Permutationally invariant potential energy surfaces in high dimensionality

Recent Advances in Development and Applications of the Mixed Quantum/Classical Theory for Inelastic Scattering

“…55,56 The corresponding effective collision energy U is defined as (13) It does not appear in any equations listed above but is used to set up the initial conditions in the actual calculations (namely, in eq 8 instead of E). Using simple algebra, one can express U, E, and E′ through each other: 55,56 (14) (15) (16) (17) Note that E′ < U < E. Also note that at the threshold for excitation, when E = ΔE and E′ = 0, we obtain U = ΔE/4.…”

“…[9][10][11][12] Second, the molecule-molecule interaction is described by a complicated multidimensional potential, represented by the potential energy surface (PES), or hypersurface to be exact, that should be constructed using the tools of electronic structure theory and the state-of-the-art methods of surface fitting. 13,14 Third, the range of scattering energies we have to cover is often broad, with different mechanisms dominating in different collision regimes (e.g., scattering resonances at low energies 2,12,15 in contrast to vibrational excitation at high energies). [16][17][18] Finally, very detailed information about the process may be needed, such as differential over scattering angle cross section for state-to-state transition.…”

Neural network‐based approaches for building high dimensional and quantum dynamics‐friendly potential energy surfaces

“…[69,71,75] Importantly for applications to materials modeling and design, this method has been shown to provide highaccuracy PESs for bulk materials as well as surfaces and nanostructures, [68,69,72,74,76,77] where other highly accurate approaches developed for molecular systems [22,67] are impractical.…”

“…We have used this strategy in combination with symmetrizing the NN output and found that it works well in as many as 15 dimensions. [26,29] Guo and coworkers [62][63][64][65][66] [22,67] of interatomic coordinates. Interatomic coordinates make this approach systematic and easy to generalize.…”

“…[18][19][20][21] In the PIP approach, PESs must be fitted using (redundant) interatomic coordinates to impose permutation invariance. [22] It has been demonstrated that NN PESs are very accurate for small molecules and as good as other methods for larger molecules. [23][24][25][26][27][28][29][30][31][32] If used with exponential neurons [108] or with product neurons, [33] they yield PESs in sum-of-products (SOP) form.…”