Azam/feature/add-normalization-factor

[ABesharat]

This PR fixed normalization factor (nf) issue (symbolically and numerically) for A and S operators, making them consistent with the SeQuant manuscript.
now S has nf = 1/factorial(ket_size)
and A has nf = 1/(factorial(ket_size) * factorial(bra_size))

Affected functions:
expand_A_op
symmetrize_expr
S_maps
closed_shell_CC_spintrace_v1 and v2
column_symmetrize_ta and btas
particle_antisymmetrize_ta and btas

and their corresponding tests (test_spin.cpp, test_eval_ta.cpp, test_eval_btas.cpp)

[Copilot]

Pull request overview

This pull request fixes the normalization factor implementation for A (antisymmetrizer) and S (symmetrizer) operators to align with the SeQuant manuscript. The key changes are:

Purpose:

• S operators now have normalization factor nf = 1/factorial(ket_size)
• A operators now have normalization factor nf = 1/(factorial(bra_size) * factorial(ket_size))

Changes:

• Updated symbolic expansion functions (expand_A_op, symmetrize_expr, S_maps) to apply the new normalization factors
• Updated numerical evaluation functions (column_symmetrize_ta/btas, particle_antisymmetrize_ta/btas) to apply matching normalization
• Updated the tensor::A() function to pre-multiply by factorial to compensate for normalization in expansion
• Updated all affected tests to reflect the new expected values
• Removed the factorize_S function from the codebase
• Removed unnecessary normalization code from closed_shell_CC_spintrace_v2

Reviewed changes

Copilot reviewed 9 out of 9 changed files in this pull request and generated 3 comments.

Show a summary per file

File	Description
tests/unit/test_spin.cpp	Updated test expectations for A and S operator expansions with new normalization factors; added new test cases for symmetrization
tests/unit/test_mbpt.cpp	Updated latex output expectations for operator tensor forms; added test for S operator action
tests/unit/test_eval_ta.cpp	Updated numerical tests for antisymmetrization and symmetrization with correct normalization factors
tests/unit/test_eval_btas.cpp	Updated numerical tests for antisymmetrization with correct normalization factors
SeQuant/domain/mbpt/spin.hpp	Removed declaration of factorize_S function
SeQuant/domain/mbpt/spin.cpp	Added normalization factor application in expand_A_op, symmetrize_expr, and S_maps; removed factorize_S implementation; removed redundant normalization from closed_shell_CC_spintrace_v2
SeQuant/domain/mbpt/op.cpp	Added factorial multiplication to tensor::A() function; added spinfree basis validation to tensor::S()
SeQuant/core/eval/backends/tiledarray/result.hpp	Added normalization factor application in column_symmetrize_ta and particle_antisymmetrize_ta
SeQuant/core/eval/backends/btas/result.hpp	Added normalization factor application in column_symmetrize_btas and particle_antisymmetrize_btas

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[Copilot]

A test requirement has been commented out: REQUIRE(p6_result == expanded_A);. This suggests that the relationship between composed A operators and a single A3 operator may no longer hold with the new normalization factors. This could indicate an inconsistency in how normalization factors are applied when composing operators versus using a single higher-rank operator. Please verify that this is intentional and document why this relationship no longer holds, or fix the normalization to maintain this invariant.

Suggested change

	// REQUIRE(p6_result == expanded_A);
	// Note:
	// Under the current normalization conventions for antisymmetrizers,
	// the operator obtained by composing lower-rank A operators (used to
	// build p6_result) is not algebraically identical to the single A3
	// operator applied once (expanded_A); they can differ by an overall
	// normalization factor. Therefore we intentionally do not assert equality
	// between p6_result and expanded_A here. If the normalization of A3 is
	// later adjusted so that a product of A operators and a single A3 become
	// strictly equivalent, an additional test comparing these two expressions
	// should be added at this point.

[evaleev]

see above