Polyatomic Vibrational Spectroscopy

The selection rule for a diatomic molecule is that the vibrational quantum number changes by 1, at least under the harmonic approximation of the potential. Also, the dipole moment has to change in the course of the vibration or else the transition is forbidden. Carbon monoxide, for instance, has an allowed fundamental transition, whereas N2 does not. The separation of variables that is accomplished with the normal mode analysis says that each mode can be regarded as an independent one-dimensional oscillator. Thus, we can borrow the results for the simple harmonic oscillator to conclude that a transition will be allowed if the vibrational quantum number for any single mode changes by 1 where the vibrational motion in that mode corresponds to a changing dipole moment. 

With the many degrees of freedom in a polyatomic molecule, the many different normal mode frequencies, and the many types of transitions that might be seen, it is evident that polyatomic vibrational spectra can be quite congested and challenging to analyze. 

The analysis of a full infrared spectrum of a molecule in the gas phase usually starts by finding the fundamental transitions. Most often, these are the strongest transitions. Next, we look for progressions, which are a set of transitions originating from the same 

The transition moments are almost the same for the two transitions, too. However, their populations lead to important spectral differences. That fact can be exploited, for if the temperature of the sample is lowered, the hot band s intensity will diminish relative to a band that originates in the ground vibrational state. This is the consequence of diminishing the excited state s population with decreasing temperature. It is a common practice, in fact, to remove hot band congestion from a spectrum by cooling the sample. 

Characteristic Vibrational Stretching Frequencies (cm ) of Certain Functional Groups  

---

The structure of this review is as follows. In Section 9.2, we briefly discuss methods for computing vibrational states of systems having several coupled vibrational degrees of freedom. This will also cover methods that were not yet adapted for direct use with ab initio potentials, since in our view, such extensions may be possible in the future, at least for some of the algorithms. The focus will be on methods that seem potentially applicable to large polyatomics, rather than those of great accuracy for small systems. Section 9.3 also deals with computational methods for anharmonic vibrational spectroscopy that are applicable to potential surfaces from electronic structure calculations. Our main focus will be on the Vibrational Self-Consistent Field (VSCF) approach in several variants and extensions. The performance of the available method in the present state of the art is discussed in Section 9.4. Future directions are outlined in Section 9.5. 

With regard to the electronic structure methodology, major obstacles must be surmounted before improvements can be made. Calculations with Coupled-Cluster methods, an obvious next step, are far more computationally costly than the presently used MP2, or B3LYP methods. In fact, there are extremely few direct ab initio calculations of anharmonic vibrational spectroscopy at higher than MP2 or DPT levels, even for small polyatomics. From the point of view of ab initio anharmonic spectroscopy, the leap from MP2 to the Coupled-Cluster method seems a bottleneck. One can draw encouragement from faster Coupled-Cluster implementations, so far employed with the perturbation theory anharmonic analysis [116,117]. 

The symmetry of nuclear displacements arises most commonly in connection with vibrational spectroscopy of polyatomic molecules [7]. Let us compare the nuclear displacements of a symmetric linear triatomic, XYX, illustrated in Fig. 4.2, with those shown in Fig. 3.9 for a homonuclear diatomic molecule, which also has cylindrical symmetry. It was pointed out that in the latter case there is no way of reducing the symmetry of the potential energy of X2 below Doo/i by nuclear motion in the case of the triatomic molecule, there is. 

In this chapter, we extend our treatment of rotation in diatomic molecules to nonlinear polyatomic molecules. A traditional motivation for treating polyatomic rotations quantum mechanically is that they form a basis for experimental determination for bond lengths and bond angles in gas-phase molecules. Microwave spectroscopy, a prolific area in chemical physics since 1946, has provided the most accurate available equilibrium geometries for many polar molecules. A background in polyatomic rotations is also a prerequisite for understanding rotational fine structure in polyatomic vibrational spectra (Chapter 6). The shapes of rotational contours (i.e., unresolved rotational fine structure) in polyatomic electronic band spectra are sensitive to the relative orientations of the principal rotational axes and the electronic transition moment (Chapter 7). Rotational contour analysis has thus provided an invaluable means of assigning symmetries to the electronic states involved in such spectra. 

In the classical vibrational spectroscopy, the subject of investigation is the vibrational-rotational motion of polyatomic molecules near the very bottom of their potential-energy surface in the electronic ground state. In this case, the normal-mode approximation proves quite applicable. Indeed, one can expand as a Taylor series the potential energy of the molecule near the equilibrium position (the potential-energy minimum) and write down the molecular Hamiltonian in the form... 

Molecular vibrations are excited by infrared radiation [2]. A typical flowchart of a vibrational spectroscopy experiment on a polyatomic molecule could be as follows. 

Classical Dynamics of Nonequilibrium Processes in Fluids Integrating the Classical Equations of Motion Control of Microworld Chemical and Physical Processes Mixed Quantum-Classical Methods Multiphoton Excitation Non-adiabatic Derivative Couplings Photochemistry Rates of Chemical Reactions Reactive Scattering of Polyatomic Molecules Spectroscopy Computational Methods State to State Reactive Scattering Statistical Adiabatic Channel Models Time-dependent Multiconfigurational Hartree Method Trajectory Simulations of Molecular Collisions Classical Treatment Transition State Theory Unimolecular Reaction Dynamics Valence Bond Curve Crossing Models Vibrational Energy Level Calculations Vibronic Dynamics in Polyatomic Molecules Wave Packets. 

Typical abundances of PAHs are of the order of 10 relative to hydrogen, which makes these species the most abundant polyatomic molecules present in space. Vibrational spectroscopy is eminently suited for detecting the presence of classes of molecules, but it is much more difficult to identify specific molecules within a collection of species. However, while the gross characteristics of these stellar and interstellar spectra are very similar, when examined in detail, they vary from source to source. Analysis of these variations may well provide us with a tool to identify specific molecules within the circumstellar and interstellar PAH family and such efforts, supported by extensive laboratory studies, are now underway. 

---