Fundamentals of Molecular Physics
The wave description of electromagnetic radiation, and the three interchangeable ways of labelling it: frequency, wavelength and wavenumber.
Why molecular energy is quantized, and the Bohr condition \(\Delta E = h\nu\) that links a level scheme to a spectrum.
The regions of the electromagnetic spectrum and which molecular motion each one excites.
Why a molecule must possess a changing dipole moment to absorb — the gross selection rule that governs every chapter of Part B.
What fixes the intensity of a spectral line: transition probability, population, and path length.
Introduction to Molecular Physics
Atomic spectroscopy, the subject of Part A, asks what an atom does when its electrons rearrange. A molecule can do that too — but it can also rotate about its centre of gravity, and its atoms can vibrate about their equilibrium separations. Each of these motions is quantized, each has its own ladder of levels, and each therefore has its own spectrum in its own region of the electromagnetic spectrum.
Molecular spectroscopy is the study of the interaction between electromagnetic radiation and matter at this level. Its central use is inferential: from the positions and intensities of the lines a molecule absorbs or emits, one deduces bond lengths, bond strengths, force constants, dipole moments, isotopic composition and molecular geometry — none of which can be measured directly.
The logic of Part B is that of a hierarchy. From Table 6.1 below, \[ E_{\text{electronic}} \;\gg\; E_{\text{vibrational}} \;\gg\; E_{\text{rotational}}, \] in the rough ratio \(10^{5} : 10^{3} : 1\). So the chapters run upward through that ladder: rotation alone (Chapter 7), then vibration with rotational fine structure on it (Chapter 8), then the scattering experiment that reaches both by a different route (Chapter 9), and finally electronic transitions carrying vibrational and rotational structure within them (Chapter 10). What is common to all four is collected here.
6.1 Fundamentals of Electromagnetic Waves
Electromagnetic radiation — of which visible light is a small part — may be treated for present purposes as a simple harmonic wave travelling in a straight line from its source. What undulates is a pair of interconnected electric and magnetic fields, and it is these fields that will interact with matter to give a spectrum.
Any simple harmonic wave has the form of a sine curve, \begin{equation} y = A\sin\theta , \label{eq:sine-wave} \end{equation} where \(y\) is the displacement and \(A\) its maximum value, the amplitude. The connection with a travelling wave is made by picturing a point \(P\) moving round a circle of radius \(A\) at uniform angular velocity \(\omega\) (\(\mathrm{rad}/\mathrm{s}\)). After a time \(t\) the point has swept an angle \(\theta = \omega t\), so its vertical displacement is \begin{equation} y = A\sin\omega t = A\sin 2\pi\nu t . \label{eq:wave-time} \end{equation}
In one second the pattern repeats \(\omega/2\pi\) times. This is the frequency \(\nu\) of the wave, measured in hertz (\(\mathrm{Hz} = \mathrm{s}^{-1}\)). For a wave travelling at speed \(c\), the distance-time relation \(x = ct\) turns \(\eqref{eq:wave-time}\) into a description in terms of distance, \begin{equation} y = A\sin 2\pi\nu t = A\sin\frac{2\pi\nu x}{c} = A\sin\frac{2\pi x}{\lambda} , \label{eq:wave-space} \end{equation} which introduces the wavelength \(\lambda\), the distance covered during one complete cycle. Since \(\nu\) waves pass in one second and each occupies \(\lambda\) metres, \begin{equation} \boxed{\;\nu\lambda = c , \qquad \lambda = \frac{c}{\nu}\;} \label{eq:nu-lambda} \end{equation}
6.1.1 Wavenumber, and why spectroscopists prefer it
There is a third label, and in practice it is the one used most. The wavenumber \(\tilde\nu\) is the reciprocal of the wavelength expressed in centimetres: \begin{equation} \tilde\nu = \frac{1}{\lambda}\ \mathrm{cm}^{-1} , \qquad\text{so that}\qquad y = A\sin 2\pi\tilde\nu x . \label{eq:wavenumber} \end{equation} It is best thought of not as a reciprocal length but as the number of complete waves contained in each centimetre of radiation. Frequency and wavenumber are strictly proportional, \begin{equation} \nu = c\,\tilde\nu , \qquad c = 3\times 10^{10}\,\mathrm{cm}/\mathrm{s} , \label{eq:nu-nutilde} \end{equation} so the two are interchangeable provided the units are kept straight.
