How do molecules vibrate? What is anharmonicity? Read this animated guide to molecular vibration.
A basic model of molecular vibration
To keep things simple we will consider the smallest possible molecule, which consists of two atoms (called a diatomic molecule). These molecules can be modelled using the concept of a simple harmonic oscillator (Figure 1). This is where we treat the two atoms as point masses connected by a ‘spring’ which represents the electrostatic force between charged atoms. The force is proportional to the distance, or bond length, between the two atomic centres. When a system like this is set in motion, the two atoms will oscillate back and forth around an equilibrium point.
Figure 1: Simple harmonic oscillator model for a diatomic molecule.
As the distance between the two atoms, r, decreases from the equilibrium length, re, the repulsive force between the positively charge nuclei is dominant, and the energy stored in the molecule increases, approaching infinity as the distance r approaches zero.
When r is increased beyond re then a force of attraction develops to pull the two atoms back together, storing further energy that again tends towards infinity. Between these two extremes the repulsive and attractive forces balance, leading to a minimum in the stored energy of the molecule. The resulting curve is described as an energy well. At temperatures above absolute zero, molecules will oscillate back and forth near the bottom of the energy well. Absorption of energy from photons increases the oscillation to higher energy levels in the well. In the quantum mechanical description of harmonic oscillation, molecules can have only discrete quantities of energy, as shown in Figure 2.
Figure 2: Quantisation of the energy levels within the harmonic energy well.
This means that molecules cannot absorb any photon, only those whose energy corresponds to the gaps between the different quantised energy levels (Figure 3). The position of the atoms is described by a probability function which is different for each quantised energy level. In the lowest vibrational energy state – called its ground state – the distance r is centred on re but with some probability of being slightly above or below it. This is represented by the lowest horizontal line on Figure 2.
Figure 3: Simple harmonic oscillator model for a diatomic molecule showing the quantised energy levels and the effect of the interaction of the molecules with photons of light with different energies. (Left) A photon with energy less than the gap between vibrational energy levels; (Middle) A photon with energy equal to the gap between vibrational energy levels; (Right) A photon with energy greater than the gap between vibrational energy levels.
The higher energy levels are known as excited states. For vibrational energy levels, the gaps between the levels correspond to photon energies in the infrared region of light. Infrared spectroscopy therefore allows us to see the change in energy required to excite the molecule from the ground state into the first excited state and higher states beyond that.
The transition from the ground state vibration to the first excited state (E0→E1) is known as the fundamental transition, while excitation to higher energy states (for instance: E0→E2) are known as overtones. The exact spacing between the different energy levels (labelled as ∆E in Figure 3) varies depending on the mass of the atoms and the bond strength (the ‘spring’ in our mass-spring model). As a result, different molecules have different IR spectra, and these spectra provide a unique fingerprint to identify the molecules present. In the simple harmonic oscillator model the spacings between these energy levels for a fixed system is constant, which would imply that the first overtone (E0→E2) has exactly double the energy of the fundamental transition.
The x-axis of the IR spectrum is a measure of the energy of the photon and typically presented in wavenumbers / cm-1, whilst the y-axis represents how much of the light is absorbed. Typically, either %-transmission or absorbance is used for this axis.
Introducing anharmonicity
So far, our simple model assumes that the spring can be stretched further and further forever without breaking. In the real world, eventually you will get to a point where the spring snaps. A more complex model that provides a better representation of reality is the anharmonic oscillator model shown in Figure 4. The critical difference is that after the atoms have grown a certain distance apart the energy of the molecule reaches a maximum, at which point the molecule dissociates into two separate atoms. The ‘spring’ has effectively broken.
The other consequence of anharmonicity is that the gaps between the higher excited states begin to narrow. In the harmonic oscillator model going from the ground to the second excited state required exactly double the energy as going from the ground to first excited state. Going from the ground state to the third excited state required 3 times as much energy as going from the ground to the first. Once anharmonicity is introduced this is no longer the case: if the energy required to transition between the ground and the first excited state is x then going from the ground to the second excite state will require slightly less than 2x.
Figure 4: Difference between the harmonic and the anharmonic oscillator model.
To use a real-world example, hydrogen chloride gas (HCl) has a fundamental IR mode at 2886 cm-1. The harmonic model would therefore predict the first overtone band (corresponding to an excitation from the ground to the second excited state) at 5772 cm-1 (twice 2886 cm-1). However, when the spectrum of HCl is measured we find that this overtone band is at 5668 cm-1, fractionally less than predicted by the harmonic model. Likewise, the second overtone band (corresponding to the ground to third excited state) is at 8348 cm-1, 11 cm-1 less than predicted by the harmonic oscillator model. Note: The exact spectrum of this compound is actually more complex due to the fact that it is a gas, however this is beyond the scope of the current article and will be covered in another primer at a later date.
Beyond diatomic molecules
What about more complex molecules than diatomic molecules? When we begin to add more atoms the number of possible vibrations increases, and more fundamental bands are observed. The total number of possible vibrations for a molecule is equal to 3N-6 (3N-5 for a linear molecule) where N is equal to the number atoms in the molecule. In practice, only the vibrations which result in a change of molecular dipole moment will be excited by photons in the infrared, which typically requires that the vibrations be asymmetrical or involve dissimilar atomic masses. This is the reason why diatomic molecules such as N2 and O2 are inactive in the IR but CO and HCl are active.
Water is a good example of a more complex molecule. It is a bent three atom molecule, with 3 fundamental vibrational modes as predicted from 3N-6. These modes are shown in Figure 5 and correspond to a symmetric stretch (where the two bonds grow and shrink together); an asymmetric stretch (where one bond is stretching whilst the other shrinks); and a bend where the two bonds vibrate towards and away from each other.
Figure 5 Animation showing the three vibrational modes of water: (i) symmetric stretch, (ii) asymmetric stretch and (iii) bending modes.
These three modes are all IR active and therefore have corresponding peaks in the IR spectrum at 3490 cm-1 (asymmetric stretch), 3247 cm-1 (symmetric stretch), and 1640 cm-1 (bending). The asymmetric and symmetric modes are both relatively broad in terms of peak width and therefore overlap, however the asymmetric band is visible as a shoulder on the symmetric band as highlighted in Figure 6 which shows an example spectrum of water collected using a Specac Quest ATR.
Figure 6: Infrared spectrum of water showing the 3 fundamental modes.
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