Grassmann extrapolation via direct inversion in the iterative subspace

We present a Grassmann extrapolation method (G-Ext) that combines the mathematical framework of the Grassmann manifold with the direct inversion in the iterative subspace (DIIS) technique to accurately and efficiently extrapolate density matrices in electronic structure calculations. By overcoming the challenges of direct extrapolation on the Grassmann manifold, this indirect G-Ext-DIIS approach successfully preserves the geometric structure and physical constraints of the density matrices. Unlike Tikhonov regularized G-Ext, G-Ext-DIIS requires no tuning of regularization parameters. Its DIIS subspace is compact, numerically stable, and independent of descriptor dimensionality, system size, and basis set, ensuring both robustness and computational efficiency. We evaluate G-Ext-DIIS using alanine dipeptide and its zwitterionic form along ϕ and ψ torsional scans, employing Coulomb, overlap, and core Hamiltonian matrix descriptors with the diffuse 6-311++G(d,p) and aug-cc-pVTZ basis sets. When using overlap or core Hamiltonian descriptors, G-Ext-DIIS achieves sub-millihartree accuracy across angular extrapolation ranges that exceed typical geometry optimization step sizes. This indicates its potential for generating high quality initial density matrices in each optimization cycle. Compared to direct extrapolation methods with or without McWeeny purification, as well as the Löwdin extrapolation from nearby geometries, G-Ext-DIIS demonstrates superior accuracy, variational consistency, and reliability across basis sets. We also explore Fock matrix extrapolation using the same DIIS coefficients, although this strategy proves less reliable for distant geometries. Overall, G-Ext-DIIS offers a robust, efficient, and transferable framework for constructing accurate density matrices, with promising applications in geometry optimization and ab initio molecular dynamics simulations.