^{1}and Wilfred F. van Gunsteren

^{1,a)}

### Abstract

The authors present a method to calculate free energy differences between two states and “on the fly” from a single molecular dynamics simulation of a reference state . No computer time has to be spent on the simulation of intermediate states. Only one state is sampled, i.e., the reference state which is designed such that the subset of phase space important to it is the union of the parts of phase space important to and . Therefore, an accurate estimate of the relative free energy can be obtained by construction. The authors applied the method to four test systems (dipole inversion, van der Waals interactionperturbation, charge inversion, and water to methanol conversion) and compared the results to thermodynamic integration estimates. In two cases, the enveloping distribution sampling calculation was straightforward. However, in the charge inversion and the water to methanol conversion, Hamiltonian replica-exchange molecular dynamics of the reference state was necessary to observe transitions in the reference state simulation between the parts of phase space important to and , respectively. This can be explained by the total absence of phase space overlap of and in these two cases.

Financial support by the National Center of Competence in Research (NCCR) Structural Biology and by Grant No. 200021-109227 of the Swiss National Science Foundation (SNSF) is gratefully acknowledged.

I. INTRODUCTION

II. METHODS

A. Theory

B. Simulation protocols

1. Dipole inversion

2. van der Waals interactionperturbation

3. Charge inversion

4. Water to methanol conversion

III. RESULTS

A. Dipole inversion

B. van der Waals interactionperturbation

C. Charge inversion

D. Water to methanol conversion

IV. DISCUSSION

A. Dipole inversion

B. van der Waals interactionperturbation

C. Charge inversion

D. Water to methanol conversion

V. CONCLUSIONS

### Key Topics

- Free energy
- 61.0
- Probability theory
- 14.0
- Molecular dynamics
- 9.0
- Entropy
- 7.0
- Perturbation methods
- 6.0

## Figures

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).

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).

(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.

(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.

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)].

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)].

(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 .

(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 .

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)].

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)].

(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.

(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.

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).

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).

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 ).

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 ).

(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).

(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).

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).

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).

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.

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.

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).

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).

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).

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).

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 .

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 .

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