The elastic collision between two ortho-positronium (e.g. $S = 1$) atoms is studied using an {\it ab-initio} static exchange model (SEM) in the centre of mass (CM) frame by considering the system as a four-body Coulomb problem where all the Coulomb interaction terms in the direct and exchange channels are treated exactly. A coupled channel methodology in momentum space is used to solve Lippman–Schwinger equation following the integral approach. A new SEM code is developed in which the Born–Oppenheimer (BO) scattering amplitude acts as input to derive the SEM amplitude adapting the partial wave analysis. The $s$-, $p$- and $d$-wave elastic phase shifts and the corresponding partial cross-sections for the spin alignment $S = 0$, i.e., singlet (+) and $S = 2$, i.e., triplet (−) states are studied. An augmented Born approximation is used to includethe contribution of higher partial waves more accurately to determine the total/integrated elastic cross-section $(\sigma)$, the quenching cross-section (σq) and ortho-to-para conversion ratio $(\sigma/\sigma q)$. The effective range theory is used to determine the scattering lengths and effective ranges in the s-wave elastic scattering. The theory includes the non-adiabatic short-range effects due to exchange.