Number of charge in `mulliken.txt` .ne. number of electrons in `running_scf.log`

[hongriTianqi]

Describe the bug

The number of electrons in running_scf.log is 336, while it is different whatever nspin calculation:

nspin 1: total charge 330.2
nspin 2: total charge 333.1
nspin 4: total charge 333.1

Expected behavior

They should be equal?

To Reproduce

see results in the attached file:
issue_FeO2.tar.gz

Environment

No response

Additional Context

No response

Task list for Issue attackers (only for developers)

• Verify the issue is not a duplicate.
• Describe the bug.
• Steps to reproduce.
• Expected behavior.
• Error message.
• Environment details.
• Additional context.
• Assign a priority level (low, medium, high, urgent).
• Assign the issue to a team member.
• Label the issue with relevant tags.
• Identify possible related issues.
• Create a unit test or automated test to reproduce the bug (if applicable).
• Fix the bug.
• Test the fix.
• Update documentation (if necessary).
• Close the issue and inform the reporter (if applicable).

[hongriTianqi]

This is normal as the Mulliken charge analysis does not ensure the sum rule.

[jinzx10]

This is normal as the Mulliken charge analysis does not ensure the sum rule.

How come? Given the density matrix $P(k) = C(k) O(k) C^{\dagger}(k)$ where $O(k) = \text{diag}(f_i(k))$ corresponds to the occupation,

$$\sum_{k}w_k\text{Tr}(P(k)S(k)) = \sum_{k}w_k\text{Tr}( O(k) C^{\dagger}(k)S(k)C(k) ) = \sum_k w_k\text{Tr}(O(k)) = N$$

Let

$$q_A \equiv \sum_{\mu\in{}A} \sum_{k}w_k[P(k)S(k)]_{\mu\mu}$$

we have the sum rule $\sum_A q_A = N$.

[hongriTianqi]

This issue is resolved by PR #3503.