  Let $L = \Delta + V$ be a Schr\"odinger operator with a non-negative
potential $V$ on a complete Riemannian manifold $M$. We prove that the vertical
Littlewood-Paley-Stein functional associated with $L$ is bounded on $L^p(M)$
{\it if and only if} the set $\{\sqrt{t}\, \nabla e^{-tL}, \, t > 0\}$ is
${\mathcal R}$-bounded on $L^p(M)$. We also introduce and study more general
functionals. For a sequence of functions $m_k : [0, \infty) \to \mathbb{C}$, we
define $$H((f_k)) := \Big( \sum_k \int_0^\infty | \nabla m_k(tL) f_k |^2 dt
\Big)^{1/2} + \Big( \sum_k \int_0^\infty | \sqrt{V} m_k(tL) f_k |^2 dt
\Big)^{1/2}.$$ Under fairly reasonable assumptions on $M$ we prove boundedness
of $H$ on $L^p(M)$ in the sense $$\| H((f_k)) \|_p \le C\, \Big\| \Big( \sum_k
|f_k|^2 \Big)^{1/2} \Big\|_p$$ for some constant $C$ independent of $(f_k)_k$.
A lower estimate is also proved on the dual space $L^{p'}$. We introduce and
study boundedness of other Littlewood-Paley-Stein type functionals and discuss
their relationships to the Riesz transform. Several examples are given in the
paper.
