Constructing global functional maps between molecular potentials and quantum observables

The relationships that connect potential energy surfaces to quantum observa bles can be complex and nonlinear. In this paper, an approach toward global ly representing and exploring potential-observable relationships using a fu nctional mapping procedure is developed. Based on selected solutions of the Schrodinger equation, it is demonstrated that an observable's behavior can be learned as a function of the potential and any other variables needed t o specify the quantum system. Once such a map for the observable is in hand , it is available for use in a host of future applications without further need for solving the Schrodinger equation. As formulated here, maps provide explicit information about the global response of the observable to the po tential. In this paper, we develop the mapping concept, estimate its scalin g behavior (measured as the number of times the Schrodinger equation must b e solved during the learning process), and numerically illustrate the techn ique's globality and nonlinearity using well-understood systems that demons trate its capabilities. For atom-atom scattering, we construct a single map capable of learning elastic cross sections (i.e., differential cross secti ons at 2 degrees intervals over angle, as well as integral, diffusion, and viscosity cross sections for scattering energies between 50 meV and 2 eV) i nvolving collisions between any pair of atoms from the Periodic Table. The map for each class of cross sections over the Periodic Table is quantitativ e with prediction errors shown to be <<1%. We also consider a (3)Sigma (+)( u) Na-2 and create a rovibrational spectral map that encompasses all of the currently proposed potentials for that system. The Na-2 map is highly accu rate with the ability to predict rovibrational spectra with errors less tha n 1x10(-3) cm(-1) over variations in the potential that exceed 130 cm(-1). (C) 2001 American Institute of Physics.