• Introduction
• Simulation methods:
- Monte Carlo
- Molecular Dynamics
- Langevin methods
• Next lab.
Outline
Chemical Physics 446 (2015) 118–126
• When solving a single or a few molecules, electronic structure methods are typically used.
- Schrödinger equation.
- Temperature of 0 K.
- Vacuum.
- Molecules are at the ground state.
Computational Challenge
• However, the majority of chemical reactions and biologically relevant processes an are carried out in solution.
• Two major techniques for generating an ensemble:
- Monte Carlo
- Molecular dynamics
- (Langevin methods)
Generating an Ensemble
• Two major techniques for generating an ensemble:
- Monte Carlo
- Molecular dynamics
- (Langevin methods)
Generating an Ensemble
Basic Monte Carlo Algorithm
Set initial geometry
Choose random particle and give it a random “kick”
Accept new configuration
Study ΔE Δ𝐸 > 0Δ𝐸 < 0
Choose next random particle and give it a random “kick”
Accept with a probability of 𝑒−Δ𝐸/𝑘𝐵𝑇0 .
Advantages:
• Possibility of “tunnelling” between energetically
separated regions of phase space.
• It’s possible to “freeze” certain degrees of
freedom
• Requires only the ability to evaluate the energy of
the system.
Monte Carlo
Chemical Physics 446 (2015) 118–126
Disadvantages:
• limΔ𝐸→∞
𝑒−Δ𝐸/𝑘𝐵𝑇0 = 0 : the step size must be fairly small.
• Lack of time dimension and atomic velocities.
→ Not suitable for studying time-dependent phenomena or properties
depending on momentum.
• Two major techniques for generating an ensemble:
- Monte Carlo
- Molecular dynamics
- (Langevin methods)
Generating an Ensemble
• Nuclei are heavy enough that they, to a good approximation, behave as classical particles.
• Molecular Dynamics (MD) methods generate a trajectory by propagating a starting set of coordinates and velocities using Newton’s second equation.
• For each atom:
- Position (𝑟).
- Momentum (𝑚 + 𝑣).
- Charge (𝑞).
- Bond information (neighbors, bond angles, etc.).
Molecular Dynamics
• To discover the trajectory, we need to know the potential affecting each atom.
- Bonded neighbors.
- Non-Bonded atoms.
From Potential to Movement
bondednonbonded EERV −+=)(
Bonded Atoms
Stretch Bend Rotate
bondalongrotatebendanglestretchbondbonded EEEE −−−− ++=
MM mimimizes the potential energy given by a force field.MD is the actual algorithmic solving of newton's equations to see motion.
• The Non-bonded interaction includes:
- Van der Waals Potential.
- Electrostatic Potential.
Non-Bonded Atoms
• Combining the van der Waals potential and the electrostatic potential gives us the non-bonded potential.
The Non-Bonded Potential
ticelectrostaWaalsdervanbondednon EEE += −−−
Basic Molecular Dynamics Algorithm
Set initial conditions: and)( 0tir )( 0tiv
Calculate new forces: )( ii rF
Solve the equations of motionnumerically over a short step, .
)()( ttt ii +→ rr)()( ttt ii +→ vvt
ttt +=Advance time:
Is ?maxtt End Simulation.No Yes
• An atom moving out of
boundary comes back
from the other side.
• Large enough to include
the molecule itself.
• The size of the
computational cell should
be larger than 2𝑅𝑐𝑢𝑡.
Periodic boundary conditions
cutR
• Solvent molecules: computational burden.
• Our simulation space has certain boundaries: cell.
• All atoms in the computational cell are replicated throughout space to form an infinite lattice.
• The name and inspiration come from annealing in metallurgy.
Simulated Annealing
Bath temperature
Time step
heating
equilibration
cooling
Simulated Annealing: Example
The total energy increases.
Dynamics simulation for Alanine (heating and run phase at 300𝐾).
The temperature plot merged with the kinetic energy.
The potential energy mirrors the kinetic energy.
• Deterministic simulations - the output is fully determined by the
parameter values and the initial conditions.
• Stochastic simulations - possess some inherent randomness.
Deterministic vs. Stochastic
3 + 3 = ?
• A stochastic process is said to be ergodic if its statistical properties can be
deduced from a single, sufficiently long, random sample of the process.
→A time average over a single particle is equivalent to an average of a large
number of particles at any given time snapshot.
• Most processes in nature are ergodic.
• Example of a non-ergodic process: two coins, one fair and the other has two
heads. Choosing first one coin and then perform a sequence of independent
tosses.
Ergodic Process
Time averaging Ensemble averaging
Molecular Dynamics
• Deterministic (in principle).
• Time averaging. (sec. 4d)
Molecular Dynamics vs. Monte Carlo
Monte Carlo
• Stochastic.
• Ensemble averaging.
Ensemble Type
• A simulation can be characterized by quantities such as volume (𝑉),
pressure (𝑃), total energy (𝐸), temperature (𝑇), number of particles (𝑁),
chemical potential (𝜇), etc., but not all of these are independent.
- For a constant 𝑁, either the 𝑉 or the 𝑃 can be fixed, but not both.
- A constant 𝜇 is incomemensurable with a constant 𝑁.
• The ensemble is labelled according to the fixed quantities, with the
remainder being derived from the simulation data, and thus displaying a
statistical fluctuation.
Ensemble Type
• An MC simulation uses 𝑇 as the parameter for deciding acceptance or
rejection of trial moves (𝑒−Δ𝐸/𝑘𝐵𝑇0)
→ naturally of the 𝑁𝑉𝑇 type.
• An MD simulation (Langevin), on the other hand, preserves 𝐸 and is
therefore naturally of the 𝑁𝑉𝐸 type.
• Two major techniques for generating an ensemble:
- Monte Carlo
- Molecular dynamics
- (Langevin methods)
Generating an Ensemble
• Molecular dynamics methods generate detailed information about all the
particles in the system.
• In some cases, the major interest is in the dynamics of a single molecule.
Then, the surrounding molecules can be modelled by only including the
average interactions.
𝑚𝑑2𝒓
𝑑𝑡2= 𝑭𝑖𝑛𝑡𝑟𝑎 − 𝜁
𝑑𝒓
𝑑𝑡+ 𝑭𝑟𝑎𝑛𝑑𝑜𝑚
→ The Langevin equation of motion gives rise to stochastic dynamics.
Langevin methods
• Explicit solvation - method that uses individual solvent molecules.
- Better physical behavior.
- Computationally demanding.
• Implicit solvation - method to represent solvent as a continuous medium instead of individual “explicit” solvent molecules
- Cheaper.
- Less physical.
Explicit vs. Implicit Solvent
• You will simulate the Dynamic and Equilibrium behavior of Alanine zwitterions.
• Step 1: creation of an isolated Alanine zwitterion and measuring its properties.
- Molecular mechanics, AMBER force field.
• Step 2: Solvating the Structure.
- Geometry optimization, MM.
- Energy: lower or higher than in vacuum?
Next lab
• Step 3: Superposition.
- What are the significant changes?
• Step 4: Molecular Dynamics: simulated annealing
- Simulated annealing.
- Reoptimizing the new structure.
- Did you get a smaller energy value?
• Step 5: Langevin and Monte Carlo Simulations
- Compare.
Next lab