Fix: add DeePKS V_delta commutator -i[V_delta, r] to rt-TDDFT velocity-gauge current

[xuan112358]

In velocity gauge (td_stype=1), out_current=1 builds the current operator
(current_term) only from the kinetic and nonlocal-pseudopotential terms, so the
DeePKS non-local potential V_delta contribution -i[V_delta, r] was missing. Since
V_delta couples different atoms, this commutator is nonzero, and its omission
produced a spurious DC component in the reported DeePKS current.

• Add TDDeePKS operator (td_deepks_lcao.{h,cpp}): a lcao_tddft_periodic sub-chain
  node inserted between TDEkinetic and TDNonlocal that accumulates -i[V_delta, r]
  into current_term, mirroring the nonlocal term's phase-free commutator. It only
  writes the current term and forwards hR_tmp; it does not modify hR.
• Add get_psi_r_alpha / init_alpha to cal_r_overlap_R for the r-weighted projection
  <phi|r|alpha> between orbitals and DeePKS projectors, reusing the existing
  Center2_Orb machinery. Add an optional lmax_extra argument to initialize_orb_table
  (default 0 keeps existing callers byte-identical) so the tables cover the higher
  descriptor Lmax.
• Register the operator in hamilt_lcao.cpp and add the new source to the build lists.

Currently supports nspin=1. Validated: <phi|alpha> matches phialpha to ~2e-5;
disabling the term reproduces the pre-fix current to ~1e-17; the DC fraction of the
DeePKS current drops from +0.5 to ~0 (PBE/HSE level).

[Copilot]

Pull request overview

This PR fixes rt-TDDFT velocity-gauge DeePKS current reporting by adding the missing DeePKS nonlocal potential commutator contribution into the velocity-operator current term, and extends the two-center integral machinery to support the needed r-weighted orbital–projector projections.

Changes:

• Introduces a new LCAO TD operator (TDDeePKS) that accumulates the DeePKS commutator contribution into TD_info::td_vel_op->current_term, placed between TDEkinetic and TDNonlocal.
• Extends cal_r_overlap_R with init_alpha() and get_psi_r_alpha() plus an initialize_orb_table(..., lmax_extra) option to cover higher-L DeePKS descriptor projectors.
• Registers/builds the new operator by wiring it into hamilt_lcao.cpp and adding sources to CMake/Makefile object lists.

Reviewed changes

Copilot reviewed 8 out of 8 changed files in this pull request and generated 3 comments.

Show a summary per file

File	Description
source/source_lcao/module_operator_lcao/td_deepks_lcao.h	Declares the new TD DeePKS operator that contributes the missing commutator term to the velocity-gauge current.
source/source_lcao/module_operator_lcao/td_deepks_lcao.cpp	Implements commutator accumulation logic and forwards the shared TD real-space buffer along the TD operator sub-chain.
source/source_lcao/module_operator_lcao/CMakeLists.txt	Adds td_deepks_lcao.cpp to the operator library build.
source/source_lcao/hamilt_lcao.cpp	Registers/inserts the new TD DeePKS operator between TDEkinetic and TDNonlocal when DeePKS is enabled.
source/source_lcao/CMakeLists.txt	Adds td_deepks_lcao.cpp to the broader LCAO build list.
source/source_io/module_hs/cal_r_overlap_R.h	Adds alpha/projector initialization + `<phi
source/source_io/module_hs/cal_r_overlap_R.cpp	Implements alpha/projector tables and get_psi_r_alpha(), and updates orb table sizing logic.
source/Makefile.Objects	Adds td_deepks_lcao.o to object build lists.

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