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Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method

The stationary points (SPs) of a potential energy landscape play a crucial role in understanding many of the physical or chemical properties of a given system. Unless they are found analytically, there is, however, no efficient method to obtain 'all' the SPs of a given potential. We introduce a novel method, called the numerical polynomial homotopy continuation (NPHC) method, which numerically finds all the SPs, and is 'embarrassingly parallelizable'. The method requires the non-linearity of the potential to be polynomial-like, which is the case for almost all of the potentials arising in physical and chemical systems. We also certify the numerically obtained SPs so that they are independent of the numerical tolerance used during the computation. It is then straightforward to separate out the local and global minima. As a first application, we take the XY model with power-law interaction which is shown to have a polynomial-like non-linearity and apply the method.