Linear response and Time dependent Hartree-Fock

χ (−)
0 (r, r ′ , ω) = 1 ∑ e i(p−h)·(r′ −r)
Ω 2 ¯hω + ɛ p − ɛ h + iη
ph
∑
= ρ Ω q
∫
ρ
≡
(2π) 3
e iq·(r′ −r) 1 N
∑
h
n(h)¯n(h + q)
¯hω + ɛ h+q − ɛ h + iη
d 3 qe iq·(r′ −r) χ (−)
0 (q, ω) . (71)
Since the functions χ (+)
0 (q, ω) and χ (−)
0 (q, ω) are invariant under q ↔ −q, we
can add the two functions to the coordinate form of the full response function
of the non-interacting system and write
χ 0 (r, r ′ ; ω) =
ρ ∫
d 3 qe iq·(r′ −r) χ
(2π) 3 0 (q, ω) (72)
If, furthermore, we approximate the Hartree-Fock spectrum by the free singleparticle
spectrum, we obtain the familiar Lindhard function. This is actually
consistent with the approximation to leave out the exchange matrix elements
in the matrices A and B. We finally transform the full equation (67) in
momentum space and solve
or
δρ(q, ω) =
χ(q, ω) =
χ 0 (q, ω)
1 − Ṽ (q)χ 0(q, ω) δ˜h(q, ω), . (73)
χ 0 (q, ω)
1 − Ṽ (q)χ 0(q, ω) . (74)
This is the beloved “RPA” equation for the density-density response function.
The present derivation is perhaps a bit complicated, but it shows very well
what kinds of approximations have been made. These are
• Assume a local, tramslationally invariant (effective) interaction
• Omit exchange
• Assume a free single particle spectrum
References
[1] D. Pines and P. Nozieres. The Theory of Quantum Liquids, volume I.
Benjamin, New York, 1966.
[2] D. J. Thouless. The quantum mechanics of many-body systems. Academic
Press, New York, 2 edition, 1972.
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