A simple, general, and rigorous scheme for adapting permutation symmetry in molecular systems is proposed and tested for fitting global potential energy surfaces using neural networks (NNs). The symmetry adaptation is realized by using low-order permutation invariant polynomials (PIPs) as inputs for the NNs. This so-called PIP-NN approach is applied to the H + H₂ and Cl + H₂ systems and the analytical potential energy surfaces for these two systems were accurately reproduced by PIP-NN. The accuracy of the NN potential energy surfaces was confirmed by quantum scattering calculations.

With advances in ab initio theory, it is now possible to calculate electronic energies within chemical (<1 kcal/mol) accuracy. However, it is still challenging to represent faithfully a large number of ab initio points with a multidimensional analytical function over a large configuration space, which is needed for accurate dynamical studies. In this Review, we discuss our recent work on a new potential-fitting approach based on artificial neural networks, which are ultra-flexible in representing any multidimensional real functions. A unique feature of our neural network approach is how the symmetries, particularly those associated with the exchange of identical atoms in the system, are enforced. To this end, symmetry functions in the form of symmetrised monomials that satisfy a particular type of symmetry possessed by the system are used in the input layer of the neural network. This approach is rigorous, accurate, and efficient. It is also simple to implement, requiring no modification of the neural network routines. Its applications to the construction of multi-dimensional potential energy surfaces in many gas phase and gas–surface systems as surveyed here.

A rigorous, general, and simple method to fit global and permutation invariant potential energy surfaces (PESs) using neural networks (NNs) is discussed. This so-called permutation invariant polynomial neural network (PIP-NN) method imposes permutation symmetry by using in its input a set of symmetry functions based on PIPs. For systems with more than three atoms, it is shown that the number of symmetry functions in the input vector needs to be larger than the number of internal coordinates in order to include both the primary and secondary invariant polynomials. This PIP-NN method is successfully demonstrated in three atom-triatomic reactive systems, resulting in full-dimensional global PESs with average errors on the order of meV. These PESs are used in full-dimensional quantum dynamical calculations.

Many gas-phase chemical reactions proceed via reaction intermediates, supported by potential wells. The characteristics of such complex-forming reactions differ drastically from those for direct reactions that involve barriers. For example, the reaction path for a complex-forming reaction is often barrierless, which results in weak and sometimes negative temperature dependence for its rate constant. The product angular and internal distributions of such reactions also bear clear signatures. Specifically, the angular distribution (i.e. differential cross-section) of a complex-forming reaction is often dominated by scattering in the forward and backward directions, and the product rotational state distribution usually peaks near the highest accessible rotational state, while vibrational state distribution often decays monotonically. While the quantum dynamics of direct reactions is well established, our understanding of complex-forming reactions is still far from complete. Given the importance of such reactions in interstellar, atmospheric and combustion chemistry, much research effort has recently been devoted to understand their dynamics. In this review, we survey the recent progress in the quantum dynamics of several prototypical complex-forming reactions, particularly those involving three or four atoms. We will focus on methodological advances in quantum scattering theory, quasi-classical trajectory methods and statistical models.