10
and a long range part using the error function and can be written as:
1
r
= erfc(!r)
r
+ erf(!r)
r
. (2.11)
The above expression shows how the splitting of the exchange interaction is
achieved. The first term on the right hand side denotes the short range (SR)
exact exchange whereas the second terms denotes the long range (LR) exchange
interaction. The strength of the screening is decided by the value
of the parameter !. The final expression for the exchange energy after the
splitting can be written as:
x = 0.25EHF,SR
EHSE
x (!) + 0.75EPBE,SR
x (!) + 0.25EHF,LR
x (!)
x (!) − 0.25EPBE,LR
+EPBE,LR
x (!). (2.12)
It turns out that for a range of ! values pertinent for real physical systems, the
EHF,LR
x (!) term cancels the −EPBE,LR
x (!) term. Thus the reduced equation
for exchange-correlation energy is:
EHSE
x (!) + 0.75EPBE,SR
xc = 0.25EHF,SR
x (!) + EPBE
x (!) + EPBE,LR
c . (2.13)
From the above equation we can see that the exchange energy has two parts,
one is screened HF exchange and the other is screened GGA exchange. The
expression for screened exact exchange in the plane-wave basis can be written
as 18:
Vk(G,G’) = hk + G| ˆ Vx|k + G’i
= −
4e2
X
mq
2wqfqm
×
X
G”
Cqm(G’ - G”)Cqm(G - G”)
|k -q + G”|2
×(1 − e|k -q + G”|2/4!2 ). (2.14)
In the above equation we can see that the exchange term does not have a
singularity at |k -q + G”| = 0. In the HSE06 functional the optimized value
of the parameter ! is 0.11 a−1
0 (where a0 is the Bohr radius). The current
implementation of HSE in GPAW is non self-consistent in which the GGA and
HF exchange interactions are calculated with PBE calculated ground state
density and wavefunctions.