  We give several versions of Siu's $\partial\bar{\partial}$-formula for maps
from a strictly pseudoconvex pseudo-Hermitian manifold $(M^{2m+1}, \theta)$
into a K\"ahler manifold $(N^n, g)$. We also define and study the notion of
pseudo-Hermitian harmonicity for maps from $M$ into $N$. In particular, we
prove a CR version of Siu Rigidity Theorem for pseudo-Hermitian harmonic maps
from a pseudo-Hermitian manifold with vanishing Webster torsion into a K\"ahler
manifold having strongly negative curvature.
