coordinates that do not involve the 3 orientation rotations and 3 orientation translations. However, as the number of atoms in a molecule increases, it becomes challenging to describe the molecular geometry using internal coordinates alone. For instance, when there are more than three atoms (N > 3) in a molecule, it is more convenient to use a set of 3-dimensional coordinates (x, y, z) to represent the molecular geometry. This is particularly important for molecular design or generation using artificial intelligence, as the datasets often include polyatomic molecules with a high number of atoms (N > 3). Therefore, by embedding the molecular geometry information in a set of 3-dimensional coordinates, accurate representation of the molecule becomes possible, facilitating computational analysis. With advanced computer simulations and other computational methods like permutationally invariant polynomial (PIP-NN), fundamental invariance neural network (FI-NN), or Gaussian process (GP) method available today, exploring the PESs of molecular systems makes it possible to predict the behaviors of the special structures under different conditions. Nowadays, it is a popular approach to describe the 3-dimesional coordinates of polyatomic molecules by SE3 group. For SE3 group , a function φ : X→Y, if all SE3 group elements 6 and all x in X, the Euclidean transformation Lx and Ly satisfying: