  Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$
over the complex numbers. For a fixed natural number $d$, let $R_d(X_0)$ be the
open sub-scheme of the Hilbert scheme, parameterizing irreducible rational
curves of degree $d$ on $X_0$. In this paper, we show that (1) $R_d(X_0)$ is
smooth and of expected dimension, \par (2) Combining the Calabi-Yau condition
on $X_0$, we further show that it consists of
  immersed rational curves.
  (3) Parts (1) and (2) imply a statement of Clemens' conjecture: if $C_0\in
R_d(X_0)$ and $c_0:\mathbf P^1\to C_0$ is the normalization, the
\par\hspace{1cc} normal sheaf
  is isomorphic to the vector bundle $$N_{c_0/X_0}\simeq \mathcal O_{\mathbf
P^1}(-1)\oplus \mathcal O_{\mathbf P^1}(-1).$$
