# 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:¶

1. Assign atom types to all the atoms
2. Specify which atoms are bonded together (either assumed using vdW spheres, or manually specified)
3. 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.