How to explain the phenomenon of optical rotation of chiral?

[Djalmao23]

I am Professor of Inorganic Chemistry of the State University of Southwest of Bahia - UESB. I have a PhD from UFMG in Natural Product Chemistry.

I use the polarimeter in my experiments and would like to understand what happens on a micro scale (chiral molecules) to provide the phenomenon of optical rotation.

 

[Greg Bernhardt]

Welcome to PF!

 

[DrDu]

Hi!

Before explaining the microscopic basis, I want to stress that optical activity is a collective property and not one of isolated molecules. Stated differently, almost any isolated molecule will rotate the polarisation of the scattered light for some orientations. The point is that for chiral molecules the rotation does not average to 0 when averaging over all orientations of the molecules. This is easy to see that a helix looks alike when rotated by 180 degrees, i.e. the helicity does not average out on rotations.
Now to the description of helicity: The response of materials to electromagnetic fields is described by the dielectric displacement $D$ or equivalently by the polarisation $P=D-E$. The relation between $D$ and $E$ is given by the material equations, whose general form is
$D(r,t)=\int dt' \int dr' \epsilon(r-r',t-t') E(r', t')$
This looks complicated. As the dimensions of the molecules is small compared to the wavelength of the electric field, it is usually an excellent approximation to assume that the dielectric response is local, i.e.
$D(r,t)=\int dt' \epsilon(t-t') E(r, t')$
(in a homogeneous medium, epsilon can only depend on the difference of r and r' or t and t')
or,
for monochromatic fields with frequency $\omega$
$D(r,\omega )=\epsilon(\omega)E(r,\omega)$.
But for chiral substances, this isn't sufficient. Optical rotation is an effect of the dielectric response being non-local.
The easiest model to see this is maybe the Kuhn model of coupled oscillators.
Consider a molecule containing two fluorophores which can be described by two dipoles which aren't parallel but are coupled.
I.e. if you excite one dipole, the other will get excited, too, and may also re-emit the radiation, with the plane of polarisation being rotated.
The spatial separation of the dipoles is necessary for this effect not to average out when averaging over all orientations of the molecule.

There exists an alternative formulation going back to Rosenfeld: The non-local dependence of the polarisation on the electric field can be replaced by a local dependence of the polarisation on both $E$ and the magnetic field $B$, as the magnetic field and the electric field are related themselves by a non-local Maxwell equation. Then the condition for chirality becomes that the molecule in question has a parallel electric and magnetic transition dipole moment. This formulation is especially well suited for routine quantum chemical calculations of optical activity.
An elementary introduction can be found in "Quantum chemistry" by Eyring, Walter, Kimball