This project aims to find zero point vibrational energy (ZPVE) for H atom adsorbed at Cu(111) hollow site through calculating Hessian matrix, which was built through second order finite difference approximation.

H adsorption optimization
The geometry optimization was performed in Material Studio with CASTEP Calculation Package [1]. The functional of Perdew Burke, and Ernzerhoff was employed [2]. Cu bulk was first optimized with medium SCF tolerance. Energy cutoff of 480 eV and 9 X 9 X 9 K points sampling were used. The pseudopotential was solved through Koelling-Harmon treatment. After Cu optimization, 111 surface was cut from bulk and H atom was place at hollow hcp site. K points of 11 X 11 X 1 was applied for slab optimization. Figure 1 shows optimized configuration, in which four layers of Cu was chosen and 20 angstrom vacuum was built above the surface to ensure no interaction between super cells.

Figure 1. Optimized H adsorption at Cu (111) hollow hcp site

Figure 2. Optimized H adsorption at Cu (111) hollow fcc site

Energy of structure with 0.05 angstrom H displacement at hcp site

Energy of structure with 0.05 angstrom H displacement at fcc site

Then finite difference approximation was used to calculate Hessian matrix elements. The diagonal element can be obtained as equation (1) and off-diagonal element can be obtained from equation (2).
\begin{equation} H_{ii} \cong \frac{E(\delta x_{i})+E(-\delta x_{i})-2E_{0}}{\delta x_{i}^2} \end{equation}
\begin{equation} H_{ij} \cong \frac{E(\delta x_{i}, \delta x_{j})+E(-\delta x_{i},-\delta x_{j})-E(\delta x_{i}, -\delta x_{j})-E(-\delta x_{i}, \delta x_{j})}{4\delta x_{i}\delta x_{j}} \end{equation}

After calculating eigenvalues of Hessian matrix, we can calculate vibrational frequency and corresponding zero point energies, which is shown in following table.

Conclusions
In this project, we applied harmonic approximation and use finite difference approximation to calculate vibrational frequency of H atom adsorbed at the Cu(111) hollow sites and further zero point energy. According to our calculation, the zero point energy of H on hcp site is 0.1893 eV and H on fcc site is 0.2001 eV. The difference of the zero point energies between H on fcc site and hcp site are less 0.011 eV.

Reference
[1] “First principle methods using CASTEP” Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)
[2] Perdew, J. P; Burke, K; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868