Laser News, Vol.10, No.2, Apr.1999

Understanding Basics

On the Molecular Infrared Absorption

Indranath Mukhopadhyay

Centre for Advanced Technology

Indore 452 013

Infrared light absorption in a molecular medium is an important phenomenon which is utilised in a wide variety of scientific experimentation. These experiments include spectroscopy, photochemistry, isotope separation, combustion chemistry, and pollution monitoring to name a few. In this note the basic mechanism of infrared absorption by a molecule is described in a purely classical manner. The basic existence of the fundamental and overtone bands has been explained. A short discussion is added for selection rules and "Fermi" resonance.

Degrees of Freedom and Fundamentals

In the simplest model a molecule is thought to be composed of atoms which are represented by point masses. The inter-atomic forces holding the molecule together are analogous to those exerted by massless springs which tend to restore bond lengths and angles to the equilibrium values. Each mass requires three co-ordinates to define its position, such as x, y and z in a rectangular co-ordinate system. Thus it has three independent degrees of freedom of motion, in the x, y and z direction. If there are N atoms in the molecule, there will be a total of 3N degrees of freedom of motion of all atoms in the molecule. The centre of mass of the molecule requires three co-ordinates to define its position and therefore has three independent degrees of freedom of motion which are translations of the centre of mass of the molecule in any three directions. When a non-linear molecule is in its equilibrium, it requires three rotational co-ordinates to specify the molecular orientation about the centre of mass. Therefore, a non-linear molecule has three independent rotational degrees of freedom. A linear molecule only has two independent rotational degrees of freedom about two axes mutually perpendicular to the molecular axis. Rotation of a linear molecule about the molecular axis is not considered a degree of freedom of motion because no displacements of atoms are involved. Thus when we exclude the translational and rotational degrees of freedom, we are left with 3N- 6 internal degrees of freedom for a non-linear molecule and 3N- 5 internal degrees of freedom for a linear molecule. This internal degrees of freedom correspond to the same number of independent normal modes of vibration. In each normal mode of vibration all the atoms oscillate at the same frequency and all atoms pass through their equilibrium positions simultaneously. If the forces holding the atoms together in molecule are directly proportional to the displacement of the atoms, then the molecular vibrations will be harmonic, but if the relation between the restoring force and the displacement is non-linear the vibrations are called anharmonic.

When infrared radiation falls on the molecule the electric field of the radiation executes forced vibration in the molecule. If the frequency of the infrared light matches with that of one of the normal modes of vibration the molecule absorbs more energy from the light and the vibrations become more vigorous. This is the resonance. At nonresonant frequencies of the light the energy transfer from the radiation will be ineffective. If the molecule is irradiated with a wide range of infrared frequencies, only the frequencies, which match, with the normal mode frequencies will be absorbed. These are the fundamental bands. When the incident light intensity is very high the displacement of the atoms from the equilibrium positions will be large to make the relationship between the restoring force and the displacements non-linear. This will result in absorption at multiples and various additive combinations of the normal mode frequencies. These are the overtone and combination bands. Then there could be weak higher order combination bands due to additive combination of multiples of various normal mode frequencies.

Dipole Moment Considerations

While the frequency of infrared absorption depends on the molecular vibrational frequency, the absorption intensity depends on how effectively the infrared photon energy can be transferred to the molecule, and this depends on the change in the dipole moment that occurs as a result of molecular vibration. The dipole moment is defined, in the case of a simple dipole, as the magnitude of either charge in the dipole multiplied by the charge spacing. In a complex molecule this simple picture can be retained if the positive particle represents the total positive charge of the protons concentrated at the center of charge of the protons, and the negative particle represents the total negative charge of the electrons concentrated at the center of charge of the electrons. The center of charge of the protons coincides with the center of mass of the protons, and the center of charge of the electrons coincides with the center of mass of the electrons. Since the wavelength of infrared radiation is far greater than the size of most molecules, the electric field of the photon in the vicinity of a molecule can be considered uniform over the whole molecule. The electric field of the photon exerts forces on the molecular charges and obviously the forces on opposite charges will be exerted in opposite directions. Therefore the oscillating electric field of the photon will exert forces tending to change the spacing between the proton and electron centers of charge, thereby tending to induce the dipole moment of the molecule to oscillate at the frequency of the photon.

