Chapter 6 Vibrational Spectroscopy As with other applications of symmetry and group theory, these techniques reach their greatest utility when applied to the analysis of relatively small molecules in either the gas or liquid phases. As the size of the molecule increases, many of these vibrations have very similar frequencies and are no longer individually distinguishable. At this level, spectroscopic assignment is usually confined to identifying frequencies associated primarily with specific chemical structure (e.g. functional group). We will confine our discussions to smaller molecules, where the power of symmetry and group theory is greatest.
Chapter 6 Vibrational Spectroscopy 6.1 Vibrational Modes and Their Symmetries 6.2 Symmetry-Based Selection Rules and Their General Consequences 6.3 Spectroscopic Activities and Structures of Nonlinear Molecules 6.4 Linear Molecules 6.5 Overtones, Combinations, and Other Complications
6.1 Vibrational Modes and Their Symmetries The individual atoms of a molecule are constantly in motion over the entire range of real temperature above absolute zero. These individual atomic motions result in three kinds of molecular motions: vibration, translation, and rotation. For diatomic molecule, suppose two atoms move apart from their equilibrium internuclear distance, r e, so as to stretch the chemical bond. As they move apart, away from the equilibrium position, they will experience a restoring force, F, in opposition to the motion. In classical mechanics, the restoring force will be proportional to the displacement from the equilibrium distance, Δr, and vary according to Hooke s law: F = -kδr
6.1 Vibrational Modes and Their Symmetries At some point the restoring force will cause the two atoms momentarily to arrest their travel away from each other, after which they will reverse their motions and begin to travel toward each other. As they approach one another they will pass through the equilibrium internuclear distance and continue to move together, until their mutual repulsions arrest them at a minimum separation and drive back in the opposite direction. This periodic series of motions constitutes one cycle of the vibrational mode of the molecule as a harmonic oscillator.
6.1 Vibrational Modes and Their Symmetries From classical mechanics, the potential energy of the system would vary as a function of displacement. However, the energy of a real vibrating molecule is subject to quantum mechanical restrictions. Where v is the vibrational quantum number, whose values may be 0, 1, 2, ;ν is the vibrational frequency. It is more convenient to define the energy of the system in wavenumber units, called term values, T.
6.1 Vibrational Modes and Their Symmetries Equation 6.3 suggest a model in which we have a series of equally spaced energy levels. The minimum energy of the system, called the vibrational ground state, is attained when v =0. Note that it does not lie at the minimum of the parabola defined for the classical oscillator. Zero point energy
6.1 Vibrational Modes and Their Symmetries Real molecules are not perfect harmonic oscillators. Have a Morse curve. The behavior of an anharmonic oscillator. By solving Schrödinger equation:
6.1 Vibrational Modes and Their Symmetries Suppose both atoms move in parallel in the same direction, resulting in a translation of the entire molecule through space. This is not a periodic motion, so it has no interaction with electromagnetic radiation; that is, it cannot be detected by infrared or Raman spectroscopy.
6.1 Vibrational Modes and Their Symmetries Suppose two atoms move in opposite directions perpendicular to the bond axis. This will cause the molecule to tumble or rotate. Unlike translation, rotation can be detected spectroscopically, because it occurs with a repeating periodic cycle. (in the microwave region)
6.1 Vibrational Modes and Their Symmetries Linear molecule 2 rotation modes Nonlinear molecule 3 rotation modes
6.1 Vibrational Modes and Their Symmetries For polyatomic molecule. The motions of each atom can be resolved into components along the three directions of a Cartesian coordinate system. Therefore, any molecule composed of n atoms possesses 3n degrees of freedom of motion including vibrations, translations, and rotations. A linear molecule possesses 3n - 5 vibrational modes (3 translations, 2 rotations) A nonlinear molecule possesses 3n - 6 vibrational modes (3 translations, 3 rotations) Like the diatomic case, each normal mode of vibration has a characteristic frequency and can assume a series of quantized energies. The frequencies recorded by infrared and Raman spectroscopy arise from transitions between these states.
