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Basic Force Field: Part 1


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


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.)