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We have avoided directly computing the integrand at q=0 because the AIMD trajectories may be too short to adequately sample the small q regions. 〈 dH (q) /dq 〉 at such q values exhibit larger statistical fluctuations. To test the extrapolation to q=0, we have conducted classical force field TI simulations with much longer trajectory lengths but otherwise identical TI protocol and compared to 〈 dH (q) /dq 〉 directly computed at q=0. These simulations indicate that a cubic fit using our set of 6q values yields a good approximation to the q=0 integrand.
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We have avoided directly computing the integrand at q=0 because the AIMD trajectories may be too short to adequately sample the small q regions. 〈 dH (q) /dq 〉 at such q values exhibit larger statistical fluctuations. To test the extrapolation to q=0, we have conducted classical force field TI simulations with much longer trajectory lengths but otherwise identical TI protocol and compared to 〈 dH (q) /dq 〉 directly computed at q=0. These simulations indicate that a cubic fit using our set of 6q values yields a good approximation to the q=0 integrand.
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43
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66749139754
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Recall that this contribution is estimated using maximally localized Wannier functions to decompose the total electron density into individual water contributions (Ref.). As an additional test, we have taken the nuclear configuration of each of the 32 individual water molecules in an AIMD snapshot, computed the individual water φq contribution in the absence of other water molecules, added them, and compared the result with the global φq correction computed with all 32 H2 O simultaneously present in the same cell. Even though the individual H2 O approach neglects many-water effects, the two φq contributions computed are within 1%, or 1 kcal/mol, of each other.
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Recall that this contribution is estimated using maximally localized Wannier functions to decompose the total electron density into individual water contributions (Ref.). As an additional test, we have taken the nuclear configuration of each of the 32 individual water molecules in an AIMD snapshot, computed the individual water φq contribution in the absence of other water molecules, added them, and compared the result with the global φq correction computed with all 32 H2 O simultaneously present in the same cell. Even though the individual H2 O approach neglects many-water effects, the two φq contributions computed are within 1%, or 1 kcal/mol, of each other.
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This behavior is not universal. Attempting to put a partial (or an entire) 2s electron on the Li+ PP to yield Liq+ in water, as opposed to globally scaling that PP by the factor q, results in the partial electron leaving the vicinity of Liq+ and becoming solvated as an excess electron in water. In other words, if we were interested in the Li Li→ Li+ half cell reaction, a more complex λ -paths would have been needed. Spontaneous ejection of electrons does not happen with Ag or Ni+, both of which are less electropositive than Li.
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This behavior is not universal. Attempting to put a partial (or an entire) 2s electron on the Li+ PP to yield Liq+ in water, as opposed to globally scaling that PP by the factor q, results in the partial electron leaving the vicinity of Liq+ and becoming solvated as an excess electron in water. In other words, if we were interested in the Li Li→ Li+ half cell reaction, a more complex λ -paths would have been needed. Spontaneous ejection of electrons does not happen with Ag or Ni+, both of which are less electropositive than Li.
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The 6-311+G (d,p) basis also yields a PBE binding energy of 15.7 eV. As discussed in Ref., this basis and the plane-wave/PAW method used in VASP yield results that are in good agreement.
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The 6-311+G (d,p) basis also yields a PBE binding energy of 15.7 eV. As discussed in Ref., this basis and the plane-wave/PAW method used in VASP yield results that are in good agreement.
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It may be argued that the trajectory length is short and the sampled configurations are correlated, which may underestimate the numerical noise. Hence we have tested the uncertainty using classical force fields and much longer trajectories. With otherwise identical parameters (32 H2 O, 0.1 ps sampling intervals, extrapolation to q=0), a 400 ps SPC/E trajectory reveals that, on average, a 40 ps segment of the trajectory exhibits 0.44 and 0.52 kcal/mol standard deviations for the six- and two-point TI scheme, respectively. Normally a six-point TI should exhibit far less noise than a two-point one; in the present case, the extrapolation to q=0 required for the six-point trapezoidal rule has introduced additional uncertainties. These uncertainties in SPC/E simulations are indeed comparable to and even smaller than the standard deviations estimated for the 40 ps AIMD trajectories.
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It may be argued that the trajectory length is short and the sampled configurations are correlated, which may underestimate the numerical noise. Hence we have tested the uncertainty using classical force fields and much longer trajectories. With otherwise identical parameters (32 H2 O, 0.1 ps sampling intervals, extrapolation to q=0), a 400 ps SPC/E trajectory reveals that, on average, a 40 ps segment of the trajectory exhibits 0.44 and 0.52 kcal/mol standard deviations for the six- and two-point TI scheme, respectively. Normally a six-point TI should exhibit far less noise than a two-point one; in the present case, the extrapolation to q=0 required for the six-point trapezoidal rule has introduced additional uncertainties. These uncertainties in SPC/E simulations are indeed comparable to and even smaller than the standard deviations estimated for the 40 ps AIMD trajectories.
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While the isolated Cl- ion as predicted by PBE may not be stable in vacuum, within our periodic boundary condition simulations, Cl- has a well defined total energy and a HOMO-LUMO gap over a large range of simulation cell sizes.
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