- Source: Dyall Hamiltonian
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
{\displaystyle {\hat {H}}^{\rm {D}}={\hat {H}}_{i}^{\rm {D}}+{\hat {H}}_{v}^{\rm {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
{\displaystyle {\hat {H}}_{i}^{\rm {D}}=\sum _{i}^{\rm {core}}\varepsilon _{i}E_{ii}+\sum _{r}^{\rm {virt}}\varepsilon _{r}E_{rr}}
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
)
{\displaystyle {\hat {H}}_{v}^{\rm {D}}=\sum _{ab}^{\rm {act}}h_{ab}^{\rm {eff}}E_{ab}+{\frac {1}{2}}\sum _{abcd}^{\rm {act}}\left\langle ab\left.\right|cd\right\rangle \left(E_{ac}E_{bd}-\delta _{bc}E_{ad}\right)}
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
{\displaystyle C=2\sum _{i}^{\rm {core}}h_{ii}+\sum _{ij}^{\rm {core}}\left(2\left\langle ij\left.\right|ij\right\rangle -\left\langle ij\left.\right|ji\right\rangle \right)-2\sum _{i}^{\rm {core}}\varepsilon _{i}}
h
a
b
e
f
f
=
h
a
b
+
∑
j
(
2
⟨
a
j
|
b
j
⟩
−
⟨
a
j
|
j
b
⟩
)
{\displaystyle h_{ab}^{\rm {eff}}=h_{ab}+\sum _{j}\left(2\left\langle aj\left.\right|bj\right\rangle -\left\langle aj\left.\right|jb\right\rangle \right)}
where labels
i
,
j
,
…
{\displaystyle i,j,\ldots }
,
a
,
b
,
…
{\displaystyle a,b,\ldots }
,
r
,
s
,
…
{\displaystyle r,s,\ldots }
denote core, active and virtual orbitals (see Complete active space) respectively,
ε
i
{\displaystyle \varepsilon _{i}}
and
ε
r
{\displaystyle \varepsilon _{r}}
are the orbital energies of the involved orbitals, and
E
m
n
{\displaystyle E_{mn}}
operators are the spin-traced operators
a
m
α
†
a
n
α
+
a
m
β
†
a
n
β
{\displaystyle a_{m\alpha }^{\dagger }a_{n\alpha }+a_{m\beta }^{\dagger }a_{n\beta }}
. These operators commute with
S
2
{\displaystyle S^{2}}
and
S
z
{\displaystyle 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.