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17. Strictly speaking, ø(τ)) describes a fictitious dynamics of an electron-hole pair, since |ø(τ = 0)) ξ|P) = H−λ J|0) involves the Hamiltonian. Viewed differently, |P) can be thought of as frequency-dependent (H−1 in the expression for |P) arises from the I/ω2 prefactor of Eq. 5).
18. The author has not found a single, simple reason for this trend. It may be a combi- nation of variations in electron density, band width (and therefore electron and hole hopping parameters), bond length, etc.
19. The quantity fo dωωτ2 (ω) is equal for non-interacting theories and conserving in- teracting theories, due to the cancellation of repulsive and attractive terms of the interaction. It should be mentioned, however, that the theory we use (characterized by Eqs. 1, 2 and known as the Tamm-Dancoff approximation) is not a conserving approximation. Whether conserving or non-conserving, an attractive term, Hattr, can be defined for which (P|Hattr|P) is non-zero, measuring the downward shift of oscillator strength.
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23. This approximation neglects contributions from virtual transitions between excited electron and hole states. Such contributions can be thought of as giving rise to a widened gap, which would alter tBG(q), particularly near q = 0. Since the free carrier terms dominate at small q, we neglect this correction. We hasten to add that the static screening approximation is not as justified here as it is in the n = 0 case, due to the presence of excited charge-carrier plasmons. Unfortunately, the difficulty of solving the Bethe-Salpeter equation with a dynamically screened interaction for an infinite system precludes the possibility of taking this important effect into account in this work.