Calculating the frequencies and intensities of strongly anharmonic modes of adsorbates on surfaces: A low-cost but accurate computational approach

We present a method for calculating the frequencies and intensities of the vibrational modes of adsorbates on surfaces. Our method is based on density functional perturbation theory (DFPT) and provides accurate estimates of the vibrational intensities even when the vibrations are strongly anharmonic. Furthermore, it does so at a negligible additional computation cost compared to conventional DFPT calculations. We illustrate our method by calculating the vibrational spectra of three example systems—ethylidyne on Rh(111), benzene on Rh(111) coadsorbed with CO, and terephthalic acid on Cu(100)—and comparing them to experimental measurements performed using high-resolution electron energy loss spectroscopy. We find excellent agreement between our predictions and the experimentally measured frequencies and intensities in all three cases.

Figure 1

HREEL spectra for $(2\times 2)\text{−}\left({\mathrm{C}}_2{\mathrm{H}}_3\right)$ on Rh(111). (a) Calculated harmonic (in orange), harmonic including anharmonic corrections (in blue), and experimental (in green) spectra. Frequency values are included for the most relevant bands (in ${\mathrm{cm}}^{-1}$). (b)–(e) Resulting energy evaluations (blue circles) for the scanned normal coordinates of the following fundamental vibrational modes of ethylidyne: (b) $\mathrm{Rh}\text{−}\mathrm{C}$ stretching, (c) $\mathrm{C}\text{−}\mathrm{C}$ stretching, (d) ${\mathrm{CH}}_3$ wagging, and (e) $\mathrm{C}\text{−}\mathrm{H}$ stretching. The Morse fit is represented with solid blue lines. Harmonic potential curves for each mode (solid orange lines) are included to highlight the difference between the harmonic and anharmonic potentials (blue solid lines). Arrows in the models are set to graphically represent the motion of each vibrational mode.

Figure 2

HREEL spectra for $c(2\sqrt{3}\times 4)\text{rect}\text{−}{\mathrm{C}}_6{\mathrm{H}}_6$/CO on Rh(111). (a) Calculated harmonic (in orange), harmonic including anharmonic corrections (in blue), and experimental (in green) spectra. Frequency values are included for the most relevant bands (in ${\mathrm{cm}}^{-1}$). (b), (c) Resulting energy evaluations (blue circles) for the scanned normal coordinates of the following fundamental vibrational modes of benzene: (b) ${\mathrm{CH}}_3$ wagging and (c) $\mathrm{C}\text{−}\mathrm{O}$ stretching. Harmonic potential curves for each mode (solid orange lines) are included to highlight the difference between the harmonic and anharmonic potentials (blue solid lines). Arrows in the models are set to graphically represent the motion of each vibrational mode. Experimental spectrum is adapted from Ref. [4].

Figure 3

HREEL spectra for $(3\times 3)$-TPA on Cu(100). (a) Calculated harmonic (in orange), harmonic including anharmonic corrections (in blue), and experimental (in green) spectra. Frequency values are included for the most relevant bands (in ${\mathrm{cm}}^{-1}$). (b), (c) Resulting energy evaluations (blue circles) for the scanned normal coordinates of the following fundamental vibrational modes of TPA: (b) $\mathrm{C}\text{−}\mathrm{H}$ bend and (c) $\mathrm{C}\text{−}\mathrm{H}$ symmetric stretching. Harmonic potential curves for each mode (solid orange lines) are included to highlight the difference between the harmonic and anharmonic potentials (blue solid lines). Arrows in the models are set to graphically represent the motion of each vibrational mode. Experimental spectrum is adapted from Ref. [3].