Softmax parameterization of the occupation numbers for natural orbital functionals based on electron pairing approaches.

Within the framework of natural orbital functional theory, having a convenient representation of the occupation numbers and orbitals becomes critical for the computational performance of the calculations. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used in Piris natural orbital functionals through the adoption of the softmax function, a pivotal component in modern deep-learning models. Our approach not only ensures adherence to the N-representability of the first-order reduced density matrix (1RDM) but also significantly enhances the computational efficiency of 1RDM functional theory calculations. The effectiveness of this alternative parameterization approach was assessed using the W4-17-MR molecular set, which demonstrated faster and more robust convergence compared to previous implementations.

Inverse molecular design and parameter optimization with Hückel theory using automatic differentiation.

Semiempirical quantum chemistry has recently seen a renaissance with applications in high-throughput virtual screening and machine learning. The simplest semiempirical model still in widespread use in chemistry is Hückel's π-electron molecular orbital theory. In this work, we implemented a Hückel program using differentiable programming with the JAX framework based on limited modifications of a pre-existing NumPy version. The auto-differentiable Hückel code enabled efficient gradient-based optimization of model parameters tuned for excitation energies and molecular polarizabilities, respectively, based on as few as 100 data points from density functional theory simulations. In particular, the facile computation of the polarizability, a second-order derivative, via auto-differentiation shows the potential of differentiable programming to bypass the need for numeric differentiation or derivation of analytical expressions. Finally, we employ gradient-based optimization of atom identity for inverse design of organic electronic materials with targeted orbital energy gaps and polarizabilities. Optimized structures are obtained after as little as 15 iterations using standard gradient-based optimization algorithms.

Fast evaluation of the adsorption energy of organic molecules on metals via graph neural networks.

Modeling in heterogeneous catalysis requires the extensive evaluation of the energy of molecules adsorbed on surfaces. This is done via density functional theory but for large organic molecules it requires enormous computational time, compromising the viability of the approach. Here we present GAME-Net, a graph neural network to quickly evaluate the adsorption energy. GAME-Net is trained on a well-balanced chemically diverse dataset with C1-4 molecules with functional groups including N, O, S and C6-10 aromatic rings. The model yields a mean absolute error of 0.18 eV on the test set and is 6 orders of magnitude faster than density functional theory. Applied to biomass and plastics (up to 30 heteroatoms), adsorption energies are predicted with a mean absolute error of 0.016 eV per atom. The framework represents a tool for the fast screening of catalytic materials, particularly for systems that cannot be simulated by traditional methods.

Multi-objective optimization for retinal photoisomerization models with respect to experimental observables.

The fitting of physical models is often done only using a single target observable. However, when multiple targets are considered, the fitting procedure becomes cumbersome, there being no easy way to quantify the robustness of the model for all different observables. Here, we illustrate that one can jointly search for the best model for each desired observable through multi-objective optimization. To do so, we construct the Pareto front to study if there exists a set of parameters of the model that can jointly describe multiple, or all, observables. To alleviate the computational cost, the predicted error for each targeted objective is approximated with a Gaussian process model as it is commonly done in the Bayesian optimization framework. We applied this methodology to improve three different models used in the simulation of stationary state cis-trans photoisomerization of retinal in rhodopsin, a significant biophysical process. Optimization was done with respect to different experimental measurements, including emission spectra, peak absorption frequencies for the cis and trans conformers, and energy storage. Advantages and disadvantages of previously proposed models are exposed.

Six-dimensional potential energy surface for NaK-NaK collisions: Gaussian process representation with correct asymptotic form.

Constructing accurate global potential energy surfaces (PESs) describing chemically reactive molecule-molecule collisions of alkali metal dimers presents a major challenge. To be suitable for quantum scattering calculations, such PESs must represent accurately three- and four-body interactions, describe conical intersections, and have a proper asymptotic form at the long range. Here, we demonstrate that such global potentials can be obtained by Gaussian Process (GP) regression merged with the analytic asymptotic expansions at the long range. We propose an efficient sampling technique, which allows us to construct an accurate global PES accounting for different chemical arrangements with <2500 ab initio calculations. We apply this method to (NaK)2 and obtain the first global PES for a system of four alkali metal atoms. The resulting surface exhibits a complex landscape including a pair and a quartet of symmetrically equivalent local minima and a seam of conical intersections. The dissociation energy found from our ab initio calculations is 4534 cm-1. This result is reproduced by the GP models with an error of less than 3%. The GP models of the PES allow us to analyze the features of the global PES, representative of general alkali metal four-atom interactions. Understanding these interactions is of key importance in the field of ultracold chemistry.

