182
12 Order Relations and Separability
etc., is size-consistent (here, more specifically, size-intensive), if for local excitations,
say on fragment A, the computed excitation energies and transition moments do not
depend on whether the method is applied to the fragment or the composite system.
Obviously, the outcome in the artificial separate fragment model is indicative for
the performance of the method in realistic systems, e.g., formed by interacting subsystems, or just extended systems beyond a certain size.
Let us begin with a few specifications of the separate fragment model. The total
hamiltonian is the sum
ˆ
H = ˆ
H A + ˆ
H B
(12.20)
of the two fragment hamiltonians, ˆ
H A and ˆ
H B , so that the ground state of the composite is given as the product
| 0
= |
A
0
|
B
0
(12.21)
of the fragment ground states, |
A
0
and |
B
0
. It should be noted that in this and other
product states inter-fragment antisymmetrization is irrelevant and can be waived.
The one-particle states (HF orbitals) of S are assumed to be local, that is, a
given orbital either belongs to fragment A or B. As a consequence, the electron
configurations in the set (11.2) can be partitioned into three different subsets, namely
local excitations I A on fragment A, local excitations I B on fragment B, and mixed (or
non-local) excitations I AB involving both fragments A and B. A mixed excitation, for
example, might consist of an ionization on A, accompanied by a neutral excitation
on B. In analogy to the physical operators (11.2), we introduce the operator set
{ ˆ
C J } =
c
†
a c k ; c
†
a c
†
b c k c l , a < b, k < l; . . .
(12.22)
associated with the neutral 1 p-2h, 2p-2h, . . . , excitations.
It should be noted that there are non-local excitations that do not conserve the
charge of the fragments, such as a double ionization on A accompanied by electron
attachment on B, resulting in an A
++ B
− structure. However, such charge-transfer
excitations need not be regarded. In the separate fragment model, they are strictly
decoupled from the fragment-charge conserving excitations to be considered in the
following.
How is the structure of separated fragments reflected in the intermediate states
| ˜
I
? For the CE states |
0
J
, forming the starting point of the ECO- IS construction,
the answer is trivial. Since the ground state is the product of the fragment ground
states (Eq. 12.21) and the physical excitation operators ˆ
C J are operator products of
local fermion operators, the CE states can be written as products of each two fragment
states. A local state simply reads
|
0
J A
= ˆ
C J A | 0
= ˆ
C J A |
A
0
|
B
0
(12.23)
while for non-local states the factorization takes on the form
12 Order Relations and Separability
etc., is size-consistent (here, more specifically, size-intensive), if for local excitations,
say on fragment A, the computed excitation energies and transition moments do not
depend on whether the method is applied to the fragment or the composite system.
Obviously, the outcome in the artificial separate fragment model is indicative for
the performance of the method in realistic systems, e.g., formed by interacting subsystems, or just extended systems beyond a certain size.
Let us begin with a few specifications of the separate fragment model. The total
hamiltonian is the sum
ˆ
H = ˆ
H A + ˆ
H B
(12.20)
of the two fragment hamiltonians, ˆ
H A and ˆ
H B , so that the ground state of the composite is given as the product
| 0
= |
A
0
|
B
0
(12.21)
of the fragment ground states, |
A
0
and |
B
0
. It should be noted that in this and other
product states inter-fragment antisymmetrization is irrelevant and can be waived.
The one-particle states (HF orbitals) of S are assumed to be local, that is, a
given orbital either belongs to fragment A or B. As a consequence, the electron
configurations in the set (11.2) can be partitioned into three different subsets, namely
local excitations I A on fragment A, local excitations I B on fragment B, and mixed (or
non-local) excitations I AB involving both fragments A and B. A mixed excitation, for
example, might consist of an ionization on A, accompanied by a neutral excitation
on B. In analogy to the physical operators (11.2), we introduce the operator set
{ ˆ
C J } =
c
†
a c k ; c
†
a c
†
b c k c l , a < b, k < l; . . .
(12.22)
associated with the neutral 1 p-2h, 2p-2h, . . . , excitations.
It should be noted that there are non-local excitations that do not conserve the
charge of the fragments, such as a double ionization on A accompanied by electron
attachment on B, resulting in an A
++ B
− structure. However, such charge-transfer
excitations need not be regarded. In the separate fragment model, they are strictly
decoupled from the fragment-charge conserving excitations to be considered in the
following.
How is the structure of separated fragments reflected in the intermediate states
| ˜
I
? For the CE states |
0
J
, forming the starting point of the ECO- IS construction,
the answer is trivial. Since the ground state is the product of the fragment ground
states (Eq. 12.21) and the physical excitation operators ˆ
C J are operator products of
local fermion operators, the CE states can be written as products of each two fragment
states. A local state simply reads
|
0
J A
= ˆ
C J A | 0
= ˆ
C J A |
A
0
|
B
0
(12.23)
while for non-local states the factorization takes on the form