# 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?¶

#### In a single dimension…¶

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

#### In multiple dimensions…¶

• We still use curve fitting algorithms, but this time, we use multidimensional algorithms

• Which one should we use? Fourier? Exponential? Polynomial?
• What would be the mosts useful coordinate system to use?

## Mature Concepts in Physical Organic Chemistry¶

• 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.)