Starting from the FRW metric (for simplicity flat space, radial direction only):
$$ds^2=-c^2dt^2+a(t)^2dr^2$$
If we take $dt=0$ then the proper distance $ds(t)$ between two spatially separated points at cosmological time $t$ is given by:
$$ds(t)=a(t)dr$$
Now at the present time $t_0$ we can define $a(t_0)=1$ so that we also have:
$$ds(t_0)=dr$$
Therefore by eliminating $dr$ in the above equations we find:
$$ds(t)=a(t)\ ds(t_0)$$
If we define $ds(t)=1$ so that a hydrogen atom has a unit proper diameter, at any time $t$, then the equivalent diameter at the present time $t_0$ is given by:
$$ds(t_0)=\frac{1}{a(t)}$$
According to quantum mechanics the mass/energy of a quantum system is inversely proportional to its size.
Therefore if the mass/energy of the hydrogen atom at time $t$ is one unit then the mass/energy of an equivalent atomic system at the present time $t_0$ is $a(t)$ units.
Thus can one infer that hydrogen atoms at time $t$ in the future have an energy that is a factor $a(t)$ higher relative to the energy of hydrogen atoms today?
