Davidson works when doubles folding in ADC(2)

[TianLin-228]

Implement a modified Davidson procedure for doubles-folding. It solves pseudo-eigenvalue problem ω_i u_i = A(ω_i) u_i, where A(ω_i) is an effective ADC matrix which depends on the eigenvalue ω_i. The modified procedure is based on the implementation stated in the paper ( J. Chem. Phys. 158, 124121 (2023)).

The algorithm consists of state-specific macro and micro iterations and further DIIS acceleration. Micro-iterations use Davidson iterations to diagonalize the effective matrix A_eff(ω_i) with the fixed eigenvalue ω_i, giving a new eigenvalue ω^'_i. After convergence, a new macro-iteration begins with the effective matrix A_eff(ω^'_i) . This process continues until the eigenvectors meet a threshold (default=1e-3), after which DIIS further converges the individual roots.

NB: This procedure only works for single excitations.

Changes:

1. In file “davidson.py”:

• New functions named “davidson_folded_DIIS“ and “eigsh_folded” are added, which implements the main procedure of the modified Davidson method. It consists of loops for macro iterations, in which the davidson_iterations are performed as micro iterations, and loops for DIIS iterations.

• Changes in function “davidson_iterations” :

  • For the folding case, when it breaks at convergence or otherwise, the eigenvectors with the number of n_block are calculated by the rvecs and guesses, and the subspace_vectors is updated for the next macro iteration;
  • A different convergence criteria for the folding case as following:
    ω⁰_i - ω¹_i > ω^n-1_i - ωⁿ_i , where ω⁰_i is the initial guess of ω_i and ωⁿ_i is the Ritz value of the current micro iteration.

• Add new Class DIIS to implement diis acceleration.

• There are two new Classes: Class FoldedDavidsonState inherits from Class DavidsonState and contains more properties about the DavidsonState in doubles-folding case; Class DIIS is used when performing extrapolation on the corrected vector.

• Add new properties for Class DavidsonState: “folded” is the flag to distinguish normal or modified davidson procedures when running “davidson_iterations”, and “DIIS_iter” sums the total number of performed DIIS iterations for all required states.

• In the function “default_print”, there are new identifiers for the folding case.

2. In the file “AdcMatrix.py”, the matrix-vector product of the folded ADC matrix with singles component is defined in the new Class AdcMatrixFolded.

3. Workflow - two options of guesses vector in doubles-folding case:
   guesses_fold = “adc1”, which takes adc1 results (excitation_vector and excitation_energy_uncorrected) as initial guess vectors and guess eigenvalues.
   Or obtaining guess vectors by inspecting the diagonal matrix elements as default.

Testing:

test_davidson_folded.py: testing davidson_folded_DIIS in two cases of given guesses and taking guesses as adc1 results.
test_workflow.py: I added two tests for new parameters “fold”, “guesses_fold” and “omegas”.