Overview of Molecular Spectroscopy

In an earlier lecture, we discussed photoelectron spectroscopy as a means of measuring the electronic energy levels of a system. The purpose of spectroscopy, in general, is to probe allowed energies, which could be electronic energy levels, or energy levels of the nuclei. Recall that, in the Born-Oppenheimer approximation, we separate the electronic and nuclear Schrödinger equations and determine energy levels separately:

$$\hat{H}_{\rm elec}\psi_{\alpha}(r,R) = \varepsilon_{\alpha}^{\rm (elec)}(R)\psi_{\alpha}^{\rm (elec)}(r,R)$$

$$\left[\hat{K}_n + V_{nn}(R) + \varepsilon_{\alpha}(R)\right]\psi^{\rm (nucl)}_{\beta}(R) = E_{\beta}\psi^{\rm (nucl)}_{\beta}(R)$$

where $\alpha$ and $\beta$ are the complete sets of quantum numbers for the electronic and nuclear subsystems, respectively. Thus, we determine the electronic energy levels at fixed nuclear configurations and then on each Born-Oppenheimer elecronic surface $\varepsilon_{\alpha}(R)$, we determine nuclear energy levels $E_{\beta}$. Pictorally, we can represent the nuclear energy levels we obtain on several of the electronic surfaces of bonding orbitals as shown in the figure below:

Figure: Bound energy levels on the Born-Oppenheimer surfaces or bonding orbitals..

Note that the surfaces of non-bonding orbitals have no boundlevels. The basic idea is that subjecting a system to electromagnetic radiation of some frequency $\nu$ induces transitions among the various energy levels. If the absorption of a photon of frequency $\nu$ causes a transition from an initial energy level $E_i$ to a final energy level $E_f$, then the energy difference $E_f-E_i$ is related to $\nu$ via

$$h\nu = E_f -E_i$$

We have already seen that X- or UV radiation is needed to induce transitions among electronic energy levels. However, on each electronic energy surface, we have a large manifold of nuclear energy levels characterized by different types of motion. Bond vibrations are generally the highest frequency and have the largest spacing between energy levels. Bending motion is also a high-frequency vibration. Rotational motion or motion of dihedral angles about single bonds is much lower frequency. Consequently, between vibrational energy levels, there might be many rotational energy levels. Finally, nuclear spins couple to the magnetic field component of the external radiation, and this causes a very small energy splitting, so between rotational levels, there will be nuclear spin states. The table below shows the frequency of the external radiation and the type of transition induced by it:

$$\underline{{\rm Radiation}}\;\;\;\;\; \;\;\;\;\;\underline{{\rm Frequency}}\;\;\;\;\;\;\underline{{\rm Transition}}$$

$${\rm Radio\ waves} \;\;\;\;\; \;\;\;\;\;10^7-10^9\;\;\;\;\;\;\;\;{\rm Nuclear\ spin}$$

$${\rm Microwave,\ Far\ IR}\;\;\;\;\; \;\;\;\;\;10^9-10^{12}\;\;\;\;\;\;\;\;{\rm Rotationl}$$

$${\rm Near\ IR}\;\;\;\;\; \;\;\;\;\;10^{12}-10^{14}\;\;\;\;\;\;\;{\rm Vibrational}$$

$${\rm Visible,\ UV}\;\;\;\;\; \;\;\;\;\;10^{14}-10^{17}\;\;\;\;\;\;\;{\rm Valence\ electrons}$$

$${\rm X-ray}\;\;\;\;\; \;\;\;\;\;10^{17}-10^{19}\;\;\;\;\;\;\;{\rm Core\ electrons}$$

In order to generate a spectrum, sweep through a range of frequencies and record the frequencies at which radiation is absorbed as well as the intensity of the absorption. This gives us a graph of absorption intensity $\alpha(\nu)$ (also denoted $I(\nu)$) at frequency $\nu$. Generally, we are interested in a limited range of frequencies, e.g. the entire IR spectrum, which gives us information about rotational and vibrational transitions only, thereby characterizing motions in the nuclear subsystem.