  We consider a nonnegative self-adjoint operator $L$ on $L^2(X)$, where
$X\subseteq \mathbb{R}^d$. Under certain assumptions, we prove atomic
characterizations of the Hardy space $$H^1(L) = \l \{f\in L^1(X) \ : \ \
{\|}\sup_{t>0} \ |\exp(-tL)f \ | \ {\|}_{L^1(X)}<\infty\ \}.$$ We state simple
conditions, such that $H^1(L)$ is characterized by atoms being either the
classical atoms on $X\subseteq \mathbb{R}^d$ or local atoms of the form
$|Q|^{-1}\chi_Q$, where $Q\subseteq X$ is a cube (or cuboid).
  One of our main motivation is to study multidimensional operators related to
orthogonal expansions. We prove that if two operators $L_1, L_2$ satisfy the
assumptions of our theorem, then the sum $L_1 + L_2$ also does. As a
consequence, we give atomic characterizations for multidimensional Bessel,
Laguerre, and Schr\"odinger operators.
  As a by-product, under the same assumptions, we characterize $H^1(L)$ also by
the maximal operator related to the subordinate semigroup $\exp(-tL^\nu)$,
where $\nu\in(0,1)$.
