a question about constrained MD, shake algorithm and the free energy calculation using Lagrange multiplier
Fangyong Yan
yanfa... at gmail.com
Sat Jan 7 22:28:16 CET 2017
constrained MD simulations belong to potential of mean force (free energy)
calculation, there are many very good reviews for free energy calculation,
for example,
"FREE ENERGY VIA MOLECULAR SIMULATION: Applications to Chemical and
Biomolecular Systems D. L. Beveridge and F. M. DiCapua", Annu. Rev.
Biophys. Biophys. Chem. 1989. 18:431-92, and many others.
And Dann Frenkel and Berend Smit's book is also helpful for understand
details of MD/MC simulations.
I have lots of fun in reading and understanding, even though I am a very
slow learner.
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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