References,
Aschroft : Solid State Phy. Ch.17
Kaxiras, Ch. 2
Atomic Hatree Unit System
Density
을 구하는 것이 중요하
다!
Manybody Schrődinger Equation
Slater Determinant
Hatree-Fock Non-local, Non
local)
Exchange interaction is only for the same-spin electrons
1
Ex     i |  j
2 i j

 i *(r ) j *(r ) j (r ) i (r )
r  r`
d 3 r d 3 r`
1
1
    i j  j i    i j  j i
2 i j
2 i j
( spin up )

( spin down)
N
 j *(r ) i (r )d 3 r `
j 1
r  r`
V x  i (r )    i |  j  j (r ) 

N

j 1
same spin
 j (r ) 
 j *(r ) i (r )d 3 r `
r  r`
• Hatree-Fock is non-local
• Let us approximate the exchange
potential with a purely local function
DFT방정식(Kohn-Sham equation)

 
 


 (r ) 3 
2
  Vatom (r )     d r   Vxc  (r )    (r )

| r  r |
 2m

2
Exchange Potential function
Here it is local function

1 ,  1 (r )

 2 ,  2 (r )

 3 ,  3 (r )

 n ,  n (r )

 n1 ,  n1 (r )

 n  2 ,  n  2 (r )
n

 2
 (r )   | i (r ) |
i 1
 2 3
 | i (r ) | d r  1
3

(
r
)
d
rN

비선형방정식과 SCF 과정

 
 


 (r ) 3 
2
  Vatom (r )     d r   Vxc  i (r )   i  i (r )

| r  r |
 2m

2
Ĥ
n


* 


 (r )   i (r ) i (r )
i 1



① Ĥ operator 가 방정식의 해  1 (r ),  2 (r ),   n (r )
를 내포하고 있는 비선형 방정식.
② 한번 만에 풀 수가 없고 SCF 과정이 필요하다.
• How to find a reasonable local function,
that is a functional of electron density ?
• The most primitive version is “SlaterDirac’s exchange energy” for spinunpolarized
Exchange Energy of the
uniform spin-unpolarized electron gas
 Salter and Diract (1928)
 E    ( r ) [  ]d r





3 3 
1/ 3
        e   (r )  
4 


SD
x
3
x
1
3
;
Bloch & Dirac (1920 ~ 1930)
2
x
Filatov & Thiel , Molecular Physics, 91, 847(1997)
Dirac, Comb. Phil. Soc. Math. Phys. Sci, 26, 376(1930)
DFT: X-α (spin-unpolarized)
 2 2
1 
e2  (r ) 3
2
  Vatom (r )  
d r   1.475 e (  (r )) 3  i (r )   i i (r )

r  r
 2m

n
 (r )    i (r )
i 1
2
Spin-polarized electron density
① Formula for the spin-compensated (      
 E    ( r ) [  ]d r





3 3 
1/ 3
        e   (r )  
4 


0
x
1
2
)
3
x
;
1
3
Bloch & Dirac (1920 ~ 1930)
2
x
② For spin  polarized cases
1
Ex
Dirac  LSD
4
4
3 6  3 2
3
3
[   ,   ]  E x [2   ]  E x [2   ]     e  [      3 ] d r
2
2
4 
1
0
1
0
Filatov & Thiel , Molecular Physics, 91, 847(1997)
Dirac, Comb. Phil. Soc. Math. Phys. Sci, 26, 376(1930)
You please understand
Ex
Dirac
[  ,  ] 
1
2
E x [2   ] 
0
1
2
E x [2   ]
0
Ex [  ,  ]  ( Exchange between spin  up orbitals ) 
( Exchange between spin  down)
ε
ε


1

2


E[   ,   ] 
ε
1
2
Ex 0 [2  ] 
up  spin 전자숫자
만큼으로 구성된
Spi n- compens at ed
cas e
1
2
Ex 0 [2  ]
down  spin 전자숫자
만큼으로 구성된
Spi n- compens at ed
cas e
1
2

Density-Functional Theory : X-α method
X
E XC
 E X  EC   E XDirac [ ( r ) ]
 Spin  compensated Case 
1
E
X
3 3  3 2
    e 
4 
1
V X
4
3
d3r
1
2
1
 3  2 13
3 3 e
3
    e   2  
(a0  (r )) 3
 
   2a0
3
1
3
3 3
   , 2    2.95 
2
 
V
X
1
1
 e2 
3
3 2
 2.95 
a

(
r
)


1.475(

(
r
))
e

 0
 2a0 
 Ashcroft , page 337 
다시 …Hartree-Fock
Slater Determinant
1
 Hˆ  
A B C 
N!
Hˆ A  B  C 
N!개의 항
N
1
1
ˆ
ˆ
H   h(ri )  
2 i j ri  rj
i 1
1-body operator
2-body operator
One body operator ;
One body operator
Particle index => state index
Indirect term 의 모든 합은 0 이다.
One body term
=>
= 0
two body operator
two body operator
Indirect term
1
  (r1 )  (r2 )  (r3 )  (r4 )
  (r2 )  (r1 )  ( r3 )  ( r4 )
r1  r2
       
  (r1 )  (r2 )  (r3 )  (r4 )

{i , j }
1
  (r2 )  (r1 )  ( r3 )  ( r4 )
ri  rj
        
{ ,  }
1 N N
        
2  1  1
Every distinct pair
Hatree-Fock
Hatree-Fock; Coulomb 항
Hatree-Fock; exchange 항
총에너지 ? 전자구름의 기저상태 에
너지
Hatree-Fock (분자의 총에너지)
Unrestricted Hatree-Fock
Unrestricted singlet frequently collapse to the corresponding restricted singlets.
Every pairs of the same spins
For example three electrons
C
B
A
“같은 spin” 하고 만 exchange interaction 한다.
zero:
No! exchange potential!