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Basic Force Field: Part 2


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



  • 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


  • 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.