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The Journal of Chemical Physics

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

Shuocang Zhang, Zhiyan Liu, Peng Bao, Qiang Shi

Published Sep 2, 2025• 9 citations
Publisher

Abstract

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.

DOI
10.1063/5.0266136

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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.
We apply the matrix product state (MPS) method to propagate the hierarchical equations of motion (HEOM). It is shown that the MPS approximation works well in different type of problems, including boson and fermion baths. The MPS method based on the time-dependent variational principle is also found to be applicable to HEOM with over one thousand effective modes. Combining the flexibility of the HEOM in defining the effective modes and the efficiency of the MPS method thus may provide a promising tool in simulating quantum dynamics in condensed phases.
We show that the dynamics of (vertical) Franck-Condon excitations in the regime where Holstein-coupled vibrational modes mix strongly with electronic degrees of freedom sharply contrasts with the known self-localized behavior of vibrationally relaxed excitations. Instead, the strongly coupled modes are found to periodically induce resonances between interacting electronic sites, during which effective excitation transfer occurs, allowing Franck-Condon excitations to attain substantial mean square displacements under conditions where relaxed excitations are essentially trapped to a single site. In demonstrating this behavior, we employ a multiset matrix product state formalism. We find this tensor network state method to be a remarkably efficient and accurate approach for the notoriously difficult problem posed by the Holstein model in the regime where the electronic coupling, the vibrational quantum, and the vibrational reorganization energy are comparable in magnitude.
We present a Hierarchical Equations of Motion (HEOM) approach in the Matrix Product State (MPS) formalism to simulate carrier transport in molecular aggregates described by an electron-phonon Hamiltonian with bosonic dissipation. Transport properties are evaluated through time-dependent population analysis and mobility calculations. The method's validity is rigorously established through benchmarking against conventional HEOM. Comparative analysis with Thermo-Field Dynamics combined with MPS (TFD + MPS) reveals fundamental similarities and differences in their effective Hamiltonians and demonstrates the superior accuracy and computational efficiency of our HEOM + MPS framework. For single-electron systems, we introduce state-vector space configurations that enhance performance beyond traditional Fock space approaches. Results confirm that our method provides a robust, nearly exact, and efficient numerical quantum dynamic approach for carrier transport in dissipative bosonic environments.
Non-adiabatic dynamics at molecule-metal interfaces govern diverse and technologically important phenomena, from heterogeneous catalysis to dye-sensitized solar energy conversion and charge transport across molecular junctions. Realistic modeling of such dynamics necessitates taking into account various charge and energy transfer channels involving the coupling of nuclear motion with a very large number of electronic states, leading to prohibitive cost using classical computational methods. In this work we introduce a generalization of the Anderson-Newns Hamiltonian and develop a highly optimized quantum algorithm for simulating the non-adiabatic dynamics of realistic molecule-metal interfaces. Using the PennyLane software platform, we perform resource estimations of our algorithm, showing its remarkably low implementation cost for model systems representative of various scientifically and industrially relevant molecule-metal systems. Specifically, we find that time evolution for models including 100100100 metal orbitals, 888 molecular orbitals, and 202020 nuclear degrees of freedom, requires only 271271271 qubits and 7.9×1077.9 \times 10^77.9×107 Toffoli gates for 100010001000 Trotter steps, suggesting non-adiabatic molecule-metal dynamics as a fruitful application of first-generation fault-tolerant quantum computers.
Hierarchical Equations of Motion (HEOM) in the Tensor-Train (TT) representation is applied to study the charge-transfer dynamics in organic semiconductors (OSCs). The theoretical formulation as well as the basic computational aspects of HEOM-TT are discussed in detail. Charge transfer in OSCs is modeled using dissipative polaronic models that incorporate the effects of both high- and low-frequency molecular vibrations, and it is simulated in a fully quantum and nonperturbative manner, which has not been studied intensively. The capability of treating complex electron-vibrational systems is examined by analyzing and comparing the numerical behavior of the time-dependent variational approach and the time-Alternating Minimal Energy methods and by calculating the current autocorrelation function and diffusivity across various models. Our results indicate that the HEOM-TT framework offers a robust tool for the detailed analysis of complex polaronic systems, suggesting its potential for broader applications.
We investigate the effect of inter-mode coupling on the vibrational relaxation dynamics of molecules in weak dissipative environments. The simulations are performed within the reduced density matrix formalism in the Markovian regime, assuming a Lindblad form for the system-bath interaction. The prototypical two-dimensional model system representing two CO molecules approaching a Cu(100) surface is adapted from an ab initio potential, while the diatom-diatom vibrational coupling strength is systematically varied. In the weak system-bath coupling limit and at low temperatures, only first order non-adiabatic uni-modal coupling terms contribute to surface-mediated vibrational relaxation. Since dissipative dynamics is non-unitary, the choice of representation will affect the evolution of the reduced density matrix. Two alternative representations for computing the relaxation rates and the associated operators are thus compared: the fully coupled spectral basis, and a factorizable ansatz. The former is well-established and serves as a benchmark for the solution of Liouville-von Neumann equation. In the latter, a contracted grid basis of potential-optimized discrete variable representation is tailored to incorporate most of the inter-mode coupling, while the Lindblad operators are represented as tensor products of one-dimensional operators, for consistency. This procedure results in a marked reduction of the grid size and in a much more advantageous scaling of the computational cost with respect to the increase of the dimensionality of the system. The factorizable method is found to provide an accurate description of the dissipative quantum dynamics of the model system, specifically of the time evolution of the state populations and of the probability density distribution of the molecular wave packet. The influence of intra-molecular vibrational energy redistribution appears to be properly taken into account by the new model on the whole range of coupling strengths. It demontrates that most of the mode mixing during relaxation is due to the potential part of the Hamiltonian and not to the coupling among relaxation operators.