Basic Force Field: Part 1¶

Abstract

From last lesson: How can we know where all the critical points are without calculate the whole PES?

Info

Practically speaking, we can fit a PES curve from the IR excitations, by mapping out each individual excitation and it’s associated oscillation

How is this curve represented mathematically?¶

We could use a polynomial equation to model this, fitting a function to the curve
\(0\) is the point that we’re expanding around, \(U(r)\) is our energy function
- We do this because we know the most about \(r_{eq}\) due to spectroscopy

\[ U(r)=0+a(r-r_{eq})+b(r-r_{eq})^2+c(r-r_{eq})^3+... \]

To do this cleanly, we should (and do) be using a Taylor expansion, with the derivatives of the functions as the coefficients
- If we stop as the second derivative, we create a harmonic oscillator (second derivative is the force constant)

\[ U(r)=U(r_{eq})+\frac{U'(r_{eq})}{1!}(r-r_{eq})+\frac{U''(r_{eq})}{2!}(r-r_{eq})^2+\frac{U'''(r_{eq})}{3!}(r-r_{eq})^3+... \]

We still use curve fitting algorithms, but this time, we use multidimensional algorithms
- Which one should we use? Fourier? Exponential? Polynomial?

Strain - Interactions within the molecule that prevent the individual oscillator from being able to take on its optimal state
- Steric hinderance
- Comes in different types; Length, Angle, Torsion
- The ideal values of these oscillators depend on not just the type of atom, but also the state of the atom (hybridisation, multiple minima (resonance), electron distribution, bond order)
A “force field” is defined by its atom types (and forms), the functions it uses to calculate the bonds/angles/torsions and any constants (force, eq distances, etc.)