Table 2.

Estimated errors for the free energy profile of the Na⁺:Cl⁻ system from VFEP andMBAR

Window (α)	x̄α	$\Delta {\widehat{\ell }}^{\alpha }(s,g)$	$\Delta {\widehat{\ell }}^{\alpha }(s,m)$	stat error (N=50)	stat error (N=100)	MBAR
1	2.629	0.022	0.060	0.018	0.018	0.043
2	2.655	0.040	0.015	0.018	0.018	0.042
3	2.686	0.125	−0.001	0.018	0.018	0.049
4	2.754	0.421	−0.034	0.018	0.018	0.070
5	3.113	0.555	−0.071	0.018	0.015	0.086
6	3.945	0.304	−0.034	0.018	0.010	0.073
7	4.275	0.160	−0.113	0.010	0.010	0.052
8	4.460	0.016	0.013	0.010	0.010	0.042
9	4.638	0.002	−0.025	0.010	0.010	0.040
10	4.797	0.002	0.007	0.009	0.009	0.040
11	4.964	0.002	0.012	0.008	0.008	0.039
12	5.127	0.003	−0.034	0.008	0.008	0.038
13	5.334	0.016	−0.008	0.009	0.008	0.039
14	5.588	0.015	−0.006	0.009	0.009	0.041
15	5.863	0.006	0.001	0.010	0.010	0.041
16	6.188	0.006	−0.012	0.010	0.011	0.038
17	6.500	0.002	−0.007	0.012	0.012	0.037
18	6.719	0.002	−0.012	0.013	0.014	0.036
19	6.955	0.000	0.010	0.015	0.015	0.035
20	7.194	0.000	−0.014	0.016	0.017	0.035
21	7.413	0.000	−0.004	0.017	0.018	0.000
RMS		0.172	0.036	0.013	0.013	0.048

The numbers here are derived from a 21-window umbrella sampling simulation on a Na⁺:Cl⁻ pair in a TIP3P water box with a biasing potential of 5 kcal/mol/Ų (see the Results section, also Figure 2). x̄^α is the average of the sampled coordinates of the αth window. $\Delta {\widehat{\ell }}^{\alpha }(s,m)$ and $\Delta {\widehat{\ell }}^{\alpha }(s,g)$ are likelihood errors defined in Eq. (8) and Eq. (9), respectively. The “stat error” is the statistical error estimated by performing bootstrap error analysis on the free energy shift term, −lnZ^α , with the same calculations performed on 50 or 100 randomly chosen data sets. The numbers reported are the standard deviations of −lnZ^α from different sets of data. MBAR errors are from the MBAR output. 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.