liquid Na at 371 K, 927 kg/m^3: r_s = 4.0483 bohr; measured speed: CRC (373 K) 2531.0 m/s; Fink & Leibowitz fit (371 K) 2508.5 m/s
measured adiabatic modulus rho c^2 = 5.938 GPa (CRC speed)
local-field slope gamma0 = 0.2813 (exchange alone: 0.25); band-structure sum over 32084 reciprocal vectors up to 40 k_F

ladder (B in GPa, speed in m/s; percent against CRC 2531):
  1. Bohm-Staver: free-electron gas pressure only
      r_c =   -     static B =  7.931   adiabatic B =  7.931   c =  2924.9  (+15.6%)
  2. + exchange, correlation, electrostatics, ion core; static; r_c from zero pressure
      r_c = 1.788 bohr   static B =  6.829   adiabatic B =  6.829   c =  2714.1  (+7.2%)
  3. + heat: ions' thermal pressure in the fit, and adiabatic (Cp/Cv)
      r_c = 1.727 bohr   static B =  5.796   adiabatic B =  6.416   c =  2630.9  (+3.9%)
  4. + band structure (screening clouds, second order; bcc arrangement)
      r_c = 1.971 bohr   static B =  5.963   adiabatic B =  6.622   c =  2672.6  (+5.6%)

  step 3 details: thermal pressure 0.516 GPa; isothermal B 5.796; Cp/Cv 1.1070 (Fink & Leibowitz 1.383/1.262 = 1.0959)
  step 4 details: thermal pressure 0.531 GPa; isothermal B 5.963; Cp/Cv 1.1104

contributions to the static bulk modulus at the liquid density, GPa (step 4 constants):
  kinetic           +7.931
  exchange          -5.325
  correlation       -0.666
  electrostatic    -10.413
  ion core         +18.604
  band structure    -4.167
  sum               +5.963   (heat then adds +0.658: adiabatic B 6.622; measured 5.938)

sensitivity of the step-4 speed (each changed alone):
  q = 0.5 (volume dependence of Grueneisen gamma)    c =  2726.5  (+53.9 m/s)
  q = 2                                              c =  2573.8  (-98.8 m/s)
  thermal expansion -5%                              c =  2673.7  (+1.1 m/s)
  thermal expansion +5%                              c =  2672.3  (-0.3 m/s)
  Cp -2%                                             c =  2675.7  (+3.0 m/s)
  screening without exchange-correlation hole (G = 0) c =  2644.2  (-28.5 m/s)
  electrostatics of a sphere (0.9) instead of bcc    c =  2684.0  (+11.3 m/s)

check, cold crystal from the step-4 model: zero pressure at r_s = 3.947 (bcc a = 4.242 A), static B there 8.07 GPa
