Huajie Chen

Numerical analysis of finite dimensional approximations of Kohn---Sham models

In this paper, we study finite dimensional approximations of Kohn–Sham models, which are widely used in electronic structure calculations. We prove the convergence of the finite dimensional approximations and derive the a priori error estimates for ground state energies and solutions. We also provide numerical simulations for several molecular systems that support our theory.

Adaptive Finite Element Approximations for Kohn--Sham Models

The Kohn-Sham equation is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanosciences. In this paper, we study the adaptive finite element approximations for the Kohn-Sham model. Based on the residual type a posteriori error estimators proposed in this paper, we introduce an adaptive finite element algorithm with a quite general marking strategy and prove the convergence of the adaptive finite element approximations. Using D{ o}rflers marking strategy, we then get the convergence rate and quasi-optimal complexity. We also carry out several typical numerical experiments that not only support our theory,but also show the robustness and efficiency of the adaptive finite element computations in electronic structure calculations.

Numerical Methods for a Kohn–Sham Density Functional Model Based on Optimal Transport

In this paper, we study numerical discretizations to solve density functional models in the strictly correlated electrons (SCE) framework. Unlike previous studies, our work is not restricted to radially symmetric densities. In the SCE framework, the exchange-correlation functional encodes the effects of the strong correlation regime by minimizing the pairwise Coulomb repulsion, resulting in an optimal transport problem. We give a mathematical derivation of the self-consistent Kohn-Sham-SCE equations, construct an efficient numerical discretization for this type of problem for N = 2 electrons, and apply it to the H2 molecule in its dissociating limit.

QM/MM Methods for Crystalline Defects. Part 2: Consistent Energy and Force-Mixing

QM/MM hybrid methods employ accurate quantum (QM) models only in regions of interest (defects) and switch to computationally cheaper interatomic potential (MM) models to describe the crystalline bulk. nWe develop two QM/MM hybrid methods for crystalline defect simulations, an energy-based and a force-based formulation, employing a tight binding QM model. Both methods build on two principles: (i) locality of the QM model; and (ii) constructing the MM model as an explicit and controllable approximation of the QM model. This approach enables us to establish explicit convergence rates in terms of the size of QM region.

QM/MM methods for crystalline defects. Part 1 : Locality of the tight binding model

The tight binding model is a minimal electronic structure model for molecular modeling and simulation. We show that for a finite temperature model, the total energy in this model can be decomposed into site energies, that is, into contributions from each atomic site whose influence on their environment decays exponentially. This result lays the foundation for a rigorous analysis of QM/MM coupling schemes.

Two-scale finite element discretizations for integro- differential equations

Some two-scale finite element discretizations are introduced for a class of linear partial differential equations. Both boundary value and eigenvalue problems are studied. Based on the two-scale error resolution techniques, several two-scale finite element algorithms are proposed and analyzed. It is shown that this type of two-scale algorithms not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.

Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry

Atomic-like basis functions provide a natural, physically motivated description of electronic states, among which Gaussian-type orbitals are the most widely used basis functions in molecular simulations. This paper aims at developing a systematic analysis of numerical approximations based on linear combinations of Gaussian-type orbitals. We derive a priori error estimates for Hermite-type Gaussian bases as well as for even-tempered Gaussian bases. Numerical results are presented to support the theory.

QM/MM Methods for Crystalline Defects. Part 1

The tight binding model is a minimal electronic structure model for molecular modeling and simulation. We show that for a finite temperature model, the total energy in this model can be decomposed into site energies, that is, into contributions from each atomic site whose influence on their environment decays exponentially. This result lays the foundation for a rigorous analysis of QM/MM coupling schemes.

Pair Densities in Density Functional Theory

The exact interaction energy of a many-electron system is determined by the electron pair density, which is not well-approximated in standard Kohn-Sham density functional models. Here we study the (complicated but well-defined) exact universal map from density to pair density. We survey how many common functionals, including the most basic version of the LDA (Dirac exchange with no correlation contribution), arise from particular approximations of this map. We develop an algorithm to compute the map numerically, and apply it to one-parameter families {a*rho(a*x)} of one-dimensional homogeneous and inhomogeneous single-particle densities. We observe that the pair density develops remarkable multiscale patterns which strongly depend on both the particle number and the width 1/a of the single-particle density. The simulation results are confirmed by rigorous asymptotic results in the limiting regimes a>>1 and a<<1. For one-dimensional homogeneous systems, we show that the whole spectrum of patterns is reproduced surprisingly well by a simple asymptotics-based ansatz which slowly smoothens out the strictly correlated a=0 pair density while slowly turning on the a=infty exchange terms as a increases. Our findings lend theoretical support to the celebrated semi-empirical idea [Becke93] to mix in a fractional amount of exchange, albeit not to assuming the mixing to be additive and taking the fraction to be a system independent constant.

Thermodynamic Limit of Crystal Defects with Finite Temperature Tight Binding

We consider a tight binding model for localised crystalline defects with electrons in the canonical ensemble (finite Fermi temperature) and nuclei positions relaxed according to the Born–Oppenheimer approximation. We prove that the limit model as the computational domain size grows to infinity is formulated in the grand-canonical ensemble for the electrons. The Fermi-level for the limit model is fixed at a homogeneous crystal level, independent of the defect or electron number in the sequence of finite-domain approximations. We quantify the rates of convergence for the nuclei configuration and for the Fermi-level.