# [CP2K:8746] Re: a question about constrained MD, shake algorithm and the free energy calculation using Lagrange multiplier

Fangyong Yan yanfa... at gmail.com
Sun May 21 01:41:43 UTC 2017

```I am not talking about using classical MD to do reactions, I am just saying
that in classical MD, there are also free energy mapping, which includes
local minima, maxima, and saddle points.

On Sat, May 20, 2017 at 9:24 PM, Fangyong Yan <yanfa... at gmail.com> wrote:

> I think in order to use constrained MD, we need to locate the reaction
> coordinate, which is not an easy task. I think lots of people have been
> working on such problems, how to map the free energy surface, not only in
> ab-initio MD calculations, but also in classical MD calculations, how to
> locate the reactants, saddle points (transition states), and products. In
> quantum mechanics calculation, we use potential energy surface, in MD, we
> use free energy surface.
>
> I am still trying to learn all these, but I think in order to understand
> our problems, we need to have a better understanding of our energy
> surfaces, which is a very complicated problem.
>
> On Fri, Mar 3, 2017 at 12:22 PM, Fangyong Yan <yanfa... at gmail.com>
> wrote:
>
>> however, at this moment, I dont think I can find a better way to simulate
>> high free energy barrier for chemical reactions. Ab-initio constrained MD
>> seems to be the only choice, with enough sampling and good guess for the
>> reaction coordinate, the trajectory made up ab-initio constrained MD will
>> be reasonable.
>>
>> On Sun, Feb 26, 2017 at 7:48 PM, Fangyong Yan <yanfa... at gmail.com>
>> wrote:
>>
>>> personally thinking, the constrained md is kind of biasing to the
>>> reaction coordinates, so I personally prefer running a 1000 ps
>>> unconstrained MD, to running the same length of constrained md. For some
>>> reactions with high free energy barrier, unconstrained MD cannot simulate
>>> the reaction. In this case, maybe I can try other methods.
>>>
>>> On Wednesday, July 6, 2016 at 12:05:25 PM UTC-4, Fangyong Yan wrote:
>>>
>>>> Hi,
>>>>
>>>> I have a question about the free energy calculation using the
>>>> constrained MD. For the simplest case, such as constraining a
>>>> inter-molecular distance between two atoms, i and j, In the constrain MD in
>>>> the NVT ensemble, CP2K uses shake algorithm to update the position and
>>>> velocity, where the constrain follows the holonomic constrain,
>>>>
>>>> sigma = (ri - rj) ** 2 - dij ** 2, where dij is the constrain distance,
>>>> and the total force is equal to F_i + G_i, G_i is the constrained force
>>>> and is equal to, lamda * the first derivative of simga versus r_i, thus,
>>>> G_i = -2 * lamda * r_i, (where these eq. borrows from the original
>>>> shake paper, JEAN-PAUL RYCKAERT, GIOVANNI CICCOTTI, AND HERMAN J. C.
>>>> BERENDSEN, JOURNAL OF COMPUTATIONAL. PHYSICS 23, 321-341 (1977)).
>>>>
>>>> In the free energy calculation, I think CP2K uses the eq. derived by
>>>> Michiel Sprik and GIOVANNI CICCOTTI, Free energy from constrained molecular
>>>> dynamics, J. Chem. Phys., Vol. 109, No. 18, 8 November 1998, where in this
>>>> paper, the free energy uses a different constrain,
>>>> where constrain is equal to |ri - rj| - dij = 0, "| |" represents the
>>>> absolute value, and in this case, the constrained force G_i = - lamda, (see
>>>> eq. 13 in the paper). The free energy is equal to
>>>>
>>>> dW / d Zeta' = < Z^(-1/2) * [ -lamda + kTG] > / < Z^(-1/2)>
>>>>
>>>> W is the free energy, Zeta is the constrained eq., in this case is
>>>> equal to |ri - rj| - dij = 0, Zeta' represent different Zeta's; < > is the
>>>> ensemble average, Z is a factor arises from the requirement that when Zeta
>>>> is equal to zero for all times, the first derivative of Zeta (the velocity
>>>> of this constrain) is also equal to zero for all times. (from E.A. CARTER,
>>>> Giovanni CICCOTTI, James T. HYNES, Raymond KAPRAL, Chem. Phys. Lett. 156,
>>>> 472 ~1989.); G is equal to
>>>> G = (1 / Z^2) * (1/m_i * 1/m_j) * the first derivative of Zeta versus
>>>> r_i * the second derivative of Zeta versus r_i and r_j * the first
>>>> derivative of Zeta versus r_j,
>>>> when Zeta = |r_i - r_j| - dij, the first derivative of Zeta versus r_i
>>>> = the first derivative of Zeta versus r_j = 1, the second derivative of
>>>> Zeta versus r_i and r_j = 0, thus, the free energy is equal to
>>>>
>>>> dW / d Zeta' = < Z^(-1/2) * [ -lamda + kTG] > / < Z^(-1/2)> = <
>>>> Z^(-1/2) * [ -lamda] > / < Z^(-1/2)>, and Z is a constant in this simple
>>>> case, thus,
>>>> dW / d Zeta'  = <-lamda>
>>>>
>>>> Now my question is, since shake uses Zeta = (r_i - r_J) ** 2 - dij**2 =
>>>> 0, in this case, G wont disappear, and the constrained force G_i = - 2 *
>>>> lamda * r_i. Since CP2K does use SHAKE algorithm, how does CP2K do the free
>>>> energy calculation, do CP2K uses Zeta = (r_i - r_j) ** 2 - dij**2 =0, or
>>>> Zeta = |r_i - r_j| - dij = 0, since these two cases the lagrange multiplier
>>>> is different.
>>>>
>>>> Thanks for your patience for reading this, and I hope someone who can
>>>> help me with this issue!
>>>>
>>>> Fangyong
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
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>>
>>
>
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