Electron correlations: I. Ground-state results in the high-density regime

The coupled-cluster or $\exp \mathrm{}\left(S\right)\mathrm{}$ formalism is applied to the problem of correlations in a many-electron system. In this first work in a series of papers we concentrate on exact results and restrict ourselves to various ring approximations applicable essentially to the high-density regime. It is shown that the well-known random-phase approximation (RPA) can be formulated as a nonlinear three-dimensional integral equation for the four-point function $S_2$ which provides a measure of the two-particle—hole pair component in the true ground-state wave function. Exact analytic solutions are presented for both $S_2$ and the three-point particle-hole vertex function in the RPA, and their properties are discussed. The Tamm-Dancoff approximation to the ring summation is also formulated, and the analagous exact solutions are presented for the first time. It is demonstrated that both solutions are unique, and a comparison of the two approximations is presented in terms of the correlation energy.