Electronic Circular Dichroism

The lowest-order absorption process involving mixed electric and magnetic perturbations, the absorptive analogue of the optical rotation, is known as electronic circular dichroism (ECD) or just CD for short.

The differential absorption of circularly polarized light, corresponding to the difference between the absorptive index of the two circular components of linearly polarized light, is proportional to the rotational strength, which is normally calculated as the residue of the linear response mixed electric dipole-magnetic dipole polarizability

\[ \lim_{\omega \rightarrow \omega_{n0}} \hbar (\omega - \omega_{n0}) \langle\langle \hat{\mu}_{\alpha};\hat{m}_{\beta} \rangle\rangle_{\omega} = \langle 0 \mid {\hat{\mu}}_{\alpha} \mid n\rangle \langle n \mid {\hat{m}}_{\beta} \mid 0\rangle. \]

Since this expression corresponds to the infinite lifetime approximation for the excited state, only a single number will be obtained at the frequency of the electronic excitation.

In general \(^nR\), the rotatory strength for the transition \(\mid 0\rangle \rightarrow \mid n\rangle \) includes an electric dipole-magnetic dipole contribution

\[ ^nR^{\rm m}_{\alpha\beta} = -\frac{3 i {e^2}}{4 m_e } (\delta_{\alpha\beta} \langle 0 \mid {\bf r} \mid n\rangle \langle n \mid {\bf{l}^{\rm T}} \mid 0\rangle - \langle 0 \mid {\hat{r}}_{\beta} \mid n\rangle \langle n \mid {\hat{l}}_{\alpha} \mid 0\rangle ), \]

and an electric dipole–electric quadrupole contribution

\[ ^nR^{\rm Q}_{\alpha\beta} = -\frac{3 \omega_{n0} {e^2}}{4} {\mbox{${\varepsilon}_{\alpha \gamma \delta}$}} \langle 0 \mid {\hat{r}}_{\gamma} \mid n\rangle \langle n \mid {{\hat{q}}}_{\delta\beta} \mid 0\rangle. \]

For randomly oriented molecules, the averaging leaves only the electric dipole-magnetic dipole contribution and the scalar rotatory strength is given by

\[ ^nR = -\frac{i {e^2}}{2 m_e} \langle 0 \mid {\bf r}^{\rm T} \mid n\rangle \langle n \mid {\bf{l}} \mid 0\rangle, \]

These expressions are given in the length gauge. In the velocity gauge

\[ ^nR^{\rm m}_{\alpha\beta} = \frac{3 {e^2}}{4 m_e^2 \omega_{n0}} (\delta_{\alpha\beta} \langle 0 \mid {\bf{p}} \mid n\rangle \langle n \mid {\bf{l}^{\rm T}} \mid 0\rangle - \langle 0 \mid {\hat{p}}_{\beta} \mid n\rangle \langle n \mid {\hat{l}}_{\alpha} \mid 0\rangle ), \]

and

\[ ^nR^{\rm Q}_{\alpha\beta} = \frac{3 {e^2}}{4 m_e^2 \omega_{n0}} {\mbox{${\varepsilon}_{\alpha \gamma \delta}$}} \langle 0 \mid {\hat{p}}_{\gamma} \mid n\rangle \langle n \mid {\hat{T}}^+_{\delta\beta} \mid 0\rangle, \]

where \({\mathbf{T}}^+\) indicates the velocity form of the electric quadrupole

\[ {\mathbf{T}}^+ = -\Big( {\mathbf{r}} {\mathbf{p}} + {\mathbf{p}} {\mathbf{r}} \Big). \]

This form has an advantage in comparison to the length form. Although with a translation of the reference frame the magnetic dipole and electric quadrupole components change, the total tensor in the velocity gauge is invariant to such a change of origin. For the length gauge, this invariance depends in addition on the fulfillment of the hypervirial relation,

\[ \langle 0 \mid \hat{p}_\alpha \mid n \rangle = i m_e \; \omega_{0n}\; \langle 0 \hat{r}_\alpha \mid n \rangle, \]