CASINO manual - Theory of Condensed Matter

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 )
, (111)
Ω
R
where R denotes the lattice translation vectors and Ω is the volume of the simulation cell. The Ewald
formula for the periodic potential corresponding to this charge density is
v E (r, r j ) = ∑ erfc ( γ 1 2 |r − (r j + R)| )
− π |r − (r j + R)| Ωγ
R
+ 4π ∑ exp ( −G 2 /4γ )
Ω
G 2 exp(iG · (r − r j )) , (112)
G≠0
where G denotes the reciprocal lattice translation vectors. The value of v E (r, r j ) is in principle
independent of the screening parameter γ and this also holds in practice provided enough vectors are
included in the sums. 23 The larger the value of γ the more rapidly convergent is the real space sum,
but the more slowly convergent is the reciprocal space sum. A compromise is required to minimize
the overall computational cost. In casino, this parameter is set to (2.8/Ω 1/3 ) 2 which approximately
minimizes the cost for a wide variety of Bravais lattices [38].
The full periodic potential of the simulation cell is obtained by adding the potentials of all the N
charges and their cancelling backgrounds (which sum to zero because the cell has no net charge),
v(r) =
N∑
q j v E (r, r j ) . (113)
j=1
The potential acting on the charge at r i is therefore
where
v(r i ) =
N∑
q j v E (r i , r j ) + q i ξ , (114)
j(≠i)
(
)
1
ξ = lim v E (r, r i ) −
r→ri |r − r i |
= ∑ R≠0
erfc(γ 1 2 R)
R
− 2γ 1 2
π 1 2
− π Ωγ + 4π Ω
∑
G≠0
(115)
exp(−G 2 /4γ)
G 2 (116)
is the potential acting on the charge at r i due to its own images and cancelling background. The full
Ewald potential energy appearing in the QMC Hamiltonian is therefore
U(r 1 , . . . , r N ) = 1 2
= 1 2
N∑
i=1 j=1
(j≠i)
N∑
i=1 j=1
(j≠i)
N∑
q i q j v E (r i , r j ) + ξ 2
N∑
qi 2 (117)
i=1
N∑
q i q j (v E (r i , r j ) − ξ) , (118)
where we have used the charge neutrality condition, q i = − ∑ j(≠i) q j. The interaction v E (r i , r j ) − ξ
approaches 1/|r i − r j | as r i → r j and is independent of the choice of the zero of potential.
The gradient of the 3D Ewald potential [Eq. (112)] is required for evaluation of the core-polarization
contribution to the total energy (Sec. 19.3). It is given by
∇v E (r, r j ) = − ∑ ( ( 1
r − (r j + R) 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 )
2
R
−
4π ∑
G exp ( −G 2 /4γ )
Ω
G 2 sin (G · (r − r j )) . (119)
G≠0
23 Increasing the input parameter ewald control will increase the number of reciprocal vectors included in the sum,
the effect of which is to increase the range of γ over which the energy is constant. The default value of γ should
normally lie in the middle of the constant energy region for the default number of reciprocal vectors, so playing with
ewald control is normally unnecessary.
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