## 5 Dielectric Spectroscopy-Tomorrow  
The (linear) interaction of electro-magnetic waves is described by convention in the optical spectral range (IR, visible and UV) by the complex index of refraction \(n^{*}\) as  
\[n^{*}=n^{\prime}+in^{\prime\prime} \tag{6}\]  
where \(n^{\prime}\) describes the real and \(n^{\prime\prime}\) the imaginary part of the index of refraction (\(i=\sqrt{-1}\) represents the imaginary unit), whereas for Tera-Hz frequencies and the spectral range of Broadband Dielectric Spectroscopy, the complex dielectric function \(\varepsilon^{*}\)  
\[\varepsilon^{*}=\varepsilon^{\prime}-ie^{\prime\prime} \tag{7}\]  
is used with \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) being the real and imaginary parts. The two quantities are related by Maxwell's relation  
\[n^{*}=\sqrt{\varepsilon^{*}} \tag{8}\]  
The various absorption processes in the IR spectral range can be described within the Lorentz-oscillator model by  
\[\varepsilon^{*}(\omega)=\varepsilon^{\prime}(\omega)-ie^{\prime\prime}( \omega)=\left[\varepsilon_{\infty}+\sum_{j}\frac{s_{j}\Big{(}\omega_{o,j}^{2}- \omega^{2}\Big{)}}{\Big{(}\omega_{o,j}^{2}-\omega^{2}\Big{)}^{2}+\big{(} \Gamma_{j}\omega\big{)}^{2}}\right]-i\left[\sum_{j}\frac{s_{j}\big{(}\Gamma_{ j}\omega\big{)}}{\Big{(}\omega_{o,j}^{2}-\omega^{2}\Big{)}^{2}+\big{(} \Gamma_{j}\omega\big{)}^{2}}\right] \tag{9}\]  
where \(\omega\) is the angular frequency, \(\varepsilon_{\infty}\) is the value of the dielectric function at a very high frequency, \(\omega_{o,j}\) is the resonance frequency of the \(j\)th oscillator, \(s_{j}=\omega_{p,j}^{2}f_{j}\) is related to the strength of the \(j\)th absorption mechanism with parameter \(f_{j}\) and \(\Gamma_{j}\) describes the damping in the \(j\)th oscillator. Hereby the real parts of the index of refraction and of the complex dielectric function have the character of a memory function, in which all absorption processes add up with decreasing frequency (Figure 5), respectively, increasing wavelength. Hence, the different contributions of the polar moieties to the net orientational polarization can be determined by measuring their response separately in the IR. In the fingerprint range of the IR spectrum, glycerol and sorbitol show several characteristic absorptions (Figure 2) representing the stretching vibrations of the (CO), (OH), (CH) and (CCO) moieties and the bending of the CH group. Applying an external, frequency-dependent electric field to a sample at widely varying temperatures and determining  
Figure 5: Schematic representation of the real (black) and imaginary parts (red) of the complex permittivity in a wide frequency range, covering vibrations and orientational polarization. the field-induced IR-dichroism of specific absorption bands enables one to measure the orientational polarization of the respective molecular units separately. The net dipole moment of a molecule \(\langle\vec{\mu}\rangle_{mol}\), which is classically considered to be determined by dielectric spectroscopy, is the result of the superposition of all atomistic polar moieties \(\vec{p}_{j}\):  
\[\langle\vec{\mu}\rangle_{mol}=\sum_{j}\vec{p}_{j} \tag{10}\]  
In the classical view, an external electric field acts on a dipole \(\vec{\mu}\), trying to orient it against the thermal fluctuations of the molecular system as a whole. The counterbalance between the two results in an ensemble-averaged orientational polarization \(\langle\vec{\mu}\rangle\) given by the Langevin equation. However, as established above, in molecules with intra-molecular degrees of freedom, a more refined view on the submolecular scale is required. For an atomistic moiety having a transition moment \(\langle\psi_{i}|\vec{p}_{j}|\psi_{e}\rangle\), the absorption \(A\) in the quantum mechanical dipole approximation [55] is proportional to  
\[A\sim\langle\psi_{i}|\vec{p}_{j}|\psi_{e}\rangle^{2} \tag{11}\]  
where \(\psi_{i}\) and \(\psi_{e}\) are the wave functions of the initial and excited states of the IR absorption event.