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Qiu-White Residual-Gaussian Route

Pseudocode

  1. Build the one-dimensional distorted gausslet bases on the three Cartesian axes: Gx, Gy, Gz. Code: src/bases.jl via build_basis(...)

  2. Form the full three-dimensional gausslet product basis G = {g_i^x(x) g_j^y(y) g_k^z(z)}. Code: src/ordinary_cartesian_ida.jl

  3. Form the added three-dimensional Gaussian orbitals A = {a_I}, typically from a standard basis set. On the current atomic QW routes, these added 3D orbitals can now come from the true atomic-centered Cartesian supplement object built from the shared named-basis loader, including active s, p_x, p_y, p_z content for lmax = 1. Code: src/legacy_basis_adapter.jl, src/ordinary_qiu_white_rg.jl

  4. Define the residual Gaussians by orthogonalizing the added 3D Gaussian orbitals to the full 3D gausslet space: a'_I = (1 - P_G) a_I, then orthonormalize the surviving {a'_I} among themselves to obtain the residual-Gaussian set R = {r_A}. Code: src/ordinary_qiu_white_rg.jl

  5. Define the final orthonormal hybrid basis as B = G ∪ R. Code: src/ordinary_qiu_white_rg.jl

  6. Construct the one-particle matrices exactly by Galerkin matrix elements in the raw gausslet-plus-GTO space: S_raw, T_raw, Vnuc_raw, ... Then transform them into the final basis B using the linear transformation from the raw basis to the orthonormal basis {G, R}. Code: src/ordinary_qiu_white_rg.jl

  7. Construct the gausslet-gausslet part of the two-electron interaction using the integral-diagonal approximation (IDA). Code: src/ordinary_qiu_white_rg.jl

  8. For all interaction terms involving one or more residual Gaussians, use a residual-Gaussian approximation. Code: src/ordinary_qiu_white_rg.jl

    8a. In the nearest-center / GGT version, assign each residual Gaussian to the nearest/core gausslet and use the corresponding diagonal interaction data.

    8b. In the matched-width-Gaussian (MWG) version, compute the exact first and second moments of each residual Gaussian: <x>, <x^2>, <y>, <y^2>, <z>, <z^2>.

    8c. From those moments, define an effective separable 3D Gaussian for each residual Gaussian, with matched centers and widths along x, y, z.

    8d. Use those effective Gaussian orbitals to evaluate the diagonal RG-gausslet and RG-RG two-electron terms, keeping the interaction in the same two-index integral-diagonal-approximation (IDA) form used for the gausslet channel. This is the form used in the work of Qiu and White, although the Qiu-White paper was unfortunately ambiguous on this point.

  9. Return the final hybrid Hamiltonian in the basis B. Code: src/ordinary_qiu_white_rg.jl

References

  • Y. Qiu and S. R. White, "Hybrid gausslet/Gaussian basis sets"
  • Residual-Gaussian definition and raw-space one-body transformation: Sec. III
  • Residual-Gaussian GGT and MWG interaction approximations: Sec. IV

What This Builds

This algorithm builds the paper-faithful Qiu-White hybrid basis and Hamiltonian route:

  • a fixed 3D Cartesian gausslet product basis
  • plus orthonormalized 3D residual Gaussians
  • with exact one-particle matrices in that final basis
  • and a diagonal / two-index interaction representation in which the residual-Gaussian terms are approximated by GGT or MWG

The key object is the final basis B = G ∪ R, where R is defined in the orthogonal complement of the full 3D gausslet space.

Where the diagonal approximation enters

The diagonal approximation enters only in the two-electron terms involving residual Gaussians.

  • The gausslet-gausslet two-electron part uses the ordinary gausslet IDA.
  • Terms involving residual Gaussians are approximated into the same two-index diagonal / IDA form by GGT or MWG.
  • The one-particle matrices are not approximated by GGT or MWG.

The very approximate residual-Gaussian treatment is justified by the very low occupancies of the RGs.

Current Repo Status

The repo now has a separate paper-faithful reference path for this algorithm.

Current split:

  • src/ordinary_qiu_white_rg.jl implements the reference path described on this page:
    • full 3D gausslet product basis first
    • 3D residual Gaussians orthogonalized to that full space
    • exact contracted 1D raw-space blocks assembled into the 3D one-body and moment matrices, then transformed into the final basis
    • RG interaction terms kept in the same two-index IDA form
  • on the atomic line, the added 3D Gaussian orbitals used in this route may now come from the explicit atomic-centered Cartesian s/p supplement object built from LegacyAtomicGaussianSupplement
  • the public hybrid path in src/ordinary_hybrid.jl is still the later COMX/localized hybrid route
  • the current :residual_gaussian_nearest and :residual_gaussian_mwg treatments on HybridMappedOrdinaryBasis1D remain useful surrogate or comparison paths, but they are not the paper-faithful Qiu-White route

This page is still the source-of-truth algorithm for the Qiu-White reference implementation.

When this route emits geometric point or path datasets for inspection, the repo viz/ viewers can be used to inspect them quickly. For the lightweight entry points and usage conventions, see Visualization utilities.

Code Map

The current code is concentrated in:

  • src/ordinary_qiu_white_rg.jl
    • ordinary_cartesian_qiu_white_operators(...) public entry points for the ordinary and nested QW routes
    • _ordinary_cartesian_qiu_white_operators_atomic_shell_3d(...) active atomic-centered 3D s/p supplement path on the ordinary line
    • _ordinary_cartesian_qiu_white_operators_nested_atomic_shell_3d(...) same active 3D s/p supplement path for the nested fixed-block consumer
    • _qwrg_interaction_matrix_nearest(...) step 8a, nearest-center / GGT residual interaction
    • _qwrg_interaction_matrix_mwg(...) steps 8b-8d, MWG residual moment matching and interaction
  • src/legacy_basis_adapter.jl
    • LegacyAtomicGaussianSupplement shared named-basis loader object for the added 3D Gaussian orbitals in step 3

The older landed comments already follow this page reasonably well, especially the # Alg QW-RG step ... comments in src/ordinary_qiu_white_rg.jl.

Implementation Notes

For this algorithm family, code comments should track the pseudocode steps.

Recommended comment style:

# Alg QW-RG step 4: Define residual Gaussians by orthogonalizing 3D GTOs
# to the full 3D gausslet space.
# See docs/src/algorithms/qiu_white_residual_gaussian_route.md.

Guidelines:

  • use the same step number as in this page
  • keep the comment wording close to the pseudocode wording
  • include the docs path exactly
  • place the comment at the code block that implements that step
  • if the implementation is only partial, say so nearby in the code or docstring