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Enveloping distribution sampling: A method to calculate free energy differences from a single simulation
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10.1063/1.2730508
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Affiliations:
1 Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
a) Author to whom correspondence should be addressed. Electronic mail: wfvgn@igc.phys.chem.ethz.ch
J. Chem. Phys. 126, 184110 (2007)
/content/aip/journal/jcp/126/18/10.1063/1.2730508
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View: Figures

## Figures

FIG. 1.

Pictorial view of the formation of the reference state Hamiltonian from two harmonic oscillators and that differ in location and value of their minima. The reference state potential energy (squares) comprises the minima of the state potential energy (solid) and state potential energy (dashed) (all in the left panel). The resulting distribution (squares) envelops the distributions of the end states (solid and dashed) (all in the right panel).

FIG. 2.

(Color) Dipole inversion: Solute-solvent energy probability densities calculated using Eq. (11) from an EDS simulation of the reference state . For comparison, the probability density obtained from the thermodynamic integration simulation is given.

FIG. 3.

Dipole inversion: Convergence of the free energy difference calculated via thermodynamic integration (dashed) and EDS [Eq. (6), solid]. The error bars indicate an error estimate obtained by block averaging [see also Eq. (9)].

FIG. 4.

(Color) van der Waals interaction perturbation: Solute-solvent energy probability densities obtained via Eq. (11) from an unbiased (, lower panel) and a biased ( and , upper panel) EDS simulation of the reference state. Probability densities obtained from the thermodynamic integration simulations are shown for comparison. The inset shows the same data in a different scaling in order to show .

FIG. 5.

van der Waals interaction perturbation: Convergence of the free energy difference obtained from an unbiased (, dashed-dotted) and a biased ( and , dashed) EDS simulation of the reference state. The thermodynamic integration result is . Error estimates were obtained by block averaging [see also Eq. (9)].

FIG. 6.

(Color) Charge inversion: Solute-solvent energy probability densities obtained using Eq. (11) from a HREMD-EDS simulation of the reference state with and . The upper panel shows the densities for the lowest replica, for which no soft-core interactions are used. The lower panel corresponds to the softest replica, for which the use of soft-core solute-solvent interactions leads to overlapping densities. The densities obtained from TI are shown for comparison.

FIG. 7.

Charge inversion: Radial dipole-orientational correlation function (rocf) of the water at a distance around the ion during a HREMD-EDS (nonsoft replica) and a TI simulation. A rocf of corresponds to a situation where all water dipoles point towards the ion, and a rocf of to water dipoles pointing away from the ion. EDS simulation configurations for which the state energy is lower than the state energy (, solid) show the rocf of water around an anion (TI: circles) and configurations with (dashed) show the rocf of water around a cation (TI: squares).

FIG. 8.

Charge inversion: Convergence of the free energy difference obtained from a HREMD-EDS simulation of the reference state ( and ). In the left panel, the EDS result is shown. Only the energy trajectory of the nonsoft replica was used. The solid line shows the free energy difference obtained when discarding the first of the of energy trajectory of the nonsoft replica. Note that the total simulation time, i.e., is shown on the abscissa. The dashed line shows the result obtained when using a reversed time series of energy values. The right panel compares the EDS results to free energy differences obtained by two different TI calculations: one without (thick solid) and one with soft-core interactions at intermediate points (dashed-dotted, and ).

FIG. 9.

(Color) Water to methanol conversion: Solute-solvent energy probability densities obtained from HREMD-EDS simulation of the reference state . ; . The upper panel shows the densities obtained from the nonsoft replica. The lower panel shows the densities obtained from the softest of the 21 replicas. The insets show the same data at different scaling in order to show (blue) and (magenta).

FIG. 10.

Water to methanol conversion: Convergence of the free energy difference obtained from a HREMD-EDS simulation of the reference state . The EDS result (solid) converges quickly to the TI result of (dashed and dashed-dotted), although the reference state switches between the two states only rarely. The error bars denote the statistical error calculated according to Eq. (9).

FIG. 11.

Dipole inversion: Solute-solvent energy probability densities and corresponding overlap integrals obtained via Eqs. (11) and (12) from a MD-EDS simulation of the reference state.

FIG. 12.

van der Waals perturbation: Solute-solvent energy probability densities and corresponding overlap integrals obtained via Eqs. (11) and (12) from a MD-EDS simulation of the reference state ( and ). Here corresponds to having in Eq. (11).

FIG. 13.

Charge inversion: Solute-solvent energy probability densities and corresponding overlap integrals obtained via Eqs. (11) and (12) from a HREMD-EDS simulation of the reference state ( and ). Here corresponds to having in Eq. (11).

FIG. 14.

Water to methanol conversion: Solute-solvent energy probability densities and corresponding overlap integrals obtained via Eqs. (11) and (12) from a HREMD-EDS simulation of the reference state .

/content/aip/journal/jcp/126/18/10.1063/1.2730508
2007-05-14
2014-04-18

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