invDFT: A CPU-GPU massively parallel tool to find exact exchange-correlation potentials from groundstate densities

TL;DR

invDFT tackles the inverse DFT problem of retrieving the exact exchange-correlation potential from a target density by casting it as a PDE-constrained optimization and solving it on a systematically convergent finite-element basis with asymptotic corrections. The framework combines advanced solvers (Chebyshev filtering for KS eigensolves and MINRES for adjoints) with HIP/CUDA-accelerated HPC strategies to achieve robust accuracy and scalable performance on CPU-GPU architectures. It validates the approach against LDA densities and produces exact XC potentials for benchmark systems spanning weak to strong correlation, demonstrating Koopmans-consistent eigenvalues and smooth potentials free from basis artifacts. The work delivers a practical, open-source tool that can inform XC functional development and enable high-quality XC potentials for machine learning and theory refinement, with future plans to extend to spin-unrestricted and periodic systems.

Abstract

Density functional theory (DFT) remains the most widely used electronic structure method. Although exact in principle, in practice, it relies on approximations to the exchange-correlation (XC) functional, which is known to be a unique functional of the electron density. Despite 50 years of active research, existing XC approximations remain far from general purpose chemical accuracy of various thermochemical and materials properties. In that light, the inverse DFT problem, of finding the exact XC potential corresponding to an accurate groundstate density, offers an insightful tool to understand the nature of the XC functional as well as aid in the development of more accurate functionals. However, solving the inverse DFT problem is fraught with several numerical challenges, such as non-uniqueness or spurious oscillations in the solution and non-convergence. We present invDFT as an open-source framework to address the outstanding challenges in inverse DFT and computed XC potentials solely from a target density. We do so by use of a systematically convergent finite-element basis and asymptotic corrections to the target density. We also employ several numerical and high-performance computing (HPC) advances that affords both efficiency and parallel scalability, on CPU-GPU hybrid architectures. We demonstrate the accuracy and scalability of invDFT using accurate full-configuration interaction (FCI) densities as well as model densities, ranging up to 100 electrons and spanning both weakly and strongly correlated molecules.

Figures (14)

• Figure 1: Illustration of three adjacent 1D quadratic finite-elements. The vertical dashed lines denote the boundary between adjacent elements. The black circles highlight the $C^0$ continuity (cusp) of the basis at the element boundary.
• Figure 2: Comparison of XC potential for H$_2$ at equilibrium bond-length with and without the $\Delta\rho$ correction to the Slater density obtained from CI calculations.
• Figure 3: Overview of the inverse DFT algorithm in invDFT
• Figure 4: Schematic for the different HPC strategies pursued in this work. (a) Cell-level operations involved in computing $\widetilde{\boldsymbol{\textbf{H}}} \boldsymbol{\textbf{X}}$. $L$ and $N_v$ denote the number of FE basis and the number of vectors in $\boldsymbol{\textbf{X}}$. The top panel shows the typical sparse cell-structure of $\widetilde{\boldsymbol{\textbf{H}}}$ matrix in an FE basis. This structure allows for decomposing $\boldsymbol{\textbf{Y}} = \widetilde{\boldsymbol{\textbf{H}}} \boldsymbol{\textbf{X}}$ into cell-level dense-dense matrix operation, followed by an assembly to get $\boldsymbol{\textbf{Y}}$. $\boldsymbol{\textbf{X}}$ has a row-major memory layout: contiguous memory for all $N_v$ vectors for a given FE basis. (b) Data transfer between two meshes with incompatible partitioning. The partitioning of mesh 1 and mesh 2 are shown in different colors (color indicate processor index). Considering the green partition of mesh 2, this regions spans parts of the green, pink, and orange partitions of mesh 1. The middle panel shows the points in that partition where we need the solution field from mesh 1. Circles denote points for which the solution is available locally on mesh 1 (from green partition of mesh 1). Crosses denote points for which the solution needs to be fetched from other processors on mesh 1 (from pink and orange partition of mesh 1).
• Figure 5: Overview of invDFT and its various APIs