Initial Guesses for Multicomponent Mean-Field Methods: Assessment and New Developments

TL;DR

This work addresses SCF-convergence challenges in mean-field nuclear–electronic orbital methods by introducing two HO-derived protonic initial guesses, HOa (anisotropic) and HOi (isotropic), and benchmarking them against existing approaches such as SAD, core, 1s, and SND. It develops a full theoretical framework including partial Hessian vibrational analysis, basis-set projections, and analytic overlap formulas, enabling HOi to be efficiently integrated via $ extbf{P}_2 = extbf{U} extbf{P}_1 extbf{U}^ op$ with $ extbf{U} = extbf{S}_{22}^{-1} extbf{S}_{21}$ and overlaps $ raket{oldsymbol{ ho} | oldsymbol{ ho}_{lmn}}$ calculated analytically. Empirical results show HOi often outperforms existing guesses in NEO-DFT with simultaneous SCF, while HOa is generally impractical; Hessian-based approximations (e.g., GFN2-xTB) yield reliable exponents with $f$-scores above 0.90, making HOi cost-effective and robust. The findings provide a practical route to improved convergence in mean-field NEO computations and point to extensions to heavier nuclei and transition-state contexts.

Abstract

The convergence of self-consistent field equations in mean-field nuclear-electronic orbital methods strongly depends on the choice of initial guesses for quantum nuclei. Although several such guesses have been proposed in the literature, a systematic comparison of their performance as well as attempts of constructing novel approximations based on model tasks of quantum mechanics were not reported to date. In this work, we address both issues by introducing novel nuclear initial guesses derived from the analytical solutions of the three-dimensional quantum harmonic oscillator and benchmarking them against existing approaches. We demonstrate that the isotropic variant of our guess outperforms existing approximations in nuclear-electronic orbital density functional theory calculations employing a simultaneous self-consistent field convergence protocol. Although our guess requires the computation of partial Hessians, we demonstrate that these can be evaluated with low-cost methods without affecting the accuracy of resulting protonic density matrices. Our results demonstrate that the proposed guess is robust and efficient and could provide a route to improved convergence in mean-field nuclear-electronic orbital computations.

Figures (3)

• Figure 1: $f$-values computed with NEO-HF and NEO-DFT using the Cartesian PB4-F2 basis set and four different initial guesses for molecules containing a single quantum proton. Bars correspond to $f$-values averaged over all molecules in the set containing a single quantum proton, whereas range bars indicate minimal and maximal $f$-values.
• Figure 2: FHF$^-$ protonic density plotted along the F--H--F direction. Results generated with the NEO-HF approach are shown on the left, whereas those from NEO-DFT are on the right. Cartesian PB4-F2 basis set was employed in both cases.
• Figure 3: Numbers of SCF iterations required to reach convergence in NEO-HF (top) and NEO-DFT (bottom) computations employing the simultaneous convergence protocol and Cartesian PB4 basis sets. Bars correspond to numbers of SCF iterations averaged over all molecules in the set, whereas range bars indicate minimal and maximal values.