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Mathematics for Physics

226 Molecular Structure

A three-dimensional spatial arrangement of a molecule’s atoms that is used to predict its physical stability, chemical reactivity, and interactions with other systems.

Definite molecular geometry. Molecules have definite shapes set by bond lengths and bond angles, not flat drawings. A bond length is the equilibrium distance between two bonded nuclei. A bond angle is the angle formed by three nuclei. This principle is used to specify the geometry that spectroscopy and calculation must recover.

The energy-minimum shape. The natural shape is the arrangement of nuclei where the electronic energy is lowest. This principle is used to find the equilibrium geometry by minimizing \(E_{\mathrm{el}}(R)\).

The equilibrium geometry condition is

\[ \dfrac{\partial E_{\mathrm{el}}}{\partial R} = 0 \]

where

  • \(E_{\mathrm{el}}\) is the electronic energy including nuclear repulsion.
  • \(R\) is a nuclear coordinate.

The balance of attractive and repulsive forces. The shape is held by a balance of forces: atoms attract as they share electrons at longer range and push apart if pressed too close. This principle is used to explain why each bond has a preferred length.

Structure-function control. That shape controls bulk behavior such as melting and how receptors recognize a molecule. This principle is used to relate molecular geometry to physical and biological function.

Spectroscopic determination of geometry. Shapes are too small to see directly, so rotation and vibration spectra recover bond lengths and angles. The rotational constant of a linear molecule falls as the inverse of the moment of inertia. This principle is used to extract internuclear distances from microwave spectra.

The rotational constant is

\[ B = \dfrac{h}{8\pi^{2}cI} \]

where

  • \(B\) is the rotational constant.
  • \(h\) is Planck’s constant.
  • \(c\) is the speed of light.
  • \(I\) is the moment of inertia about the rotation axis.

226.1 References

  1. Atkins, P., de Paula, J., & Keeler, J. Atkins’ Physical Chemistry. Focus 9 — molecular structure. Focus 14 — molecular interactions.
  2. Levine, I. N. Physical Chemistry. Ch. 19 — molecular electronic structure. §20.5 — spectroscopy of structure.
  3. Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics, Vol. I. Ch. 1 and Ch. 12 — molecular shape and forces.