Wavelengths are quoted in whatever unit avoids large powers of ten: centimetres or millimetres in the microwave region, micrometres in the infrared, nanometres in the visible and ultraviolet. The older ångström survives in places: \begin{equation} 1\,\mathrm{\mu m} = e-6\,\mathrm{m}, \qquad 1\,\mathrm{Å} = e-10\,\mathrm{m}, \qquad 1\,\mathrm{nm} = e-9\,\mathrm{m} = 10\,\mathrm{Å} . \label{eq:length-units} \end{equation}
The symbols \(\nu\) (frequency) and \(\tilde\nu\) (wavenumber) look alike and are routinely confused. Keep the units in view rather than the symbol: wavenumber is always in \(\mathrm{cm}^{-1}\), frequency always in \(\mathrm{Hz}\). If a question quotes “an energy of \(10\,\mathrm{cm}^{-1}\)” it means a wavenumber separation, not a frequency. Throughout Part B the symbol \(\varepsilon\) denotes an energy already expressed in \(\mathrm{cm}^{-1}\).
6.2 The Quantization of Energy
Towards the end of the nineteenth century experiments began to conflict with the accepted view that matter takes up energy continuously. In 1900 Planck proposed instead that the energy of an oscillator is discontinuous, and can change only by a jump between two distinct energy states.
The idea extended rapidly to every form of molecular energy. A molecule in space possesses rotational energy by virtue of bodily rotation about its centre of gravity; vibrational energy by virtue of the periodic displacement of its atoms from their equilibrium positions; and electronic energy from the motion of the electrons in its bonds. Each of these is quantized: the molecule may exist in one of a set of discrete levels and can move from one to another only by a sudden jump involving a finite amount of energy.
Consider two such levels, \(E_{1}\) and \(E_{2}\), with \(E_{2} > E_{1}\). A transition between them may occur provided the appropriate amount of energy \begin{equation} \Delta E = E_{2} - E_{1} \label{eq:deltaE} \end{equation} is absorbed from, or emitted into, the radiation field. Planck's proposal was that this energy travels as electromagnetic radiation of frequency \begin{equation} \boxed{\;\nu = \frac{\Delta E}{h}\ \mathrm{Hz} , \qquad\text{equivalently}\qquad \Delta E = h\nu \ \text{joules} \;} \label{eq:planck} \end{equation} with \(h = 6.626\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\) per molecule. This single equation is the bridge between a level diagram and a spectrum, and every chapter of Part B uses it.
Direct a monochromatic beam of frequency \(\nu = \Delta E/h\) at molecules in state 1: they jump to state 2 and energy is removed from the beam. Use instead a beam containing a wide range of frequencies — “white” radiation — and a detector will show that only the frequency \(\nu = \Delta E/h\) has been attenuated, all others passing undiminished. That is an absorption spectrum. Alternatively, molecules already in state 2 may fall to state 1 and radiate; the emission spectrum so obtained is the complement of the absorption spectrum.
A common misreading: the frequency associated with an energy change does not imply that the transition occurs that many times per second. An electronic transition may involve radiation of frequency \(\sime15\,\mathrm{Hz}\), but the transition itself does not occur \(10^{15}\) times a second. It may occur once or many times, and on each occurrence it absorbs or emits one quantum of the appropriate frequency.
