In this project a hydrogen atom will be adsorbed to a Cu(1 1 1) surface and the vibration modes of the hydrogen atom will be calculated. There are two sites where the hydrogen atom can be stably adsorbed, the fcc site and hcp site, as such both sites will be considered.
For these calculations the functional PBE was use, along with OTFG ultrasoft pseudopotentials, the Koelling-Hamon relativistic treatment, a k-point grid of 11x11x1 and a energy cutoff of 600eV. The dipole correction was not used for these calculations as the corrections were smaller that the level of precision guaranteed by the k-point grid and energy cut selection.
Pseudo atomic configuration Cu: 3d10 4s1
Pseudo atomic configuration H: 1s1
Optimization
To calculate this first the hydrogen atom was place on either the hcp or fcc lattice site of a Cu(111) surface and then the geometry of the system was optimized holding the bottom three layers fixed and allowing the top layer of the Cu surface and the hydrogen atom move freely. For the convergence, the energy convergence was set to 10-5 eV/atom, the max force was set to 0.03 eV/A, the max stress was set to 0.05 GPa, and the max displacement was set to 0.001A. The results of the optimization can be seen in Figures 1 and 2.
Figure 1: Optimized structure for hydrogen adsorbed to fcc site.
Figure 2: Optimized structure for hydrogen adsorbed to hcp site.
Construction of the Hessian Matrix
In this work, only the vibrations of the hydrogen are wanted, so an approximation that can be used is to only allow the hydrogen atom to move while constructing the Hessian. This will simplify the Hessian to a 3×3 matrix. From the following equation the Hessian matrix can be calculated.
Using a displacement of 0.1A the Hessian matrix was computed and then divided by the mass of a hydrogen atom to get the mass-weighted Hessian matrix whose matrix elements are given by the following,
Then the eigenvalues and eigenvectors of the mass-weighted Hessian, A, were calculated using the following relation.
Once the eigenvalues, 
The results of these calculations for the hydrogen at the fcc site and hcp site are given in Tables 1 and 2 respectively.
Table 1: Normal mode frequency and Eigenvector for H atom is fcc site.| Eigenvectors(Å) | ||||
|---|---|---|---|---|
| Normal Mode | Frequency (cm-1) | x | y | z |
| 1 | 742.7 | -0.188 | 0.769 | -0.611 |
| 2 | 375.9 | 0.787 | -0.254 | -0.562 |
| 3 | 366.7 | -0.588 | -0.586 | -0.558 |
Table 2: Normal mode frequency and Eigenvector for H atom is hcp site.
| Eigenvectors(Å) | ||||
|---|---|---|---|---|
| Normal Mode | Frequency (cm-1) | x | y | z |
| 1 | 740.9 | -0.182 | -0.763 | -0.620 |
| 2 | 372.7 | 0.788 | 0.264 | -0.557 |
| 3 | 363.2 | -0.589 | 0.590 | -0.553 |
Zero-point Energy Correction
The zero point energy for a quantum mechanical system is not that of the classical system, the corrected form is as the following,
This correction is 0.0921eV for the the system with the hydrogen at the fcc site and 0.0915eV for the system with the hydrogen at the hcp site.
Conclusion
The difference in the energy of the hydrogen adsorbing to the fcc site and the hydrogen adsorbing to the hcp site is 0.0154eV. Taking into account the zero-point energy correction the difference shrinks to 0.0149eV
