Electronic Structure Calculations for Strongly Correlated Electron Systems

According to the celebrated density functional theory (DFT)¹, the ground state of a many-electron system can be determined by the single-particle density, effectively eliminating the curse of dimensionality in principle. The variational argument leads to the universal Levy-Lieb functional, comprising both kinetic and electron-electron interaction contributions, yet lacking a tractable expression.

The popular Kohn-Sham DFT (KSDFT)² uses a non-interacting reference system and focuses on approximating the kinetic contribution. The approximation error for the Levy-Lieb functional is called the exchange-correlation functional, which has no closed-form expression. Because of its starting point, KSDFT can fail to accurately capture the behavior of strongly correlated electron systems, where interaction energy dominates, even with state-of-the-art approximations of exchange-correlation functionals³. Since strongly correlated materials have broad applications (e.g., high-temperature superconductivity, thermoelectric materials, and magnetic storage), alternative models and numerical approaches are urgently needed.

A plethora of models have emerged, often contending with the curse of dimensionality. My interests revolve around analyzing their theoretical properties and developing scalable numerical approaches. Currently, my collaborators and I are primarily working on:

• Strictly-correlated-electrons density functional theory⁴.

• Variational Monte Carlo methods⁵.

Strictly-correlated-electrons density functional theory

Collaborators (in the alphabetic order):

• Huajie Chen (Beijing Normal University);
• Mengyu Li (Renmin University of China);
• Xin Liu (Chinese Academy of Sciences);
• Cheng Meng (Renmin University of China).

This model runs from the opposite face of KSDFT and builds on a multimarginal optimal transport problem with Coulomb cost (MMOT). Leveraging a low-dimensional ansatz for the square modulus of wave function $|\Psi|^2$, which takes roots in optimal transport and encodes two-electron couplings, one could derive a nonconvex reformulation for the MMOT. The number of unknowns therein scales linearly with respect to the number of electrons. Due to the intricacies of the nonconvex problem, existing works in this aspect consider systems with only two electrons.

Our contributions include:

• Establishing the theoretical properties of the nonconvex formulation. The nonconvex formulation (denoted as (P)) is not suitable for numerical resolution due to violating conventional constraint qualifications. We establish the equivalence between (P) and a nonconvex quadratic programming (denoted as (P’)) without additional assumptions from the literature. Interestingly, we numerically observe that (P’) is “nearly” amenable to any local solvers.

  
  

• Proposing a multigrid optimization framework balancing accuracy and efficiency. Inspired by the multigrid methods and optimal transport, we starts with solving (P’) accurately over a coarse mesh and proceeds with local solvers for (P’) over finer meshes. The local solvers are fed with initial points constructed from the previous solutions. In simulations, electron positions mappings are first visualized in two/three-dimensional contexts.

  
  

• Devising sampling-based methods as local solvers without cubic complexities. To reduce the cubic computational burden in gradient calculations, we adopt Kullback-Leibler divergence and incorporate importance sampling-based matrix sparsification into iterative schemes. These methods avoid storing and operating on dense matrices. Compared with conventional stochastic algorithms, our methods add randomness to constraints.

  
  

• Providing the first convergence results for the local solvers under realizable conditions. To enhance efficiency, local solvers may use infeasible methods for subproblems. With infeasible iterates, nonmonotonicity of objective values is inevitable, complicating convergence analyses. Unlike existing works, we demonstrate convergence under realizable conditions and derive novel convergence rates.

List of publications

1. H., H. Chen, and X. Liu. A global optimization approach for multimarginal optimal transport problems with Coulomb cost. SIAM Journal on Scientific Computing, 2023, 45(3): A1214-A1238. (link)

2. H. and X. Liu. The convergence properties of infeasible inexact proximal alternating linearized minimization. Science China Mathematics, 2023, 66(10): 2385-2410. (link)

3. H. and X. Liu. The exactness of the $\ell_1$ penalty function for a class of mathematical programs with generalized complementarity constraints. Fundamental Research, published online. DOI: 10.1016/j.fmre.2023.04.006. (link)

4. H., M. Li, X. Liu, and C. Meng. Sampling-based methods for multi-block optimization problems over transport polytopes. Mathematics of Computation, 2024+, accepted. (link)

Ongoing works

• Local solvers with better scaling with respect to the number of electrons and discretization size.
• Landscape analysis for the nonconvex reformulation of the MMOT.
• Convergence analysis for the multigrid optimization framework.
• Numerical approaches for the other reformulations of the MMOT.