Vibrational Spectroscopy Book Pdf
Hildegard Mccauley
Ro-vibrational spectroscopy concerns molecules in the gas phase. There are sequences of quantized rotational levels associated with both the ground and excited vibrational states. The spectra are often resolved into lines due to transitions from one rotational level in the ground vibrational state to one rotational level in the vibrationally excited state. The lines corresponding to a given vibrational transition form a band.[1]
The appearance of rotational fine structure is determined by the symmetry of the molecular rotors which are classified, in the same way as for pure rotational spectroscopy, into linear molecules, spherical-, symmetric- and asymmetric- rotor classes. The quantum mechanical treatment of rotational fine structure is the same as for pure rotation.
The strength of an absorption line is related to the number of molecules with the initial values of the vibrational quantum number ν and the rotational quantum number J \displaystyle J , and depends on temperature. Since there are actually 2 J + 1 \displaystyle 2J+1 states with rotational quantum number J \displaystyle J , the population with value J \displaystyle J increases with J \displaystyle J initially, and then decays at higher J \displaystyle J . This gives the characteristic shape of the P and R branches.
Diatomic molecules with the general formula AB have one normal mode of vibration involving stretching of the A-B bond. The vibrational term values G ( v ) \displaystyle G(v) ,[note 3] for an anharmonic oscillator are given, to a first approximation, by
When the molecule is in the gas phase, it can rotate about an axis, perpendicular to the molecular axis, passing through the centre of mass of the molecule. The rotational energy is also quantized, with term values to a first approximation given by
The first two terms in this expression correspond to a harmonic oscillator and a rigid rotor, the second pair of terms make a correction for anharmonicity and centrifugal distortion. A more general expression was given by Dunham.
where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated when the B values for two different vibrational states can be found. For carbon monoxide req = 113.0 pm.[8]
The quantum mechanics for homonuclear diatomic molecules such as dinitrogen, N2, and fluorine, F2, is qualitatively the same as for heteronuclear diatomic molecules, but the selection rules governing transitions are different. Since the electric dipole moment of the homonuclear diatomics is zero, the fundamental vibrational transition is electric-dipole-forbidden and the molecules are infrared inactive.[10] However, a weak quadrupole-allowed spectrum of N2 can be observed when using long path-lengths both in the laboratory and in the atmosphere.[11] The spectra of these molecules can be observed by Raman spectroscopy because the molecular vibration is Raman-allowed.
The spectra of centrosymmetric molecules exhibit alternating line intensities due to quantum state symmetry effects, since rotation of the molecule by 180 about a 2-fold rotation axis is equivalent to exchanging identical nuclei. In carbon dioxide, the oxygen atoms of the predominant isotopic species 12C16O2 have spin zero and are bosons, so that the total wavefunction must be symmetric when the two 16O nuclei are exchanged. The nuclear spin factor is always symmetric for two spin-zero nuclei, so that the rotational factor must also be symmetric which is true only for even-J levels. The odd-J rotational levels cannot exist and the allowed vibrational bands consist of only absorption lines from even-J initial levels. The separation between adjacent lines in the P- and R- branches is close to 4B rather than 2B as alternate lines are missing.[20] For acetylene the hydrogens of 1H12C12C1H have spin-1/2 and are fermions, so the total wavefunction is antisymmetric when two 1H nuclei are exchanged. As is true for ortho and para hydrogen the nuclear spin function of the two hydrogens has three symmetric ortho states and one antisymmetric para states. For the three ortho states, the rotational wave function must be antisymmetric corresponding to odd J, and for the one para state it is symmetric corresponding to even J. The population of the odd J levels are therefore three times higher than the even J levels, and alternate line intensities are in the ratio 3:1.[21][22]
These molecules have equal moments of inertia about any axis, and belong to the point groups Td (tetrahedral AX4) and Oh (octahedral AX6). Molecules with these symmetries have a dipole moment of zero, so do not have a pure rotation spectrum in the infrared or microwave regions.[24]
Tetrahedral molecules such as methane, CH4, have infrared-active stretching and bending vibrations, belonging to the T2 (sometimes written as F2) representation.[note 7] These vibrations are triply degenerate and the rotational energy levels have three components separated by the Coriolis interaction.[25] The rotational term values are given, to a first order approximation, by[26]
The fact that the selection rules are different is the justification for the classification and it means that the spectra have a different appearance which can often be immediately recognized.An expression for the calculated wavenumbers of the P- and R- branches may be given as[30]
The C-Cl stretching vibration of methyl chloride, CH3Cl, gives a parallel band since the dipole moment change is aligned with the 3-fold rotation axis. The line spectrum shows the sub-structure of this band rather clearly;[6] in reality, very high resolution spectroscopy would be needed to resolve the fine structure fully. Allen and Cross show parts of the spectrum of CH3D and give a detailed description of the numerical analysis of the experimental data.[31][32]
Overtones of a degenerate fundamental vibration have components of more than one symmetry type. For example, the first overtone of a vibration belonging to the E representation in a molecule like ammonia, NH3, will have components belonging to A1 and E representations. A transition to the A1 component will give a parallel band and a transition to the E component will give perpendicular bands; the result is a hybrid band.[34]
Asymmetric top molecules have at most one or more 2-fold rotation axes. There are three unequal moments of inertia about three mutually perpendicular principal axes. The spectra are very complex. The transition wavenumbers cannot be expressed in terms of an analytical formula but can be calculated using numerical methods.
When measuring the spectra of gases it is relatively easy to obtain very long path-lengths by using a multiple reflection cell.[50] This is important because it allows the pressure to be reduced so as to minimize pressure broadening of the spectral lines, which may degrade resolution. Path lengths up to 20m are commercially available.
The method of combination differences uses differences of wavenumbers in the P- and R- branches to obtain data that depend only on rotational constants in the vibrational ground or excited state. For the excited state
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This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-24-1-0006. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Air Force.
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