Potentials and Integration methods Flashcards
What decides the structure of biomolecules?
Intramolecular interactions and intermolecular interactions with the solvent.
What does an atom-centered force field mean opposed to an electron-centered?
In an electron-centered force field does one consider the electrons as a basis for the interactions between the atoms. But when considering atom-centered force field one applies the Born-Oppenheimer approximation, which means that the electrons are only considered as an average electron distribution. Thus the force field does only depend on the coordinates of the nucleus of the atom.
Explain the hard sphere potential.
Simple model for gas particles.
From the center of the atom and out to some distance sigma is the the potential for an other atom to stay there infinity. Then after this distance sigma is the potential 0, meaning that outside sigma does the atom don’t interact with other particles.
This is not very physically correct and hard to simulate because of the discontinuity in the potential.
Explain the square-well potential.
Similar to the hard sphere potential.
There are two distances from the center of the atom defining this potential. First, from the center and out to some distance sigma_1 is the potential infinity, no other atom can be there. At sigma_1 it changes to a negative value, ie. -eps, meaning that it is favourable for another atom to stay there, between sigma_1 and sigma_2, where it goes to 0, meaning that the particle does not interact anymore with other particles.
This potential is also discontinuous and therefore hard to use in MD simulations.
Describe the soft-sphere potential.
This potential starts at infinity for r=0, and then decays as (sigma/r)^v, to the power of v, how fast the decay is depends on v. This describes the repulsion of a particle, here there is no attraction, the potential is never negative.
Describe the Lennard-Jones potential.
The Lennard-Jones potential have two different parts, one repulsive due to the Pauli exclusion principle and one attractive part due to van der Waals interactions (dipoles).
The repulsive part is a soft-sphere potential with v=12, which means it decays fast.
The van der Waals interaction originates from dipole-dipole interactions, and is proportional to 1/r^6.
The total LJ-potantial is:
V_LJ(r) = 4*eps ((sigma/r)^12 - (sigma/r)^6)
eps is the depth of the potential well, that is how much particles are attracted to each other, and sigma is where the potential crosses the x-axis, at this distance it starts to become attractive.
Write down the Verlet integration method. Derived with Taylor.
r(t+dt) = 2r(t) - r(t-dt) + F(t)/mdt^2
It depends on two former positions to calc. the new position. Need r(t) and v(t) as starting param. and can calc. r(t-dt) from these as r(t-dt) = r(t) - dt*v(t)
The velocities are not included in the formula, but they can be calc. as difference ratios:
v(t) = (r(t+dt) - r(t-dt)) / 2*dt
The global error of verlet is O(dt^2) and it is calc. by doing n time steps and calc. this as a geometrical sum, which gives n^2 with n=T/dt, and the error in one step is of O(dt^4)
–> O(dt^4) / dt^2
What are the pros and cons of Verlet integration method?
+ Fast and simple
+ Good short-term energy conservation
+ Small long-term energy drift
+ time reversible
- Velocities don’t enter directly
- Need to coordinates to start with
Explain the Leap-frog method.
It is based on the Euler method, and calc. the position and velocity at different time steps. The position is given first at t=0, then the velocity at t=dt/2 is calc. and next the position at t=dt and so on.
r(t+dt) = r(t) + dt*v(t+dt/2) v(t+3/2*dt) = v(t+dt/2) - F(t+dt)/m*dt
Starting with r(0) and v(0), do a half-step first:
v(dt/2) = v(0) + dt/2*F(0)/m.
The half-step method is also used if one needs the velocities at the normal time steps, ie. dt, 2dt, etc.
Explain the Velocity Verlet method.
Velocity Verlet is like leap-frog method but the velocity is also calc. at every ndt step.
The algorithm is:
v(t+dt/2) = v(t) + dt/2F(t)/m
r(t+dt) = r(t) + dtv(t+dt/2)
v(t+dt) = v(t+dt/2) + dt/2F(t+dt)/m = v(t) + dt/2m*(F(t+dt) + F(t))
What are the advantages of using leap-frog or velocity verlet compared to normal verlet method?
+ The leap-frog and velocity verlet are less sensitive to round-ff errors, because adding terms of O(dt) instead of dt^2.
+ All three algorithms are time reversible.
+ The both conserve the angular momentum.
+ The algoritms are symplectic, that is the area in phase space is conserved as time evolves.
