Fix Incorrect Expressions for Hamiltonian and Overlap Matrices with Complex Wavefunctions

[rcclay]

Please review the developer documentation
on the wiki of this project that contains help and requirements.

Proposed changes

For the linear method, mainline QMCPACK builds the Umrigar/Toulouse Hamiltonian and Overlap matrices from d logpsi/dc, E_L, and dE_L/dc, where derivatives are taken w.r.t. variational parameters c. Before the code actually builds the matrices, it takes the real part of all components. For real wave functions, this is fine, but is obviously incorrect for complex wave functions, the first smoking gun being the lack of complex conjugation in the wave function overlap matrices.

For what follows notationally, refer to: https://pubs.aip.org/aip/jcp/article/126/8/084102/309065

If one rederives the Umrigar/Toulouse 2007 expressions under the assumptions of:
1.) Complex wave function
2.) Real local energy
3.) Real variational parameters (enforced by PR#4819, which implies real dE_L/dc derivatives),

We arrive at the following expressions for the Hamiltonian and Overlap matrices respectively.

This PR implements the above expressions by changing Dsaved type from Real to ValueType, adding in appropriate complex conjugation, and deferring taking the real part until we finally build the Hamiltonian and overlap matrices.

What type(s) of changes does this code introduce?

Delete the items that do not apply

• Bugfix

Does this introduce a breaking change?

• No

What systems has this change been tested on?

Intel Xeon, RHEL8, Gnu compilers.

Checklist

Update the following with a yes where the items apply. If you're unsure about any of them, don't hesitate to ask. This is
simply a reminder of what we are going to look for before merging your code.

• Yes. This PR is up to date with current the current state of 'develop'
• No. Code added or changed in the PR has been clang-formatted
• No. This PR adds tests to cover any new code, or to catch a bug that is being fixed
• No. Documentation has been added (if appropriate)

[ye-luo]

How hard it is to fix the non-batched code path?

[rcclay]

As easy as remembering to git commit.

[rcclay]

There's a nonzero chance that this will trash whatever complex optimization integration tests we have.

[ye-luo]

If no other comments come, will merge tomorrow.

[ye-luo]

Test this please

[prckent]

Thanks for working through this. Apparently we will need tougher integration tests of complex optimization :-) It would be good to put more of the references and links to analysis in the source code so that we can find the formal formulas in future.

[rcclay]

deterministic-unit_test_new_driver failure in mixed precision complex builds. I'll take a look.

[ye-luo]

deterministic-unit_test_new_driver failure in mixed precision complex builds. I'll take a look.

I have a fix for this and will PR separately.

[rcclay]

Thank you!!!

[ye-luo]

Test this please