Switching on the electron-electron interaction connects the Kohn-Sham to the physical system. The correlation energy, the only unknown energy component in this process, is determined at fixed density, using a technique based on the Lieb Legendre transform definition of the universal density functional. Results are shown for this adiabatic coupling process for He, Ne8+, Be, Ne6+ as well as for the exponential densities n(N,zeta)(r) = N(zeta(3)/pi)e(-2 zeta r) (N = 2 or 4; zeta greater than or equal to 1; for N = 4 degeneracy is present and D-1 and P-3 are analyzed). The data are fitted to a rational approximant and appear to be in good agreement with those given by the less computationally demanding Harris-Jones adiabatic connection.