Improved version of simplified method for including tensor effect in cluster models

The tensor contribution can be directly incorporated in the cluster model in a simplified way. In conventional $\alpha$ cluster models, the contribution of the non-central interactions exactly cancels because of the antisymmetrization effect and spatial symmetry of $\alpha$ clusters. The mixing of breaking components of $\alpha$ clusters to take into account the spin-orbit and tensor effects is needed. Previously we proposed a simplified method to include the spin-orbit effect, and also for the tensor part, a simplified model to directly take into account the contribution of the tensor interaction (called SMT) was introduced; however the contribution of the tensor interaction was quite limited. Here we improve SMT, which is called $i$SMT. Using newly proposed $i$SMT, the contribution of the tensor interaction in $^4$He is more than $-40$ MeV, four times larger than the previous version. The method is applied to four-$\alpha$ cluster structure of $^{16}$O. In $^{16}$O, the tensor contribution is also large, and this is coming from the finite size effect for the distances among $\alpha$ clusters with a tetrahedral configuration.