Comparison of integral equations for correlations in liquid metallic hydrogen

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Abstract
The density-functional formalism leads to several integral equations for the pair correlations in a liquid metal as a coupled electron-ion mixture, with the aid of the hypernetted-chain (HNC) approximation and the information from electrons in the jellium model: the quantal HNC equation, the integral equations of Dharma-wardana and Perrot, and those of Ichimaru and collaborators. The latter two are extended to include the effects of the electron-ion correlation, which are neglected in their earlier treatments. By solving these five integral equations numerically for the case of liquid-metallic hydrogen with fully degenerate electrons, the limitations of their approximations are investigated. Furthermore, the approximation to use the exchange-correlation effect in the jellium model in place of that for the electrons coupled with the protons is examined on the basis of the quantal HNC equation.