TOMBO Ver.2 Manual

1.3 All-electron charge density and potential 6
the usual PW expansion method:
⟨k + G|H|k + G ′ ⟩ = ¯h2 (k + G) 2
δ
2m GG ′ +Ṽ (G − G ′ ). (1.12)
PW-AO matrix elements can be accurately calculated by 1D integration of the product of
the spherical Bessel function j l ′(Gr) and the radial AO function R AO
jnl (r):
⟨k + G|H|ϕ Bloch e−iG·R ∫
jnlm ⟩ = j
√ Y lm ( k + ˆ G)
Ω cell
[ ¯h 2 (k + G) 2 ]
j l (|k + G|r)
+V (r)
2m
× R jnl (r)r 2 dr. (1.13)
AO-AO matrix elements can be accurately calculated by 1D integration of the product of
two radial AO functions:
∫ [
⟨ϕ Bloch Bloch
jnlm |H|ϕ j ′ n ′ l ′ m ′⟩ = δ j j ′ ϕ AO
jnlm (r) − ¯h2
cell 2m ∇2 +V (r)
]
ϕ A0
jn ′ l ′ m ′(r)d3 r. (1.14)
The overlap matrix elements are given by those of (1.13) and (1.14) without [...] inside the
integrands.
1.3 All-electron charge density and potential
In the all-electron mixed basis approach, all-electron charge density ρ(r) is made of
three contributions: PW-PW, AO-PW,and AO-AO.
ρ(r) = ρ PW−PW (r) +∑ρ AO−PW
j
j (r) +∑
j
ρ AO−AO
j (r). (1.15)
In the all-electron mixed basis approach, the charge density is made of the three contributions,
ρ AO−PW
j (r) = √ 2 occ
Ω
occ
j (r) = 2
ρ AO−AO
∑
v
ρ PW−PW (r) = 2 Ω
∑ ∑
v
nlm
∑ ∑
n ′ l ′ m ′ nlm
∑
G
occ
∑
v
∑∑
G
c PW∗
v
G ′
Here the prefactor 2 denotes the spin duplicity, ∑ occ
ν
(G ′ )c PW
v (G)e i(G−G′ )·r , (1.16)
c AO∗
v ( jnlm)c PW
v (G) × ϕ jnlm (r − R j )e i(G)·r + c.c., (1.17)
c AO∗
v ( jn ′ l ′ m ′ )c AO
v ( jnlm) × ϕ jn ′ l ′ m ′(r − R j)ϕ jnlm (r − R j ). (1.18)
means the sum over all occupied states