A generalization of the optimized random phase approximation in the thermodynamic perturbation theory of liquids is proposed. By introducing a new method, we can derive an approximate analytical expression for the free energy of simple liquids. This is achieved by invoking a direct correlation function more general than the mean spherical approximation. By correcting the unphysical behavior of the corresponding radial distribution function in the hard core region, we attain a new perturbation expansion as a natural generalization of the optimized random phase approximation. In order to assess its accuracy and reliability, the theory is employed to find the structure and thermodynamics of hard-core Yukawa fluids. The new resulted optimized second order random phase approximation (OSORPA) is shown to be a little more accurate than the optimized random phase approximation in finding the thermodynamic properties, but it performs much better in predicting the static structure factor of hard-core Yukawa potentials. (C) 2014 Elsevier B.V. All rights reserved.