  The ionic atmospheres around nucleic acids play important roles in biological
function. Large-scale explicit solvent simulations coupled to experimental
assays such as anomalous small-angle X-ray scattering (ASAXS) can provide
important insights into the structure and energetics of such atmospheres but
are time- and resource-intensive. In this paper, we use classical density
functional theory (cDFT) to explore the balance between ion-DNA, ion-water, and
ion-ion interactions in ionic atmospheres of RbCl, SrCl$_2$, and CoHexCl$_3$
(cobalt hexammine chloride) around a B-form DNA molecule. The accuracy of the
cDFT calculations was assessed by comparison between simulated and experimental
ASAXS curves, demonstrating that an accurate model should take into account
ion-ion correlation and ion hydration forces, DNA topology, and the discrete
distribution of charges on DNA strands. As expected, these calculations
revealed significant differences between monovalent, divalent, and trivalent
cation distributions around DNA. About half of the DNA-bound Rb$^+$ ions
penetrate into the minor groove of the DNA and half adsorb on the DNA strands.
The fraction of cations in the minor groove decreases for the larger Sr$^{2+}$
ions and becomes zero for CoHex$^{3+}$ ions, which all adsorb on the DNA
strands. The distribution of CoHex$^{3+}$ ions is mainly determined by Coulomb
and steric interactions, while ion-correlation forces play a central role in
the monovalent Rb$^+$ distribution and a combination of ion-correlation and
hydration forces affect the Sr$^{2+}$ distribution around DNA.
