Ab_init stress perturbation theory - Department of Physics and ...

DFPT for elastic and piezoelectric tensors
• Mixed 2 nd derivatives of the energy with respect to two perturbations are
needed.
– By the “2n+1” theorem, these only require one set of 1 st order wave functions,
occ
( λλ 1 2) ( λ 2) ( λ1) ( λ1) ( λ1) (0)
el
= ∑ ψα
( +
ext
+
Hxc0)
ψα
α
E T V H
occ
2
+ (0) (0)
∑ ψ + + α
α
2 ∂ ∂ 1 2
• Including atomic relaxation, we need
( λλ 1 2) ( λλ 1 2)
Hxc
( T Vext
) ψα
,
(0)
1
∂ E
λ λ
n
– Clamped-atom elastic tensor ------------
– Internal strain tensor -----------------------
– Interatomic force constants --------------
– Clamped-atom piezoelectric tensor ----
– Born effective charges ---------------------
2
∂ Eel
∂ηαβ∂
2
∂ Eel
∂ηαβ∂
∂
2
2
∂ Eel
∂ηαβ∂
∂
2
E
E
∂! τ′
∂!
τ
el i j
η
! τ
!
γδ
el
∂! τ
i∂E
j !
j
E!
j
Available
Available