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Basis Sets on Radial Grids

Basis set files in CHAMP define atomic orbitals on numerical radial grids. These files store the radial parts of the basis functions evaluated on a discrete grid, which CHAMP uses to construct molecular orbitals for the trial wavefunction. The radial grid representation provides high accuracy for the orbital representation while maintaining computational efficiency.

Overview

CHAMP represents atomic orbitals by evaluating their radial parts on a logarithmic grid and combining them with angular functions (spherical harmonics). This approach:

  • Provides accurate representation of orbital shapes, including near-nucleus behavior
  • Enables efficient calculation of orbital values and derivatives
  • Supports various basis set types (Gaussian, Slater, numerical)
  • Works seamlessly with pseudopotentials

Key Concepts

  • Radial shells: Groups of basis functions with the same angular momentum quantum number
  • Grid points: Discrete radial distances where basis functions are evaluated
  • Logarithmic spacing: Grid points distributed to capture both short and long-range behavior

File Format

Basis set files have a fixed format and are typically stored in the pool/ directory. Files generated from the trex2champ converter can be used directly.

Naming Convention

Basis files follow the naming pattern:

<basis_name>.basis.<element_symbol>

Examples: - BFD-T.basis.C - BFD-T basis for carbon - ccpVTZ.basis.Si - cc-pVTZ basis for silicon - VDZ.basis.O - VDZ basis for oxygen

File Structure

9 3 2000 1.003000 20.000000 0
 0.000000000000e+00  5.469976184517e-01  2.376319920758e+00  5.557936498748e-01  3.412818210005e+00  2.206803021951e-01  8.610719484857e-01  3.738901952004e-01  3.289926074834e+00  1.106692909826e+00
 1.508957441883e-04  5.469976454488e-01  2.376319870895e+00  5.557936481942e-01  3.412817957941e+00  2.206803015581e-01  8.610719410992e-01  3.738901923954e-01  3.289925989316e+00  1.106692890335e+00
 3.018821756935e-04  5.469976724431e-01  2.376319821040e+00  5.557936465134e-01  3.412817705877e+00  2.206803009210e-01  8.610719337127e-01  3.738901895905e-01  3.289925903799e+00  1.106692870844e+00
 ... (additional grid points)

Formula

The radial grid \(r_i\) is generated based on the gridtype parameter. The radial distance \(r_i\) for the \(i\)-th grid point (where \(i = 0, \dots, N-1\)) is calculated as follows:

Type 3 (Default):

\[r_i = r_0 \times (\alpha^i - 1)\]

Type 2:

\[r_i = r_0 \times \alpha^i\]

Type 1:

\[r_i = r_0 + i \times \alpha\]

where the standard parameters are:

  • \(N = 2000\) (gridpoints)
  • \(\alpha = 1.003\) (gridarg)
  • \(r_0 = 20.0\) bohr (gridr0)

Format Specification

Line 1: Header information

9 3 2000 1.003000 20.000000 0
  • 9 = Number of radial shells (basis functions with different angular momenta)
  • 3 = Grid type
  • 2000 = Number of radial grid points
  • 1.003000 = Grid argument
  • 20.000000 = Grid radius (in bohr)
  • 0 = Flag (default 0)

Lines 2+: Radial function values

Each subsequent line contains values for all radial shells at a specific grid point: 

r_i  phi_1(r_i)  phi_2(r_i)  phi_3(r_i)  ...  phi_N(r_i)

where: - r_i = Radial distance for this grid point (in bohr) - phi_j(r_i) = Value of the j-th radial shell at distance r_i - Number of values per line = 1 + number of shells

The grid points are typically logarithmically spaced to provide high resolution near the nucleus while covering large distances efficiently.

Examples

Example 1: Carbon with BFD-T Basis

Basis file (BFD-T.basis.C): 

9 3 2000 1.003000 20.000000 0
 0.000000000000e+00  5.469976184517e-01  2.376319920758e+00  5.557936498748e-01  3.412818210005e+00  2.206803021951e-01  8.610719484857e-01  3.738901952004e-01  3.289926074834e+00  1.106692909826e+00
 1.508957441883e-04  5.469976454488e-01  2.376319870895e+00  5.557936481942e-01  3.412817957941e+00  2.206803015581e-01  8.610719410992e-01  3.738901923954e-01  3.289925989316e+00  1.106692890335e+00
 3.018821756935e-04  5.469976724431e-01  2.376319821040e+00  5.557936465134e-01  3.412817705877e+00  2.206803009210e-01  8.610719337127e-01  3.738901895905e-01  3.289925903799e+00  1.106692870844e+00
 ... (1997 more lines)

