The ground-state energy of a quantum electron-hole liquid in the three cases (i) a single isotropic maximum for the hole band and a single minimum for the conduction band (called the model system), (ii) Ge under a large [111] strain, and (iii) Si under a large [100] strain, has been calculated in the random-phase (RPA), Hubbard (HA), and fully-self-consistent (FSC) approximations. The last approximation takes into account multiple scatterings to infinite order between all components of the plasma. Effects of anisotropy of bands have also been fully considered. Besides the ground-state energy, we have also calculated the partial-pair-correlation functions and enhancement factors. For the model system, calculations have also been made for different mass ratios. Two important results of this paper are (i) the electron-hole liquid in both Ge and Si under a large uniaxial strain should exhibit a metallic phase relative to a free excitonic phase. The calculated binding energies for Ge[111] and Si[100] are, respectively, 4.9 °K and 21.8 °K, corresponding to equilibrium densities of 1.11 ×

10^{16}

and 4.47 ×

10^{17}

{cm}^{- 3}

. (ii) The enhancement factors at the equilibrium density for Ge[111] and Si[100] are, respectively, 6.8 and 7.4. We suggest that experiments should be done to test these predictions of the FSC theory because of their bearing on the validity of different many-body approximations.

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