Theor Chem Acc (2015) 134:86
1 3
This is a Hermitian Hamiltonian that does not contain
explicitly three- and four-center integrals any more. It
requires some further regrouping in order to present it as a
sum of terms that can be assigned to the individual atoms
and pairs of atoms. For that reason, we introduce the atomic
one-electron Hamiltonian ˆ
h A
and perform projections of all relevant quantities to atomic
subspaces in order to separate out effective atomic Hamiltonians. However, the error of projecting two-center quantities
on the one-center ones is not neglected but is assigned to the
diatomic terms of the Hamiltonian. In this manner we get
(31)
ˆ
h
A
= −
1
2
−
Z A
r A
.
where
and
(32)
ˆ
H =⇒
𚶀
A
ˆ
H A +
A ˆ
H AB ,
(33)
ˆ
H A =
1
2
ν,τ ∈A
μ
A
A
μτ h
A
τ ν ˆ
ϕ
+
μ ˆ
ϕ
−
ν + h
A
ντ A
A†
τ μ ˆ
ϕ
+
ν ˆ
ϕ
−
μ
+
1
4
η,κ,τ ,ρ∈A
μ,ν
A
A
μτ A
A
νη [τ η|κρ] ˆ
ϕ
+
μ ˆ
ϕ
+
ν ˆ
ϕ
−
ρ ˆ
ϕ
−
κ
+[κρ|τ η]A
A†
τ μ A
A†
ην ˆ
ϕ
+
κ ˆ
ϕ
+
ρ ˆ
ϕ
−
ν ˆ
ϕ
−
μ
(30)
ˆ
H =⇒
A
μ,ν∈A
h μν ˆ
ϕ +
μ ˆ
ϕ −
ν +
A,B
A =B
μ∈A
ν∈B
⎡
⎢
⎢
⎣ h AB
μν −
1
2
C
C =A,B
⎛
⎝
ρ∈BC
A BC
μρ ρ|
Z C
r C
|ν +
ρ∈AC
μ|
Z C
r C
|ρA AC†
ρν
⎞
⎠
⎤
⎥
⎥
⎦ ˆ
ϕ +
μ ˆ
ϕ −
ν
+
1
2
A
μ,ν,κ,ρ∈A
[μν|κρ] ˆ
ϕ +
μ ˆ
ϕ +
ν ˆ
ϕ −
ρ ˆ
ϕ −
κ +
1
4
A,B
A =B
⎡
⎢
⎢
⎢
⎣
κ,ρ∈A
μ,ν∈AB
(μ /
∈A)∨(ν /
∈A)
[μν|κρ] ˆ
ϕ +
μ ˆ
ϕ +
ν ˆ
ϕ −
ρ ˆ
ϕ −
κ + [ρκ|νμ] ˆ
ϕ +
κ ˆ
ϕ +
ρ ˆ
ϕ −
ν ˆ
ϕ −
μ
+
κ∈A
ρ∈B
μ,ν∈AB
[μν|κρ] ˆ
ϕ +
μ ˆ
ϕ +
ν ˆ
ϕ −
ρ ˆ
ϕ −
κ + [ρκ|νμ] ˆ
ϕ +
κ ˆ
ϕ +
ρ ˆ
ϕ −
ν ˆ
ϕ −
μ
+
κ,ρ,τ ,η∈A
μ∈B
ν
(ν /
∈AB)
A A
μτ A A
νη [τ η|κρ] ˆ
ϕ +
μ ˆ
ϕ +
ν ˆ
ϕ −
ρ ˆ
ϕ −
κ + [ρκ|ητ ]A A†
ην A A†
τ μ ˆ
ϕ +
κ ˆ
ϕ +
ρ ˆ
ϕ −
ν ˆ
ϕ −
μ
+
κ∈A
ρ∈B
τ ,η∈AB
μ,ν
(μ /
∈AB)∨(ν /
∈AB)
A AB
μτ A AB
νη [τ η|κρ] ˆ
ϕ +
μ ˆ
ϕ +
ν ˆ
ϕ −
ρ ˆ
ϕ −
κ + [ρκ|ητ ]A AB†
ην A AB†
τ μ ˆ
ϕ +
κ ˆ
ϕ +
ρ ˆ
ϕ −
ν ˆ
ϕ −
μ
⎤
⎥
⎥
⎥
⎦
(34)
ˆ
H AB =
Z A Z B
R AB
−
1
2
τ ∈AB
ν
⎡
⎣
μ∈A
A
AB
ντ τ |
Z B
r B
|μ ˆ
ϕ
+
ν ˆ
ϕ
−
μ + +μ|
Z B
r B
|τ A
AB†
τ ν ˆ
ϕ
+
μ ˆ
ϕ
−
ν
+
μ∈B
A
AB
ντ τ |
Z A
r A
|μ ˆ
ϕ
+
ν ˆ
ϕ
−
μ + +μ|
Z A
r A
|τ A
AB†
τ ν ˆ
ϕ
+
μ ˆ
ϕ
−
ν
⎤
⎦
+
1
2
κ∈A
ρ∈B
η,τ ∈AB
μ,ν
A
AB
μτ A
AB
νη [τ η|κρ] ˆ
ϕ
+
μ ˆ
ϕ
+
ν ˆ
ϕ
−
ρ ˆ
ϕ
−
κ + [κρ|τ η]A
AB†
τ μ A
AB†
ην ˆ
ϕ
+
κ ˆ
ϕ
+
ρ ˆ
ϕ
−
ν ˆ
ϕ
−
μ
+
1
2
ν∈A
μ∈B
h
A
μν −
τ ∈A
A
A
μτ h
A
τ ν + h
B
μν −
τ ∈B
h
B
μτ A
B†
τ ν
ˆ
ϕ
+
μ ˆ
ϕ
−
ν +
h
A
νμ −
τ ∈A
h
A
ντ A
A†
τ μ + h
B
νμ −
τ ∈B
A
B
ντ h
B
τ μ
ˆ
ϕ
+
ν ˆ
ϕ
−
μ
+
1
2
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
κ,ρ∈A
μ,ν∈AB
(μ /
∈A)∨(ν /
∈A)
⎡
⎣
⎛
⎝ [μν|κρ] −
η,τ ∈A
A
A
μτ A
A
νη [τ η|κρ]
⎞
⎠ ˆ
ϕ
+
μ ˆ
ϕ
+
ν ˆ
ϕ
−
ρ ˆ
ϕ
−
κ +
⎛
⎝ [κρ|μν] −
η,τ ∈A
[κρ|τ η]A
A†
τ μ A
A†
ην
⎞
⎠ ˆ
ϕ
+
κ ˆ
ϕ
+
ρ ˆ
ϕ
−
ν ˆ
ϕ
−
μ
⎤
⎦
+
κ,ρ∈B
μ,ν∈AB
(μ /
∈B)∨(ν /
∈B)
⎡
⎣
⎛
⎝ [μν|κρ] −
η,τ ∈B
A
B
μτ A
B
νη [τ η|κρ]
⎞
⎠ ˆ
ϕ
+
μ ˆ
ϕ
+
ν ˆ
ϕ
−
ρ ˆ
ϕ
−
κ +
⎛
⎝ [κρ|μν] −
η,τ ∈B
[κρ|τ η]A
B†
τ μ A
B†
ην
⎞
⎠ ˆ
ϕ
+
κ ˆ
ϕ
+
ρ ˆ
ϕ
−
ν ˆ
ϕ
−
μ
⎤
⎦
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
36
Reprinted from the journal