Solving the schrödinger equation : deep-learnt exchange-correlation potential and efficient quantum transport calculation with selected inversion

Abstract

The Schröodinger equation is one of the foundations to describe quantum phenomena. The solving of Schrödinger's equation is improved from the aspect of accuracy and efficiency. To improve the accuracy, an exchange-correlation potential of density-functional theory is constructed via the convolutional neural network. Instead of requiring the global electron density function, the electron density distribution of a finite volume around a spatial point is used to determine the local exchange-correlation potential at the point, based on the holographic electron density theorem. The highly accurate electron density function is calculated using CCSD method, and the corresponding exchange-correlation potential is determined through the optimized effective potential method. The exact electron density and near-exact potential form a large dataset for training and testing. A three-dimensional convolutional neural network is constructed and trained against the dataset to learn the mapping from discrete electron density distribution of a finite volume to the potential at the volume centre. The optimal neural network is then integrated into the self-consistent calculation to serve as an exchange-correlation functional. Two systems, namely H2/HeH+ and helium dimer, have been tested using the neural network based functional. The optimal electron density distribution and related molecular properties are obtained. Comparison to those calculated with the B3LYP functional is made to confirm the accuracy and effectiveness of the new method. Additional test on He-H-H-He2+ molecule, which is beyond the dataset, shows the out-of-sample transferability of the neural network functional. The simulation on helium dimer further demonstrates the functional's capability in correct description of van der Waals interaction, which is usually hard for conventional functionals.

To improve the efficiency, the selected inversion (SelInv) algorithm is introduced and applied to the quantum transport simulation. The feasibility of integrating SelInv into the non-equilibrium Green's function-density functional tight binding (NEGF-DFTB) framework is established by comparing the selective components generated from the algorithm and required by the NEGF method. The NEGF-DFTB-SelInv method is implemented subsequently. It shows less time cost compared to the existing recursive Green's function method. The method is then employed to simulate an ultra-thin three-terminal transistor at atomic level. The obtained $I-V$ curve and potential distribution is consistent with the experimental results, confirming that such method can be used in quantum transport problems fully following the experimental setup.