support for printing soc matrix in LR-TDDFT calculation

[liusss324]

Background

Dear,
Does LR-TDDFT allow printing the SOC matrix? I have noticed that CP2K can output the SOC matrix in LR-TDDFT calculations. This matrix serves as a useful tool for analyzing singlet and triplet states, in particular the oscillator strengths of spin-flip transitions.

Describe the solution you'd like

https://manual.cp2k.org/trunk/CP2K_INPUT/FORCE_EVAL/PROPERTIES/TDDFPT/PRINT/SOC_PRINT.html
https://manual.cp2k.org/trunk/CP2K_INPUT/FORCE_EVAL/PROPERTIES/TDDFPT/SOC.html
Like CP2K, provide a tag to control whether the SOC matrix is printed. Here, SOC is treated as a perturbation.
Including this feature would allow a comparison between the perturbative SOC treatment and the full SOC effects in LR-TDDFT calculations.

Task list only for developers

• Notice possible changes of behavior
• Explain the changes of codes in core modules of ESolver, HSolver, ElecState, Hamilt, Operator or Psi

Notice Possible Changes of Behavior (Reminder only for developers)

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Notice any changes of core modules (Reminder only for developers)

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Notice Possible Changes of Core Modules (Reminder only for developers)

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Additional Context

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Task list for Issue attackers (only for developers)

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• Research on the existing solutions and relevant research articles/resources.
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• Create a design document outlining the proposed solution and implementation details.
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• Develop the feature following the agreed design.
• Write unit tests and integration tests for the feature.
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