At certain characteristic frequencies a forced dipole moment oscillation will tend to activate a nuclear vibration. These are molecular vibrational frequencies where the nuclear vibration causes a change in dipole moment. The more the dipole moment changes during a vibration, the more easily the photon electric field can activate that vibration. If a molecular vibration should cause no change in dipole moment then a forced dipole moment oscillation cannot activate that vibration. This gives rise to the selection rule that in order to absorb infrared radiation, a molecular vibration must cause a change in the dipole moment of the molecule. It can be shown that the intensity of an infrared absorption band is proportional to the square of the change in dipole moment with respect to the normal coordinate of the corresponding molecular vibration giving rise to the absorption band. Another way to picture the dipole moment is to consider the atoms in the molecule as charged particles rather than the individual electrons and protons. If an atom has the same number of electrons and protons it is electrically neutral and does not contribute to the dipole moment. However, there are chemical forces in the molecule that tend to redistribute the electrons somewhat, so that a given atom may have a little electron excess or deficiency, and may be considered as a particle with a small residual charge. The positive particle of the molecular dipole represents the total excess positive charge on the atoms, concentrated at the center of the excess positive charge, and the negative particle of the dipole represents the total excess negative charge on the atoms, concentrated at the center of the excess negative charge.

We take the example of the hydrochloric acid (HCl) molecule for illustration. In the HC1 molecule, chlorine atom has a slight excess of negative charge and the hydrogen atom will have a slight excess of positive charge. These two atoms form a simple dipole. During the vibration of the HC1 molecule, the dipole spacing changes and the excess charge distribution also changes. Both of these effects change the dipole moment, which means that this vibration is infrared active. Now let us talk about the hydrogen molecule (H2). In the H2 molecule a centre of symmetry is present. The centre of symmetry can be explained as follows: at equilibrium, every atom, which is not at the center of the molecule, has an exactly equivalent atom on the direct opposite side of the center. This arrangement guarantees that the center of positive charge and the center of negative charge will coincide at the center of the molecule, which specifies a zero dipole moment. During the vibration of the H atom oscillation the vibrationally distorted molecule retains the center of symmetry so the dipole moment does not change, and it remains zero. This vibration is infrared inactive since there is no change in dipole moment caused by the vibration. The electric field of the photon cannot pull the two equivalently charged hydrogen atoms in opposite directions as required in a diatomic vibration. This type of vibration is infrared inactive. In general, if a molecule has a center of symmetry at equilibrium, then vibrations during which the center of symmetry is retained will be infrared inactive. However, molecules that do not have a center of symmetry may also have infrared inactive vibrations if other types of symmetry such as planes or axes of symmetry are present.

Anharmonicity and Overtones

If a plot is made of dipole moment versus time for a classical vibration, a periodic but nonsinusoidal wave will result if either mechanical or electrical anharmonicity is present. However, any such periodic function can be resolved into simple harmonic components where the frequencies are integral multiples of the fundamental vibrational frequencies. This means that if the molecular vibration is anharmonic, the dipole moment will oscillate with the fundamental frequency and integral multiples thereof. These are called the fundamental, first overtone, second overtone, etc., as discussed above.

Raman Effect

While infrared and Raman spectra both involve vibrational and rotational energy levels, they are not duplicates of each other but rather complement each other. This is because the intensity of the spectral band depends on how effectively the photon energy is transferred to the molecule and the mechanism for photon-energy transfer differs in the two techniques. This is discussed in the following: If a molecule is placed in the electric field of electromagnetic radiation then the electrons and protons will experience oppositely directed forces exerted by the electric field. As a result the electrons are displaced relative to the protons and the polarized molecule has an induced dipole moment caused by the external field. The induced dipole moment, divided by the strength of the electric field, is the polarizability. The polarizability can be looked on as the deformability of the electron cloud of the molecule by the electric field. In order for a molecular vibration to be Raman active, the vibration must be accompanied by a change in the polarizability of the molecule.

