We consider a nonnegative self-adjoint operator L on L^2(X), where Xsubseteq {{mathbb {R}}}^d. Under certain assumptions, we prove atomic characterizations of the Hardy space H1(L)=f∈L1(X):supt>0exp(-tL)fL1(X)<∞.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} H^1(L) = \\left\\{ f\\in L^1(X) \\ : \\ \\left\\| \\sup _{t>0} \\left| \\exp (-tL)f \\right| \\right\\| _{L^1(X)}<\\infty \\right\\} . \\end{aligned}$$\\end{document}We state simple conditions, such that H^1(L) is characterized by atoms being either the classical atoms on Xsubseteq {mathbb {R}^d} or local atoms of the form |Q|^{-1}chi _Q, where Qsubseteq 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 Schrödinger 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).