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IPR — Inverse Participation Ratio

Compute the Inverse Participation Ratio (IPR) from a VASP PROCAR file to quantify the spatial localization of electronic wavefunctions.

valyte ipr

What is IPR?

For a wavefunction at band \(n\) and k-point \(\mathbf{k}\), the atomic IPR is defined as:

\[\text{IPR}_{n\mathbf{k}} = \sum_I \left( \frac{p_I}{\sum_{I'} p_{I'}} \right)^2\]

where \(p_I\) is the total orbital projection weight on atom \(I\).

IPR value N_eff = 1/IPR Interpretation
≈ 1 ≈ 1 Fully localized on a single atom
≈ 1/N ≈ N Uniformly delocalized across all N atoms
intermediate intermediate Partially localized (e.g. defect state)

When to use IPR

IPR is particularly useful for identifying localized defect states in supercell calculations — for example, vacancy levels, impurity states, or polaron wavefunctions. Delocalized bulk-like bands will have a small IPR and large N_eff; localized in-gap states will have IPR approaching 1.

Interactive workflow

Running valyte ipr starts an interactive session:

  1. Reads PROCAR from the current directory
  2. Prints system info — number of k-points, bands, and atoms
  3. Prompts for band indices to analyze
  4. Optionally prints per-k-point IPR values
  5. Saves results to ipr_procar.dat

Band index input formats

5           → single band
5 6 7       → multiple bands
5-7         → range (inclusive)
5 8-10 13   → mixed

Output file

Results are written to ipr_procar.dat with four columns:

Column Description
Band Band index (1-based)
Energy Average energy across k-points (eV)
IPR k-averaged Inverse Participation Ratio
N_eff Effective number of atoms = 1/IPR

Example:

# Band  Energy(eV)   IPR   N_eff
    5    -0.312400  0.0023   434.78
    6     0.018200  0.9812     1.02
    7     0.124500  0.4901     2.04

In this example, band 6 (IPR ≈ 0.98, N_eff ≈ 1) is a strongly localized defect state sitting near the Fermi level.

Requirements

  • A PROCAR file in the current directory, generated by VASP with LORBIT = 11 (or ≥ 10)
  • The PROCAR must include atom-projected orbital weights (not just total charge)

Related commands

  • valyte band — Visualize the band structure where defect states appear
  • valyte effmass — Extract effective masses at band extrema