Tree Tensor Networks Methods for Efficient Calculation of Molecular Vibrational Spectra

TL;DR

This paper develops a generalized tree tensor network framework for vibrational spectroscopy, representing Hamiltonians with TTNOs and wavefunctions with TTNSs to capture long-range couplings beyond MPO limitations. It implements block LOBPCG and block inverse iteration within the TTN setting, employing zip-up contractions, variational fitting, and ALS to efficiently solve the eigenvalue problem. Benchmarking on a 64-dimensional bilinearly coupled oscillator and acetonitrile demonstrates sub-wavenumber accuracy across TTN topologies, with T3NS often outperforming MPS in accuracy and efficiency, while leaf-only trees incur higher costs. The work provides open-source PyTreeNet tools for reproducible vibronic calculations and sets the stage for automated tree-structure optimization and transition-intensity calculations for IR/Raman spectra.

Abstract

We develop and employ general Tree Tensor Networks (TTNs) to compute the vibrational spectra for two model systems: a set of 64-dimensional coupled oscillators and acetonitrile. We explore various tree architectures, ranging from the simple linear structure of Matrix Product States (MPS), to trees where only the leaf nodes carry a physical leg -- as seen in the underlying ansatz of the Multilayer Multiconfiguration Time-Dependent Hartree (ML-MCTDH) method -- and further to more general trees in which all nodes are allowed to possess a physical leg. In addition, we implement Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) methods and Inverse Iteration methods as eigensolvers. By means of comprehensive benchmarking of runtime and accuracy, we demonstrate that sub-wavenumber accuracy in vibrational spectra is achievable with all TTN structures. MPS and three-legged tree tensor network states (T3NS) have similar runtimes, whereas leaf-only trees require significantly more time. All numerical simulations were performed using PyTreeNet, a Python package designed for flexible tensor network computations.

Figures (8)

• Figure 1: Examples of a TTNS and TTNO with the same tree topology, differing in the number of physical legs per tensor. Adapted from Cakir2025.
• Figure 2: Illustration of the zip-up method. Circle tensors represent the state, square tensors represent the operator, and ellipses are intermediate tensors, for example, the $R$ tensor. When contracting a local site tensor of the state and a local site tensor of the operator into an intermediate ellipse tensor, as shown in steps (C) to (D), one can optimize the contraction order to minimize the computational cost.
• Figure 3: Variational fitting. Circle tensors are the given state $\phi_i$, square tensors are the given operator $O_i$, and triangle tensors are the target state $\psi$. The tensors in the shaded area on the left represent the effective Hamiltonian $A$. The dashed triangle tensor is the solution of the linear system $A x = b$. The right-hand side represents the vector $b$ after contraction.
• Figure 4: Alternating least squares. Circle tensors are the given state $\phi$, square tensors are the given operator $H$, and triangle tensors are the target state $\psi$. The tensors in the shaded area on the left represent the effective Hamiltonian $A$. The dashed triangle tensor is the solution of the linear system $A x = b$. The right-hand side represents the vector $b$ after contraction.
• Figure 5: (a) Convergence curves over six iterations for the lowest 30 eigenstates of the 64-dimensional coupled-oscillator model, computed using the MPS and T3NS ansatz. Color intensity encodes the energy level, with lighter shades corresponding to lower energies. (b) Final absolute error for both structures.