Neural networks vs Gaussian process regression for representing potential energy surfaces: A comparative study of fit quality and vibrational spectrum accuracy.

For molecules with more than three atoms, it is difficult to fit or interpolate a potential energy surface (PES) from a small number of (usually ab initio) energies at points. Many methods have been proposed in recent decades, each claiming a set of advantages. Unfortunately, there are few comparative studies. In this paper, we compare neural networks (NNs) with Gaussian process (GP) regression. We re-fit an accurate PES of formaldehyde and compare PES errors on the entire point set used to solve the vibrational Schrödinger equation, i.e., the only error that matters in quantum dynamics calculations. We also compare the vibrational spectra computed on the underlying reference PES and the NN and GP potential surfaces. The NN and GP surfaces are constructed with exactly the same points, and the corresponding spectra are computed with the same points and the same basis. The GP fitting error is lower, and the GP spectrum is more accurate. The best NN fits to 625/1250/2500 symmetry unique potential energy points have global PES root mean square errors (RMSEs) of 6.53/2.54/0.86 cm-1, whereas the best GP surfaces have RMSE values of 3.87/1.13/0.62 cm-1, respectively. When fitting 625 symmetry unique points, the error in the first 100 vibrational levels is only 0.06 cm-1 with the best GP fit, whereas the spectrum on the best NN PES has an error of 0.22 cm-1, with respect to the spectrum computed on the reference PES. This error is reduced to about 0.01 cm-1 when fitting 2500 points with either the NN or GP. We also find that the GP surface produces a relatively accurate spectrum when obtained based on as few as 313 points.

Assessing Gaussian Process Regression and Permutationally Invariant Polynomial Approaches To Represent High-Dimensional Potential Energy Surfaces.

The mathematical representation of large data sets of electronic energies has seen substantial progress in the past 10 years. The so-called Permutationally Invariant Polynomial (PIP) representation is one established approach. This approach dates from 2003, when a global potential energy surface (PES) for CH5+ was reported using a basis of polynomials that are invariant with respect to the 120 permutations of the five equivalent H atoms. More recently, several approaches from "machine learning" have been applied to fit these large data sets. Gaussian Process (GP) regression is such an approach. Here, we consider the implementation of the (full) GP due to Krems and co-workers, with a modification that renders it permutationally invariant, which we denote by PIP-GP. This modification uses the approach of Guo and co-workers and later extended by Zhang and co-workers, to achieve permutational invariance for neural-network fits. The PIP, GP, and PIP-GP approaches are applied to four case studies for fitting data sets of electronic energies: H3O+, OCHCO+, and H2CO/ cis-HCOH/ trans-HCOH with the goal of assessing precision, accuracy in normal-mode analysis and barrier heights, and timings. We also report an application to (HCOOH)2, where the full PIP approach is possible but where the PIP-GP one is not feasible. However, by replicating data, which is feasible in this case, the GP approach is able to represent the data with precision comparable to that of the PIP approach. We examine these assessments for varying sizes of data sets in each case to determine the dependence of properties of the fits on the training data size. We conclude with some comments on the different aspects of computational effort of the PIP, GP, and PIP-GP approaches and also challenges these methods face for more "rugged" PESs, exemplified here by H2CO/ cis-HCOH/ trans-HCOH.

Extrapolating Quantum Observables with Machine Learning: Inferring Multiple Phase Transitions from Properties of a Single Phase.

We present a machine-learning method for predicting sharp transitions in a Hamiltonian phase diagram by extrapolating the properties of quantum systems. The method is based on Gaussian process regression with a combination of kernels chosen through an iterative procedure maximizing the predicting power of the kernels. The method is capable of extrapolating across the transition lines. The calculations within a given phase can be used to predict not only the closest sharp transition but also a transition removed from the available data by a separate phase. This makes the present method particularly valuable for searching phase transitions in the parts of the parameter space that cannot be probed experimentally or theoretically.