In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows:
H
^
D
=
H
^
i
D
+
H
^
v
D
+
C
H
^
i
D
=
∑
i
c
o
r
e
ϵ
i
E
i
i
+
∑
r
v
i
r
t
ϵ
r
E
r
r
H
^
v
D
=
∑
a
b
a
c
t
h
a
b
e
f
f
E
a
b
+
1
2
∑
a
b
c
d
a
c
t
⟨
a
b
|
c
d
⟩
(
E
a
c
E
b
d
−
δ
b
c
E
a
d
)
C
=
2
∑
i
c
o
r
e
h
i
i
+
∑
i
j
c
o
r
e
(
2
⟨
i
j
|
i
j
⟩
−
⟨
i
j
|
j
i
⟩
)
−
2
∑
i
c
o
r
e
ϵ
i
h
a
b
e
f
f
=
h
a
b
+
∑
j
(
2
⟨
a
j
|
b
j
⟩
−
⟨
a
j
|
j
b
⟩
)
where labels
i
,
j
,
…
,
a
,
b
,
…
,
r
,
s
,
…
denote core, active and virtual orbitals (see Complete active space) respectively,
ϵ
i
and
ϵ
r
are the orbital energies of the involved orbitals, and
E
m
n
operators are the spin-traced operators
a
m
α
†
a
n
α
+
a
m
β
†
a
n
β
. These operators commute with
S
2
and
S
z
, therefore the application of these operators on a spin-pure function produces again a spin-pure function.
The Dyall Hamiltonian behaves like the true Hamiltonian inside the CAS space, having the same eigenvalues and eigenvectors of the true Hamiltonian projected onto the CAS space.
