muVT and NPT ensembles mismatch?

[Andrey Frolov]

Dear ERmod-users,

I would appreciate very much if someone can clarify/comment on the following issue.

The classical DFT in both representations (coordinate and energy ones) is formulated in grand canonical ensemble (muVT). However, molecular simulations are usually performed in NVT or NPT ensembles. Therefore, when one uses distribution functions (in energy representation) obtained from MD simulations as input for the chemical potential expression in DFT, one automatically makes an assumption that distribution functions from NVT (NPT) MD are equivalent to distribution functions in muVT ensemble, which is rigorously not true. Is this correct?

1) Could the mismatch between the ensembles for the MD and the DFT theory lead to any artifacts in free energy calculations?
2) Could DFT in energy representation be formulated for NPT ensemble in order to keep the MD distribution function and chemical potential expression totally consistent? 

A similar problem with the ensemble mismatch arises when one tries to calculate the Kirkwood-Buff integral based on g(r) estimated form NVT (NPT) MD simulations [see e.g. J. Phys. Chem. Lett. 2013, 4, 235−238].

Thank you very much.
With kindest regards,
Andrey

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Andrey I. Frolov, PhD
Junior researcher
Institute of Solution Chemistry
Russian Academy of Sciences

[MATUBAYASI Nobuyuki]

Dear Andrey

one automatically makes an assumption that distribution functions from NVT (NPT) MD

are equivalent to distribution functions in muVT ensemble, which is rigorously not true. Is this correct?

Roughly speaking, you are right.

Actually, the distribution functions from different ensembles differ by O(1/N),

where N denotes the system size, when seen point-wise.

What I mean is, for example, that g(r) from different ensembles differ by O(1/N) at each point.

The ensemble-dependent value then arises when the integral over the whole space is taken.

The KB integral is an example.

¥int_{whole space} (g – 1) provides the compressibility in muVT ensemble, while it is trivial in NVT.

The point is that the “whole space” is of O(N).

Even when the distribution functions are different only by O(1/N) when seen point-wise,

an integration over the space gives rise to an O(N) X O(1/N) operation

and leads to an ensemble dependence, since the O(1/N) term depends on the ensemble.

To be more specific, the distribution of solvent far from the solute is O(1/N)

while the number of solvent far from the solute is O(N).

The O(N) X O(1/N) effect arises from the solvent far from the solute.

For this matter, I have one paper

On the local and nonlocal components of solvation thermodynamics and their relation to solvation shell models,

N. Matubayasi, E. Gallicchio, and R. M. Levy, J. Chem. Phys. 109, 4864-4872 (1998).

1) Could the mismatch between the ensembles for the MD and the DFT theory

lead to any artifacts in free energy calculations?

For the free energy, the point-wise contribution of solvent far from the solute is o(1/N), instead of O(1/N).

The integration is then o(1) and vanishes in the thermodynamic limit.

So if the system is not too small, there is no worry.

2) Could DFT in energy representation be formulated for NPT ensemble

in order to keep the MD distribution function and chemical potential expression totally consistent?

Yes.

One point about DFT in a fixed-N ensemble is the zero of the external (solute-solvent) potential.

When N is fixed, a constant shift of the potential does nothing on the distribution.

So the one-to-one correspondence between potential and distribution is valid

with an allowance of the constant shift of the potential.

Actually, the constant can be fixed easily when the system is large enough.

When the potential is not zero at large distance between solute and solvent.

the solvation free energy will have a term of the non-zero constant times the number of solvent molecules.

This term is of O(N) and scales with the size of the system.

Such a behavior breaks a thermodynamic requirement that the chemical potential is an intensive property.

Thus to keep the chemical potential intensive,

the potential needs to be set zero at large distances between solute and solvent.

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