Simultaneous Rotation and Vibration

Let us take a linear molecule vibrating along its axis and also rotating about its center of mass. The dipole moment, m , along the bond axis direction is oscillating due to the vibration, and hence

m µ a cos w vt

where a is the amplitude, w v, is the vibrational angular frequency, and t is time. Now imagine that this molecule is approached by a plane-polarised infrared light where the electric field is oscillating in the z direction which lies in the plane defined by the rotating dipole moment. At any time t the molecular axis makes an angle f with the z axis so that f = w r t, where w r, is the rotational frequency. The dipole moment component, m z, in the photon e field direction z, is

m z = m cos f µ cos (w vt) cos (w rt)

= [cos (w v + w r) t + cos (w v - w r) t]

Therefore, the dipole moment oscillates with the frequencies (w v + w r) and (w v - - w r) and can interact with electromagnetic radiation which has these same frequencies. If we consider a large number of identical molecules in the gaseous state, all will vibrate at the same frequency w v which depends on the masses and force constant. However, a collection of molecules will have a variety of rotational frequencies w v. The number of molecules with in a given rotational frequency will reach a maximum at some intermediate rotational frequency. Thus the absorption band will have a broad doublet structure. This classical prediction for band shape is well realised in a low resolution infrared spectrum of a gaseous asymmetric diatomic molecule where the dipole moment change is along the molecular axis.

A linear polyatomic molecule such as C02 has two kinds of infrared active vibrations. When the dipole moment changes along the axis, as in the anti-symmetric stretch vibration, then a broad doublet is predicted using the previous arguments. When the dipole moment changes perpendicular to the molecular axis as in the bending vibration, a different situation arises. The molecule can rotate about two mutually perpendicular axes. When the rotational axis is perpendicular to the plane in which the molecule is bending, then the molecular rotation changes the orientation of the oscillating dipole moment as before with similar spectral results. However, when the rotational axis direction is parallel to the direction of the dipole moment change, the molecular rotation does not affect the orientation of the changing dipole moment which simply oscillates at the frequency w v, and gives rise to a single spectral peak. Since the molecule can rotate with equal probability about both axes, the total band is a combination of a central peak and two broad wings. In the out-of-phase anti-symmetric stretch vibration it is not possible to rotate the molecule without changing the orientation of the oscillating dipole moment, so a central peak is forbidden. These classical structures are fairly well realised in a low resolution spectrum of CO2, where the two strong bands both have similar broad doublet structures but only one band has a central peak. It is seen that the vibrational-rotational band shape can be used to tell whether the dipole moment change is along or across the axis of a linear molecule, which in this case is sufficient to assign the bands unambiguously to the bending and stretching vibrations. If the resolution is increased sufficiently, the spectrum is into a series of discrete lines because the angular momentum is quantised and not continuously variable as in the classical picture.

Symmetry and the Spectra

Symmetry of a molecule plays an important role in the structure of the exhibited spectra i.e. which bands are permitted and which are forbidden. It is thus important to set up some system whereby molecules could be classified by their symmetry characteristics. The molecules can be classified using five symmetry elements. For each symmetry element there is a corresponding symmetry operation which produces a configuration indistinguishable from the original molecule. We have seen that a molecular vibration will be infrared active only if the vibration causes a change in the dipole moment m and in quantum mechanical terms the intensity is proportional to the square of the transition moment matrix element whose x, y, and z components are

á fô m xô iñ , á fô m yô iñ and á fô m zô iñ

where i and f represent the initial and final states, respectively and m x, m y and m z are the dipole moment components. In order for a transition f ß i to be infrared active at least one of the above elements must be nonzero. For certain vibrational symmetry species all three integrals may be zero as a necessary consequence of the symmetry of a molecule. Since the transition moment components are definite integrals over the whole space, they will be unchanged by a symmetry operation. Furthermore, the band intensity, which is proportional to the square of the transition moment must have the same value for all indistinguishable orientations of the molecule. This means that if the integrals have nonzero values the product (y fm y i) must be totally symmetric.

