Yes, guys! I am back with the second part of my Classical Molecular Dynamics (CMD) series. For those who missed the first part, please see it here.
Summary of the first part :
CMD simulations are techniques used to mimic the motions of molecules assuming they obey Newton's laws. Since the force assumed is of the conservative nature(please see the first article) we can write the force as below:
Where 'U' is the total potential energy.
Force Field formulation
So we have the equation. Now solving this equation is the focus of any MD simulation software or program. In this part, I will explain how you do you find the potential energy function 'U'. We are yet to know the potential energy, right? So let us see that. The potential energy function defines the interactions among particles in the system. The force field comprises of the functional form and parameters used to find the potential function U. The potential energy term comes from 2 kinds of interactions:
- Bonded interactions
- Non bonded interactions
The bonded interactions are modeled like a Hooke's linear spring model. In case you want a refresher about Hooke's law, see below. (Or maybe you can see this Khan Academy video):
In the above figure I have derived the work done for compressing the spring to X distance from the equilibrium position of the uncompressed spring. That work done is equal to the potential energy stored in the compressed spring. So now if we are going to model our bonded interactions assuming that the bonds behave like springs the form of potential energy for these terms would be identical to the work equation in the above figure.
The bonded interactions are essentially trying to model the covalent bonds(like a carbon-carbon bond). But we assume interactions between two particles, interactions between angles of 3 consecutive particles and interactions between planes comprising of 4 consecutive particles as in the figure below.
The nonbonded interactions are basically comprised of Lennard Jones potential for Van-der Waal's interactions (which model hydrogen bonds etc) and the coulombic aka electrostatic interactions.
A quick introduction to Lennard-Jones Potential: Also known as LJ potential actually approximate the interactions between neutral atoms. LJ potential has a mathematical form as below(Image Source):
The (1/r)^12 is the repulsion term and the (1/r)^6 term is the long-range attraction term. The LJ potential looks like below(Image source):
The plot intuitively means that the particles cannot touch each other because of the repulsive force and they fall back to equilibrium bond length 'r_m'. If the atoms are far away there is an attraction which dominates although the slope is small.
Electrostatic interaction is the interaction between two particles due to its coulombic charges. Like charges repel and unlike charges attract.
Now we are ready to formulate our basic force field. The very basic assumption is that the resultant potential U is the summation of the above-mentioned interaction terms. Below is the equation(Adapted from this Image):
Below is an illustration showing the forms of bonded interactions. The dihedral potential term has 2 minima. The 2 structures at respective minima show a "cartoonish" illustration of attaining 2 different configurations but similar stability.
The parameters like "spring constants" comes from different experiments like from Quantum chemistry. But don't forget that our system is Classical in nature as mentioned in the previous article.
Hurray! So now we have a very basic force field.
Assuming that we have every parameter from empirical observations(experiments) we can substitute this Force field equation in
So the next task is to properly solve the system. That means creating a nice integrator for this differential equation. So that will be the third part in the series.
COMING NEXT: Designing a proper integrator to solve the MD differential equation.
Note:
I am focusing on very basic aspects how to minimally get started in MD simulations. Advanced topics and discussions are not completely out of the picture. Please reply to me here. I will be happy to discuss those with you people.
In case you want a textbook for reference I recommend these books:
- "Statistical Mechanics: Theory and Molecular Simulations" by Mark E. Tuckerman
- "Molecular Modelling : Principles and Applications" by Andrew R. Leach
- "Computer Simulation of Liquids" by D. J. Tildesley and M.P. Allen [Classic! and its little bit hard for beginners]
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I am Devanand from Chennai, India. And my steemit handle is @dexterdev
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