6.1 Vibrational Modes and Their Symmetries One of the rotational modes of a nonlinear molecule becomes a bending vibrational mode when the molecule is made linear. Relative to the overall molecular symmetry, all of the 3n degrees of freedom-normal modes of vibrations, translations, and rotationshave symmetry relationships consistent with the irreducible representations of molecule s point group. In other words, we can catalogue all degrees of freedom according to the appropriate Mulliken symbols for their corresponding irreducible representations.
6.1 Vibrational Modes and Their Symmetries The entire set of 3n vectors as a basis for a representation in the molecule s point group. For SO 2, the identity operation 9
6.1 Vibrational Modes and Their Symmetries C 2 rotation -1 σ(xy) 1
6.1 Vibrational Modes and Their Symmetries σ(yz) 3
6.1 Vibrational Modes and Their Symmetries Γ 3n = 3A 1 + A 2 + 2B 1 + 3B 2
6.1 Vibrational Modes and Their Symmetries We have found nine nondegenerate species for the nine degrees of freedom possible for SO 2. We can find the symmetry species of the normal modes of vibration by identifying and removing the three translations and three rotations from Γ 3n. Where the species comprising trans and rot are the same as those of the three unit vector and three rotational vector transformations. From character table, Γ trans = A 1 + B 1 + B 2, Γ rot = A 2 + B 1 + B 2. Finally, Γ 3n-6 = 2A 1 + B 2
6.1 Vibrational Modes and Their Symmetries Two of the modes preserve this symmetry relationships implied by the A 1 representation. From the characters of B 2 we know that this mode is antisymmetric with respect to both C 2 rotation and σ(xy) reflection.
6.1 Vibrational Modes and Their Symmetries Γ rot = T 1 Γ 3n-6 = A 1 + E + 2T 2 If the molecule belongs to a point group that has doubly or triply degenerate irreducible representations, some vibrational modes may be degenerate and therefore have identical frequencies. For example CH 4, Γ 3n = A 1 + E + T 1 + 3T 2 Γ trans = T 2
6.1 Vibrational Modes and Their Symmetries Γ 3n-6 = A 1 + E + 2T 2 We predict that there should be four frequencies: ν 1 (A 1 ), ν 2 (E), ν 3 (T 2 ), ν 4 (T 2 )
6.2 Symmetry-Based Selection Rules and Their General Consequences Now we know that the various normal modes of a molecule can be catalogued and analyzed in terms of their symmetry with respect to the overall molecular symmetry. The question now arises whether or not these vibrations can be observed in the infrared and Raman spectra. These selection rules indicate which normal modes are active (allowed) or inactive (forbidden) in each kind of spectrum. A normal mode belonging to the same symmetry species as any of the unit vector transformations x, y, or z will be active in the infrared spectrum. A normal mode will be Raman active if it belongs to the same symmetry species as one of the binary direct products of vectors listed in the character table.
6.2 Symmetry-Based Selection Rules and Their General Consequences For SO 2, we can find the following unit vector and direct product transformation from the character table. Γ 3n-6 = 2A 1 + B 2 Both A 1 and B 2, the species of the three normal modes, have listing for unit vectors and direct products. Therefore, all three modes are active in both the infrared and Raman spectra.
6.2 Symmetry-Based Selection Rules and Their General Consequences For CH 4, we can find the following unit vector and direct product transformation from the character table. Γ 3n-6 = A 1 + E + 2T 2 All three unit vectors transform degenerately as T 2, so only normal modes with T 2 symmetry can be infrared active.
