  We investigate Hardy spaces $H^1_L(X)$ corresponding to self-adjoint
operators $L$. Our main aim is to obtain a description of $H^1_L(X)$ in terms
of atomic decompositions similar to such characterisation of the classical
Hardy spaces $H^1(\mathbb{R}^d)$. Under suitable assumptions, such a
description was obtained by Yan and the authors in [Trans. Amer. Math. Soc. 375
(2022), no. 9, 6417-6451], where the atoms associated with an $L$-harmonic
function are considered. Here we continue this study and modify the previous
definition of atoms.
  The modified approach allows us to investigate settings, when the generating
operator is related to a system of linearly independent harmonic functions. In
this context, the cancellation condition for atoms is adjusted to fit this
system. In an explicit example, we consider a symmetric manifold with ends
$\mathbb{R}^d \# \mathbb{R}^d$. For this manifold the space of bounded harmonic
functions is two-dimensional. Any element from the Hardy space $H^1_L(X)$ has
to be orthogonal to all of the harmonic functions in the system.
