10.3 Properties of the non-Dyson ADC Schemes
159
This means that Tr( f
† f ) = N + O(n + 1) at the ADC(n) level, which can be used
as a check for the correctness of the ADC(n) expressions for f .
The ADC form (10.8) can also be applied to Eq. (3.44), relating the ground-state
energy to the electron propagator, which allows us to write E 0 in the form
E 0 =
1
2
Tr( f
†
(K + C) f + T
t f
† f )
(10.41)
As above, this establishes a PT expansion of E 0 which, at the ADC(n) level, recovers
the original RSPT series through order n.
Exercises
10.1 Evaluate the time-orderings (6) and (12) obtained by turning diagrams (6) and
(12) in Fig. 7.3 upside down and derive therefrom the ADC(2) expression for
f ka .
10.2 Apply the direct ADC(2) scheme to ionization in the 2E-2O model (Exercise 2.4).
10.3 (a) Use Eq. (10.37) and the ADC(2) expressions (10.34) to expand the h-h and
p- p matrix elements of the density matrix, ρ kk , ρ aa , through second order.
(b) Verify that Tr(ρ) = N + O(3).
10.4 (a) Analyze the matrix-times-vector product for the ADC(2) secular matrix
and establish that the direct diagonalization can be designed to scale as m
4 .
(b) Perform a similar analysis for the four third-order contributions to the
ADC(3) secular matrix listed in Appendix A.9.
References
1. Schirmer J, Trofimov AB, Stelter G (1998) J Chem Phys 109:4734
2. Durand S, Malrieu J-P (1987) Adv Chem Phys 67:321
3. Trofimov AB, Stelter G, Schirmer J (2002) J Chem Phys 117:6402
159
This means that Tr( f
† f ) = N + O(n + 1) at the ADC(n) level, which can be used
as a check for the correctness of the ADC(n) expressions for f .
The ADC form (10.8) can also be applied to Eq. (3.44), relating the ground-state
energy to the electron propagator, which allows us to write E 0 in the form
E 0 =
1
2
Tr( f
†
(K + C) f + T
t f
† f )
(10.41)
As above, this establishes a PT expansion of E 0 which, at the ADC(n) level, recovers
the original RSPT series through order n.
Exercises
10.1 Evaluate the time-orderings (6) and (12) obtained by turning diagrams (6) and
(12) in Fig. 7.3 upside down and derive therefrom the ADC(2) expression for
f ka .
10.2 Apply the direct ADC(2) scheme to ionization in the 2E-2O model (Exercise 2.4).
10.3 (a) Use Eq. (10.37) and the ADC(2) expressions (10.34) to expand the h-h and
p- p matrix elements of the density matrix, ρ kk , ρ aa , through second order.
(b) Verify that Tr(ρ) = N + O(3).
10.4 (a) Analyze the matrix-times-vector product for the ADC(2) secular matrix
and establish that the direct diagonalization can be designed to scale as m
4 .
(b) Perform a similar analysis for the four third-order contributions to the
ADC(3) secular matrix listed in Appendix A.9.
References
1. Schirmer J, Trofimov AB, Stelter G (1998) J Chem Phys 109:4734
2. Durand S, Malrieu J-P (1987) Adv Chem Phys 67:321
3. Trofimov AB, Stelter G, Schirmer J (2002) J Chem Phys 117:6402