16.1 Equations-of-Motion (EOM) Method for N ± 1 Electrons
243
where ω n = E
N −1
− E 0 denotes the excitation (ionization) energy of the cationic
energy eigenstate ˆ
n | 0 . In a related way, the killer condition can be written in the
form
0 |[ ˆ
H , ˆ
n ] = 0
(16.8)
Now we “multiply” (form a scalar product) the first equation from the left with
0 | ˆ
O
†
I and the second equation from the right with ˆ
O
†
I | 0 ,
0 | ˆ
O
†
I [ ˆ
H , ˆ
n ]| 0 = ω n 0 | ˆ
O
†
I
ˆ
n | 0
(16.9)
0 |[ ˆ
H , ˆ
n ] ˆ
O
†
I | 0 = 0
(16.10)
These two equations can be added to give
0 |{ ˆ
O
†
I , [ ˆ
H , ˆ
n ]}| 0 = ω n 0 |{ ˆ
O
†
I , ˆ
n }| 0
(16.11)
where { ˆ
A, ˆ
B} = ˆ
A ˆ
B + ˆ
B ˆ
A denotes the anticommutator of the operators ˆ
A, ˆ
B. Note
that the killer condition has been used on the right-hand side to replace 0 | ˆ
O
†
I
ˆ
n | 0
with 0 |{ ˆ
O
†
I , ˆ
n }| 0 . Inserting the expansion (16.3) for ˆ
n yields the secular
equations,
J
(M I J − ω n S I J )X J n = 0
(16.12)
where
M I J = = 0 |{ ˆ
O
†
I , [ ˆ
H , ˆ
O J ]}| 0
(16.13)
S I J = = 0 |{ ˆ
O
†
I , ˆ
O J }| 0
(16.14)
are the elements of the EOM secular matrix M and the metric (or overlap) matrix S,
respectively. Using matrix notation, Eq. (16.12) takes on the form
MX = SX
(16.15)
The normalization of the eigenvectors is obtained according to
X
† SX = 1
(16.16)
which follows from the orthonormalization of the final states,
0 |{ ˆ
†
n , ˆ
m }| 0 = δ nm
(16.17)
invoking here again the killer condition.
In the EOM secular equations, the operators ˆ
O I and ˆ
O
†
I are treated on an equal
footing, and since the latter generate (N +1)-electron states when acting on | 0 , one