16.2 A State Representation of the EOM Secular Equations
249
which supposes a real-valued one-particle representation. Note that the operator ˆ
H
and the corresponding resolvent simplify if applied to an (N +1)- or (N −1)-electron
state, e.g., ˆ
Hc
†
q | 0 = (− ˆ
H + E 0 )c
†
q | 0 . The representation (16.39) means that the
EOM secular equations, which according to Eqs. (16.29), (16.35) represent secular
equations for ˆ
H and, in extension, the resolvent operator (ω + ˆ
H)
−1 , allow one to
determine the poles and residues of the electron propagator. Another way of relating
the EOM secular problem to algebraic propagator equations is the superoperator
formalism reviewed in Appendix A.8.
16.3 EOM Treatment of N-Electron Excitations
While the derivation of the EOM method for N -electron excitations is essentially
analogous to the (N ± 1)-case, there emerge some distinct features which need to
be addressed.
Excited states | n of the N -electron system are represented in the form of an
operator expansion
ˆ
n =
J
X J n ˆ
O J
(16.41)
acting on the exact ground state,
| n = ˆ
n | 0
(16.42)
In addition, there is the killer condition,
ˆ
†
n | 0 = 0
(16.43)
which here also implies that the excited state is orthogonal to the ground state,
n | 0 = 0.
The Manne–Dalgaard operator basis required for a unique solution of Eqs. (16.42),
(16.43),
{ ˆ
O J } =
c
†
a c k ; c
†
a c
†
b c k c l , a < b, k < l; . . .
∪
c
†
k c a ; c
†
l c
†
k c b c a , a < b, k < l; . . .
(16.44)
is formed by the set (14.10) of physical N -electron excitation operators, { ˆ
C I }, and the
set { ˆ
C
†
I } of their hermitian adjoints, also referred to as de-excitation operators. Like
the 1 p-1h, 2 p-2h, . . . , classes of physical excitations (I ∈ {+}), one may distinguish
classes 1h-1 p, 2h-2 p, . . . of de-excitation operators, (I ∈ {−}).
In the derivation of the secular equations one proceeds along Eqs. (16.7)–(16.10),
but then uses commutators rather than anticommutators in Eq. (16.11), as a consequence of subtracting Eq. (16.10) from Eq.(16.9) The resulting secular equations
read
249
which supposes a real-valued one-particle representation. Note that the operator ˆ
H
and the corresponding resolvent simplify if applied to an (N +1)- or (N −1)-electron
state, e.g., ˆ
Hc
†
q | 0 = (− ˆ
H + E 0 )c
†
q | 0 . The representation (16.39) means that the
EOM secular equations, which according to Eqs. (16.29), (16.35) represent secular
equations for ˆ
H and, in extension, the resolvent operator (ω + ˆ
H)
−1 , allow one to
determine the poles and residues of the electron propagator. Another way of relating
the EOM secular problem to algebraic propagator equations is the superoperator
formalism reviewed in Appendix A.8.
16.3 EOM Treatment of N-Electron Excitations
While the derivation of the EOM method for N -electron excitations is essentially
analogous to the (N ± 1)-case, there emerge some distinct features which need to
be addressed.
Excited states | n of the N -electron system are represented in the form of an
operator expansion
ˆ
n =
J
X J n ˆ
O J
(16.41)
acting on the exact ground state,
| n = ˆ
n | 0
(16.42)
In addition, there is the killer condition,
ˆ
†
n | 0 = 0
(16.43)
which here also implies that the excited state is orthogonal to the ground state,
n | 0 = 0.
The Manne–Dalgaard operator basis required for a unique solution of Eqs. (16.42),
(16.43),
{ ˆ
O J } =
c
†
a c k ; c
†
a c
†
b c k c l , a < b, k < l; . . .
∪
c
†
k c a ; c
†
l c
†
k c b c a , a < b, k < l; . . .
(16.44)
is formed by the set (14.10) of physical N -electron excitation operators, { ˆ
C I }, and the
set { ˆ
C
†
I } of their hermitian adjoints, also referred to as de-excitation operators. Like
the 1 p-1h, 2 p-2h, . . . , classes of physical excitations (I ∈ {+}), one may distinguish
classes 1h-1 p, 2h-2 p, . . . of de-excitation operators, (I ∈ {−}).
In the derivation of the secular equations one proceeds along Eqs. (16.7)–(16.10),
but then uses commutators rather than anticommutators in Eq. (16.11), as a consequence of subtracting Eq. (16.10) from Eq.(16.9) The resulting secular equations
read