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Consider the subring $\mathcal{R}_cL$ of continuous real-valued functions defined on a frame $L$, comprising functions with a countable pointfree image. We present some useful properties of $\mathcal{R}_cL$. We establish that both $\mathcal{R}_cL$ and its bounded part, $\mathcal{R}_c^*L$, are clean rings for any frame $L$. We show that, for any completely regular frame $L$, the $z_c$-ideals of $\mathcal{R}_cL$ are contractions of the $z$-ideals of $\mathcal{R}L$. This leads to the conclusion that maximal ideals (or prime $z_c$-ideals) of $\mathcal{R}_cL$ correspond precisely to the contractions of those of $\mathcal{R}L$. We introduce the ${\bf O}_c$- and ${\bf M}_c$-ideals of $\mathcal{R}_cL$. By using ${\bf M}_c$-ideals, we characterize the maximal ideals of $\mathcal{R}_cL$, drawing an analogy with the Gelfand-Kolmogoroff theorem for the maximal ideals of $C_c(X)$. We demonstrate that fixed maximal ideals of $\mathcal{R}_cL$ have a one-to-one correspondence with the points of $L$ in the case where $L$ is a zero-dimensional frame. We describe the maximal ideals of $\mathcal{R}_c^*L$, leading to a one-to-one correspondence between these ideals and the points of $\beta L$, the Stone-\v{C}ech compactification of $L$, when $L$ is a strongly zero-dimensional frame. Finally, we establish that $\beta_0L$, the Banaschewski compactification of a zero-dimensional $L$, is isomorphic to the frames of the structure spaces of $\mathcal{R}_cL$, $\mathcal{R}_c(\beta_0L)$, and $\mathcal{R}(\beta_0L)$.

In this work, we investigate the excited-state solute and solvation structure of $\mathrm{[Ru(bpy)_3]^{2+}}$, $\mathrm{[Fe(bpy)_3]^{2+}}$, $\mathrm{[Fe(bmip)_2]^{2+}}$ and $\mathrm{[Cu(phen)_2]^{+}}$ (bpy=2,2'-pyridine; bmip=2,6-bis(3-methyl-imidazole-1-ylidine)-pyridine; phen=1,10-phenanthroline) transition metal complexes (TMCs) in terms of solute-solvent radial distribution functions (RDFs) and evaluate the performance of some of the most popular partial atomic charge (PAC) methods for obtaining these RDFs by molecular dynamics (MD) simulations. To this end, we compare classical MD of a frozen solute in water and acetonitrile (ACN) with quantum mechanics/molecular mechanics Born-Oppenheimer molecular dynamics (QM/MM BOMD) simulations. The calculated RDFs show that the choice of a suitable PAC method is dependent on the coordination number of the metal, denticity of the ligands, and type of solvent. It is found that this selection is less sensitive for water than ACN. Furthermore, a careful choice of the PAC method should be considered for TMCs that exhibit a free direct coordination site, such as $\mathrm{[Cu(phen)_2]^{+}}$. The results of this work show that fast classical MD simulations with ChelpG/RESP or CM5 PACs can produce RDFs close to those obtained by QM/MM MD and thus, provide reliable solvation structures of TMCs to be used, e.g. in the analysis of scattering data.

The aim of this article is to survey ring-theoretic properties of Kasch, the regularity and the injectivity of the ring of real-continuous functions on a topoframe $L_{ \tau}$, i.e., $\mathcal{R}L_\tau$. In order to study these properties, the concept of $P$-spaces and extremally disconnected spaces are extend to topoframes. For a $P$- topoframe $L_{ \tau}$, the ring $\mathcal{R}L_\tau$ is $\aleph_0$-Kasch ring. $P$- topoframes are characterized in terms of ring-theoretic properties of the regularity and injectivity of the ring of real-continuous functions on a topoframe. It follows from these characterizations that the ring $\mathcal{R}L_\tau$ is regular if and only if it is $\aleph_0$-selfinjective. For a completely regular topoframe $L_\tau$, we show that $\mathcal{R}L_\tau$ is a Bear ring if and only if it is a $CS$-ring if and only if $L_\tau$ is extremally disconnected and also prove that it is selfinjective ring if and only if $L_{ \tau}$ is an extremally disconnected $P$-topoframe.