Table 2.

Average electrostatic potential $⟨\mathit{\varphi }({\mathbf{r}}_I)⟩$ and associated root-mean-square fluctuation (rmsf) at a sodium ion site, average physical potential energy per water atom in the restraint region $⟨{\tilde{u}}_s⟩$ and associated rmsf, as well as total average restraint energy $⟨V^{\mathrm{restr}}⟩$ and average restraint energy per water atom in the restraint region $⟨V^{\mathrm{restr}}⟩$ monitored during 1 ns simulation of a hydrated sodium ion without (“unres.”) and with application of a polarization restraint (Eq. 26) to the Born polarization (Eq. 30, using ${\mathit{ϵ}}_S^{\prime }=66.6$) within the reported ranges of 1.0–1.4, 1.0–1.9, 1.1–1.9 and 1.2–1.9 nm from the ion. ${\tilde{u}}_s$ was calculated for the range 0.0–1.4 nm for unrestrained simulations or for the restraining range in restrained simulations (Eqs. 45 and 47)

Scheme	Res. range (nm)	$⟨\mathit{\varphi }({\mathbf{r}}_I)⟩$ (rmsf) [$\text{kJ}{\text{mol}}^{-1}{e}^{-1}]$	$⟨{\tilde{u}}_s⟩$ (rmsf) [$\text{kJ}{\text{mol}}^{-1}]$	$⟨V^{\mathrm{restr}}⟩$ [$\text{kJ}{\text{mol}}^{-1}$]	$⟨v^{\mathrm{restr}}⟩$ [$\text{kJ}{\text{mol}}^{-1}$]
BM	Unres.	−657.72 (43.65)	−27.67 (0.21)	–	–
	1.0–1.4	−655.89 (44.90)	−27.19 (0.31)	4,985.8	7.1
	1.0–1.9	−656.43 (44.83)	−27.19 (0.15)	10,895.8	4.6
	1.1–1.9	−660.72 (43.97)	−27.32 (0.17)	9,984.7	4.5
	1.2–1.9	−660.41 (43.85)	−27.41 (0.18)	8,822.9	4.3
CM	Unres.	−812.22 (45.66)	−27.95 (0.22)	–	–
	1.0–1.4	−788.98 (47.58)	−27.28 (0.32)	5,944.1	8.5
	1.0–1.9	−792.51 (47.39)	−27.49 (0.17)	13,284.2	5.6
	1.1–1.9	−798.73 (46.89)	−27.61 (0.17)	11,919.1	5.3
	1.2–1.9	−803.56 (46.58)	−27.69 (0.17)	10,658.6	5.1
BA	Unres.	−654.57 (43.04)	−27.71 (0.32)	–	–
	1.0–1.4	−651.95 (44.30)	−26.97 (0.56)	4,988.1	7.2
	1.0–1.9	−651.78 (45.22)	−27.22 (0.25)	11,093.5	4.7
	1.1–1.9	−654.68 (44.87)	−27.34 (0.25)	9,924.6	4.4
	1.2–1.9	−654.32 (43.97)	−27.44 (0.26)	8,990.5	4.3