Energy differences between molecular levels are minute in joules per molecule, so three other units are in constant use. Multiplying by the Avogadro number gives joules per mole; dividing by \(hc\) gives wavenumbers; and dividing by \(e\) gives electron-volts: \begin{equation} \varepsilon\,[\mathrm{cm}^{-1}] = \frac{\Delta E}{hc} , \qquad 1\,\mathrm{cm}^{-1} = 1.24\times 10^{-4}\,\mathrm{eV} = 11.96\,\mathrm{J}/\mathrm{mol} . \label{eq:energy-units} \end{equation}
6.3 Regions of the Spectrum
Different regions of the electromagnetic spectrum excite different kinds of molecular motion.
| Region | Frequency | Wavelength | Spectroscopy | Energy change |
|---|---|---|---|---|
| Radiofrequency | \(3\times 10^{6}\text{--}3\times 10^{8}\,\mathrm{Hz}\) | \(10\text{--}\,\mathrm{m}\)–\(1\,\mathrm{cm}\) | NMR, ESR | \(0.001\text{--}10\,\mathrm{J}\,\mathrm{mol}^{-1}\) |
| Microwave | \(3\times 10^{10}\text{--}3\times 10^{12}\,\mathrm{Hz}\) | \(1\,\mathrm{cm}\)–\(100\,\mathrm{\mu m}\) | Rotational | \(10\text{--}100\,\mathrm{J}\,\mathrm{mol}^{-1}\) \ or \ \(1\text{--}10\,\mathrm{cm}^{-1}\) |
| Infrared | \(3\times 10^{12}\text{--}3\times 10^{14}\,\mathrm{Hz}\) | \(100\text{--}1\,\mathrm{\mu m}\) | Vibrational | \(\sime4\,\mathrm{J}\,\mathrm{mol}^{-1}\) \ or \ \(\sime3\,\mathrm{cm}^{-1}\) |
| Visible & UV | \(3\times 10^{14}\text{--}3\times 10^{16}\,\mathrm{Hz}\) | \(1\,\mathrm{\mu m}\)–\(10\,\mathrm{nm}\) | Electronic | \(\sim100\,\mathrm{kJ}\,\mathrm{mol}^{-1}\) |
Two of the ranges as printed were internally inconsistent with their own stated wavelengths:
The microwave region was given as “\(3\times10^{0} - 3\times10^{2}\) Hz”. Since \(\nu = c/\lambda\), the stated range \(1\,\mathrm{cm}\)–\(100\,\mathrm{\mu m}\) corresponds to \(3\times 10^{10}\text{--}3\times 10^{12}\,\mathrm{Hz}\).
The visible/UV wavelength range was given as “\(10\,\mathrm{\mu m}\)–\(10\,\mathrm{nm}\)”. The stated frequency range \(3\times 10^{14}\text{--}3\times 10^{16}\,\mathrm{Hz}\) corresponds to \(1\,\mathrm{\mu m}\)–\(10\,\mathrm{nm}\).
The ordering \[ E_{\text{electronic}} \gg E_{\text{vibrational}} \gg E_{\text{rotational}} \] — roughly \(e5: e3: 1\) in \(\mathrm{J}\,\mathrm{mol}^{-1}\) — underlies every molecular spectrum. It is why rotational structure appears as fine detail within vibrational bands (Chapter 8) and why vibrational structure appears within electronic bands (Chapter 10).
6.4 Why a Dipole is Needed: the Gross Selection Rule
A molecule shows a pure rotational spectrum only if it possesses a permanent electric dipole moment.
| Rotationally active | Rotationally inactive |
|---|---|
| HCl, CO, HF, H2O, NH3 | H2, N2, O2, Cl2 (homonuclear) |
| CO2, CH4, SF6 (symmetric, no net dipole) |
6.4.1 The mechanism
Place a heteronuclear molecule between capacitor plates driven by a microwave source. In the first half cycle the upper plate is positively charged and the lower negatively charged, so the two ends of the molecular dipole feel opposite Coulomb forces — a couple, which exerts a torque and rotates the molecule.
In the next half cycle the plates reverse polarity. If the molecule has meanwhile rotated through \(180^{\circ}\), the torque acts again in the same rotational sense. The result is a resonant coupling between the field and the molecular rotation.
Classical electromagnetic theory says an oscillating dipole radiates at the frequency of its oscillation, and absorbs most strongly when driven at that same frequency. Hence the molecule exchanges energy with the microwave field. The resulting spectrum is discrete, because molecular rotation is quantized.
