10.3 Properties of the non-Dyson ADC Schemes
157
schemes, for example, the error caused by omitting the 3h-2 p and higher excited
configurations is of fourth order.
– Finally, the direct ADC(n) schemes are size-consistent, which is an indispensable
requirement in applications to larger molecules.
The justification of both the compactness and size-consistency properties, as given in
Sect. 9.2 for the Dyson-ADC approach, was based on diagrammatic arguments. While
the latter apply here as well, the non-Dyson ADC variant allows for an alternative,
more stringent formulation. Here, the key is that the direct ADC secular matrix can
be identified as the representation of the (subtracted) hamiltonian with respect to
a specific set of intermediate states presented in the following Chap. 11. The basic
features of that intermediate state representation (ISR), ensuring the compactness
and size-consistency of the ISR-ADC(n) computational schemes, will be addressed
in Chap. 12.
It is instructive to compare the direct ADC scheme to the Dyson-ADC version
presented in Chap. 9 at the lowest non-trivial, that is, second-order level. The matrix
K + C of the direct scheme (Eq. 10.33) can be largely retrieved in the (N −1)electron parts of the Dyson secular matrix B (Eq. 9.32). For example, the coupling
block C
(1)
21 can be identified with the corresponding matrix elements of U
(1)− . The
notable difference arises in the 1h diagonal block, where the C
(2)
11 second-order
contributions are absent in B 11 (the PT expansion of (∞) begins in third order). The
counterpart to the C
(2)
11 contribution is effected in the Dyson scheme by the coupling
of the 1h configurations via U
+ to the 2 p-1h configurations of N +1 electrons.
This can be put on a more rigorous foundation by applying a QDPT-type procedure
(quasi-degenerate perturbation theory [2]) to the 1h/2 p-1h coupling in B.
An important indicator of the efficiency of a computational method is the scaling
of the computing effort with the size of the system or, more specifically, with the
number m of the one-particle basis states (molecular orbitals) entering the computation. At the ADC(2) level for example, the construction of the C 11 block scales as m
5 ,
as there are m
2 matrix elements, each requiring m
3 integral multiplications. (Obviously, a more refined scaling expression could have been obtained by distinguishing
occupied and virtual orbitals.) The matrix-times-vector step in an iterative diagonalization procedure scales as m
4 , as there are m
4 non-vanishing secular matrix elements
(arising in the C 12 block). This would mean that the overall scaling behaviour of the
ADC(2) scheme is m
5 . Actually, the overall scaling can be reduced to m
4 by using
the so-called direct diagonalization procedure, in which, rather than computing and
storing the secular matrix, the matrix elements are computed (and re-computed) “on
the fly” as needed in the matrix-vector product cycles. This allows one to form suitable intermediates, thereby reducing the computational cost (see Exercise 10.4). The
corresponding scaling of the ADC(3) scheme is m
5 .
The computational performance of the direct third-order ADC scheme has been
examined in model applications [3] to a series of small molecules, allowing for
the comparison with full CI results. As expected, there is good mutual agreement
between the results obtained using the direct and the Dyson-ADC variants. For the
ionization energies of outer valence 1h states, the discrepancies of the ADC(3) and
157
schemes, for example, the error caused by omitting the 3h-2 p and higher excited
configurations is of fourth order.
– Finally, the direct ADC(n) schemes are size-consistent, which is an indispensable
requirement in applications to larger molecules.
The justification of both the compactness and size-consistency properties, as given in
Sect. 9.2 for the Dyson-ADC approach, was based on diagrammatic arguments. While
the latter apply here as well, the non-Dyson ADC variant allows for an alternative,
more stringent formulation. Here, the key is that the direct ADC secular matrix can
be identified as the representation of the (subtracted) hamiltonian with respect to
a specific set of intermediate states presented in the following Chap. 11. The basic
features of that intermediate state representation (ISR), ensuring the compactness
and size-consistency of the ISR-ADC(n) computational schemes, will be addressed
in Chap. 12.
It is instructive to compare the direct ADC scheme to the Dyson-ADC version
presented in Chap. 9 at the lowest non-trivial, that is, second-order level. The matrix
K + C of the direct scheme (Eq. 10.33) can be largely retrieved in the (N −1)electron parts of the Dyson secular matrix B (Eq. 9.32). For example, the coupling
block C
(1)
21 can be identified with the corresponding matrix elements of U
(1)− . The
notable difference arises in the 1h diagonal block, where the C
(2)
11 second-order
contributions are absent in B 11 (the PT expansion of (∞) begins in third order). The
counterpart to the C
(2)
11 contribution is effected in the Dyson scheme by the coupling
of the 1h configurations via U
+ to the 2 p-1h configurations of N +1 electrons.
This can be put on a more rigorous foundation by applying a QDPT-type procedure
(quasi-degenerate perturbation theory [2]) to the 1h/2 p-1h coupling in B.
An important indicator of the efficiency of a computational method is the scaling
of the computing effort with the size of the system or, more specifically, with the
number m of the one-particle basis states (molecular orbitals) entering the computation. At the ADC(2) level for example, the construction of the C 11 block scales as m
5 ,
as there are m
2 matrix elements, each requiring m
3 integral multiplications. (Obviously, a more refined scaling expression could have been obtained by distinguishing
occupied and virtual orbitals.) The matrix-times-vector step in an iterative diagonalization procedure scales as m
4 , as there are m
4 non-vanishing secular matrix elements
(arising in the C 12 block). This would mean that the overall scaling behaviour of the
ADC(2) scheme is m
5 . Actually, the overall scaling can be reduced to m
4 by using
the so-called direct diagonalization procedure, in which, rather than computing and
storing the secular matrix, the matrix elements are computed (and re-computed) “on
the fly” as needed in the matrix-vector product cycles. This allows one to form suitable intermediates, thereby reducing the computational cost (see Exercise 10.4). The
corresponding scaling of the ADC(3) scheme is m
5 .
The computational performance of the direct third-order ADC scheme has been
examined in model applications [3] to a series of small molecules, allowing for
the comparison with full CI results. As expected, there is good mutual agreement
between the results obtained using the direct and the Dyson-ADC variants. For the
ionization energies of outer valence 1h states, the discrepancies of the ADC(3) and