  Machine learning has revolutionized the high-dimensional representations for
molecular properties such as potential energy. However, there are scarce
machine learning models targeting tensorial properties, which are rotationally
covariant. Here, we propose tensorial neural network (NN) models to learn both
tensorial response and transition properties, in which atomic coordinate
vectors are multiplied with scalar NN outputs or their derivatives to preserve
the rotationally covariant symmetry. This strategy keeps structural descriptors
symmetry invariant so that the resulting tensorial NN models are as efficient
as their scalar counterparts. We validate the performance and universality of
this approach by learning response properties of water oligomers and liquid
water, and transition dipole moment of a model structural unit of proteins.
Machine learned tensorial models have enabled efficient simulations of
vibrational spectra of liquid water and ultraviolet spectra of realistic
proteins, promising feasible and accurate spectroscopic simulations for
biomolecules and materials.
