Self-Consistent Equations Including Exchange and Correlation Effects
Physical Review, 1965 · paper · Physics
Builds on the Hohenberg-Kohn theory to derive self-consistent equations, analogous to Hartree and Hartree-Fock, for inhomogeneous interacting electrons including exchange and correlation. Finds Slater's exchange potential too large by a factor of 3/2, and treats finite temperature and magnetic fields.
Checked: PDF from courses.physics.ucsd.edu, fetched 5 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance Eq. (3.13) prints "S[n]=S_s[n]+∫n(r)(∂f_xc(n)/∂τ)_{n(r),V}dr". Combining (3.11) S=−(∂G/∂τ), (3.12) S_s=−(∂G_s/∂τ), (3.2) G=G_s+F_xc and (3.4) F_xc=∫n f_xc dr gives a minus sign before the integral instead.
- p. 4 Eqs. (3.11)–(3.13)
Combining Eqs. (3.11), (3.12), (3.2), and (3.4), we obtain
Only the authors can say what is right here.
Judge's note: Checked Eqs. (3.2)–(3.15) on page 4. The derivation the text cites yields S = S_s − ∫ n ∂f_xc/∂τ dr. Eq. (3.15) appears to carry the compensating opposite sign, since ∂f/∂τ = −entropy per particle, so (3.17) may still come out right. Only the authors can say which equation carries the intended sign; no alternative sign convention is stated in the paper.
Second check: another model agreed
Report a problem with this finding
Substance Eq. (4.6) prints "χ[n]=(1/V)(∂/∂H)∫m(r)dr=∫G⁻¹(r,r';[n])dr dr'". Substituting (4.5) for a uniform field gives (1/V)∫∫G⁻¹ dr dr', so the factor 1/V is missing from the right-hand side.
- p. 5 Eq. (4.6)
For a uniform field this gives for the susceptibility
Judge's note: The crop of Eq. (4.6) shows 1/V on the middle expression only. Eqs. (4.8)–(4.9) keep the explicit 1/V and N/V factors, so the paper does not set V to 1.
Second check: another model agreed