Geometry Optimization with Machine Trained Topological Atoms

Abstract

The geometry optimization of a water molecule with a novel type of energy function called FFLUX is presented, which bypasses the traditional bonded potentials. Instead, topologically-partitioned atomic energies are trained by the machine learning method kriging to predict their IQA atomic energies for a previously unseen molecular geometry. Proof-of-concept that FFLUX's architecture is suitable for geometry optimization is rigorously demonstrated. It is found that accurate kriging models can optimize 2000 distorted geometries to within 0.28 kJ mol^-1 of the corresponding ab initio energy, and 50% of those to within 0.05 kJ mol^-1. Kriging models are robust enough to optimize the molecular geometry to sub-noise accuracy, when two thirds of the geometric inputs are outside the training range of that model. Finally, the individual components of the potential energy are analyzed, and chemical intuition is reflected in the independent behavior of the three energy terms [Formula: see text](intra-atomic), [Formula: see text] (electrostatic) and [Formula: see text] (exchange), in contrast to standard force fields.

Figure 1

Flowchart of FFLUX’s training (first four steps) and execution (DL_POLY), detailing the programs involved and summaries of their tasks.

Figure 2

S-curves for the 100, 300, 500, T500 and TE500 water models described using the three energies given in Eqn. 1 ($E_{\mathrm{intra}}^{\mathrm{A}}$, $V_{\mathrm{cl}}^{\text{AA}\text{'}}$ and $V_{\mathrm{x}}^{\text{AA}\text{'}}$). The label “T” stands for the tighter scrubbing threshold of 0.00005 Hartrees, while “TE” stands for this tight model using single total atomic energies, $E_{\mathrm{IQA}}^{\mathrm{A}}$.

Figure 3

SP1 (left, +15.05 kJ mol⁻¹), SP2 (middle, +47.97 kJ mol⁻¹) and SP3 (right, +126.18 kJ mol⁻¹) water geometries. Bond distances are in Å, and bond angles in degrees.

Figure 4

T500 molecular model geometry optimization trajectory steps with SP1 (blue), SP2 (red) and SP3 (green) starting points: (a) Set 1 (0 K and 1 fs timestep) truncated at 500 steps where the energy fluctuation is <0.0001 kJ mol⁻¹ and (b) Set 3 (GC and 1 fs timestep) with no truncation. The x-axis marks the timestep number. In the left panels, the y-axes denote molecular energy; in the right panels the y-axes denote ΔE (current energy – previous energy). All energies are in kJ mol⁻¹.

Figure 5

Performance of the T500 model using the 0 K optimization: (a) aggregated plot of the molecular energy evolution in time for each of the 2000 starting geometries considered (runs are coloured from dark to light blue allowing tracking); (b) magnified energy evolution between the 900^th and 1000^th timesteps; (c) distribution of energies at the 1000^th timestep, relative to the ab initio energy.

Figure 6

Single-energy optimized water geometries using the individual $E_{\mathrm{intra}}^{\mathrm{A}}$, $V_{\mathrm{x}}^{\text{AA}\text{'}}$ and $V_{\mathrm{cl}}^{\text{AA}\text{'}}$ energies. Initialization geometry is the QM minimum, and the optimizations are performed using the T500 model with parameter Set 1.