My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX A. THE QUASI-2D EWALD SUM
Finally, the third charge distribution is
ρ 3 (r) = 1
A √ πσ e−z2 /σ 2 .
(A.15)
Because this function only depends on z, Poisson’s equation reduces to a onedimensional
problem; the appropriate Green’s function is
G(z, z ′ ) = 1 2 |z − z′ |.
The potential is therefore given by
∫ ∞
( )( )
1 1
φ 3 (r) = −4π
−∞ 2 |z − z′ |
A √ /σ 2
πσ e−z′2 dz ′
= − 2π ( ( z
)
z erf + σ )
√ e −z2 /σ 2 .
A σ π
(A.16)
(A.17)
Combining the three previous results gives the following expression for the potential
due to a charge at r = 0 and the corresponding images:
v E (r) = ∑ ( )
1 |r − R|
|r − R| erfc − 2π σ A
R
+ ∑ [ (
π
σk
e −kz erfc
kA
2 − z )
σ
k
[ ( z
)
z erf + σ ]
√ e −z2 /σ 2
σ π
( σk
+ e kz erfc
2 + z )]
e ik·r ‖
.
σ
(A.18)
The self-interaction energy is the energy associated with the interaction between
this charge and its images:
ξ = lim
(v E (r) − 1 )
r→0 r
( 1
( r
)
= lim
r→0 r erfc − 1 )
+ ∑ σ r
R≠0
= − 2
σ √ π + ∑ ( )
1 R
R erfc σ
R≠0
A.2 Expansion
( )
1 R
R erfc − 2σ√ π
σ A
+ ∑ k≠0
− 2σ√ π
A
+ ∑ k≠0
(
2π σk
kA erfc 2
The function appearing in the exchange-correlation energy (U EW
XC
)
.
( )
2π σk
kA erfc 2
(A.19)
in equation (4.24))
is v E (r) − ξ. The exchange-correlation hole is normally short-ranged; the extent to
187