A multiset matrix product state approach to hierarchical equations of motion and its application to vibrational relaxation on metal surfaces.

We develop a multiset matrix product state (MPS) approach based on the time-dependent variational principle to solve the hierarchical equations of motion (HEOM) for the fermionic bath and apply it to inelastic vibrational scattering on metal surfaces. By using a Newns-Anderson model with two nuclear degrees of freedom, we investigate the vibrational energy relaxation of NO scattering on Au(111) and Ag(111) surfaces. Our results show that the extent of vibrational relaxation depends strongly on both incident energy and molecule-surface coupling strength. Vibrational relaxation on the Au(111) surface is enhanced with increasing incident energy due to greater transient electron transfer. The dependence on molecule-surface coupling strength and effective metal bandwidth is more complex. In the case of scattering on the Au(111) surface with low initial vibrational excitation, the landscape of the adiabatic potential energy surface plays an important role. Stronger coupling enhances transient electron transfer and leads to more pronounced vibrational relaxation. However, for high initial vibrational excitation, the extent of transient electron transfer remains similar. Larger coupling strength increases adiabaticity and reduces vibrational relaxation. The vibrational relaxation on the Ag(111) surface is found to be more pronounced than on the Au(111) surface, consistent with experimental observations. Our study provides detailed insights into the nonadiabatic dynamics during molecule-surface scattering and demonstrates the utility of the multiset MPS-HEOM approach for studying such processes.