We develop a Gaussian process regression enhanced line integral string method to accelerate ring polymer instanton calculations of tunneling rates and tunneling splittings in molecular proton transfer reactions. By exploiting uncertainty estimates from the surrogate representation, we show that the number of force evaluations required to converge an instanton path becomes effectively independent of the number of beads used to discretize the pathway. To reduce the computational overhead associated with training, particularly when Hessian information is included, we implement graphics processing unit accelerated black box matrix matrix multiplication, achieving an order of magnitude speedups relative to standard implementations. For rate calculations, we introduce a selective Hessian training strategy that distinguishes flexible modes strongly coupled to the transferring proton from more rigid modes weakly coupled to the reaction coordinate. This enables the construction of accurate surrogate potential energy surfaces with substantially fewer Hessian evaluations. Applications to malonaldehyde and Z-3-aminopropenal demonstrate that tunneling rates can be predicted within 20% of exact values while reducing force and Hessian evaluations. The approach is further extended to tunneling splitting calculations for the formic acid dimer and malonaldehyde, yielding splittings in reasonable agreement with experiment and high level theoretical results.
Intramolecular proton transfer is a fundamental chemical process that involves coupled electronic and atomic motion. Including nuclear quantum effects, such as quantum tunneling, is crucial for an accurate description of proton transfer through a barrier, even at room temperature. Direct solution of the Schrödinger equation can provide the exact tunneling rate. However, its computational cost scales exponentially with the number of degrees of freedom (DOF), which makes this approach impractical for large molecules. Advanced wave function based methods, including the Multi-Configuration Time-Dependent Hartree (MCTDH) method [1][2][3] and tensor-train methods, 4,5 have shown promise as practical alternatives. Ring polymer instanton theory [6][7][8][9][10][11][12] provides an efficient way to study tunneling rates in complex molecular systems with sufficient accuracy. In Feynman's path integral formulation, the instanton represents the path that gives the dominant contribution to the tunneling rate. In instanton theory, the tunneling rate is obtained from a path integral that is approximated by the instanton and the harmonic fluctuations surrounding it. This approach achieves computational efficiency by avoiding explicit sampling of the large number of molecular geometries required in wave function based treatments and enables the inclusion of tunneling effects in atomistic simulations where classical transition state theory fails.
Instanton theory has been an active area of research. A derivation from first principles has been proposed. 13,14 Extensions to non-adiabatic systems have been developed, [15][16][17][18][19] and several studies have introduced corrections to address rate calculation errors near the crossover temperature. [20][21][22][23][24][25][26] Perturbative corrections up to fourth derivatives of the potential have also been incorporated. 27,28 In addition, microcanonical rate theory [29][30][31] has been formulated within the instanton framework.
On the algorithmic side, the application of ring polymer instanton theory requires locating the instanton path in a high-dimensional quantum system, followed by evaluation of the fluctuation factor through Hessians of the potential at the ring polymer beads. Although instanton theory is more efficient than full-dimensional wave function methods, it remains more computationally demanding than classical transition state theory, especially when combined with on-the-fly electronic structure calculations. This computational cost has limited its broader application to complex molecular systems.
Gaussian Process Regression (GPR) based methods [32][33][34] and neural network approaches 35 have been developed to accelerate instanton calculations. A chain-of-states approach called the Line Integral Nudged Elastic Band (LI-NEB) method, 36,37 in the same spirit as the NEB method, 38,39 has been introduced for instanton path optimization. In our previous work, 40 we combined LI-NEB with GPR to locate the instanton path and achieved an order of magnitude reduction in the number of force evaluations required. The LI-NEB and string methods have also been implemented for zero-temperature instanton path searches and tunneling splitting calculations. 41,42 In this follow-up study to our previous work, 40 we develop a GPR enhanced Line Integral String (LI-String) method for efficient instanton path optimization. The string method 43,44 is used to minimize the abbreviated action to obtain the instanton path. 41,42 We further show that, when using the surrogate model generated by GPR, the cost of converging the instanton path no longer scales with the number of beads representing the path. 45 Hyperparameter optimization in GPR based on Cholesky decomposition is inefficient, which limits the applicability of GPR to larger systems. 34,46,47 We demonstrate that the Blackbox Ma-trix Matrix Multiplication (BBMM) approach and its GPU implementation 48 can accelerate GPR model training. For intramolecular proton transfer reactions, the selective Hessian training strategy 34 further reduces the cost of Hessian evaluations required for computing ring polymer instanton rates. We then apply the GPR enhanced LI-String method to compute the ground state tunneling splitting of two prototypical systems, malonaldehyde and the formic acid dimer (FAD).
The paper is organized as follows. In Section 2, for completeness, we briefly introduce ring polymer instanton theory, the LI-String method, and GPR-based surrogate modeling used for path optimization and rate calculations, including aspects of our implmentation. Section 3 focuses on algorithmic acceleration of instanton calculations, including low-scaling path optimization, GPU-accelerated GPR hyperparameter training, and adaptive regression strategies with selective Hessian modeling. In Section 4, we apply the proposed approach to representative intramolecular proton transfer system and finally, we summarize the main conclusions and di
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