Update a new compute angular rdf

[Binxu-Stack]

Add Angular-dependent Radial Distribution Function (Angular RDF)

This PR adds functionality to calculate angular-dependent radial distribution functions (Angular RDF), which extends the traditional RDF by incorporating angular information.

Key Features

1. Calculates both radial and angular distributions simultaneously
2. Supports both:
   • Total Angular RDF (all atom pairs)
   • Partial Angular RDF (specific atom type pairs)

Implementation Details

• Added new class AngularRDF for computing angular-dependent RDF
• Uses both radial bins (r) and angular bins (θ) for 2D histogram
• θ range: [-π, π] using atan2 for angle calculation
• Proper normalization considering:
  • Shell volume: 4π/3(r_outer³ - r_inner³)
  • Angular sector: Δθ/(2π)
  • Number density of each atom type
  • Double counting correction for same atom types

Usage

The Angular RDF can be computed using the following command in the input script:

compute_angular_rdf Rcut Nbins Ntheta interval [atom type1 type2] ...

Parameters:

• Rcut: Radial cutoff distance
• Nbins: Number of radial bins
• Ntheta: Number of angular bins
• interval: Sampling interval
• Optional atom type pairs for partial RDF calculation

Output

Results are written to angular_rdf.out with columns:

1. Radial distance (r)
2. Angle (θ)
3. Total RDF
4. Partial RDFs for each atom type pair (if needed)

Validation

• Tested with simple atomic configurations
• Verified normalization against analytical expectations
• Checked consistency with traditional RDF when integrating over angles

[brucefan1983]

Could you change Chinese comment within the code to English?

[Binxu-Stack]

I think it is ready.

• Chinese comment to english
• doc compiling check
• code compiling check and test.