Linear response and Time dependent Hartree-Fock

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and therefore, for ω >= Imχ(r, r ′ ; ω) = π ∑ n>0 〈0| ˆρ(r) |n〉〈n| ˆρ(r ′ ) |0〉 δ(¯hω − ¯hω n0 ) (20) (The definition of common factors and signs differs in different textbooks !) In scattering experiments, one observes the momentum transfer, so the ∫ χ(q, ω) = d 3 rd 3 r ′ e iq·(r−r′) χ(r, r ′ ; ω) = ∑ [ |〈0| ˆρq |n〉| 2 n>0 ¯hω − ¯hω n0 + iη − |〈0| ˆρ q |n〉| 2 ] . (21) ¯hω + ¯hω n0 + iη 2 Excited state wave function and the action integral It comes now to the point where we must some definite model for the excited states of a physical system. This can not be done in general; we must make some approximations. With all the reservations one can have about the Hartree-Fock approximation, we start with that one. We approximate the wave function of excited states by |ψ(t)〉 = 1 ∑ N (t) e−iE HF t/¯h e 1 2 ph c ph(t)a † pa h |φHF 〉 (22) where |φ HF 〉 is a Hartree-Fock ground state. E HF the Hartree-Fock energy expectation value, and the c ph (t) “particle–hole amplitudes”. N 2 (t) = 〈φ HF | e 1 2 ∑ ph c∗ ph (t)a† h a p e 1 2 ∑ ph c ph(t)a † pa h |φHF 〉 (23) is the normalization integral. Note that states that are labeled with h, h i , h ′ , . . . are understood to be occupied (“hole”) states (i.e. states that are present in the Slater determinant φ HF and states labeled with p, p i , p ′ , . . . are unoccupied “particle” states. As above, assume that the system is described by a Hamiltonian that is written in coordinate space as H = H 1 + V (24) H 1 = ∑ h(r i ) = ∑ ] [− ¯h2 i i 2m ∇2 i + U ext (r i ) (25) V = ∑ v(r i , r j ) (26) i
or, in second quantized form H 1 = ∑ αβ 〈α| h ext |β〉 a † αa β (27) V = 1 ∑ 〈αβ| v |γδ〉 a † 2 αa † β a δ a γ . (28) αβγδ We furthermore subject the system to a weak, scalar external field δH ext (t) = ∑ i δh ext (r i ; t) (29) or, in second quantized form, δH 1 (t) = ∑ αβ 〈α| δh(t) |β〉 a † αa β . (30) Similar to the determination of the single-particle wave functions of the Hartree-Fock theory by a variational principle, the “particle-hole amplitudes” are determined by the “Kerman-Koonin” stationarity principle ∫ t1 δS = δ dtL(t) (31) t 0 L(t) = 〈ψ(t)| H + δH ext − i¯h ∂ |ψ(t)〉 . (32) ∂t We want to derive linear equations of motion, hence we must keep all second order terms in the Lagrangian. The driving quantity is the external perturbing field δH 1 (t) which is, by assumption, of first order. Linear equations of motion will lead to c ph (t) that are of the same order. Thus, to get linear equations of motion, we must keep the linear terms in c ph (t) with the perturbing field, and the quadratic terms in all others. Thus: 〈ψ(t)| δH ext |ψ(t)〉 ≈ 1 ∑ [ c ∗ 2 ph 〈φ HF | a † h a pδH ext (t) |φ HF 〉 + c ph 〈φ HF | δH ext (t)a † pa h |φ HF 〉 ] = 1 2 = 1 2 = 1 2 ph ∑ phαβ ∑ phαβ ∑ ph 〈α| δh ext (t) |β〉 [ c ∗ ph 〈φ HF | a † h a pa † αa β |φ HF 〉 + c ph 〈φ HF | a † αa β a † pa h |φ HF 〉 ] 〈α| δh ext (t) |β〉 [ c ∗ phδ α,p δ β,h + c ph δ β,p δ α,h ] [ c ∗ ph 〈p| δh ext (t) |h〉 + c ph 〈h| δh ext (t) |p〉 ] . (33) 5
Page 1 and 2: Linear response and Time dependent
Page 3: with ω n0 = ω n − ω 0 . We can
Page 7 and 8: + 1 ∑ c ∗ 4 phc p ′ h ′ 〈
Page 9 and 10: = 1 ∑ 〈pβ| v |γδ〉〈φ HF
Page 11 and 12: first order in the particle-hole am
Page 13: χ (−) 0 (r, r ′ , ω) = 1 ∑

Linear response and Time dependent Hartree-Fock

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