Unsupervised Machine Learning Neural Gas Algorithm for Accurate Evaluations of the Hessian Matrix in Molecular Dynamics

Abstract

The Hessian matrix of the potential energy of molecular systems is employed not only in geometry optimizations or high-order molecular dynamics integrators but also in many other molecular procedures, such as instantaneous normal mode analysis, force field construction, instanton calculations, and semiclassical initial value representation molecular dynamics, to name a few. Here, we present an algorithm for the calculation of the approximated Hessian in molecular dynamics. The algorithm belongs to the family of unsupervised machine learning methods, and it is based on the neural gas idea, where neurons are molecular configurations whose Hessians are adopted for groups of molecular dynamics configurations with similar geometries. The method is tested on several molecular systems of different dimensionalities both in terms of accuracy and computational time versus calculating the Hessian matrix at each time-step, that is, without any approximation, and other Hessian approximation schemes. Finally, the method is applied to the on-the-fly, full-dimensional simulation of a small synthetic peptide (the 46 atom N-acetyl-L-phenylalaninyl-L-methionine amide) at the level of DFT-B3LYP-D/6-31G* theory, from which the semiclassical vibrational power spectrum is calculated.

Applicability of the Thawed Gaussian Wavepacket Dynamics to the Calculation of Vibronic Spectra of Molecules with Double-Well Potential Energy Surfaces

“…However, the cost of evaluating Hessians using ab initio electronic structure can be prohibitive in certain cases, e.g., for larger molecules or for highly accurate but costly electronic structure methods. To avoid these limitations, a number of efficient updating, , interpolation, , or machine-learning , schemes were proposed. To this end, two of us recently proposed a crude but practical approach that replaces the Hessians evaluated along the trajectory by a single Hessian evaluated at a reference geometry, i.e., This single-Hessian method was shown to be almost as accurate as the original thawed Gaussian approximation in several model and realistic systems, and was used to simulate both steady-state , and time-resolved spectra. , As in the harmonic approximation, there are different possible choices for choosing the reference Hessian, including the adiabatic and vertical Hessians of the final state or the adiabatic Hessian of the initial state (labeled “initial Hessian”).…”

“…However, the cost of evaluating Hessians using ab initio electronic structure can be prohibitive in certain cases, e.g., for larger molecules or for highly accurate but costly electronic structure methods. To avoid these limitations, a number of efficient updating, 73,74 interpolation, 15,60 or machine-learning 75,76 schemes were proposed. To this end, two of us recently proposed a crude but practical approach 57 that replaces the Hessians evaluated along the trajectory by a single Hessian evaluated at a reference geometry, i.e.,…”

Recent Advances toward Efficient Calculation of Higher Nuclear Derivatives in Quantum Chemistry

“…Gandolfi and Ceotto adapted a neural gas (NGas) algorithm for approximating Hessians of structures that constitute AIMD trajectories . Neural gas is a robust alternative to k-means clustering of data that adapts a finite number of feature vectors to the data .…”

Efficient Method for Numerical Calculations of Molecular Vibrational Frequencies by Exploiting Sparseness of Hessian Matrix

“…In this context, early attempts develop different parametrization schemes to approximate the second derivative matrix, which are widely used during geometry optimizations. − In addition, if the studied system is separable, the whole Hessian matrix is divided into sub-blocks, and different strategies are used to treat each block. − Another route to approximate Hessian is to use the compact finite difference (CFD) scheme for systems in which adequately smooth potential energy surfaces are available. − Recently, the compressed sensing technique has been used to construct correlations of Hessian matrices obtained separately at high-level and low-level theories. , The Gaussian process regression uses a different route to estimate the Hessian matrix, which is based on the construction of a potential energy surface . In recent years, the machine learning technique has been employed to approximate the Hessian matrix over a group of configurations with similar geometric properties . One can also obtain the Hessian matrix by fitting full-dimensional potential energy surfaces, but such multivariable fitting becomes more and more challenging when the number of atoms grows.…”