A homonuclear molecule such as H2 has no dipole for the field to grip, so no torque and no absorption — however fast it rotates. This is why the atmosphere's N2 and O2 are transparent at microwave frequencies while its H2O is not.
6.5 Intensity of Spectral Lines
The gross selection rule of 6.4 decides whether a line appears at all. Its intensity, once it does, is governed by three separate factors, and questions in this area almost always turn on distinguishing them.
Transition probability — how likely the molecule is to change from one state to the other.
Population — how many molecules are in the state from which the transition starts.
Concentration and path length — how much material the beam traverses.
6.5.1 Transition probability
Calculating an absolute transition probability requires the exact wavefunctions of both states, and is rarely possible with accuracy. At a much lower level of sophistication, however, one can decide whether a particular transition is forbidden or allowed — that is, whether its probability is zero or non-zero. This is the business of the selection rules, which are derived for each kind of spectroscopy in the chapters that follow: \(\Delta J = \pm1\) for rotation, \(\Delta v = \pm1\) for the harmonic oscillator, \(\Delta J = 0,\pm2\) for rotational Raman scattering, and so on.
6.5.2 Population of states: the Boltzmann distribution
If two levels are equally able to make a transition to a third, the more intense line will come from the level with the greater population. At thermal equilibrium that population is fixed by the Boltzmann distribution: for \(N\) molecules distributed between a lower level of energy \(E_{\text{lower}}\) and an upper level \(E_{\text{upper}}\), \begin{equation} \boxed{\; \frac{N_{\text{upper}}}{N_{\text{lower}}} = \exp\!\left(-\frac{\Delta E}{k_{\mathrm B}T}\right), \qquad \Delta E = E_{\text{upper}} - E_{\text{lower}} \;} \label{eq:boltzmann} \end{equation} with \(k_{\mathrm B} = 1.381\times 10^{-23}\,\mathrm{J}/\mathrm{K}\). Since \(\Delta E > 0\), the exponential is always less than one: the lower state is always the more populated, as intuition demands.
At room temperature \(k_{\mathrm B}T = 25.85\,\mathrm{meV}\), equivalently \(k_{\mathrm B}T/hc = 208.5\,\mathrm{cm}^{-1}\). Compare this with the level spacing before doing any arithmetic:
Rotational spacings are a few \(\mathrm{cm}^{-1}\), far below \(208\,\mathrm{cm}^{-1}\), so many rotational levels are appreciably populated — which is why a rotational spectrum shows many lines at once.
Vibrational spacings are \(\sime3\,\mathrm{cm}^{-1}\), well above it, so essentially every molecule sits in \(v=0\) — which is why the fundamental band dominates and hot bands are weak.
Electronic spacings are larger still, so excited electronic states are empty at equilibrium.
This single comparison explains the qualitative appearance of all three kinds of spectrum, and it recurs in the Stokes / anti-Stokes intensity ratio of Chapter 9.
Note that (11.9) gives the ratio of populations of individual states. Where a level is degenerate — as every rotational level is, with degeneracy \(2J+1\) — the population of the level carries that factor as well: \begin{equation} \frac{N_{J}}{N_{0}} = (2J+1)\exp\!\left(-\frac{E_{J}}{k_{\mathrm B}T}\right). \label{eq:boltzmann-degenerate} \end{equation} The competition between the rising factor \((2J+1)\) and the falling exponential produces a maximum at intermediate \(J\), which is the origin of the characteristic intensity envelope of a rotational spectrum (7.4).
6.5.3 Concentration and path length
The third factor is experimental rather than molecular. The absorbance of a sample follows the Beer–Lambert law, \begin{equation} I = I_{0}\,e^{-\epsilon c \ell} , \label{eq:beer-lambert} \end{equation} where \(c\) is the concentration, \(\ell\) the path length and \(\epsilon\) the molar absorption coefficient. A weak line can always be strengthened by using a longer cell or a more concentrated sample — which is why microwave spectrometers use metre-long absorption cells, and why trace atmospheric gases are detectable at all.