266
17 Coupled-Cluster Methods for Generalized Excitations
the right eigenvectors are separable. The problem is due to the use of the dual ground
state (17.30) in the right basis set transition moments (17.43). The dual ground state,
being essentially a CI expansion, is not separable, that is, a factorization
0 | = =
A
0 ||
B
0 |
(17.57)
of the composite ground state is attained only in the exact (full) BCC treatment.
It should be noted that in the CCLR formulation this shortcoming is avoided, as
here a separable, if more elaborate expression is employed for the right transition
moments [14, 15].
Exercises
17.1 (a) Compare the CC expansion | 0 = e
ˆ
T 1 + ˆ
T 2 + ˆ
T 3 +...
| 0 for the ground state
with the CI expansion | 0 = (1 + ˆ
S 1 + ˆ
S 2 + ˆ
S 3 + . . . )| 0 , where ˆ
S ν =
[I ]=ν x I ˆ
C I denotes the excitations of class ν in the CI expansion. Express the
four lowest CI operators ˆ
S 1 , . . . , ˆ
S 4 , in terms of the CC operators ˆ
T 1 , . . . , ˆ
T 4 .
(b) Deduce the order relations for ˆ
T 1 , ˆ
T 2 , and ˆ
T 3 from those for ˆ
S 1 , ˆ
S 2 , and ˆ
S 3 ;
what is the situation for ˆ
T 4 ?
17.2 (a) Consider the excitation operator ˆ
C K for a quadruple excitation K ≡
abcd; i jkl and establish that this operator can be written in 18 distinct ways as
a product of two “disjunct” double excitations, such as ˆ
C abcd;i jkl = ˆ
C acil ˆ
C bd jk .
(b) Inspect the PT expression for the ground-state amplitude x
(2)
abcd;i jkl =
abcd;i jkl |
(2)
0 in second order (see Appendix A.1), and show that x
(2)
abcd;i jkl
can be written as a sum over products x
(1)
I x
(1)
J of first-order amplitudes for
(disjunct) double excitations, I, J , where ˆ
C abcd;i jkl = ˆ
C I ˆ
C J .
(c) Use the results of (b) together with Exercise 17.1 to show that the CC
amplitude t abcd;i jkl is (at least) of third order.
17.3 Consider the BCC representation for an (N −1)-electron system (IP- EOMCC) and expand the coupling matrix elements M
cc
i,abjkl and M
cc
abjkl,i for a 1h
state and a 3h-2 p state through first order.
17.4 Establish the block structure of the BCC secular matrix with respect to separate
fragment partitioning (Fig. 17.2).
17.5 Apply the ground-state CC concept to the 2E-2O model considered in Exercise 2.4. Here, the CC operator consists of a single double excitation, ˆ
T =
t c
†
uα c
†
uβ c gβ c gα so that |
cc
0 = e
ˆ
T
| 0 = | 0 + t| 1 . Project the Schrödinger
equation for |
cc
0 onto | 0 and | 1 and determine t and E 0 .
17.6 CCD(doubles) treatment of the multiple 2E-2O system (see Exercise 12.2):
Consider the CC operator ˆ
T =
i t i ˆ
C i where ˆ
C i = c
†
iuα c
†
iuβ c igβ c igα , i = 1, . . . ,
M, are double excitation operators for the M sub-systems. Derive the groundstate CC equations and verify that the amplitudes are all equal, t i = t. Determine t, and establish that the CCD results reproduce the exact ground-state
energy and wave function.
17 Coupled-Cluster Methods for Generalized Excitations
the right eigenvectors are separable. The problem is due to the use of the dual ground
state (17.30) in the right basis set transition moments (17.43). The dual ground state,
being essentially a CI expansion, is not separable, that is, a factorization
0 | = =
A
0 ||
B
0 |
(17.57)
of the composite ground state is attained only in the exact (full) BCC treatment.
It should be noted that in the CCLR formulation this shortcoming is avoided, as
here a separable, if more elaborate expression is employed for the right transition
moments [14, 15].
Exercises
17.1 (a) Compare the CC expansion | 0 = e
ˆ
T 1 + ˆ
T 2 + ˆ
T 3 +...
| 0 for the ground state
with the CI expansion | 0 = (1 + ˆ
S 1 + ˆ
S 2 + ˆ
S 3 + . . . )| 0 , where ˆ
S ν =
[I ]=ν x I ˆ
C I denotes the excitations of class ν in the CI expansion. Express the
four lowest CI operators ˆ
S 1 , . . . , ˆ
S 4 , in terms of the CC operators ˆ
T 1 , . . . , ˆ
T 4 .
(b) Deduce the order relations for ˆ
T 1 , ˆ
T 2 , and ˆ
T 3 from those for ˆ
S 1 , ˆ
S 2 , and ˆ
S 3 ;
what is the situation for ˆ
T 4 ?
17.2 (a) Consider the excitation operator ˆ
C K for a quadruple excitation K ≡
abcd; i jkl and establish that this operator can be written in 18 distinct ways as
a product of two “disjunct” double excitations, such as ˆ
C abcd;i jkl = ˆ
C acil ˆ
C bd jk .
(b) Inspect the PT expression for the ground-state amplitude x
(2)
abcd;i jkl =
abcd;i jkl |
(2)
0 in second order (see Appendix A.1), and show that x
(2)
abcd;i jkl
can be written as a sum over products x
(1)
I x
(1)
J of first-order amplitudes for
(disjunct) double excitations, I, J , where ˆ
C abcd;i jkl = ˆ
C I ˆ
C J .
(c) Use the results of (b) together with Exercise 17.1 to show that the CC
amplitude t abcd;i jkl is (at least) of third order.
17.3 Consider the BCC representation for an (N −1)-electron system (IP- EOMCC) and expand the coupling matrix elements M
cc
i,abjkl and M
cc
abjkl,i for a 1h
state and a 3h-2 p state through first order.
17.4 Establish the block structure of the BCC secular matrix with respect to separate
fragment partitioning (Fig. 17.2).
17.5 Apply the ground-state CC concept to the 2E-2O model considered in Exercise 2.4. Here, the CC operator consists of a single double excitation, ˆ
T =
t c
†
uα c
†
uβ c gβ c gα so that |
cc
0 = e
ˆ
T
| 0 = | 0 + t| 1 . Project the Schrödinger
equation for |
cc
0 onto | 0 and | 1 and determine t and E 0 .
17.6 CCD(doubles) treatment of the multiple 2E-2O system (see Exercise 12.2):
Consider the CC operator ˆ
T =
i t i ˆ
C i where ˆ
C i = c
†
iuα c
†
iuβ c igβ c igα , i = 1, . . . ,
M, are double excitation operators for the M sub-systems. Derive the groundstate CC equations and verify that the amplitudes are all equal, t i = t. Determine t, and establish that the CCD results reproduce the exact ground-state
energy and wave function.