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Free complement method with Gaussian expanded complements: hierarchical decontraction to mitigate the exponential wall before selection

Cong Wang

Abstract

The previous work (arXiv:2508.04635 [physics.chem-ph]) of the free complement (FC) method with Gaussian expanded complement functions adopts the Slater initial wavefunction. This may introduce an exponential complexity of the variational coefficients associated to the Gaussian complement functions with respect to the number of electrons at a given order before the overlap matrix based selection, for more than one Gaussian function used in the expansion. The present work uses decontractions via the distinct exponents introduced by the $g$ functions to avoid this scenario at low order of the FC method. The exponential number of the variational parameters is postponed to higher orders of the FC expansion.

Free complement method with Gaussian expanded complements: hierarchical decontraction to mitigate the exponential wall before selection

Abstract

The previous work (arXiv:2508.04635 [physics.chem-ph]) of the free complement (FC) method with Gaussian expanded complement functions adopts the Slater initial wavefunction. This may introduce an exponential complexity of the variational coefficients associated to the Gaussian complement functions with respect to the number of electrons at a given order before the overlap matrix based selection, for more than one Gaussian function used in the expansion. The present work uses decontractions via the distinct exponents introduced by the functions to avoid this scenario at low order of the FC method. The exponential number of the variational parameters is postponed to higher orders of the FC expansion.
Paper Structure (11 sections, 17 equations, 1 figure, 4 tables)

This paper contains 11 sections, 17 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Hierarchical decontractions of Gaussian expanded complements from a Slater initial wavefunction
  4. Single Gaussian initial wavefunction
  5. Algorithms
  6. Implementations
  7. Numerical results and discussions
  8. Hierarchical decontraction with Slater initial wavefunction
  9. Decontraction with a single Gaussian initial wavefunction
  10. Exponent distributions for the energy-based selection for the decontracted approach
  11. Summary and outlook

Figures (1)

  • Figure 1: Distributions of the exponents $\{ \alpha_1, \alpha_2, \alpha_{12} \}$ for the helium ground state with the decontracted Gaussian expanded FC method wang2025variational at $n=1$ and $n\mathrm{G}=10$ level, after the overlap wang2025variational and energy selections nakashima2020freenakatsuji2020solving. Parameters $\zeta = 1.6875$ and $\gamma_1 = \gamma_2 = 0.3125, \gamma_{12} = 0.5$ are used with Eqs. \ref{['initial_wf']} and \ref{['g_function_adopted']}, respectively nakatsuji2022accurate. Exponents before the symmetrizations from the Gaussian expansion Eq. \ref{['gauss_expansion']} of the complement functions associated to $n_{12} = 1$ in expression \ref{['cf_form']} are plotted.