The vibrational wavefunction y symmetry properties related to those of the normal co-ordinate. If a symmetry operation reverses an anti-symmetrical normal co-ordinate (q Ù - q) then, the wavefunction is left unchanged if the quantum number is even. In the ground state the total vibrational wavefunction is a product of normal co-ordinate wavefunctions where all the quantum numbers are v = 0. This ground state wavefunction is not changed by any symmetry operation and must belong to the totally symmetric species. In a fundamental transition, the molecule goes from the ground vibrational state to an excited state where the quantum number is v = 1 (which is odd) for one normal co-ordinate only and v = 0 for all the other normal co-ordinates. In this case the wavefunction of the excited state must have the same symmetry properties as that one excited normal co-ordinate. In order to investigate the symmetry properties of the product (y fm y i) one starts with the symmetry properties of the product (y fy i) and then compare these to the symmetry properties of the dipole moment m . In order to do this it is necessary to define the character of the direct product of representations. Therefore, in order for the above transition moments to be nonzero the product (y fm y i) must belong to the totally symmetric species. This product will be totally symmetric only if the wavefunction product y fy i belongs to the same species as at least one component of the dipole moment m . It follows that a vibrational transition between energy levels is allowed in the infrared spectrum when at least one component of the dipole moment vector has the same species as the product of the wavefunctions which characterize the molecular state for the two levels involved For example, in order for a fundamental transition to be infrared active, at least one component of the dipole moment should belong to the same species as (y fy i) which is the same as that of the excited normal coordinate. In order for a transition to be Raman active, an integral of the type á fô a llô iñ must be nonzero where a ll is one component of the polarizability (a ) and l and l' are x, y, or z. This leads to a selection rule for Raman activity identical to the one above for infrared activity, but with the polarizability substituted for the dipole moment. It can showed that a vibrational species will be infrared active if one or more of the three components of translation are listed in the row for that species in the conventional character table. This is because both the dipole moment and the translation of the centre of mass of the molecule have similar symmetry transformation. For details discussions the readers are referred to Ref. [1]

Fermi and Coriolis Resonance

In a molecule, each normal mode has one infrared active transition corresponding to the fundamental associated with it. If the vibration is anharmonic, new weak bands involving overtone and combination transitions can be observed. Sometimes the frequency of an overtone or combination band may have almost same frequency as another fundamental. In such a case two strong bands may be observed where only one strong fundamental is expected. This is the Fermi resonance. Both bands involve the fundamental state and the overtone or combination state. In other words, the upper sates in these transitions are "Fermi" - mixed. The intensity of the overtone band is increased at the expense of the intensity of the fundamental band - the so called intensity borrowing (or stealing) effect. It should be noted that the "Fermi" perturbation occurs through the potential energy operator. Similar resonance can occur between two vibrational levels through the kinetic energy operator and it is called Coriolis resonance. The simplest way to differentiate between "Fermi" and "Coriolis" resonance as follows. If the molecular rotation is reduced to zero the Coriolis perturbation goes to zero and the "Fermi" perturbation is essentially independent of rotation. The simplest possible example (in fact the first observation) of "Fermi" resonance is the interaction between the pair of levels (020) and (100) in the CO2 molecule [2]. These two sets of energy levels are involved in the CO2 laser operation in the 10 m wavelength region. In the absence of Fermi interaction the laser transition (001) Ù (020) would be too weak to be observed. Hence although perturbation almost always make a spectroscopist’s life difficult, it may make a laser scientist’s life easy.

References

[1] G. Herzberg, "Infrared and Raman Spectra of Polyatomic Molecules," Van Nostrand-Reinhold, NJ (1945).

[2] E. Fermi, Z. Physik, 71, 250 (1931).