Prediction of Density Matrix with Machine Learning

Abstract

Kohn-Sham Density functional theory (KS-DFT) has become an essential tool in computational chemistry. It provides accurate simulations of the electronic structure for a wide range of periodic or molecular systems. Nevertheless, there is computational overhead associated with the iterative self-consistent field (SCF) process, which remains a significant bottleneck for large-scale DFT calculations. The convergence of an SCF approach depends on various factors; one of them is associated with the choice of initial guesses for the density matrix (DM). If the initial guess DM is close to the final converged one, then the SCF approach yields fast convergence; hence, the DFT calculations can be accelerated. However, a poor guess can lead a DFT calculation towards nonconverged outcomes. This thesis aims to accelerate the SCF scheme in DFT by implementing machine learning (ML) governed approaches that predict DMs and serve the needs of better initial guesses for the SCF process. Finally, this may enable efficient, non-self-consistent molecular dynamics (MD) simulations. For this thesis, two different ML-based schemes were developed; one using neural network architecture to predict initial guesses for the DMs and another employing atomic descriptors to predict real-space electron density and then constructing KS-Hamiltonian to compute the DMs. These two schemes have been implemented with the PySCF (1, 2) electronic structure package, which always starts DFT calculation from the guess DM and uses the Gaussian type orbitals for molecular calculations. In PySCF, the standard choice of initial guess includes the superposition of the atomic densities (3, 4), one electron guess (5), atom guess (6), etc. Our neural network (NN) predicted initial guess for DM can be an excellent choice, which demonstrates a significant reduction in SCF iterations compared to any other familiar guesses, and for the DM prediction atomic positions are used as the inputs. Furthermore, we applied this ML-based approach to generate the inter-atomic forces from non-self-consistent DFT calculations for conducting MD simulations using LAMMPS (7) simulation software. The major disadvantage of this NN-predicted approach is that the density matrices are rotationally contravariant. Since our NN does not explicitly enforce this property, it may generate density matrices that are inconsistent with physically expected transformations, limiting its flexibility and generalization. As a result, we applied these NN-based predictions only to molecules oriented in a specific direction. Alternatively, in this study, we show that an ML density-based approach offers a new way to predict the DM. Thereafter, this matrix serves as an optimized initial guess for the DFT self-consistent process, requiring only a single iteration to compute atomic forces and energies necessary for molecular dynamics simulations at various temperatures. Compared to traditional alternatives for the initial guess, our density matrix exemplifies a better choice. Furthermore, PySCF does not directly provide the Kohn- Sham Hamiltonian from the electron density, and consequently, the DM, since the DM is obtained through the diagonalization of the KS Hamiltonian, which is crucial for our study. This limitation arises because the Coulomb matrix can not be computed from the density in PySCF. Here we detail the incorporation of the Coulomb repulsion matrix over Gaussian-type orbitals in PySCF, which enables the construction of the Kohn-Sham Hamiltonian directly from the electron density. These advancements pave the way for more efficient DFT-based workflows, making high-accuracy simulations accessible for complex molecular systems and real-time MD studies via PySCF and LAMMPS.