Force Fields in Molecular Dynamics
Molecular Mechanics (MM) is a computational method that uses a potential energy equation to retrieve the model of a given molecular system. The equation is a sum of terms that calculates the energy based on bond stretching, angle bending, dihedral angles, Van der Waals forces, and Coulombic interactions.
$$ E_{MM}(R)=E_{str}+E_{bend}+E_{tors}+E_{cross}+E_{el}+E_{vdW} $$
We already saw that the terms contained in this equation can be divided in two distinct groups bonding terms, and non-bonding terms, which have already been elucidated in other articles.
In this article, we will focus on the concept of force field.
Functional forms and parameters
Each term contained in the above equation for $E_{MM}(R)$ can be expressed by different mathematical expressions or functional forms. For instance, considering the bonding energy term $E_{str}$, we may express it as follows:
$$ E_{str} = \sum_{i}\frac{k_i}{2}(l_i - l_{i,0})^2 $$
This equation contains two parameters ($k_i$, $l_{i,0}$) that we can select, while $l_i$ is the variable of the equation, and it is allowed to change. If this is not clear I strongly suggest you read this article before proceeding.
The functional form used to express the different contributions in the Molecular Mechanics potential energy equation, as well as the parameters contained in it, define the so-called Force field (FF).
The parameters of each contribution can be derived both experimentally and in-silico. The ultimate goal is to develop parameters that are able to reproduce experimental data.
The development of parameters for all the different atoms in a molecule can be a huge computational task and, in most cases, is something we can not afford.
To overcome this problem we may decide to use the available parameters to describe different systems. Indeed, one of the main features of MM is that, within a specific FF, the parameters are assumed to be transferable. This practice is possible because the same functional group can be found in distinct molecules and its behavior is approximated to be substantially the same in different environments.
In summary, the force field mainly consists of three components:
- A list of atom types
- A set of empirical energy functions (functional forms)
- A collection of parameters able to fit the experimental data.
This is the main idea behind force fields. An example may help us to make it more intuitive.
Let’s consider the $C=C$ double bond.
We know that a $C$ atom involved in this specific bond has a similar equilibrium bond length in a countless number of molecules. So we may define an atom type $C_{sp^2}$, add it to our force field, and then use it to describe the same atom type in different systems. In this way, we don’t have to compute the parameters for each $C_{sp^2}$.
The force field is built up by adding more atom types and every possible interaction among them. So we can add other atom types of interest such as $C$, $H$, $Br$, and every bond between them $C_{sp^3}-H$, $C_{sp^2}-Br$, and so on. If a given functional group behaves differently in certain configurations, such as the $C=C$ bond in aromatic rings, we just need to create an additional atom type for the specific case.
Which is the correct force field?
It is fundamental to underline that the force field is not unique and none of the available ones can be considered “correct”. They are purely evaluated based on their performance.
Some FFs may provide accurate results for a class of systems, while results could be extremely poor for other systems.
We need to choose the FF that is more suitable for our problem. As usual, the ultimate goal in computational chemistry is to find the right compromise between accuracy and computational cost.
In large systems composed of several atoms, we have to simplify the $E_{MM}$ expression by choosing basic functional forms containing few parameters. In such cases, we may decide to avoid using cross terms, to use the simple harmonic approximation to model $E_{str}$ and $E_{bend}$, and to use the Lennard-Jones potential for the $E_{vdW}$ contribution.
On the contrary, when the system is small enough we may choose to use more complex force fields including cross terms and non-basic functional forms.
Nowadays, we generally have the possibility to use force fields that are specific for a given class of molecules. For example, some force fields are specifically designed for proteins, considering the atom types of different amino acids, and reducing the number of parameters allowing the treatment of large systems.
Common Force Fields
As you may know, many FFs are currently available. This comes extremely useful to beginners that are approaching the world of computational chemistry for the first time. Mainly because they don’t have the urgency to develop their own FF, since the already existing ones work fine for the majority of the cases.
Despite this, when choosing a FF it is crucial to understand its capabilities and its limitations, to make sure that we made the right decision.
Here you can find a list of the most common Molecular Mechanics FF that you may come across during your journey in computational chemistry:
- AMBER (Assisted Model Building and Energy Refinement)
- CHARMM (Chemistry at Harvard Macromolecular Mechanics)
- GROMOS (GROningen MOlecular Simulation)
- OPLS (Optimized Potentials for Liquid Simulations)
You should go through the scientific literature to decide which one is the most suitable to your specific case.