9.2 Dyson-ADC Secular Equations
143
Let us state some general properties of the Dyson-ADC approximation schemes
for the electron propagator.
1. Infinite partial summation: The ADC(n) approximation (in the form of the
Dyson-ADC secular problem) represents an infinite partial summation of the
diagrammatic perturbation expansion, being complete through order n of perturbation theory. The ionization energies (electron affinities) of 1h (1 p) main states
are treated consistently through order n.
2. Diagonalization and perturbation theory: The resulting computational method
combines the eigenvalue problem (diagonalisation) of a hermitian secular matrix
with (finite) perturbation expansions of the secular matrix elements.
3. Regularity: With respect to convergence, the PT expansions of the secular matrix
elements behave essentially like the RSPT expansion of the ground-state energy.
The energy denominators are of the type
ν × virt − ν × occ ≥ ν , ν = 1, 2, . . .
(9.36)
where is the energy gap between occupied and virtual HF orbitals. There are
no “dangerous” denominators (with small or even zero absolute values) provided
the energy gap is sufficiently large. This is referred to as the regularity of the PT
expansions of the secular matrix elements.
4. Compact configuration spaces: At the second-order level, the explicit configuration space of the Dyson-ADC secular matrix comprises the 1h/1 p and 2h-1 p/2 p1h configurations. At each even order n = 4, 6, . . . , the explicit configuration
space grows by the next higher class of (N −1)- and (N +1)-electron configurations. Accordingly, for the order levels n = 2m and n = 2m + 1, m = 1, 2, . . . ,
the explicit configuration space comprises the classes 1h, . . . , (m +1)h-mp and
1 p, . . . , (m +1) p-mh.
5. Size-consistency: The Dyson-ADC approach is size-consistent. For a system S
consisting of two separated fragments A and B, the results obtained for a “local”
ionization, say, on fragment A, do not depend on whether the method is applied
to S or to fragment A. As is well known, this does not apply to the CI treatment
based on restricted (as opposed to full) CI expansions (see Sect. 12.3).
As a consequence of the compactness property, the ADC(n) configuration spaces
are smaller (more compact) than those of comparable CI expansions. The consistent treatment of 1h main states through second (and third order) requires the CI
expansions to extend through the 3h-2 p configurations. By contrast, the ADC(2)
(and ADC(3)) configuration spaces are restricted to the 2h-1 p configurations of
N −1 particles, but also comprise the 2h-1 p configurations of N +1 particles. With
increasing order n, the CI expansions required for a consistent treatment grow twice
as fast as their ADC counterparts: At each even order two further excitation classes
rather than one have to be taken into account (see Table 9.1). For example, at the
fourth-order level, the 1h, 2h-1 p, . . . , 5h-4 p CI expansion is contrasted with the
ADC(4) configuration space spanned by the 1h/1 p, 2h-1 p/2 p-1h, and 3h-2 p/3 p2h configurations.
143
Let us state some general properties of the Dyson-ADC approximation schemes
for the electron propagator.
1. Infinite partial summation: The ADC(n) approximation (in the form of the
Dyson-ADC secular problem) represents an infinite partial summation of the
diagrammatic perturbation expansion, being complete through order n of perturbation theory. The ionization energies (electron affinities) of 1h (1 p) main states
are treated consistently through order n.
2. Diagonalization and perturbation theory: The resulting computational method
combines the eigenvalue problem (diagonalisation) of a hermitian secular matrix
with (finite) perturbation expansions of the secular matrix elements.
3. Regularity: With respect to convergence, the PT expansions of the secular matrix
elements behave essentially like the RSPT expansion of the ground-state energy.
The energy denominators are of the type
ν × virt − ν × occ ≥ ν , ν = 1, 2, . . .
(9.36)
where is the energy gap between occupied and virtual HF orbitals. There are
no “dangerous” denominators (with small or even zero absolute values) provided
the energy gap is sufficiently large. This is referred to as the regularity of the PT
expansions of the secular matrix elements.
4. Compact configuration spaces: At the second-order level, the explicit configuration space of the Dyson-ADC secular matrix comprises the 1h/1 p and 2h-1 p/2 p1h configurations. At each even order n = 4, 6, . . . , the explicit configuration
space grows by the next higher class of (N −1)- and (N +1)-electron configurations. Accordingly, for the order levels n = 2m and n = 2m + 1, m = 1, 2, . . . ,
the explicit configuration space comprises the classes 1h, . . . , (m +1)h-mp and
1 p, . . . , (m +1) p-mh.
5. Size-consistency: The Dyson-ADC approach is size-consistent. For a system S
consisting of two separated fragments A and B, the results obtained for a “local”
ionization, say, on fragment A, do not depend on whether the method is applied
to S or to fragment A. As is well known, this does not apply to the CI treatment
based on restricted (as opposed to full) CI expansions (see Sect. 12.3).
As a consequence of the compactness property, the ADC(n) configuration spaces
are smaller (more compact) than those of comparable CI expansions. The consistent treatment of 1h main states through second (and third order) requires the CI
expansions to extend through the 3h-2 p configurations. By contrast, the ADC(2)
(and ADC(3)) configuration spaces are restricted to the 2h-1 p configurations of
N −1 particles, but also comprise the 2h-1 p configurations of N +1 particles. With
increasing order n, the CI expansions required for a consistent treatment grow twice
as fast as their ADC counterparts: At each even order two further excitation classes
rather than one have to be taken into account (see Table 9.1). For example, at the
fourth-order level, the 1h, 2h-1 p, . . . , 5h-4 p CI expansion is contrasted with the
ADC(4) configuration space spanned by the 1h/1 p, 2h-1 p/2 p-1h, and 3h-2 p/3 p2h configurations.