17.1 Ground-State Coupled-Cluster Formulation
257
It should be noted that the CC equations do not constitute a variational approach
to the CC amplitudes. This has implications to the performance of the truncated CC
schemes, which deteriorates when the ground state is no longer adequately described
by a single dominant (reference) configuration. To cope with such situations, multireference (MR) CC schemes have been developed (see Refs. [9, 12]), which are much
more complicated though.
As a point of particular interest, the CC amplitudes fulfill the order relations
t I ∼ O([I ] − 1), [I ] > 1
(17.8)
which are a consequence of the exponential ansatz [13]. The CC order relations can
be contrasted with the CI order relations (A.1.24),
x I ∼
1
2
O([I ]), [I ] even
1
2
O([I ] + 1), [I ] odd, > 1
(17.9)
Here, x I are the coefficients in the ground-state CI expansion (see Appendix A.1),
| 0 = | 0 +
I
x I ˆ
C I | 0
(17.10)
The 2 p-2h and 3 p-3h amplitudes are of first and second order, respectively, both in
the CC and CI expansion; but beginning with the 4 p-4h excitations, where x I ∼ O(2)
and t I ∼ O(3), the orders of the CC amplitudes are increasingly higher than those
of their CC counterparts.
As analyzed by Hubbard [13], the exponential ansatz (17.1) rests on a generalized linked-cluster theorem, which implies that, in contrast to the CI coefficients, the
t-amplitudes do not exhibit “non-linking” PT contributions, that is, terms involving
coupling matrix elements of the type H I J , where I and J differ by a double excitation, [I ] − [J ] = ±2. An elaboration of these findings is given in the first section
of Appendix A.6.
17.2 Biorthogonal Coupled-Cluster Representation
Now we can turn to the BCC approach to electronic excitations, where we consider
specifically the case of N -electron excitations. Of course, the BCC concept is quite
general and can easily be adapted to ionization (IP- EOM- CC), electron attachment
(EA- EOM- CC), and other cases of interest.
The BCC secular matrix M
cc is obtained as a non-hermitian representation of the
(shifted) hamiltonian ˆ
H − E 0 ,
M
cc
I J = = I | ˆ
H − E 0 |
0
J
(17.11)
257
It should be noted that the CC equations do not constitute a variational approach
to the CC amplitudes. This has implications to the performance of the truncated CC
schemes, which deteriorates when the ground state is no longer adequately described
by a single dominant (reference) configuration. To cope with such situations, multireference (MR) CC schemes have been developed (see Refs. [9, 12]), which are much
more complicated though.
As a point of particular interest, the CC amplitudes fulfill the order relations
t I ∼ O([I ] − 1), [I ] > 1
(17.8)
which are a consequence of the exponential ansatz [13]. The CC order relations can
be contrasted with the CI order relations (A.1.24),
x I ∼
1
2
O([I ]), [I ] even
1
2
O([I ] + 1), [I ] odd, > 1
(17.9)
Here, x I are the coefficients in the ground-state CI expansion (see Appendix A.1),
| 0 = | 0 +
I
x I ˆ
C I | 0
(17.10)
The 2 p-2h and 3 p-3h amplitudes are of first and second order, respectively, both in
the CC and CI expansion; but beginning with the 4 p-4h excitations, where x I ∼ O(2)
and t I ∼ O(3), the orders of the CC amplitudes are increasingly higher than those
of their CC counterparts.
As analyzed by Hubbard [13], the exponential ansatz (17.1) rests on a generalized linked-cluster theorem, which implies that, in contrast to the CI coefficients, the
t-amplitudes do not exhibit “non-linking” PT contributions, that is, terms involving
coupling matrix elements of the type H I J , where I and J differ by a double excitation, [I ] − [J ] = ±2. An elaboration of these findings is given in the first section
of Appendix A.6.
17.2 Biorthogonal Coupled-Cluster Representation
Now we can turn to the BCC approach to electronic excitations, where we consider
specifically the case of N -electron excitations. Of course, the BCC concept is quite
general and can easily be adapted to ionization (IP- EOM- CC), electron attachment
(EA- EOM- CC), and other cases of interest.
The BCC secular matrix M
cc is obtained as a non-hermitian representation of the
(shifted) hamiltonian ˆ
H − E 0 ,
M
cc
I J = = I | ˆ
H − E 0 |
0
J
(17.11)