  Using an angular momentum projected single particle basis, a pnQRPA approach
is used to study the $2\nu\beta\beta$ properties of ten isotopes, exhibiting
various quadrupole deformations. The mother and daughter nuclei exhibit
different quadrupole deformations. Since the projected basis enables a unified
description of deformed and spherical nuclei, situations where the nuclei
involved in the double beta decay process are both spherical, both deformed or
one spherical and another deformed, can be treated through a sole formalism.
Dependence of single $\beta^-$ and $\beta^+$ strength distribution on atomic
mass number and nuclear deformation is analyzed. For the double beta decay
process, the Gamow-Teller transition amplitudes and half lives are calculated.
Results are compared with the experimental data as well as with the predictions
of other theoretical approaches. The agreement between the present results and
experimental data is fairly good.
