CASINO manual - Theory of Condensed Matter

19.4.2 2D Ewald interaction
Infinite Coulomb sums in systems which are periodic in two dimensions (the xy-plane, according to
casino) are performed using the standard 2D Ewald method originally developed by Parry [39]. One
way to derive the relevant formula is to take the infinite separation limit of the 3D sum for a periodic
stack of finite-width slabs. casino uses this algorithm when treating two-dimensional slabs of atoms
with local Gaussian basis sets (useful in modelling surfaces) and also when treating 2D electron and
electron–hole phases (either as strict 2D planes, 2D slabs with finite thickness, or strict 2D bilayers). In
the case of periodic arrays of slabs separated by a finite vacuum gap (necessary when using plane-wave
basis sets), the regular 3D algorithm is used.
Consider a finite width slab with a charge density periodic in two dimensions consisting of a unit point
charge at r j in every simulation cell plus a uniform cancelling background.
ρ j (r) = ∑ (
δ(r − r j − R) − 1 )
, (120)
A
R
where R now denotes the 2D lattice translation vectors in the xy-plane, and A is the area of the
simulation cell in that plane. The Parry formula for the periodic potential corresponding to this
charge density is
vE 2D (r, r j ) = ∑ erfc ( γ 1 2 |r − (r j + R)| )
|r − (r j + R)|
R
+ π A
−
π A
∑
G≠0
exp(iG · (r − r j ))
G
[
erf(zγ 1 2 )z +
exp(−γz 2 )
(γπ) 1 2
[
( G
exp(zG)erfc
2γ 1 2
+ zγ 1 2
)
( G
+ exp(−zG)erfc
2γ 1 2
− zγ 1 2
]
, (121)
where G denotes the reciprocal lattice translation vectors in the xy-plane, and z is the z-component
of the r − r j vector. In two dimensions casino sets the screening parameter γ to (2.4/A 1/2 ) 2 which
again should approximately minimize the cost over different Bravais lattices.
The full periodic potential of the simulation cell is obtained by following a procedure analogous to
that described for the 3D case, with the self term ξ given by
(
)
1
ξ = lim v E (r, r i ) −
(122)
r→ri |r − r i |
( )
( )
erfc γ 1 G
2 R
∑ erfc
= ∑ R≠0
R
− 2γ 1 2
π 1 2
+ 2π A
G≠0
G
2γ 1 2
− π 1 2
Aγ 1 2
)]
. (123)
The first derivatives of the 2D Ewald potential, required for the evaluation of the core-polarization
energy in 2D slabs, are different in directions parallel and perpendicular to the plane of the slab. The
x and y derivatives are given by
∂v 2D
E (r, r j)
∂λ
= − ∑ [ ( 1
(r − (r j + R)) µ erfc γ 2 |r − (r j + R)| )
]
|r − (r j + R)| 2 + 2γ 1 2
exp(−γ|r − (r
|r − (r j + R)| π 1 j + R)| 2 ) (124)
2
R
−
π ∑
[
( )
( )]
G µ sin(G · (r − r j ))
G
exp(zG)erfc + zγ 1 G
A
G
2γ 1 2 + exp(−zG)erfc − zγ 1
2
2γ 1 2 .
2
G≠0
where λ = x or y, and the z derivative is given by
∂vE 2D(r,
r j)
= − ∑ [ ( 1
z erfc γ 2 |r − (r j + R)| )
]
∂z
|r − (r j + R)| 2 + 2γ 1 2
exp(−γ|r − (r
|r − (r j + R)| π 1 j + R)| 2 )
2
R
+ π ∑
[
( )
( )]
G
cos(G · (r − r j )) exp(zG)erfc + zγ 1 G
2 + exp(−zG)erfc − zγ 1 2
A
G≠0
2γ 1 2
− 2π A erf(zγ 1 2 ) . (125)
Note finally that in things such as 2D bilayer systems (electrons in one layer, holes in the other, say)
there is an additional ‘capacitor term’ due to interaction of the backgrounds.
2γ 1 2
142