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## Re: [ESPResSo-users] Bjerrum Length for Explicit Solvent

 From: Peter Košovan Subject: Re: [ESPResSo-users] Bjerrum Length for Explicit Solvent Date: Sun, 6 Oct 2013 11:22:38 +0200

Dear all,

first of all, thanks for the stimulating discussion. Especially for the remark by Marcus about the need to distinguish the "bjerrum" parameter of Espresso and the physical parameter "Bjerrum length". Just to keep the terminology clear, even though this may seem obvious to many, I add, that it is not the full potential of mean force but the electrostatic contribution to it.

I think Stefan is right that the Bjerrum length is defined as
$$\nabla^2 \Phi = - 4 \pi l_B \rho$$
for two point charges in a dielectric continuum. If we add some excluded volume or polarizability to our point charges, the Bjerrum length becomes ion-specific even in a dielectric continuum. We should observe this when we try to measure the potential of mean force due to electrostatic interactions in an implicit solvent in Espresso where we set the bjerrum parameter to the same value but use polarizable ions.

Similar trouble arises when we use point charges but replace the continuum solvent with an explicit one. We can measure the dielectric constant epsilon_model of our solvent model. This will probably differ from the experimental value epsilon_exp since the model is not exact.

To my understanding, the solution is to adjust our bjerrum parameter such that
1/epsilon_exp = 1/(epsilon_model * epsilon_bjerrum) where epsilon_bjerrum is obtained by substituting the value of bjerrum parameter we provided to Espresso into the common formula for Bjerrum length.

To this end, I do not see inconsistency between what Stefan and Marcus have written.

Btw: does anyone know how users of other software deal with the problem that epsilon_model does not match epsilon_exp? Do they use corrections to epsilon similar to what I described or simply ignore the difference?

With regards,

peter

2013/10/6 Stefan Kesselheim
Hi,
thanks for you long mail Marcus! I'm however not sure that your our conceptions fit..

You say, that the Bjerrum length is the distance where the potential of mean force of two ions in water equals k_B T.
But this would mean, that the Bjerrum length ion (pair) specific. Would that be exactly what we want?

To my understanding the Bjerrum length is a definition with which we theory and simulation guys can write
$$\nabla^2 \Phi = - 4 \pi l_B \rho$$
so that we have single variable determining the strength of electrostatics at a given temperature.

So who is right? It's sunday morning her, and my books don't know about the Bjerrum length.
Cheers
Stefan

On Oct 6, 2013, at 4:07 AM, Markus Deserno <address@hidden> wrote:

