Ab initio tensorial electronic friction for molecules on metal surfaces: Nonadiabatic vibrational relaxation

Molecular adsorbates on metal surfaces exchange energy with substrate phonons and low-lying electron-hole pair excitations. In the limit of weak coupling, electron-hole pair excitations can be seen as exerting frictional forces on adsorbates that enhance energy transfer and facilitate vibrational relaxation or hot-electron-mediated chemistry. We have recently reported on the relevance of tensorial properties of electronic friction [M. Askerka et al., Phys. Rev. Lett. 116, 217601 (2016)] in dynamics at surfaces. Here we present the underlying implementation of tensorial electronic friction based on Kohn-Sham density functional theory for condensed phase and cluster systems. Using local atomic-orbital basis sets, we calculate nonadiabatic coupling matrix elements and evaluate the full electronic friction tensor in the Markov limit. Our approach is numerically stable and robust, as shown by a detailed convergence analysis. We furthermore benchmark the accuracy of our approach by calculation of vibrational relaxation rates and lifetimes for a number of diatomic molecules at metal surfaces. We find friction-induced mode-coupling between neighboring CO adsorbates on Cu(100) in a $c(2\times 2)$ overlayer to be important for understanding experimental findings.

Figure 1

Vibrational motion of an adsorbate molecule [shown here is CO on a Cu(100) surface] induces low-lying electronic excitations. This leads to charge fluctuations and energy flow between substrate and adsorbate degrees of freedom. Especially for charge fluctuations that occur in molecular electronic states such as the lowest unoccupied molecular orbital (LUMO), nonadiabatic energy transfer can be large.

Figure 2

Depiction of vibrational normal-mode displacements of CO adsorbed at an atop site of a Cu(100) surface. (a) Internal stretch mode (IS), (b) frustrated rotation mode (FR), (c) surface-adsorbate mode (SA), and (d) frustrated translation (FT) mode.

Figure 3

Projected lifetimes of atop CO on Cu(100) along the IS and SA modes (see Fig 2). (a) Convergence with respect to an $a\times a\times a\mathbf{k}$ grid, with $a$ being the labeling on the $x$ axis. (b) Convergence with respect to basis set: sz, dz, and tz correspond to single, double, and triple valence; sp, dp, and tp correspond to single, double, and triple polarization functions. (c) Convergence with respect to metal layers in the substrate. All lifetimes are given in picoseconds.

Figure 4

(a) Dependence of the IS lifetime of CO on Cu(100) on the $\delta$-function broadening $\sigma$ for various $k$-point sets with (solid) and without (dashed) $\delta$-function normalization as described in Eq. (23). (b) Dissipation rate along the IS mode as a function of $\hslash \omega$ [Eq. (24)]; the Gaussian broadening of 0.05 eV is used to smoothen the spectrum.

Figure 5

Vibrational lifetimes as a function of electronic temperature due to electron-hole pairs for four different vibrational modes of CO adsorbed atop Cu(100). The vibrational modes are abbreviated as in Fig. 2.

Figure 6

CO adsorbed on (100) bulk-truncated Cu clusters of size (a) $(7\times 7)$, (b) $(5\times 5)$, and $(3\times 3)$ in (c) side and (d) top views. Odd and even layers of substrate are shown in different color tones.

Figure 7

(a)–(c) Top view of CO adsorbed on Cu(100) in (a) a $p(2\times 2)$, (b) a $p(3\times 3)$, and (c) a $p(4\times 4)$ overlayer. (d)–(f) Top view of CO adsorbed on Cu(110) in (d) a $p(1\times 2)$, (e) a $p(2\times 2)$, and (f) a $p(2\times 3)$ overlayer. Odd and even layers of substrate are shown in different color tones.

Figure 8

(a) Top view of $c(2\times 2)$ CO on a Cu(100) overlayer. Two different equivalent unit cells, each containing one or two molecules, are also shown. (b)–(e) Symmetric and antisymmetric adsorbate phonon combinations for internal stretch (IS) and surface-adsorbate stretch (SA). Odd and even layers of substrate are shown in different color tones.