Page 6 - 71 the abc of dft_opt

CONTENTS 11 12 CONTENTS
Finite systems
Kato’s cusp at nucleus: dn/dr| = −2Z α n(R α )
r=R α
% √ " √
β − 2Ir
Large r in Coulombic system: n(r) → Ar e , β = α Z α − N + 1/ 2I
Exchange potential: v X (r) → −1/r
4
Correlation potential: v C (r) → −α(N − 1)/2r , where α(N − 1) is the polarizability of the
N − 1 electron system.
3
2
Von-Weisacker: T VW [n] = d r |∇n| /(8n) (exact for N = 1, 2)
!
S
Exchange: E X = −U/N for N = 1, 2
Correlation: E C = 0 for N = 1
(2) (3)
High-density limit: E C [n γ ] = E C [n] + E C [n]/γ + . . . as γ → ∞
Low-density limit: E C [n γ ] = γB[n] + γ 3/2 C[n] + . . . as γ → 0.
Uniform gas and LSD
Measures of the local density Wigner-Seitz radius: r s (r) = (3/(4πn(r)) 1/3
2 1
Fermi wave vector: k F (r) = (3π n(r)) 3
%
Thomas-Fermi wavevector: k s (r) = 4k F (r)/π
Measure of the local spin-polarization:
Relative polarization: ζ(r) = (n ↑ (r) − n ↓ (r))/n(r)
2
Kinetic energy: t unif (n) = (3/10)k (n)n
F
S
exchange energy: e unif (n) = n& unif (n), where & unif (n) = (3k F (n)/4π),
X X X
correlation energy: & unif (r s ) → 0.0311 ln r s − 0.047 + 0.009r s ln r s − 0.017r s (r s → 0)
C
3
Thomas-Fermi: T S TF [n] = A S d r n 5/3 (r) where A s = 2.871.
!
3
LSD: E X LDA [n] = A X d r n 4/3 (r) where A X = −(3/4)(3/π) 1/3 = −0.738.
!
3
!
E LSD [n ↑ , n ↓ ] = d r n(r)& unif (r s (r), ζ(r)
C C
Gradient expansions '
3
2
&
!
Gradient expansion: A[n] = d r a(n(r)) + b(n(r)|∇n(r)| . . .
Gradient expansion approximation: A GEA [n] = A LDA [n] + ∆A GEA [n]
Reduced density gradient: s(r) = |∇n(r)|/(2k F (r)n(r)
Correlation gradient: t(r) = |∇n(r)|/(2k s (r)n(r)
Polarization enhancement: φ(ζ) = ((1 + ζ) 2/3 + (1 − ζ) 2/3 )/2
3
(
2
)
!
Kinetic energy: T S [n] = A S d r n 5/3 1 + 5s /27 or ∆T S GEA [n] = T VW [n]/9.
2
3
!
Exchange energy: E X [n] = A X d r n 4/3 ( 1 + 10s /81 )
2
3
2
High-density correlation energy: ∆E GEA = (2/3π ) d r n(r)φ(ζ(r))t (r)
!
C
3
Generalized gradient approximation: A GGA [n] = d r a(n, |∇n|)
!
3
Enhancement factor: E GGA = d r e unif (n(r)) F XC (r s (r), s(r))
!
XC X