  In this paper we introduce, according to one of the main ideas of
$\tau$-tilting theory, the $\tau$-tilting Hochschild cohomology in degree one
of a finite dimensional $k$-algebra $\la$, where $k$ is a field. We define the
excess of $\la$ as the difference between the dimensions of the $\tau$-tilting
Hochschild cohomology in degree one and the dimension of the usual Hochschild
cohomology in degree one.
  One of the main results is that for a zero excess bound quiver algebra
$\la=kQ/I$, the Hochschild cohomology in degree two $HH^2(\la) $ is isomorphic
to the space of morphisms $\Hom_{kQ-kQ}(I/I^2, \la).$ This may be useful to
determine when $HH^2(\la)=0$ for these algebras.
  We compute the excess for hereditary, radical square zero and monomial
triangular algebras. For a bound quiver algebra $\la$, a formula for the excess
of $\la$ is obtained. We also give a criterion for $\la$ to be $\tau$-rigid.
