254
16 Algebraic Propagator Methods
complex operators ˆ
O I are defined via the coordinate space representation (as
in Exercise 16.2).
(a) Consider the 1 p/1h states | p = (c
∗†
p + c p )| 0 and show that
p | q = = 0 |{c
†
p , c q }| 0 =:S pq ( = δ pq )
and
p |( ˆ
H − E 0 )(N − ˆ
N )| q = = 0 |{c
†
p , [ ˆ
H , c q ]}| 0 =:M pq
Here the hamiltonian is assumed to be real, and so is 0 (possibly up to an
irrelevant phase factor).
(b) Establish analogous results for arbitrary | I states.
16.4 Derive the RPA equations (15.5), (15.6) as an approximation to the EOM
expressions (16.46), (16.47) obtained by restricting the EOM operator manifold
to the 1 p-1h and 1h-1 p operators and replacing | 0 with the HF ground
state | 0 . Note that the relative phase of the 1h-1 p operators implied by the
definition (16.44) differs from that underlying Eq. (15.5). Verify that this is
consistent with the transition moments D
R P A
rs
= = 0 |[ ˆ
O
†
rs , ˆ
D]| 0 deriving
from Eq. (16.57).
References
1. Pickup BT, Goscinski O (1973) Mol Phys 26:1013
2. Öhrn Y, Born G (1981) Adv Quantum Chem 13:1
3. Linderberg J, Öhrn Y (2004) Propagators in quantum chemistry. Wiley, New York
4. Goscinski D, Lukman B (1970) Chem. Phys Lett 7:573
5. McCurdy CW, Rescigno TN, Yeager DL, McKoy V (1977) In: Schaefer HF (ed) Methods of
electronic structure theory. Plenum, New York, p 339
6. Herman MF, Freed KF, Yeager DL (1981) Adv Chem Phys 48:1
7. Rowe DJ (1968) Rev Mod Phys 40:153
8. Mertins F, Schirmer J, Tarantelli A (1996) Phys Rev A 53:2153
9. Dalgaard E (1979) Int J Quantum Chem 15:169
10. Herman MF, Freed KF, Yeager DL (1980) J Chem Phys 72:602
11. Mertins F (1995) Dissertation, Universität Heidelberg
12. Nielsen ES, Jørgensen P, Oddershede J (1980) J Chem Phys 73:6238
13. Oddershede J, Jørgensen P, Yeager DL (1984) Comput Phys Rep 2:33
16 Algebraic Propagator Methods
complex operators ˆ
O I are defined via the coordinate space representation (as
in Exercise 16.2).
(a) Consider the 1 p/1h states | p = (c
∗†
p + c p )| 0 and show that
p | q = = 0 |{c
†
p , c q }| 0 =:S pq ( = δ pq )
and
p |( ˆ
H − E 0 )(N − ˆ
N )| q = = 0 |{c
†
p , [ ˆ
H , c q ]}| 0 =:M pq
Here the hamiltonian is assumed to be real, and so is 0 (possibly up to an
irrelevant phase factor).
(b) Establish analogous results for arbitrary | I states.
16.4 Derive the RPA equations (15.5), (15.6) as an approximation to the EOM
expressions (16.46), (16.47) obtained by restricting the EOM operator manifold
to the 1 p-1h and 1h-1 p operators and replacing | 0 with the HF ground
state | 0 . Note that the relative phase of the 1h-1 p operators implied by the
definition (16.44) differs from that underlying Eq. (15.5). Verify that this is
consistent with the transition moments D
R P A
rs
= = 0 |[ ˆ
O
†
rs , ˆ
D]| 0 deriving
from Eq. (16.57).
References
1. Pickup BT, Goscinski O (1973) Mol Phys 26:1013
2. Öhrn Y, Born G (1981) Adv Quantum Chem 13:1
3. Linderberg J, Öhrn Y (2004) Propagators in quantum chemistry. Wiley, New York
4. Goscinski D, Lukman B (1970) Chem. Phys Lett 7:573
5. McCurdy CW, Rescigno TN, Yeager DL, McKoy V (1977) In: Schaefer HF (ed) Methods of
electronic structure theory. Plenum, New York, p 339
6. Herman MF, Freed KF, Yeager DL (1981) Adv Chem Phys 48:1
7. Rowe DJ (1968) Rev Mod Phys 40:153
8. Mertins F, Schirmer J, Tarantelli A (1996) Phys Rev A 53:2153
9. Dalgaard E (1979) Int J Quantum Chem 15:169
10. Herman MF, Freed KF, Yeager DL (1980) J Chem Phys 72:602
11. Mertins F (1995) Dissertation, Universität Heidelberg
12. Nielsen ES, Jørgensen P, Oddershede J (1980) J Chem Phys 73:6238
13. Oddershede J, Jørgensen P, Yeager DL (1984) Comput Phys Rep 2:33