The transition mechanism in high temperature cuprate superconductors is an outstanding puzzle. A previous suggestion on the role of non-linear local lattice instability modes on the microscopic pairing mechanism in high temperature cuprate superconductors cite{Lee:JSNM09} is re-examined to provide a viable mechanism for superconductivity in these cuprates via an unusual lattice vibration in which an electron is predominantly interacting with a nonlinear $Q_2$ mode of the oxygen clusters in the CuO$_2$ planes. It is shown that the interaction has explicit d-wave symmetry and leads to an indirect coupling of d-wave symmetry between electrons. As a follow-up of cite{Lee:JSNM09}, in this paper, we report detailed derivation of the superconducting gap equation and numerical solutions for the transition temperature as inherently integrated into the so-called Extended Hubbard Model (EHM). A unique feature in the EHM is that the transition temperature has an inherent k-dependence. In addition, superconducting gap solutions are restrained to specific regions in the first Brillouin zone (1BZ). It is very feasible to expect that the EHM naturally inherits a huge parameter space in which experimentally measured results, such as the well-known superconducting dome and the phase diagram from electronic Raman scattering cite{Sacuto:RPP13} can be accommodated. The EHM model hence offers a viable venue to search for or confirm any signature in k-point-sensitive experimental measurements.
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