[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 03:24:55 CEST 2017

```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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