Optical Transitions in Molecules: Introduction

The origin of spectral lines in molecular spectroscopy is the absorption, emission, and scattering of a proton when the energy of a molecule changes. In atomic spectroscopy, all transitions reflect changes in the configuration of electrons. In molecules, in addition, there are energy states corresponding to nuclei vibrations and rotations. In consequence, the molecular spectra are more complex than atomic spectra and contain information on the molecular structure and the bond strength. They also provide a way of determining a variety of molecular properties, like dipole and quadrupole moments and the quantum numbers characterizing all molecular degrees of freedom. The molecular spectroscopy is also important for astrophysical and environmental science, for investigation of chemical reactions, for biology, and in many other areas of science and technology which needs detailed investigation of properties of microscopic atomic and molecular objects.

In the Born-Oppenheimer approximation the energy of a molecule can be presented as sum of electronic energy $E_{el}$, vibrational energy $E_{vib}$ and rotational energy $E_{rot}$ energy.

$$E = E_{el} + E_{vib} + E_{rot}$$ (1)

The energy difference corresponding to the excitation of electrons $\Delta E_{el} = h \nu_{el}$ in this approximation is much larger that the energy difference corresponding to the molecular vibration $\Delta E_{vib} = h \nu_{vib}$ which is large that the energy difference corresponding to the molecular rotations $\Delta E_{rot} = h \nu_{rot}$:

$$h \nu_{el} \gg h \nu_{vib}\gg h \nu_{rot}$$ (2)

Usually, transitions within the rotation energy levels manifold belong to the far infrared and microwave spectral region, transitions within the vibrational energy levels manifold belong to the infrared spectral region, and the transitions between the electronic energy levels belong to the visible, or ultraviolet spectral region. In general, the vibrational transitions result in changes in the rotational mode and the electronic transitions result in changes in the rotational and vibrational modes as well. Selection rules determine whether transitions are allowed or not.

As discussed above the Time-Dependent Perturbation Theory shows that the probability of absorbing a light photon by a molecule is proportional to

$$W_{k \leftarrow 0} \propto \left\vert\langle k \vert\,\mu_{Z}\vert\rangle \right\vert^2 E^2_z,$$ (3)

where $\langle k \vert\,\mu_{Z}\vert\rangle$ is the matrix element of the transition dipole moment:

$$\langle k \vert\,\mu_{Z}\vert\rangle = \int {\Psi_k}^* \mu_z \Psi_0 dq$$ (4)

and $\Psi_0$ and $\Psi_k$ denote the wave functions of the molecular initial and excited states, respectively.

Absorbing the photon a molecule passes from its initial (ground) state to the excited state. Here we assume that the electric vector of light $\bf E$ (the light polarization vector) is parallel to the laboratory $Z$ axis. As shown in eq. (3) the rate of the transition is proportional to $E_Z^2$, and therefore, to the intensity $I$ of the incident radiation and is also proportional to the squire of the $Z$ component of the dipole moment matrix element $\langle k \vert\,\mu_{z}\vert\rangle$. In general, the total molecular dipole moment is a vector which can be presented as

$$\vec{\mu} = \sum_i q_i \vec{r}_i,$$ (5)

where $q_i$ is the charge of the $i$-th particle (electron, or nucleus), $\vec{r}_i$ is the radius-vector of the $i$-the particle, and summation is proceeded over all electrons and nuclei of the molecule.

This matrix element in eq. (4) depends on the symmetry of both molecular states which are indicated by the corresponding quantum numbers. For some particular quantum numbers of the initial and excited molecular states the matrix element $\langle k \vert\,\mu_{Z}\vert\rangle$ can be equal to zero, the corresponding optical transition is called forbidden transition. The relationship between the quantum numbers of the initial and excited molecular states for which the transition matrix element is not zero are known as transition selection rules.

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