Table 1.

Estimated free energy shifts of the Na⁺:Cl⁻ system from VFEP, MBAR, and WHAM.

Window (α)	x̄ α	$\Delta f_{\text{VFEP}}^{\alpha }$	$\Delta f_{\text{MBAR}}^{\alpha }$	$\Delta f_{\text{WHAM}}^{\alpha }$	$\Delta \Delta f_{\text{VFEP/MBAR}}^{\alpha }$ (×103)	$\Delta \Delta f_{\text{VFEP/WHAM}}^{\alpha }$ (×103)
1	2.629	-	-	-	-	-
2	2.655	−0.241	−0.240	−0.240	−0.840	−0.900
3	2.686	−0.021	−0.017	−0.017	−3.340	−3.350
4	2.754	0.629	0.633	0.633	−4.060	−4.030
5	3.113	1.476	1.477	1.479	−1.140	−3.010
6	3.945	1.512	1.511	1.519	0.790	−7.150
7	4.275	0.694	0.694	0.705	−0.410	−11.070
8	4.460	−0.084	−0.083	−0.071	−1.100	−13.410
9	4.638	−0.662	−0.660	−0.646	−1.520	−15.450
10	4.797	−1.048	−1.047	−1.031	−1.840	−17.370
11	4.964	−1.264	−1.262	−1.245	−2.050	−19.250
12	5.127	−1.317	−1.315	−1.297	−2.120	−20.870
13	5.334	−1.225	−1.223	−1.204	−2.040	−21.840
14	5.588	−1.056	−1.054	−1.033	−1.860	−22.610
15	5.863	−0.938	−0.937	−0.914	−1.720	−24.320
16	6.188	−0.970	−0.968	−0.943	−1.770	−26.700
17	6.500	−1.114	−1.112	−1.083	−1.930	−30.880
18	6.719	−1.285	−1.283	−1.235	−2.110	−49.640
19	6.955	−1.431	−1.429	−1.308	−2.130	−122.670
20	7.194	−1.532	−1.530	−1.206	−2.050	−325.970
21	7.413	−1.588	−1.586	−0.832	−2.500	−756.610
RMS					1.990	182.73

The numbers here are derived from the same set of simulation as in Table 2. x̄^α is the average of the sampled coordinates of the α^th window. For the VFEP approach, the $\Delta f_{\text{VFEP}}^{\alpha }$’s are calculated from −ln Z^α and the values relative to the first window are listed. $\Delta f_{\text{MBAR}}^{\alpha }$ and $\Delta f_{\text{WHAM}}^{\alpha }$ values were taken directly from the MBAR and WHAM output, respectively. The last two columns are differences (multiplied by 10³) between these three types of Δf’s. The last row (RMS) is the root-mean-square values of the corresponding column. All values are in units of k_BT, except for x̄^α, which is in units of Å.