> Folks,
>
> this is a beautiful example of the confusion that arises if one does not
> distinguish between the physics one wants to describe and the computer
> variables one types into one's favorite simulation engine.
>
> The Bjerrum length is the distance at which the electrostatic part of the
> free energy of interaction between two unit charges is equal to the thermal
> energy. I concede that my previous email didn't have the important word "free"
> in it. But anyways, this is how it is defined, and no computer simulation
> software in the world will change that. Whether one uses implicit solvent
> or explicit solvent, at the end of the day, it must be true that if you put
> two unit charges at that distance into your simulation, their potential of
> mean force in that solvent is equal to the thermal energy.
>
> Now, in ESPResSo the Bjerrum length is apparently used as an input
> parameter for electrostatics. That is partially convenient, but also partially
> confusing. Stefan is right that if you use an implicit solvent water model,
> you would type in 7 Angstrom at that point, while if you have an explicit
> water model there, there is no need to account for a dielectric constant,
> since it will be created explicitly by the solvent. Electrostatic interactions
> then don't have a dielectric permittivity in them, and one can account that
> by using a value for the Bjerrum length corresponding to vacuum, and
> that is about 56nm (not Angstrom!). But that only means that the parameter
> in the script is set to that value. The Bjerrum length in your simulation should
> be 7A. If it is not, you are simply not simulating water. (In fact, it probably
> is not, since hardly any model is perfect in giving the right dielectric constant,
> and you certainly wouldn't know just by guessing it.)
>
> I also disagree that it's impossible to measure the Bjerrum length. Of course
> it is possible. All you need to do is to measure the potential of mean force
> between two unit charges in your system as a function of distance, and look
> for the distance where it is equal to the thermal energy. This has been done.
> As Stefan suggests, you could alternatively calculate the dielectric constant
> of your system and calculate the Bjerrum length from there.
>
> So again: My main point here is: Please don't lose the physics out of sight.
> It is all fair and square if your simulating day in and day out to think about
> the values of your input parameters. But it is all too easy to forget that the
> value of the input parameters and the physical meaning of the things they
> are supposed to represent are not necessarily the same. I have witnessed
> too often to count that people waste weeks of research by that. In this case:
> Yes, if you want to simulate an explicit water model, you cannot set the value
> of the Bjerrum length input parameter in the Espresso Script to 7 Angstrom.
> And yet, that does not mean that suddenly the Bjerrum length is different.
> It means that the physics is the same but the model is different, and what
> is called "Bjerrum length" in the input script is no longer the Bjerrum length.
> ESPResSo was developed in a time where nobody wanted to simulate
> explicit water with it, and hence it was natural to call the strength parameter
> of the electrostatic interaction the Bjerrum length, since people always used
> implicit models. Now that explicit models are used as well, you simply have
> to jury-rig the code and make it do what you want. You crank up the input
> parameter of the Bjerrum length to its vacuum value. But that doesn't mean
> that the value of the Bjerrum length depends on the model. The Bjerrum
> length is a physics concept and must be model independent. Instead, the
> input parameters of the model must be adapted such as to reflect the right
> physics. Sadly, the input parameters were given physical names, and this
> now may or may not cause confusion.
>
> To be fair: Jezreel probably asked about the input parameter in the first
> place. But the language was confusing. Maybe I'm just a bit too pedantic
> to point all of this out, but in my experience it is helpful to keep a very clean
> line between the physics one wishes to describe and how to algorithmically
> implement this. Using a clear language is never a mistake.
>
> Best,
>
> Markus Deserno
>
>
> --
> Dr. Markus Deserno                    ++1-412-268-4401 (office)
> Associate Professor of Physics        ++1-412-681-0648 (fax)
> Carnegie Mellon University    ++1-412-268-8367 (Amanda Bodnar)
> 5000 Forbes Avenue                    www.cmu.edu/biolphys/deserno/
> Pittsburgh, PA 15213, USA     address@hidden
>
> On Oct 5, 2013, at 9:19 PM, Jezreel Castillo <address@hidden> wrote:
>
>> Thank you very much. It has been very informative.
>>
>> Respectfully,
>> Jezreel
>>
>>
>> On Sun, Oct 6, 2013 at 12:33 AM, Stefan Kesselheim <address@hidden> wrote:
>> Dear Jezreel, Marcus and Espressis,
>>
>> On Oct 5, 2013, at 6:20 PM, Markus Deserno <address@hidden> wrote:
>>
>> > Jezreel,
>> >
>> > the Bjerrum length is the distance at which two unit charges experience
>> > an electrostatic interaction energy equal to the thermal energy. For water,
>> > this is about 7 Angstrom, independent of whether the solvent is treated
>> > explicitly or implicitly. It would be rather unphysical if this physical quantity
>> > depended on the way you implement a solvent in your computer program.
>> > It is rather the other way around: Whatever you do to your computer model,
>> > you must make sure that physical parameters are reproduced, and the
>> > Bjerrum length is one such.
>>
>> I'm not sure if should agree on that. When you create a solvent model with dipoles, partial charges, or whatever you do not know its dielectric constant. You know however the dielectric constant of the medium, you embed it in, typically this is vacuum. In vacuum at room temperature the Bjerrum length is around 56 Angstrom. This is the number you would put into you electrostatics algorithm.
>> The dielectric constant of the medium is a result of a simulation, and can be used to define (!) a Bjerrum length in the medium.
>> I think this is important: l_B is not the distance where the electrostatic energy is unity, but the distance at which it would be unity, if the medium were a homogeneous dielectric with the measured dielectric constant. This also means: there is no way of measuring the Bjerrum length, except for in vacuum.
>> If you treat a solvent implicitly, the Bjerrum length is an input parameter, of course.
>>
>> To make an example: If you want say TIP3P water in Espresso, you create all particles and bonds and such things, and use the vacuum Bjerrum length for the electrostatics algorithm. You can measure a dielectric constant and from that determine the Bjerrum length of the solution. Next you can create an implicit solution model, where you plug in the Bjerrum length you have obtained before, and compare its properties to the original explicit model.
>>
>> I hope I caused not more confusion and that that helps.
>> Cheers
>> Stefan
>>
>>
>

--
Dr. Peter Košovan

Departmtent of Physical and Macromolecular Chemistry
Faculty of Science, Charles University in Prague, Czech Republic

Katedra fyzikální a makromolekulární chemie
Přírodovědecká fakulta Univerzity Karlovy v Praze

www.natur.cuni.cz/chemistry/fyzchem/
Tel. +420221951290
Fax +420224919752