Explanation:

  • 9 radial shells for carbon
  • 2000 grid points from ~0 to 20 bohr
  • Logarithmic spacing captures both core and valence regions
  • Compatible with BFD pseudopotential

Example 2: Multi-element System (Water)

For a water molecule (H₂O), you need two basis files:

Directory structure: 

pool/
  ├── ccpVTZ.basis.O
  └── ccpVTZ.basis.H

Oxygen basis (ccpVTZ.basis.O): 

14 3 2000 1.003000 20.000000 0
 0.000000000000e+00  1.234567890123e+00  ...  (14 shell values)
 1.508957441883e-04  1.234567890123e+00  ...  (14 shell values)
 ... (remaining grid points)

Hydrogen basis (ccpVTZ.basis.H): 

5 3 1500 1.003000 15.000000 0
 0.000000000000e+00  9.876543210987e-01  ...  (5 shell values)
 2.011929889229e-04  9.876543210987e-01  ...  (5 shell values)
 ... (remaining grid points)

Explanation:

  • Oxygen has 14 shells (larger basis), hydrogen has 5 shells
  • Different grid sizes appropriate for each element
  • Both use logarithmic grids
  • CHAMP loads both files automatically based on atom types

Example 3: Pseudopotential Compatibility

When using pseudopotentials, ensure basis sets match:

BFD pseudopotential system: 

pool/
  ├── BFD-T.basis.C      # Valence basis for carbon
  ├── BFD-T.basis.H      # All-electron basis for hydrogen
  └── BFD.gauss_ecp.dat.C  # Carbon pseudopotential

The basis must be consistent with the pseudopotential (same valence space).

Loading Basis Sets in CHAMP Input

Specify the basis set name in the %module general section:

%module general
    title  'VMC calculation for molecule'
    pool   './pool/'
    basis  'BFD-T'        # Basis set prefix
%endmodule

load trexio   $pool/molecule.hdf5
load jastrow  $pool/jastrow.jas

%module electrons
    nup    5
    nelec  10
%endmodule

CHAMP will automatically load: - BFD-T.basis.C for carbon atoms - BFD-T.basis.H for hydrogen atoms - etc., based on atom types in geometry

Alternative: Explicit Basis Loading

You can also load basis files explicitly:

load basis_num_info  $pool/BFD-T.basis.C   # For carbon
load basis_num_info  $pool/BFD-T.basis.H   # For hydrogen

This is useful when: - Using non-standard naming conventions - Debugging basis set issues - Mixing different basis sets for different atoms

Generating Basis Files

From TREXIO Files

Trexio files contain basis sets in a format that is compatible with CHAMP. There is no need to generate basis files from TREXIO files.

From Quantum Chemistry Calculations

The trex2champ converter automatically generates basis files:

#!/bin/bash
python trex2champ.py    --trex      benzene.hdf5 \
            --basis_prefix  "cc-VDZ" \
            --geom \
            --basis

This creates: - Basis files for each element: <basis_name>.basis.<element> - Properly formatted for CHAMP - Consistent with basis set used in TREXIO file

Best Practices

Basis Set Selection

  1. Match pseudopotential: Use basis sets designed for your ECP
  2. Balance accuracy/cost: Larger basis → better accuracy but slower

File Management

  • Keep all basis files in pool/ directory
  • Verify basis files match geometry atom types

Common Issues

Missing basis file: 

ERROR: Cannot find basis file: BFD-T.basis.Si
- Solution: Ensure file exists in pool/ directory with exact naming

Grid mismatch: 

WARNING: Orbital tail not converged at r_max
- Solution: Increase maximum radius or number of grid points

Angular momentum issues: 

ERROR: Insufficient angular momentum in basis
- Solution: Use higher angular momentum basis or check molecular orbital requirements

Getting Help

  • Verify basis file format matches specification exactly
  • Check that grid extends sufficiently for all orbitals
  • Ensure one basis file exists for each unique atom type
  • Consult Preparation Guide for complete workflow

Required Files

Basis set files must be present in the pool/ directory for all unique atom types in your system. CHAMP cannot proceed without the appropriate basis files.