Path Integral Molecular Dynamics with Fourth-Order Actions: Application to Para-Hydrogen and Liquid ⁴ He at Ultralow Temperatures

Path integral molecular dynamics (PIMD) is a powerful method for simulating the thermodynamic properties of quantum systems, particularly those exhibiting strong nuclear quantum effects (NQEs) in the condensed phase. However, its applicability to ultralow-temperature systems is hindered by the rapidly increasing Trotter number (<i>P</i>) required to achieve convergence. To overcome this limitation, we present a comprehensive performance assessment of primitive (PA) and three higher-order approaches based on Takahashi and Imada (TIA), Suzuki and Chin (SCA), and Chin (CA) actions. These methods are further enhanced by the incorporation of projected Hessian and multiple-time stepping (MTS) acceleration techniques. Using para-hydrogen at 25 K and liquid ⁴He at 5.1 K as benchmark systems, we demonstrate that the higher order approaches significantly improve both convergence behavior and computational efficiency. The optimal Trotter number (<i>P</i>) ratios (PA : TIA : SCA : CA) were found to be 1.0 : 5.8 : 4.0 : 26.7 for para-hydrogen and 1.0 : 6.8 : 5.2 : 34.8 for liquid ⁴He, where a larger value indicates a faster convergence. When combined with the projected Hessian method, these higher-order PIMD schemes achieved up to a 10-fold speedup relative to the conventional method, with a negligible loss of accuracy. Our findings indicate that higher-order PIMD simulations using TIA, SCA, and CA actions, when accelerated with the projected Hessian technique, offer an efficient and accurate framework for investigating condensed-phase systems at ultralow temperatures.