Energy terms: bonded and non-bonded

We are going to assume that the energy of the system is separable in different terms. The usual separation is the following

$$E_{bonded} = E_{stretching}+E_{bending}+E_{torsion}+E_{improper}$$ (2.50)

$$E_{non-bonded} = E_{van der Waals}+E_{electrostatic}$$ (2.51)

$$E_{total} = E_{bonded} + E_{non-bonded}$$ (2.52)

There are many expressions for every term, some force fields incorporate crossing terms to account for the coupling between two different interaction types. However, we are interested in those force fields that are specialized in the treatment of biomolecules. In this case the common energy terms are

$$E_{stretching} = \sum_{i}^{bonds} k_{ri}(r-r_{eq})^2$$ (2.53)

$$E_{bending} = \sum_{i}^{angles} k_{\theta i}(\theta-\theta_{eq})^2$$ (2.54)

$$E_{torsion} = \sum_{i}^{dihedr} \frac{V_{i}}{2}[1+cos(n\phi-\gamma)]$$ (2.55)

$$E_{improper} = \sum_{i}^{impropers} k_{\gamma i} (\gamma - \gamma_{eq} )^2$$ (2.56)

where $n$ in equation 1.55 is the multiplicity of the conformation. And for the non-bonded interactions

$$E_{van der Waals} = \sum_i^{atoms}\sum_{j>i}^{atoms} \epsilon_{ij} \biggl[\biggl(\fra... ...}{r_{ij}}\biggr)^{12}-2\biggl(\frac{r_{min_{ij}}}{r_{ij}}\biggr)^6\biggr]w_{ij}$$

$$with \quad r_{min}=\sigma 2^{1/6}$$

$$= \sum_i^{atoms}\sum_{j>i}^{atoms} 4\epsilon_{ij} \biggl[\biggl(\fr... ...j}}{r_{ij}}\biggr)^{12}-\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^6\biggr]w_{ij}$$

$$or \quad simply$$

$$= \sum_i^{atoms}\sum_{j>i}^{atoms} \biggl[\biggl(\frac{A_{ij}}{r_{ij}}\biggr)^{12}-\biggl(\frac{B_{ij}}{r_{ij}}\biggr)^6\biggr]w_{ij}$$ (2.57)

$$E_{electrostatic} = \sum_i^{atoms}\sum_{j> i}^{atoms} \frac{q_i q_j}{r_{ij}}w_{ij}$$ (2.58)

the weighting function $w_{ij}$ is typically set to zero for atoms $i$ and $j$ connected by a bond or angle and $w_{ij}$ ranges from $\sim0.4$ to $1$ for 1-4 interactions.

The general combination rules for Van der Waals parameters between two atoms $i$ and $j$ are

$$\epsilon_{ij}=\sqrt{\epsilon_i \epsilon_j} \quad;\quad \sigma_{ij}=\frac{\sigma_i\sigma_j}{2}$$ (2.59)

Assuming this separation of the energy the set of parameters $k_{ri}$, $k_{\theta i}$, $V_i$, $q_i$, $\epsilon_i$ define a specific force field. If every different atom had its own set of parameters the amount of parameters to optimize would be huge. To simplify this task the concept of atom type is used. The atom type idea is based on the common chemical intuition of transferability. For example, all the sp3 carbon atoms in an alkylic chain will have the same set of parameters because its chemical behavior is approximately the same in different length chains. However every force field defines its own set of atom types, so the parameters for a certain atom in a force field are not usually transferable to another force field.