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