Researchers Enhance Instanton Accuracy with Third and Fourth Derivatives, Capturing Anharmonic Effects

Understanding how chemical reactions occur at a molecular level requires accurately calculating reaction rates, a challenge when quantum mechanical effects like tunneling become important. Jindra Dušek, Joseph E. Lawrence, and Jeremy O. Richardson, from ETH Zürich and New York University, present a new method that significantly improves these calculations by accounting for the complex shapes of molecular energy landscapes. Their approach, which builds upon existing ‘instanton’ theory within the ring-polymer framework, incorporates subtle distortions of molecules during reactions, capturing ‘anharmonicity’ that previous methods often missed. This advancement allows researchers to predict reaction rates with greater precision, even for reactions involving deep tunneling, and opens the door to more accurate modelling of chemical processes in diverse fields from atmospheric chemistry to materials science.

Instanton rate theory within the ring-polymer framework, enhanced by potential corrections (RPI+PC), significantly improves the accuracy of instanton theory by incorporating third and fourth derivatives of the potential to capture anharmonic effects. Instanton theory represents a rigorous semiclassical method that extends transition-state theory by including quantum tunneling along a well-defined optimal tunneling pathway. Standard instanton theory neglects anharmonicity perpendicular to this tunneling path, limiting its precision for complex systems. The RPI+PC method addresses this limitation by utilising only local information along the same instanton trajectory as the leading-order theory, avoiding the need for a global potential energy surface description.

Reaction Rate Calculations for Hydrogen and Deuterium

Researchers have analysed computational results for the rates of chemical reactions involving hydrogen, deuterium, and another atom represented by ‘T’. These calculations explore the association of atoms and molecules, specifically hydrogen with hydrogen, deuterium with deuterium, and the atom ‘T’ with its diatomic form. The goal is to assess the accuracy of different computational methods for predicting these reaction rates. The study compares several methods for calculating rate constants, including a basic instanton approach, instanton methods with varying levels of tunneling correction, and a highly accurate reference calculation used for comparison.

The data reveals the importance of accounting for quantum mechanical tunneling, where reactants can overcome energy barriers even without sufficient classical energy. Results demonstrate that including tunneling corrections consistently improves the accuracy of the calculated rate constants. The most refined correction method provides the most accurate predictions, particularly at lower temperatures where tunneling effects are more pronounced. The errors in the calculations generally increase at lower temperatures, as accurately capturing tunneling becomes more challenging. These findings confirm that tunneling is a crucial factor in these reactions and that incorporating tunneling corrections significantly improves the reliability of predicted reaction rates.

Anharmonicity Improves Instanton Theory Accuracy

Researchers have significantly enhanced the accuracy of instanton theory, a method for calculating reaction rates, by developing a perturbatively-corrected approach termed RPI+PC. The team addressed the limitation of standard instanton theory, which neglects anharmonicity perpendicular to tunneling pathways, by incorporating third and fourth derivatives of the potential energy landscape. This advancement allows for more accurate calculations, particularly when combined with electronic-structure methods. The development of RPI+PC represents a systematic improvement over existing instanton theory, demonstrated through analysis of its behaviour in the semiclassical limit.

Tests on model systems, including the collinear hydrogen plus hydrogen reaction and its isotopic variants, confirm the enhanced performance of the new method. Results demonstrate that RPI+PC accurately predicts reaction rates even in scenarios where standard instanton theory struggles, offering a more reliable approach for complex chemical systems. This breakthrough combines the strengths of both semiclassical transition-state theory and instanton theory, providing a rigorous description of deep tunneling with anharmonic effects. The method can be implemented as a post-processing step following a standard instanton optimization, requiring only potentials, gradients, Hessians, third derivatives, and fourth derivatives along the instanton pathway. Compared to alternative methods like ring-polymer molecular dynamics, RPI+PC leverages local information along the tunneling pathway, avoiding the need for extensive sampling of the entire potential energy surface, a significant advantage for complex systems.

Perturbative Corrections Enhance Reaction Rate Accuracy

Researchers have refined a method for calculating reaction rates, building upon existing instanton theory within the ring-polymer framework. They developed a perturbatively-corrected approach, termed RPI+PC, which improves accuracy by accounting for anharmonicity along the reaction pathway. This correction utilises local derivatives of the potential energy, making it computationally efficient and compatible with advanced electronic structure calculations. The RPI+PC method represents a systematic improvement over standard instanton theory, as demonstrated through tests on model systems including hydrogen and its isotopes.

Importantly, the researchers confirmed the rigorous nature of this expansion, meaning it provides a reliable approximation as the system approaches classical behaviour. Furthermore, the magnitude of the perturbative correction itself can indicate the importance of anharmonicity in the reaction, offering insight into the accuracy of simpler calculations. Future research will focus on applying this method to more complex chemical reactions, including those occurring on surfaces, and combining it with other theoretical advancements to address limitations at specific temperature ranges. Addressing rotational symmetry and extending the theory to nonadiabatic reactions are also planned extensions.