# Basic Force Field: Part 2¶

## Abstract

## Examples of force fields¶

If we use the equation below, and substitute in \(r_{eq}=0\) (the system at equilibrium),

\[
U(r)=U(r_{eq})+\frac{U'(r_{eq})}{1!}(r-r_{eq})+\frac{U''(r_{eq})}{2!}(r-r_{eq})^2
\]

- The first term remains zero
- All subsequent terms will be \(r_{eq}-r_{eq}\)

\(\therefore\) there is no Force

- If the distance changes any amount, because the final term is squared, the force will be positive and increase accordingly

Simplifying (removing zero terms) this, we get

\[
U(r)=\frac{1}{2}k(r-r_{eq})^2
\]

- \(k\) is our force constant, specific to the bond (e.g. alkene carbon to alkene carbon double bond)
- \(r_{eq}\) is the known equilibrium bond length for this specific bond type
- This simplification, however only produces a harmonic oscillator, not the Morse potential

- Since the force tends towards infinity as \(|r|\) increases, there is no way to break bonds using these approximation… Consider this a limitation of the method

- Note that the alkane has a much smaller force constant \(k\) and will therefore have a much wider oscillator

## How many constants?¶

- For every form of every type of every atom that you want to consider in your force field, you’ll need to specify specific constants
- For every form of every atom that you use, the complexity of the system will increase at a rate \(N^2\), as each of these atom types needs constants for bonding with every other atom type…

## How to obtain these constants?¶

- Traditionally, IR spectra were used, however increasingly, high level QM calculations are used to calculate these to a very high accuracy

## Angle Bending¶

- Since angle bending is calculated agains two other atoms, the force constants increase faster (\(N^3\)) than bond length, as new atom types are increased

## Torsion¶

- Here we use a Fourier series to describe these forces, since the rotation of a bond will be periodic \(360^\circ = 0^\circ\)
- The triangle curve is the sum of the three other curves

- The three curves that it’s summing could represent
- The constants here increase at a rate \(N^4\) with atom types introduced

## Steps to calculate strain:¶

- Assign atom types to all the atoms
- Specify which atoms are bonded together (either assumed using vdW spheres, or manually specified)
- look up all the force constants/equilibrium value/phase angles required for the specified system (and decide what to do if any are missing)

- Specify a force constant based on a similar atom type
- Error and tell you what’s missing

- Fail to run and not tell you what’s missing

## Caveats¶

- Two compare two different molecules, the two molecules need to have the exact same atom type, or they’ll be compared against different constants

- In the image below, because the two molecules use different atom types, we can only compare their energies to the hypothetical unstrained version, we cannot compare them to each other

- To compare them to each other, we’d need to calculate \(\Delta H^\circ _f\) for each molecule to be able to compare the unstrained versions. From there, we can compare the relative, strained energies.