190
12 Order Relations and Separability
local ionization, say on A, is of course given as the sum E
N−1
n
= E
A
n + E
B
0 of the
fragment state energies. Here, subtraction of the FCI result for the composite ground
state, E 0 = E
A
0 + E
B
0 , would cancel E
B
0 and lead to the correct ionization energy
for the ionization of the local fragment A. In realistic applications, one has to resort
to truncated CI expansions, and here, an energy computed for the composite system differs from the sum of the CI results for the respective fragment states. Due to
this well-known size-consistency error, the CI method does not qualify as a genuine
many-body method.
Exercises
12.1 (a) Consider the symmetrical orthonormalization (SO) of the CE states according to Eq. (11.5). Construct the 1h and 3h-2 p states | i
, | abjkl
through
first order and verify Eq. (12.10). Hint: The overlap matrix S (Eq. 11.4)
of the CE states can be expanded as S = 1 + S
(1)
+ . . . so that S
−1/2
=
1 −
1
2
S
(1)
+ O(2).
(b) Sketch a proof for a factorization theorem analogous to the factorization of
the ECO-ISR states according to Eq. (12.25). Discuss the separability properties of the SO-CE secular matrix.
12.2 Multiple non-interacting 2E-2O systems (see Exercise 2.4):
A well-known model (see for example Ref. [3]) to demonstrate the sizeconsistency problem in the (restricted) CI treatment is a system composed of
M (> 1) mutually non-interacting 2E-2O sub-systems, such as, for example,
an array of separate H 2 molecules in the minimal basis (two orbital) approximation.
(a) Apply the double excitation CI scheme (CID) to the ground state of
the 2E-2O array. Establish that the configuration manifold comprises the
reference state, |
M
0
= |
(1)
0
|
(2)
0
. . . |
(M)
0
, where |
(i)
0
denotes the HF
ground state of the ith subsystem; and M doubly excited states of the
form |
M
i
= ˆ
C i |
M
0
, i = 1, . . . , M. Here, ˆ
C i = c
†
iuα c
†
iuβ c igβ c igα generates
a double excitation in the subsystem i, and ig, iu denote the orbitals of
subsystem i.
(b) Evaluate the (M + 1)-dimensional hamiltonian matrix, H, and solve the
corresponding eigenvalue equations for the ground state. Hint: the particular
form of H allows for a simple analytical solution. Determine and characterize
the remaining M solutions too.
(c) Compare the CID result, E
C I
0 , with the exact ground-state energy of the
model. How does the resulting correlation energy E
C I
c = H 00 − E
C I
0 scale
with M?
(d) Compare the exact ground state, |
M
0
, with the CID result; consider here
in particular the coefficient x 0 of the reference configuration (HF ground state),
|
M
0
, in the normalized exact ground state, |
M
0
.
12.3 (a) Expand the CID result E
C I
0 in a perturbation series in powers of V //, (see
Exercise 2.4c) and verify that the wrong scaling with M sets in with the quartic
terms.
12 Order Relations and Separability
local ionization, say on A, is of course given as the sum E
N−1
n
= E
A
n + E
B
0 of the
fragment state energies. Here, subtraction of the FCI result for the composite ground
state, E 0 = E
A
0 + E
B
0 , would cancel E
B
0 and lead to the correct ionization energy
for the ionization of the local fragment A. In realistic applications, one has to resort
to truncated CI expansions, and here, an energy computed for the composite system differs from the sum of the CI results for the respective fragment states. Due to
this well-known size-consistency error, the CI method does not qualify as a genuine
many-body method.
Exercises
12.1 (a) Consider the symmetrical orthonormalization (SO) of the CE states according to Eq. (11.5). Construct the 1h and 3h-2 p states | i
, | abjkl
through
first order and verify Eq. (12.10). Hint: The overlap matrix S (Eq. 11.4)
of the CE states can be expanded as S = 1 + S
(1)
+ . . . so that S
−1/2
=
1 −
1
2
S
(1)
+ O(2).
(b) Sketch a proof for a factorization theorem analogous to the factorization of
the ECO-ISR states according to Eq. (12.25). Discuss the separability properties of the SO-CE secular matrix.
12.2 Multiple non-interacting 2E-2O systems (see Exercise 2.4):
A well-known model (see for example Ref. [3]) to demonstrate the sizeconsistency problem in the (restricted) CI treatment is a system composed of
M (> 1) mutually non-interacting 2E-2O sub-systems, such as, for example,
an array of separate H 2 molecules in the minimal basis (two orbital) approximation.
(a) Apply the double excitation CI scheme (CID) to the ground state of
the 2E-2O array. Establish that the configuration manifold comprises the
reference state, |
M
0
= |
(1)
0
|
(2)
0
. . . |
(M)
0
, where |
(i)
0
denotes the HF
ground state of the ith subsystem; and M doubly excited states of the
form |
M
i
= ˆ
C i |
M
0
, i = 1, . . . , M. Here, ˆ
C i = c
†
iuα c
†
iuβ c igβ c igα generates
a double excitation in the subsystem i, and ig, iu denote the orbitals of
subsystem i.
(b) Evaluate the (M + 1)-dimensional hamiltonian matrix, H, and solve the
corresponding eigenvalue equations for the ground state. Hint: the particular
form of H allows for a simple analytical solution. Determine and characterize
the remaining M solutions too.
(c) Compare the CID result, E
C I
0 , with the exact ground-state energy of the
model. How does the resulting correlation energy E
C I
c = H 00 − E
C I
0 scale
with M?
(d) Compare the exact ground state, |
M
0
, with the CID result; consider here
in particular the coefficient x 0 of the reference configuration (HF ground state),
|
M
0
, in the normalized exact ground state, |
M
0
.
12.3 (a) Expand the CID result E
C I
0 in a perturbation series in powers of V //, (see
Exercise 2.4c) and verify that the wrong scaling with M sets in with the quartic
terms.