  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.
