Using a neural network based method to solve the vibrational Schrِdinger equation for H2O

A neural network (NN) algorithm is used to solve the vibrational Schrِdinger equation for a molecule. Previous NN methods computed one level at a time, optimized all of the parameters using non-linear optimization methods, and were tested only on model potentials. Our approach combines non-linear optimization of neuron parameters with a linear matrix method. This improves dimensionality scaling and permits computing many levels. We use composite, flexible shape, radial basis function neurons. The algorithm avoids the calculation of integrals and of a potential energy function. We demonstrate that only a few dozen neurons are needed to compute five levels of water from a small set of potential points.