  Let $I$ be ideal of an $n$-dimensional local Gorenstein ring $R$. In this
paper we will describe several necessary and sufficient conditions such that
the ideal $I$ becomes cohomologically complete intersections. In fact, as a
technical tool, it will be shown that the vanishing $H^i_{I}(R)= 0$ for all
$i\neq c= \grade (I)$ is equivalent to the vanishing of the Betti numbers of
$H^c_{I}(R)$. This gives a new characterization to check the cohomologically
complete intersections property with the homological properties of the
vanishing of Tor modules of $H^c_{I}(R)$.
