^{1}and Wilfred F. van Gunsteren

^{1,a)}

### Abstract

A recently proposed method to obtain free energy differences for multiple end states from a single simulation of a reference state which was called enveloping distribution sampling (EDS) [J. Chem. Phys.126, 184110 (2007)] is expanded to situations where the end state configuration space densities do not show overlap. It uses a reference state Hamiltonian suggested by Han in 1992 [Phys. Lett. A165, 28 (1992)] in a molecular dynamics implementation. The method allows us to calculate multiple free energy differences “on the fly” from a single molecular dynamics simulation. The influence of the parameters on the accuracy and precision of the obtained free energy differences is investigated. A connection is established between the presented method and the Bennett acceptance ratio method. The method is applied to four two-state test systems (dipole inversion, van der Waals perturbation, charge inversion, and water to methanol conversion) and two multiple-state test systems [dipole inversion with five charging states and five (dis-)appearing water molecules]. Accurate results could be obtained for all test applications if the parameters of the reference state Hamiltonian were optimized according to a given algorithm. The deviations from the exact result or from an independent calculation were at most . An accurate estimation of the free energy difference is always possible, independent of how different the end states are. However, the convergence times of the free energy differences are longer in cases where the end state configuration space densities do not show overlap [charge inversion, water to methanol conversion, (dis-)appearing water molecules] than in cases where the configuration space densities do show some overlap [(multiple) dipole inversion and van der Waals perturbation].

The authors would like to thank Philippe Hünenberger for helpful discussions. 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. THEORY

A. Hamiltonian and calculation of the relative free energy

B. Equations of motion

C. Optimization of parameters

D. Generalization to multiple states

E. Estimation of errors

III. SIMULATION PROTOCOLS

A. Two-state perturbations

1. Dipole inversion

2. van der Waals interactionperturbation

3. Charge inversion

4. Water to methanol conversion

B. Multiple-state perturbation

1. Dipole inversion

2. (Dis-)appearing water molecules

IV. RESULTS AND DISCUSSION

A. Two-state perturbations

1. Dipole inversion

2. van der Waals interactionperturbation

3. Charge inversion

4. Water to methanol conversion

5. Automatic parameter optimization

B. Multiple-state perturbation

1. Dipole inversions

2. (Dis-)appearing water molecules

V. CONCLUSIONS

### Key Topics

- Free energy
- 72.0
- Phase space methods
- 6.0
- Solvents
- 5.0
- Water energy interactions
- 5.0
- Probability theory
- 4.0

## Figures

Pictorial representation of the effect of the constant in Eq. (17) for (circles) and (squares). The (symbol free) solid and dashed lines correspond to the reference state Hamiltonian . The dashed lines correspond to (thin), (medium), and (thick). The solid lines correspond to (thin), (medium), and (thick).

Pictorial representation of the effect of the constant in Eq. (17) for (circles) and (squares). The (symbol free) solid and dashed lines correspond to the reference state Hamiltonian . The dashed lines correspond to (thin), (medium), and (thick). The solid lines correspond to (thin), (medium), and (thick).

Radial orientational correlation function (ROCF) of the water dipoles at a distance around the ion . The solid curves correspond to (i.e., ) and the dashed curves to (i.e., ). The circles and squares are the ROCFs obtained from simulating at state and state , respectively (taken from Ref. 38).

Radial orientational correlation function (ROCF) of the water dipoles at a distance around the ion . The solid curves correspond to (i.e., ) and the dashed curves to (i.e., ). The circles and squares are the ROCFs obtained from simulating at state and state , respectively (taken from Ref. 38).

## Tables

Relative free energies [Eq. (31)] (in kJ/mol) for the dipole inversion. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the dipole inversion. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the van der Waals perturbation. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the van der Waals perturbation. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the van der Waals perturbation with optimized parameters. For every set of and parameters, two simulations starting from a different initial condition were performed. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the van der Waals perturbation with optimized parameters. For every set of and parameters, two simulations starting from a different initial condition were performed. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the charge inversion. For , the variance is zero and no error estimate could be calculated. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the charge inversion. For , the variance is zero and no error estimate could be calculated. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the charge inversion with optimized parameters (, ). Two simulations with different initial conditions were run. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the charge inversion with optimized parameters (, ). Two simulations with different initial conditions were run. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the conversion of water into methanol. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the conversion of water into methanol. The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the conversion of water into methanol with optimized parameters (, ). The weighted average over the 20 simulations is . The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies [Eq. (31)] (in kJ/mol) for the conversion of water into methanol with optimized parameters (, ). The weighted average over the 20 simulations is . The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies (in kJ/mol) for the dipole inversion with multiple states (states , , , , and ). The lower left part of the matrix shows the results obtained with and the upper right the ones obtained with . The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies (in kJ/mol) for the dipole inversion with multiple states (states , , , , and ). The lower left part of the matrix shows the results obtained with and the upper right the ones obtained with . The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies (in kJ/mol) for the five (dis-)appearing water molecules. The lower left part of the matrix shows the results obtained with and the upper right the ones obtained with . The calculated overlap integrals are given in parentheses (see Sec. II E).

Relative free energies (in kJ/mol) for the five (dis-)appearing water molecules. The lower left part of the matrix shows the results obtained with and the upper right the ones obtained with . The calculated overlap integrals are given in parentheses (see Sec. II E).

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