Energy Units in Spectroscopy

One quantity, many units

Photon energy, frequency, wavenumber, and wavelength are linked by exact constants:

Since the 2019 SI redefinition, \(h\), \(c\), \(e\), \(k_B\), and \(N_A\) are exact, so conversions between eV, cm⁻¹, Hz, J, K, and kJ/mol have no uncertainty. The anchors worth memorizing are 1 eV = 8065.54 cm⁻¹ = 96.485 kJ/mol = 23.061 kcal/mol = 11,604.5 K, and 1 cm⁻¹ = 29.979 GHz. The energy & wavelength converter handles every conversion live.

Which unit for which job

Quantum chemists' "chemical accuracy" target of 1 kcal/mol is 1.59 mE_h, 350 cm⁻¹, or 43 meV. That is coarse by spectroscopic standards, where line positions are routinely known to 0.01 cm⁻¹ or better.

The energy ladder

Why spectroscopists use wavenumbers

Wavenumber is proportional to energy, so level spacings add and subtract linearly. A vibrational fundamental at 3000 cm⁻¹ has its first overtone near 6000 cm⁻¹ (slightly lower, because of anharmonicity). In wavelength the same pair sits at 3.33 µm and 1.67 µm, which is much harder to compare at a glance. Wavenumbers are also what interferometers measure directly: an FTIR's natural axis is the inverse of the optical path difference, in cm⁻¹. And the cm⁻¹ scale happens to have convenient magnitudes for molecules. Rotational constants run from below 0.01 cm⁻¹ for large molecules to about 60 cm⁻¹ for H₂, vibrational fundamentals from about 100 to 4000 cm⁻¹, and electronic transitions from 10,000 cm⁻¹ upward.

Converting shifts and linewidths

The relationship between wavelength and energy is reciprocal, not linear, so differences and widths need care. A Raman shift is a difference in wavenumber, and the scattered wavelength follows from

With 532 nm excitation, a 1000 cm⁻¹ Stokes line appears at 561.9 nm and the anti-Stokes line at 505.1 nm. These are not symmetric about 532 nm. A 3000 cm⁻¹ C–H stretch appears at 633.0 nm. With 785 nm excitation, the same 1000 cm⁻¹ shift lands at 851.9 nm.

For small intervals, differentiate \(\tilde\nu = 10^7/\lambda\):

A 1 nm spectrometer bandpass therefore corresponds to 268 cm⁻¹ at 193 nm, 40 cm⁻¹ at 500 nm, and only 8.8 cm⁻¹ at 1064 nm. A 0.1 cm⁻¹ laser linewidth is 3.0 GHz at any wavelength, but it is 0.0025 nm at 500 nm and 0.011 nm at 1064 nm. When comparing resolutions across a spectrum, convert to cm⁻¹ or GHz first. The grating & spectrometer calculator and the line broadening calculator report widths in several units for this reason.

Thermal energy and populations

The Boltzmann factor sets the fraction of molecules in an excited level at temperature \(T\):

• A 2000 cm⁻¹ vibration has \(e^{-2000/207} \approx 6\times10^{-5}\) of its population in \(v = 1\) at room temperature. At 1000 K this rises to 5.6 %, and "hot bands" appear in the spectrum.
• A 500 cm⁻¹ low-frequency mode is 9 % excited at room temperature. Floppy molecules have congested room-temperature spectra for this reason.
• Rotational levels spaced by a few cm⁻¹ are thermally populated over many levels. That is why room-temperature rotational envelopes are broad.
• A 2 eV electronic excitation has a Boltzmann factor of about 10⁻³⁴ at 298 K. Electronically excited states are populated only by light, collisions in plasmas, or chemistry.

Supersonic expansions and cryogenic ion traps cool molecules to a few kelvin, where \(k_BT\) is only a few cm⁻¹. Almost everything is then in the ground vibrational level and the lowest rotational levels, so spectra simplify dramatically. This is the main reason gas-phase spectroscopy goes to the trouble of cooling.

Photon energies and chemical bonds

Comparing photon energies with bond energies explains a lot about laser–material interactions. Most common single bonds in organic and inorganic materials have dissociation energies of roughly 3–5 eV (about 300–500 kJ/mol).

A single deep-UV photon carries enough energy to break a typical bond directly. That is the basis of "cold" photochemical ablation of polymers with excimer lasers. A 1064 nm photon carries about a quarter of a bond energy, and a CO₂ photon about a fortieth. Infrared processing is therefore thermal: energy is absorbed into vibrations and heat before anything breaks, unless intensities are high enough for multiphoton absorption.

Common pitfalls

• Air versus vacuum wavelengths. Tabulated visible lines are often given in standard air, which differs from vacuum by about 0.03 % (0.18 nm at 633 nm). This matters for line identification and spectrometer calibration.
• Averaging in the wrong unit. The mean of two wavelengths is not the wavelength of the mean energy. Average or difference in energy units.
• Ambiguous "per molecule" and "per mole". Convert kJ/mol to eV with the factor 96.485, not 100. The 3.5 % error from the shortcut is larger than many effects people are trying to measure.
• Mixing calories. Thermochemistry uses the thermochemical calorie, 4.184 J. The international-table calorie, 4.1868 J, appears in older engineering data.
• Dropping the factor of 2π. Angular frequency \(\omega = 2\pi\nu\) and \(\hbar\omega = h\nu\). Mixing rad/s and Hz gives a 6.28× error that looks plausible at first glance.

References

1. E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2018,” Rev. Mod. Phys. 93, 025010 (2021).
2. E. R. Cohen et al., Quantities, Units and Symbols in Physical Chemistry (IUPAC Green Book), 3rd ed., RSC Publishing (2007).