6.2 Symmetry-Based Selection Rules and Their General Consequences The general conclusions given by the symmetry-based selection rules
6.2 Symmetry-Based Selection Rules and Their General Consequences Normal modes that are totally symmetric can be identified experimentally in the Raman spectrum by measuring the depolarization ratio, ρ. The scattered Raman radiation has a polarization that can be resolved into two intensity components, I and I. Totally symmetric band has a value in the range 0< ρ <3/4 (polarized). ρ = ¾ for any mode that is not totally symmetric (depolarized). Highly symmetric molecules often has ρ 0 for polarized bands. Γ 3n-6 = A 1 + E + 2T 2
6.3 Spectroscopic Activities and Structure of Nonlinear Molecules The number and spectroscopic activity of normal modes depends upon the molecule s symmetry. On this basis we might expect that infrared and Raman spectroscopy could be used to distinguish between two or more possible structures that a particular molecule might have. Nonetheless, within the limitations of relatively small molecules in gas and liquid sample, predictions from symmetry and group theory can be effective tools to interpret vibrational spectra and deduce structures.
6.3 Spectroscopic Activities and Structure of Nonlinear Molecules For XY 4 complex which can be tetrahedral or square planar. For square planar
6.3 Spectroscopic Activities and Structure of Nonlinear Molecules Two possible structures
6.3 Spectroscopic Activities and Structure of Nonlinear Molecules If one or more atoms in a compound are substituted by some other element, the symmetry often changes and with it the vibrational selection rules. CH 4 and CH 3 D
6.3 Spectroscopic Activities and Structure of Nonlinear Molecules CH 4 and CH 2 D 2
6.3 Spectroscopic Activities and Structure of Nonlinear Molecules CH 4 and CH 2 D 2
6.4 Linear Molecules C 3 O 2, Carbon suboxide, O=C=C=C=O This is a centrosymmetric linear molecule, so the point group is D h. To avoid the problem of the group s infinite order, we will use D 2h.
6.5 Overtones, Combinations, and Other Complications We have seen that it is possible to predict the number and spectroscopic activities of the fundamental transitions of the normal modes of a polyatomic molecule. When applying these predictions to actual spectra a number of complications can arise. On the one hand, weak intensities or instrumental limitations can result in fewer observable frequencies than predicted. But more often there are more peaks in the spectrum than one would predict. The most common being the presences of the overtone bands and combination bands.
6.5 Overtones, Combinations, and Other Complications Although the fundamental selection rule for a harmonic oscillator allows only transitions for which Δv = ±1, anharmonicity in the oscillations of real molecules gives rise to weak spectroscopic bands from transitions for which Δ v = ± 2, ± 3,, ± n. For a normal mode ν i, such transitions represent the first, second, and succeeding overtones of the fundamental, customarily designated 2ν i Overtone frequencies are almost always slightly less than the whole-number multiple values. Combination bands of the type ν k + ν i. Also possible, combinations with and among overtones of the type nν k + nν i. Another way, subtractive combinations ν k - ν i and nν k -nν i.
6.5 Overtones, Combinations, and Other Complications Generally only a few have sufficient intensity to be observed. Recall that the direct product of any nondegenerate irreducible representation with itself is the totally symmetric representation. From this we may conclude that the first overtone of any nondegernerate normal mode will belong to the totally symmetric representation and will be Raman allowed. The direct product of any degenerate representation with itself contains the totally symmetric representation, so all first overtones are Raman allowed.
6.5 Overtones, Combinations, and Other Complications For CH 4, Infrared allowed Γ(T 2 T 2 )= A 1 + E + T 1 + T 2 Raman allowed Infrared allowed Γ(ET 2 )= T 1 + T 2 Raman allowed
6.5 Overtones, Combinations, and Other Complications Recall that for a planar XY 4 there is one normal mode ν 5 (B 2u ), which is not active in either infrared or Raman spectra. However, the first overtone of this mode should be totally symmetric(a 1g ) and allowed in Raman spectrum. Fermi resonance; the two bands mix and split, losing their individual identities. One feature moves to higher frequency and the other to lower.