3r n(r ) U (r ) ' U (r ) 
same argument also gives
EG  EG '  d 3r n(r ) U (r )  U '(r ) 
 EG ' EG  EG  EG '

In principle, given n(r), one
can uniquely determine U(r).
Since n(r) determines Uion(r), which determines everything else (EG, |G>… etc),
one can say that, EG is a functional of n(r):
EG [n]  T [n]  U [n]  Vee [n]
The 2nd Hohenberg-Kohn theorem
The true ground state density n minimizes the energy functional EG[n],
with the following constraint,
 d r n( r )  N .
3
Pf:
If n’ is a density different from the ground-state density n in potential U(r),
then the U’(r) (and Ψ’), that produce this n’ are different from the ΨG in U(r).
According to the variational principle,
E[n ']   ' H  '  G H G  EG [n]
Thus, for potential U(r), E [n’] is minimized by the ground-state density n.
• The energy functional
EG [n]  T [n]  Vee [n]  U [n]
= F [ n ]  U [ n]
U [ n]   d 3 r n( r ) U ( r )
The F [n] functional is the same for all electronic systems.
“No ones knows the true F [n], and no one will, so it is replaced by various
uncontrollable approximations.”
(Marder, p.247)
F[n]  T [n]  VH [n]  VXC [n]
• Kinetic energy functional
2 5
kF
k2
V
2m
10 2 m
d 3k
2
For free electron gas
T [ n]  V 
Thomas-Fermi approx.
kF   3 n(r ) 
 2 
 T TF [n] V
• Hartree energy functional
(exact)
1/3
2
(good for slow density variation)
e2
VH [n] 
2
3
3
2
3 
2
10 m
2
3
3
d
rd
r'

2/3
n(r )5/3
n(r )n(r ')
| r r '|
• Exchange-correlation functional

Vxc[n] calculated
with QMC methods
(Ceperley & Alder)
Local density approx.
VxcLDA [n]
d 3r n(r )  xc (n(r ))
(LDA)
where εxc[n] is the xc-energy (per particle) for free electron gas with local density n(r).
For example,
Generalized
gradient approx.
(GGA)
1/3
3 e2 kF
3 e2
 x ( n)  

3 2 n(r ) 

4 
4
VxcLDA [n]
3
d
 r n(r )  xc n(r ), n(r )
No approx. yet
Kohn-Sham theory
e2
E[n]  T [n]   d r n(r )U (r ) 
2
3
3
3
d
rd
r'

n(r )n(r ')
VXC [n]
| r r '|
1. KS ansatz: Parametrize the particle density in terms of a set of
one-electron orbitals representing a non-interacting reference system
n(r )   i (r )
2
i
2. Calculate non-interacting kinetic energy in terms of the i ' s
2


T [n]  T0 [n]    d r  (r )  
 2  i (r )
i
 2m 
if you don't like this approximation, then keep
3
*
i
T [n]  T0 [n]  T [n]  T0 [n]
3. Determine the optimal one-electron orbitals using the variational
method under the constraint

i  j   ij

  E[n]   i  i i  1  0

i

From J. Hafner’s slides
not valid for
excited states
Kohn-Sham equation
→
 2 2
n(r ')  Vtxc  n  
2
3
  U (r )  e  d r '


 i (r )  ii (r )
2
m
|
r

r
'
|

n
(
r
)


where Vtxc  n   Vxc  n   T [n]  T0 [n]
• Similar in form to the Hartree equation, and much simpler than HF eq.
However, here everything is exact, except the Vtxc term. (exact but unknown)
• Neither KS eigenvalues λi , nor eigenstates, have accurate physical meaning.
• However,
N
n(r )   i (r )
2
 the density is physical
i 1
Also, the highest occupied λi relative the vacuum is the ionization energy.
• If one approximates T～T0, and use LDA,
then
 Vtxc  n(r )
  xc  n(r )
 n( r )
VxcLDA [n]
3
d
 r n(r )  xc  n(r )
 2 2

n(r ')
  U (r )  e2  d 3r '
  xc  n(r )i (r )  

i i (r )
2
m
|
r

r
'
|


Self-consistent Kohn-Sham equation, an ab initio theory
1. choose initial i 
Parameter
free

2. construct n( r )   | i ( r ) |  VKS ( r )  e 2  d 3r '
2
i
n(r ')
  xc  n( r ) 
| r r '|
(for LDA)

2


3. solve  
 2  U (r )  VKS (r )   'i (r )  i 'i (r )
 2m


4. construct n '( r )   |  'i ( r ) |
2
i

5. if n '(r )  n( r )   , then STOP.
else let i ( r )   'i ( r ), GOTO 2
• Total energy
e2
E   i 
2
i
3
3
d
rd
r'

n(r )n(r ')
| r r '|
Double-counting correction.
Recall similar correction in HFA.
Strength of DFT
From W.Aulbur’s slides
Weakness of DFT
• Band-gap problem: HKS theorem not valid for excited states.
Band-gaps in semiconductors and insulators are usually underestimated.
LDA DFT Calcs.
(dashed lines)
Metal in “LDA”
calculations
Can be fixed by
GWA…etc
• Neglect of strong correlations
Rohlfing and Louie PRB 1993
- Exchange-splitting underestimated for narrow d- and f-bands.
- Many transition-metal compounds are Mott-Hubbard or charge-transfer
insulators, but DFT predicts metallic state.
LDA, GGAs, etc. fail in many cases with strong correlations.
From J. Hafner’s slides