My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX A. THE QUASI-2D EWALD SUM
which v E (r) − ξ deviates from 1/r at small r may be used to estimate the finite-size
error associated with the Ewald interaction. The expansion of this function for small
r and large lattice parameter (L) follows.
Combining equations (A.19) and (A.18) gives the function to be expanded:
v E (r) − ξ = 2
σ √ π − ∑ ( )
1 R
R erfc + ∑ ( )
1 |r − R|
σ |r − R| erfc σ
R≠0
R
− 2π [ ( z
)
z erf + σ ( ) ]
√ e −z2 /σ 2 − 1
A σ π
+ ∑ {[ (
π
σk
e −kz erfc
kA
2 − z ) ( σk
+ e kz erfc
σ
2 + z )] (A.20)
cos k · r ‖
σ
k≠0
( )} σk
−2 erfc .
2
The first line of this expression reduces quickly to
1
r + 2r2
3σ 3√ π + O [ r 4] [
+ O e −L2 /σ 2] .
(A.21)
The second line is also simply expanded, giving
− 2√ πz 2
σA + O [ z 4] . (A.22)
The sum in k-space is slightly more involved. To begin, we note that
Applying this result,
( σk
e −kz erfc
2 − z )
σ
erfc(x 0 + x) = erfc(x 0 ) + 2 √ π
(x 2 x 0 − x)e −x2 0 + O
[
x
3 ] .
( σk
+ e kz erfc
2 + z )
= ( 2 + k 2 z 2) erfc
σ
( ) σk
− 2z2 k
σ √ π e−(σk/2)2 + O [ z 4] .
2
(A.23)
(A.24)
The error is of order z 4 rather than z 3 because any terms involving odd powers of z
must cancel out. The next step is to expand the cosine to O [r 2 ]; the k-space sum
of equation (A.20) then becomes
∑
[ πk
(z 2 − (k · r ) ( )
‖) 2 σk
erfc − 2√ ]
πz 2
A k 2 2 σA e−(σk/2)2 + O [ r 4] (A.25)
k≠0
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