feat: picture change correction for dipole integrals

[brianz98]

This PR implements picture change corrections for properties, currently only dipole integrals can be picture changed. Picture change here means consistently applying the one-electron decoupling transformation that was applied to the Dirac Hamiltonian to other one-electron operators, such as the dipole operator here. Currently only even operators (those that don't couple large and small components, i.e., all non-relativistic operators) can be picture changed.

The X2C routines are refactored into the X2CHelper class, with the hcore_x2c method now a cached_property so no recalculation is needed.

[Copilot]

Pull request overview

This PR implements picture change corrections for dipole integrals in X2C (exact two-component) relativistic calculations. Picture change corrections ensure that property operators are transformed consistently with the one-electron decoupling transformation applied to the Dirac Hamiltonian. The implementation refactors X2C routines into a X2CHelper class with hcore_x2c as a cached property to avoid redundant computations.

Key Changes:

• Refactored X2C implementation from module-level functions into an X2CHelper class with cached computation
• Added picture change correction support for dipole operators via new electric_dipole_moment and picture_change_even_operator methods
• Introduced cint_sprsp integral function for small-component dipole integrals needed for picture change corrections

Reviewed changes

Copilot reviewed 11 out of 11 changed files in this pull request and generated 6 comments.

Show a summary per file

File	Description
forte2/x2c/x2c.py	Refactored X2C code into X2CHelper class; added methods for picture-changed dipole integrals; converted hcore_x2c to cached property
forte2/x2c/__init__.py	Updated export from get_hcore_x2c function to X2CHelper class
forte2/system/system.py	Integrated X2CHelper instance and updated to use x2c_helper.hcore_x2c property
forte2/props/props.py	Added skip_picture_change parameter; implemented picture change logic for dipole calculations
forte2/integrals/integrals.py	Added cint_sprsp function for computing small-component dipole integrals
forte2/integrals/libcint_two_center.h	Added declaration for int1e_sprsp integral with 12 components
forte2/api/integrals_api.cc	Exposed cint_int1e_sprsp_sph to Python API
forte2/helpers/matrix_functions.py	Added i_sigma_dot function for constructing spin-orbit coupling matrices; improved block_diag_2x2 logic
forte2/helpers/__init__.py	Exported new i_sigma_dot function
tests/props/test_props.py	Added tests for dipole calculations with sfx2c1e and sox2c1e
tests/integrals/test_libcint.py	Added test for cint_sprsp with shifted origin

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[codecov]

Codecov Report

❌ Patch coverage is 93.98148% with 13 lines in your changes missing coverage. Please review.
✅ Project coverage is 92.55%. Comparing base (056f299) to head (5a7871c).

Files with missing lines	Patch %	Lines
forte2/x2c/x2c.py	93.33%	4 Missing and 6 partials ⚠️
forte2/props/props.py	76.92%	2 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #132      +/-   ##
==========================================
+ Coverage   92.48%   92.55%   +0.06%     
==========================================
  Files         141      141              
  Lines       11647    11735      +88     
  Branches      943      949       +6     
==========================================
+ Hits        10772    10861      +89     
+ Misses        659      658       -1     
  Partials      216      216              

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