## 3 Results and Discussion  
Calculated equilibrium bond lengths (A) for the CO-C_i_, CO-O_i_, CO-CO_i_, Pd-ZGO_i_, and CO-Pd-ZGO_i_ are shown in Figures 2 and 3. In models CO-C_i_ and CO-O_i_, the equilibrium bond lengths are ranged from 2.16 A to 3.53 A. The lowest bond length occurs in the CO-C2, where the carbon atom of a CO molecule perpendicular to the surface is bonded to the zinc atom. The bond length of the CO-C_i_ is smaller than the CO-O_i_ for each model number \(i\). The smallest bond length of model CO-O_i_ is 2.51 A. In models CO-CO_i_, the bond lengths are ranged from 2.18 A to 2.65 A. The equilibrium bond lengths for models CO-CO_i_ are remarkably similar to those of the CO-C_i_ and CO-O_i_. For example, the equilibrium bond length for the CO-C_i_ is 2.17 A, where the carbon atom of a CO molecule is bonded to the Ga atom of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface. Similarly, the bond length for the CO-CO1 is 2.18 A, showing the adsorption of CO molecule from being parallel to the surface to being perpendicular to the surface. In the case of the CO-CO4, our calculations show that the corresponding bond length for C-Ga and O-Zn bonds are 2.65 A and 2.60 A, respectively, suggesting that an increased bond length means less attraction between atoms. In our  
Figure 3: Ball and stick structural representations of the optimized Pd-ZGO_i_ and CO-Pd-ZGO_i_ considered in this study. Atoms are represented by spheres: Pd (dark green, very large), Zn (gray, large), Ga (brown, large), C (gray, medium-sized), and O (red, small). Bond lengths are given in Å.  
preliminary work [5], a Pd atom on the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface shows that the calculated Pd-Ga, Pd-Zn, and Pd-O equilibrium bond lengths are 2.32 A, 2.57 A, and 2.04 A for Pd-ZGO1, Pd-ZGO2, and Pd-ZGO3, respectively. In contrast with the Pd atoms adsorbed on the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface, the calculated equilibrium O-Pd bond lengths for models CO-Pd-ZGO\(i\) are between 2.08 A and 2.50 A. The adsorption energies \(\Delta\)E of a CO molecule on the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface without Pd atoms are shown in Table 1. The calculated adsorption energies ranged from \(-\)0.04 eV to \(-\)1.88 eV, indicating that all adsorption reactions occurred spontaneously. Note that spontaneous reactions occur in the direction of decreasing Gibbs free energy change \(\Delta\)G. The Gibbs free energy change consists of the change in enthalpy, \(\Delta\)H, and the change in entropy, \(\Delta\)S, with the following formula: \(\Delta\)G = \(\Delta\)H \(-\) T\(\Delta\)S. The enthalpy change is defined as the sum of the change in internal energy, \(\Delta\)E, and the product of the pressure, P, and the change in volume, \(\Delta\)V, as follows: \(\Delta\)H = \(\Delta\)E + P\(\Delta\)V. Therefore, the Gibbs free energy change can be expressed as: \(\Delta\)G = \(\Delta\)E + P\(\Delta\)V \(-\) T\(\Delta\)S. In our study, we ignored the contribution of entropy changes and the volume changes in the ZnGa\({}_{2}\)O\({}_{4}\) structure. The Gibbs free energy change can be further simplified to the internal energy change caused by the kinetic, potential, and chemical energy of the material system. Negative Gibbs free energy change, negative internal energy change, or negative adsorption energy \(\Delta\)E provides a means by which spontaneous physical and chemical changes can occur without any external help. Conversely, positive adsorption energies suggest that the corresponding endothermic process can be supported by increasing the temperature. It can be clearly seen that the CO-C1 has the lowest adsorption energy of \(-\)1.88 eV. Consistently, this adsorption site Ga\({}_{3c}\) exists a low C-Ga bond length of 2.17 A, i.e., a strong interaction, suggesting that that the Ga\({}_{3c}\) site is more favored if CO molecules are adsorbed on the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface. It is therefore also of significant interest to carry out a detailed comparison of the adsorption energies of the CO-O\(i\) with those obtained from the CO-C\(i\).