Chemistry Reference
In-Depth Information
The homonuclear system, such as N
2
, has an inversion symmetry,
gerade
and
ungerade
. This does not mean that the symmetry-adapted 1
σ
g
and 1
σ
u
MOs,
φ
g
=
)
1
/
2
, are
indispensable to describe the total wave function for the core electrons by using
a normalized Slater determinant,
0
=
φ
g
φ
u
)
1
/
2
, and
φ
u
=
(
φ
L
+
φ
R
)
/
(2
+
2
φ
L
|
φ
R
(
φ
L
−
φ
R
)
/
(2
−
2
φ
L
|
φ
R
, where
φ
R
is a localized (right)
atomic 1s orbital. We can also describe it by
local
0
=
φ
L
φ
R
not only for the
φ
L
φ
R
heteronuclear system, but also for the homonuclear one, where
is trans-
φ
g
φ
u
formed to
with no change in energy. The situation is changed in the core-
hole state. There are two core-hole configurations
L
=
φ
−
L
0
=
φ
L
φ
R
and
R
=
φ
−
R
0
=
φ
L
φ
R
. In the homonuclear system, they are degenerate and
ought to satisfy the inversion symmetry based on multi-configuration description
[81, 83, 84, 86-88] as follows:
local
g
)
1
/
2
=
(
L
+
R
)
/
(2
+
2
L
|
R
local
u
)
1
/
2
=
(
L
−
R
)
/
(2
−
2
L
|
R
This description based on the core-electron localization is essential to avoid a
misleading discussion based on unphysical core-hole delocalization in description
with the “bonding” 1s
σ
g
and “antibonding” 1s
σ
u
orbitals,
φ
−
g
0
=
φ
g
φ
u
and
φ
−
u
0
=
φ
g
φ
u
. Upon the core-hole creation, the valence electron is greatly
rearranged to stabilize the localized hole through the core-hole screening by in-
tramolecular charge transfer (CT); therefore, the valence reorganization (relax-
ation) in the symmetry-adapted MO picture (e.g.,
φ
−
g
0
and
φ
−
u
0
) is less suffi-
cient than in the broken-symmetry MO picture (e.g.,
φ
−
L
0
and
φ
−
R
0
) [79]. The
intramolecular CT or core-hole screening effect is efficiently taken into account
within the broken-symmetry approximation. On the other hand, on the level of
the symmetry-adapted orbital, the CT effect is essentially taken into account by
adding symmetry-breaking type configurations, such as
φ
−
g
0
(
gerade
) coupled
with [(
π
u
)
−
1
(
π
g
)
+
1
](
ungerade
) shakeup-like configurations, to a main configu-
ration,
φ
−
u
0
(
ungerade
) [62, 81, 86-91]. This configuration-mixing mechanism
explains observable double and triple excitations as already discussed.
The energy splitting
gu
between
gerade
and
ungerade
core-hole states
E
(
local
g
) and
E
(
local
u
) is predicted to be
∼
0.10 eV in N
2
and C
2
H
2
. The order
estimate of
gu
is given by
gu
=
E
(
local
−
E
(
local
≈
L
|
H
|
L
L
|
R
)
)
2(
g
u
∼
O
(S
cv
)
−
L
|
H
|
R
)
where S
cv
is the interatomic core-valence overlap [84]. Thus, the
gu
of other
symmetric molecules can be smaller than in N
2
and C
2
H
2
, considering that the
