134
8 Self-Energy and the Dyson Equation
8.3 (a) Consider the model functions (8.20) and (8.21) in the simple two-pole case
(m = 2) and verify the corresponding Eq. (8.22).
(b) Perform the corresponding analysis for m = 3. (Use partial fraction decomposition without explicitly solving the cubic or quadratic equations).
(c) Consider the general case (m arbitrary) and establish that s(ω) can be written
in the form of Eq. (8.22).
8.4 Adapt the matrix representation of Abrikosov diagrams (Sect. 6.3) to the diagrams for the dynamical self-energy part M(ω) and test that concept for the
second- and third-order diagrams.
8.5 Show that the definitions (8.57) and (8.59) for the Dyson orbitals are equivalent.
(Proceed here as indicated in Sect. 8.3.)
8.6 Use (formal) RSPT for the (N −1)-electron state deriving from c k | 0 and verify
the second-order expressions (8.71)–(8.73).
References
1. Dyson FJ (1949) Phys Rev 75:1736
2. Ethofer S, Schuck P (1969) Z Phys 228:264
3. Winter J (1972) Nucl Phys A 194:535
4. Cederbaum LS (1975) J Phys B 8:290
5. Cederbaum LS, Domcke W (1977) Adv Chem Phys 36:205
6. von Niessen W, Schirmer J, Cederbaum L (1984) Comput Phys Rep 1:57
7. Ortiz JV, Zakrzewski VG, Dolgounitcheva O (1997) In: Kryachko E (ed) Conceptual trends in
quantum chemistry. Kluwer, Dordrecht, p 3
8 Self-Energy and the Dyson Equation
8.3 (a) Consider the model functions (8.20) and (8.21) in the simple two-pole case
(m = 2) and verify the corresponding Eq. (8.22).
(b) Perform the corresponding analysis for m = 3. (Use partial fraction decomposition without explicitly solving the cubic or quadratic equations).
(c) Consider the general case (m arbitrary) and establish that s(ω) can be written
in the form of Eq. (8.22).
8.4 Adapt the matrix representation of Abrikosov diagrams (Sect. 6.3) to the diagrams for the dynamical self-energy part M(ω) and test that concept for the
second- and third-order diagrams.
8.5 Show that the definitions (8.57) and (8.59) for the Dyson orbitals are equivalent.
(Proceed here as indicated in Sect. 8.3.)
8.6 Use (formal) RSPT for the (N −1)-electron state deriving from c k | 0 and verify
the second-order expressions (8.71)–(8.73).
References
1. Dyson FJ (1949) Phys Rev 75:1736
2. Ethofer S, Schuck P (1969) Z Phys 228:264
3. Winter J (1972) Nucl Phys A 194:535
4. Cederbaum LS (1975) J Phys B 8:290
5. Cederbaum LS, Domcke W (1977) Adv Chem Phys 36:205
6. von Niessen W, Schirmer J, Cederbaum L (1984) Comput Phys Rep 1:57
7. Ortiz JV, Zakrzewski VG, Dolgounitcheva O (1997) In: Kryachko E (ed) Conceptual trends in
quantum chemistry. Kluwer, Dordrecht, p 3