Page 5 - 71 the abc of dft_opt

CONTENTS 9 10 CONTENTS
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Definitions and notation Density matrix: γ(x, x ) = N dx 2 . . . dx N Ψ (x, x 2 , . . . , x N )Ψ(x , x 2 , . . . , x N )
∗
#
#
Coordinates properties: γ(x, x) = n(x), γ(x , x) = γ(x, x )
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#
2
3
Position vector: r = (x, y, z), r = |r|. kinetic energy: T = − 1 2 ! d r∇ γ(r, r )| r=r !
#
Spin index: σ =↑ or ↓ = α or β. N σ
#
# φ (r)φ iσ (r )
∗
"
Kohn-Sham: γ S (x, x ) = δ σσ !
Space-spin vector: x = (r, σ). i=1 iσ 2
3
!
!
#
#
Sums: ! dx = " ! d r Pair Density: P(x, x ) = N(N − 1) dx 3 . . . dx N |Ψ(x, x , x 3 , . . . , x N )|
σ ! # # # # #
Operators properties: dx P(x, x ) = (N − 1)n(x), P(x , x) = P(x, x ), P(x, x ) ≥ 0
N 1 ! 3 ! 3 # # #
ˆ 1 " 2 potential energy:V ee = 2 d r d r P(r, r )/|r − r |
Kinetic energy: T = − 2 ∇ .
i
# 2
i Kohn-Sham: P X (x, x ) = n(x)n(x ) − |γ S (x, x )|
#
#
ˆ ˆ ˆ
Potential energy: V = V ee + V ext Exchange-correlation hole around r at coupling constant λ:
ˆ
Coulomb repulsion: V ee = 1 " 1/|r i − r j |. P (r, r + u) = n(r) [n(r + u) + n (r, r + u)]
λ
λ
2
i!=j XC
# 2
Kohn-Sham: n X (x, x ) = −|γ S (x, x )| /n(x)
#
ˆ
N "
External potential: V ext = v ext (r i ) ! 3 ! 3 λ
i=1 properties: n X (r, r + u) ≤ 0, d u n X (r, r + u) = −1, d u n (r, r + u) = 0
C
Wavefunctions Pair correlation function: g (x, x ) = P (x, x )/(n(x)n(x ))
λ
λ
#
#
#
ˆ ˆ
Physical wavefunction: Ψ[n](x 1 ...x N ) has density n and minimizes T + V ee Electron-electron cusp condition: dg (r, u)/du| u=0 = λg (r, u = 0)
λ
λ
Kohn-Sham: Φ[n](x 1 ...x N ) has density n and minizes T ˆ Uniform coordinate scaling
p
Φ(x 1 ...x N ) = " p (−1) φ 1 (x p 1 )...φ N (x p N ) Density: n(r) → n γ (r) = γ n(γr)
3
p
Φ(x 1 ...x N ) = " p (−1) φ 1 (x p 1 )...φ N (x p N ) Wavefunction: Ψ γ (r 1 . . . r N ) = γ 3N/2 Ψ(γr 1 . . . γr N )
where φ i (x) and & i are the i-th KS orbital and energy, with i = α, σ.
Ground states: Φ[n γ ] = Φ γ [n], but Ψ[n γ ] *= Ψ γ [n]
Energies 2
ˆ ˆ ˆ Fundamental inequality: F[n γ ] ≤ γ T[n] + γV ee [n]
Universal functional: F[n] = min%Ψ|T + V ee |Ψ& = %Ψ[n]|T|Ψ[n]& 2
Ψ→n Non-interacting kinetic energy: T S [n γ ] = γ T S [n]
ˆ
Kinetic energy: T[n] = %Ψ[n]|T|Ψ[n]&. Exchange and Hartree energies: E X [n γ ] = γE X [n], U[n γ ] = γU[n]
2
3
ˆ
ˆ
Non-interacting kinetic energy: T s [n] = min%Φ|T|Φ& = %Φ[n]|T|Φ[n]& = N " ! d r|∇φ i (r)| 2 Kinetic and potential: T[n γ ] < γ T[n], V ee [n γ ] > γV ee [n] (γ > 1)
2
Φ→n i=1 Correlation energies: E C [n γ ] > γE C [n], T C [n γ ] < γ T C [n] (γ > 1)
ˆ
Coulomb repulsion energy: V ee [n] = %Ψ[n]|V ee |Ψ[n]&. ˆ
3
Hartree energy: U[n] = 1 2 ! d r ! d r n(r) n(r )/|r − r | Virial theorem: 2T = %r · ∇V & 3
#
3 #
#
!
1 " " ! 3 ! 3 # ∗ ∗ # # # N electrons: 2T + V ee = d r n(r) r · ∇v ext (r)
Exchange: E X = %Φ|V ee |Φ& − U = − d r d r φ (r) φ (r ) φ iσ (r ) φ jσ (r)/|r − r | ! 3
iσ
jσ
2 σ i,j XC: E XC + T C = d r n(r) r · ∇v XC (r)
occ
Kinetic-correlation: T C [n] = T[n] − T S [n] Spin scaling
1
Potential-correlation: U XC [n] = V ee [n] − U[n] Kinetic: T S [n ↑ , n ↓ ] = (T S [2n ↑ ] + T S [2n ↓ ])
2
ˆ ˆ 1
Exchange-correlation: E XC [n] = T[n] − T S [n] + V ee [n] − U[n] = %Ψ[n]|T + V ee |Ψ[n]& − Exchange: E X [n ↑ , n ↓ ] = (E X [2n ↑ ] + E X [2n ↓ ])
2
ˆ ˆ
%Φ[n]|T + V ee |Φ[n]& Adiabatic connection
λ
λ
λ
Potentials Hellmann-Feynman: E = E λ=0 + ! 0 1 dλ%Ψ |dH /dλ|Ψ &
3
ˆ
ˆ
!
λ
Functional derivative: F[n + δn] − F[n] = d r δn(r) δF[n]/δn(r) Wavefunction: Ψ [n] has density n and mininimizes T + λV ee ;
λ
Kohn-Sham: v S (r) = −δT S /δn(r) + µ relation to scaling: Ψ [n] = Ψ λ [n 1/λ ]
! λ 2
#
3 #
#
Hartree: v H (r) = δU/δn(r) = d r n(r )/|r − r | Energies: E [n] = λ E[n 1/λ ]
Exchange-correlation: v ext (r) = v S (r) + v H (r) + v XC (r) kinetic: T [n] = T S [n]
λ
S
λ
λ
Densities and density matrices exchange: E [n] = λE X [n], U [n] = λU[n]
X # $
(3)
(2)
2
2
λ
!
Spin density: n(x) = n σ (r) = N dx 2 . . . dx N |Ψ(x, x 2 . . . , x N )| = N " |φ i (x)| 2 correlation: E [n] = λ E C [n 1/λ ] = λ 2 E C [n] + λE S [n] + . . . for small λ
i=1 C
λ
!
properties: n(x) ≥ 0, dx n(x) = N ACF: E XC = ! 0 1 dλ U XC (λ), where U XC (λ) = U /λ
XC