Inhomogeneous Electron Gas

• p. 4 Eq. (42)

  kT≡(4kF)1/2

• p. 4 Eq. (42)

  kT ≡ (4kF)^{1/2}

• p. 4 Eq. (42)

  k_T ≡ (4k_F)^{1/2}

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Judge's note: My own working, from the page images (pp. 4–6). Eq. (50) gives g0 = (3/10)(3π²)^{2/3} n^{5/3}. With Eq. (51), g0'' = (1/3)(3π²)^{2/3} n^{-1/3} = k_F²/(3n) = π²/k_F, so g0''/4π = π/(4k_F). From Eq. (43), S(q)→1 as q→0, because ln|(q+2k_F)/(q−2k_F)| ≈ q/k_F. Eq. (41) then gives α ≈ 1 − q²/k_T², so c2 = −1/k_T². Eq. (78), g0''/4π = −c2, therefore requires 1/k_T² = π/(4k_F), i.e. k_T² = 4k_F/π, not the printed 4k_F. Eq. (58) carries the 2^{7/2}/3π factor, which is consistent with ħ = m = e = 1, so a unit convention does not absorb the π. The image shows no π in Eq. (42).

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• p. 7 Eq. (88)

  α(q)=2π/kT² [1+ q²/kT² S(q)]^-1

• p. 4 Eq. (41)

  α(q)=[1+(q²/kT²)S(q)]^-1

• p. 7 Eq. (88)

  α(q) = 2π/k_T² [1 + q²/k_T² S(q)]^-1

• p. 4 Eq. (41)

  α(q)=[1+(q²/k_T²)S(q)]^-1

Judge's note: Compared Eq. (41) on p. 4 with Eq. (88) on p. 7. At q→0, where S→1, Eq. (41) gives α→1, consistent with Eq. (38), α = 1 + c2q² + …, and with Fig. 1 (α(0) = 1.0). Eq. (88) gives 2π/k_T² instead. The text explicitly says Eq. (88) is what Eq. (41) gives, so this is one quantity printed in two inconsistent forms.

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• p. 7 Eq. (85)

  2.21

• p. 7 Eq. (87)

  0.88

• p. 2 Footnote 10

  Atomic units are used.

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Judge's note: Eq. (50) gives a kinetic energy per electron of (3/10)kF², with kF=1.919/rs, which is 1.105/rs² in the declared units. The printed 2.21 is twice this value, i.e. the Rydberg value. The unit label is missing or inconsistent.

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• p. 4 Text following Eq. (49), right column

  the gradient expansion (49) of G[n]

• p. 4 Sec. II.3, paragraph after Eq. (49)

  gradient expansion (49) of G[n]

Judge's note: On the page-4 image, Eq. (48) is G[n] = G[n0] + 2π[−c2∫ñ² + (c2²−c4)∫|∇ñ|² + …], followed by 'i.e., a gradient expansion'. Eq. (49) is r→∞: K(r) ~ const cos(2k_F r+δ)/r³. The sentence cites (49) for the gradient expansion, which is a wrong equation pointer.

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• p. 4 Text before Eq. (47)

  The power-series expansion of K(q), (43), leads to

Judge's note: Checked on the page-4 image. Eq. (43) defines S(q), and Eq. (44) is q→0: K(q) = 2π[−c2 + (c2²−c4)q² + …]. Later in the same column the paper itself writes 'power-series expansion (44) of K(q)'.

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• p. 7 Footnote 15

  E. P. Wigner, Phys. Rev. 40, 1002 (1934).

• p. 7 footnote 15

  E. P. Wigner, Phys. Rev. 40, 1002 (1934)

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Judge's note: Rests on outside knowledge: Wigner's 1934 paper is Phys. Rev. 46, 1002, and Phys. Rev. volume 40 is from 1932. The page-7 image confirms that footnote 15 prints volume 40 with year 1934 and page 1002.

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• p. 4 Eq. (34)

  λ³/Ω Σ K(q)|b1(q)|²

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Judge's note: On the page-4 image the last term of Eq. (34) appears with λ³ over Ω, while the other terms show λ². My working: substituting ñ = (λ/Ω)Σ b1 e^{-iq·r} (Eq. 28) into ∫K(r−r')ñ(r)ñ(r')drdr' with Eq. (25) gives (λ²/Ω)ΣK(q)|b1(q)|². Equating the λ² terms of Eqs. (33) and (34) then gives Eq. (35), K = (2π/q²)(1/α − 1). The exponent should be 2, and Eq. (35) is unaffected. The text layer for this term is garbled ('+—2 K(q)'), so the reading rests on the image.

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• p. 4 Eq. (47)

  K(r)=2π[−c2+(c2²−c4)∇²+···]δ(r)

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Judge's note: My working: ∇²e^{-iq·r} = −q²e^{-iq·r}, so (1/Ω)Σ q² e^{-iq·r} = −∇²δ(r). Eq. (44) therefore gives K(r) = 2π[−c2 − (c2²−c4)∇² + …]δ(r). Integrating by parts, ∫ñ(−∇²ñ) = ∫|∇ñ|², which reproduces the printed +(c2²−c4)∫|∇ñ|² in Eq. (48). With the printed +∇² in Eq. (47), Eq. (48) would carry a minus sign. Eq. (48) is consistent with Eq. (44); only the intermediate Eq. (47) disagrees.

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