Chemistry Reference
In-Depth Information
1
ux
states through the configuration mixing or
excitation CSFs describing the
interaction are as follows:
(1
1
ux
)
S
(1
σ
u
,
x
∗
)
=
(2
1
ux
)
S
(1
σ
g
,
y
∗
)
S
(y
,
x
∗
)
S
(1
σ
g
,
x
∗
)
S
(x
,
x
∗
)
=
−
(3
1
ux
)
S
(1
σ
g
,
y
∗
)
S
(x
,
y
∗
)
=
(4
1
ux
)
S
(1
σ
g
,
x
∗
)
S
(y
,
y
∗
)
S
(x
,
x
∗
)
=
{
−
}
(5
1
ux
)
S
(1
σ
u
,
x
∗
)(y
∗
)
2
(3
σ
g
)
0
=
for the shorter distance
S
(1
σ
u
,
x
∗
)
S
(y
,
x
∗
)
S
(x
,
y
∗
)
=
for the longer distance
where x, y, x
∗
, and y
∗
denote the 1
π
ux
,1
π
uy
,1
π
gx
,1
π
gy
orbitals, respectively, and
the
1
ux
states are degenerate with the
1
uy
states. Now,
(2
1
ux
)to
(4
1
ux
)
can be regarded as singlet-type
π
u
→
π
g
shake-up excitation from
(
1
g
)
=
S
(1
σ
g
,π
g
), but we have to note that
S
(a
,
b)
|
T
(a
,
b)
=
ab[
αβ
+
βα
]
=|
(a
↑
)(b
↓
)
|+|
(a
↓
)(b
↑
)
|=|
(a
↑
)(b
↓
)
|−|
(b
↑
)(a
↓
)
|
S
(a
,
c)
S
(b
,
c)
=
ab[
αβ
−
βα
]
=|
(a
↑
)(b
↓
)
|−|
(a
↓
)(b
↑
)
|=|
(a
↑
)(b
↓
)
|+|
(b
↑
)(a
↓
)
T
(a
,
c)
T
(b
,
c)
S
(a
,
b)(c)
2
=−
=−
where the superscripts T and S denote triplet and singlet spin couplings of
α
(
↑
)
and
β
(
↓
) spins, respectively, and the normalization factor is neglected here. That
is,
(2
1
ux
)to
(4
1
ux
) contain triplet-type
π
u
→
π
g
shake-up components
from
(
3
g
)
T
(1
σ
g
,π
g
). Here,
(2
1
u
)to
(5
1
u
) are dipole forbidden,
but get intensity through configuration mixing with the strong 1s
=
−
π
∗
resonance
state
(1
1
u
).
On the other hand, the two double excitations for the
1
u
states using the outer-
valence MOs (2
σ
u
,3
σ
g
, and 1
π
u
) into the lowest unoccupied 1
π
g
(
π
∗
) orbital are
described by using the following CSFs:
(1
1
u
)
S
(1
σ
u
,
3
σ
g
)[(x
∗
)
2
(y
∗
)
2
]
=
+
(2
1
u
)
S
(1
σ
g
,
2
σ
u
)[(x
∗
)
2
(y
∗
)
2
]
=
+
where these configurations are strongly mixed with each other and can get intensity
through configuration mixing with the dominant valence component in the
σ
∗
shape
resonance,
(
1
u
)
S
(1
σ
g
,
3
σ
u
).
=
