Table 3.

Quasi-chemical components (in kcal/mol) in the calculation of hydration free energy of Na⁺ and K⁺ ions.

A	n	$\Delta G_{n,\Gamma }^A\left(g_W\right)$a$[\mathrm{Exp}.]$b	$n.\Delta G_{\mathit{conc}}^W$c	$\Delta G_{n,\Gamma }^A\left(g_W\right)+n.\Delta G_{\mathit{conc}}^W$	$\Delta G_{n,\Gamma }^{A{W}_n}\left(l_W\right)$d	n.ΔGW(lW)e	$\Delta G_n^A{\left(l_W\right)}^f$f
Na+	1	-18.6 [-18.8]	-4.3	-22.8	-61.2	-8.4	-75.6
	2	-34.2 [-31.9]	-8.5	-42.7	-60.2	-16.7	-86.2
	3	-43.3 [-40.9]	-12.8	-56.1	-59.8	-25.1	-90.9
	4	-49.4 [-46.9]	-17.1	-66.5	-60.2	-33.5	-93.3
	5	-49.0 [ - ]	-21.4	-70.4	-59.6	-41.9	-88.1
	6	-45.6 [ - ]	-25.6	-71.3	-59.1	-50.2	-80.1
K+	1	-12.0 [-11.8]	-4.3	-16.3	-52.9	-8.4	-60.8
	2	-21.7 [-20.7]	-8.5	-30.3	-53.7	-16.7	-67.2
	3	-27.7 [-27.0]	-12.8	-40.6	-55.0	-25.1	-70.4
	4	-32.0 [ - ]	-17.1	-49.0	-56.3	-33.5	-71.8
	5	-29.8 [ - ]	-21.4	-51.1	-56.1	-41.9	-65.4
	6	-26.9 [ - ]	-25.6	-52.5	-54.3	-50.2	-56.6
	7	-28.8 [ - ]	-29.9	-58.7	-54.9	-58.6	-55.0
	8	-30.6 [ - ]	-34.2	-64.8	-56.2	-67.0	-53.9

^a$\Delta G_{n,\Gamma }^A\left(g_W\right)$ is the intrinsic Gibbs free energy of ion-ligand association reaction at T = 298 K and P = 1 atm, and is estimated using equation 13. Computed values are compared to the experimental estimates of Tissandier et al.,⁴⁰ but the comparison is limited to lower order clusters, n ≤ 4 for Na⁺ and n ≤ 3 for K⁺. This is because separate calculations⁶⁷ at the MP2 level of theory show that the experimental estimates for higher order gas-phase clusters correspond to configurations where waters do not directly coordinate the ion, a scenario entirely different from the one considered for this quasi-chemical analysis.

^bReference ⁴⁰

^c$\Delta G_{\mathit{conc}}^W$ refers to the favorable gain in reaction free energy associated with a higher concentration of water molecules in the liquid phase as compared to the gas phase.

^d$\Delta G_{n,\Gamma }^{A{W}_n}\left(l_W\right)$ refers to the solvation free energy of an n-fold ion-water cluster in liquid water.

^eΔG^W (l^W) is the solvation free energy of a single water molecule W in liquid water.

^f$\Delta G_{n,}^A\left(l_W\right)$ is the solvation free energy of the ion coordinating with n water molecules in liquid water and is equal to the summation given by equation 12. The lowest most favorable value of $\Delta G_{n,}^A\left(l_W\right)$ corresponds to the ion’s absolute hydration free energy, as it is the dominant term.