16.3 EOM Treatment of N -Electron Excitations
251
where the first equation is a consequence of the KC relation (16.43). In the case of the
de-excitation solutions, n ∈ {−}, where ˆ
n =
X J n ˆ
O
†
J , the KC relation eliminates
the first term of the [ ˆ
†
n , ˆ
m ] commutator,
0 |[ ˆ
†
n , ˆ
m ]| 0 = −− 0 | ˆ
m ˆ
†
n | 0 = −δ nm , n, m ∈ {−}
(16.53)
which explains the minus sign in the “normalization” of the de-excitation eigenvectors.
The state representation of an excitation solution, n ∈ {+}, explicitly reads
| n =
I
X I n ˆ
O I | 0 =
I ∈{+}
X I n ˆ
C I | 0 +
I ∈{−}
X I n ˆ
C
†
I
| 0
(16.54)
The two terms in the second equation relate to excitation and de-excitation components, respectively, of the eigenvector X n ; note that in the second term the subscripts
I are related to I ∈ {−} as implied in (16.44), e.g., ka ≡ ak. In a similar way, the
KC relation (16.43) can be written as
0 =
I
X
∗
I n
ˆ
O
†
I | 0 =
I ∈{+}
X
∗
I n
ˆ
C
†
I | 0 +
I ∈{−}
X
∗
I n
ˆ
C I | 0
(16.55)
Transition moments (for excitation solutions) are obtained according to
T m = = m | ˆ
D| 0 = = 0 |[ ˆ
†
m , ˆ
D]| 0 =
I
X
∗
I m D I
(16.56)
where the use of the commutator is justified because of the KC relation (16.55); D I
are transition matrix elements for the basis states,
D I = = 0 |[ ˆ
O
†
I , ˆ
D]| 0
(16.57)
In a similar way, properties of excited states and, more general, transition moments
for different excited states can be obtained according to
T mn = = m | ˆ
D| n = = 0 |[ ˆ
†
m , ˆ
D ˆ
n ]| 0 =
I,J
X
∗
I m X J n D I J
(16.58)
Here the basis state matrix elements are given by
D I J = = 0 |[ ˆ
O
†
I , ˆ
D ˆ
O J ]| 0
(16.59)
The RPA and Other EOM Approximations
Approximation schemes to the (in principle) exact EOM equations can be devised
by truncating the excitation operator manifold and adopting finite PT expansions for