MPPT Symmetry And Convergence#
MPPT is not generally invariant under unitary changes of the functions used to construct its multipartitioning. Rebuilding the approximation in a changed input basis can change energies and lift non-Kramers multiplet degeneracies. Such splitting alone does not establish an implementation defect.
Interpret spectra together with model-space completeness in S and, where the physical symmetry permits, L and J. Determinant selection and retained-state truncation can affect this completeness. Splittings can decrease or increase as selection proceeds; their trend must be measured through convergence.
Rebuilding MPPT Versus Changing Matrix Coordinates#
Changing orbitals, model functions, state-dependent partitions or the selected space and then rebuilding MPPT is a change to the approximate construction. Do not demand identical spectra or observables across these calculations without establishing invariance for that specific transformation.
A passive unitary change of basis within the same already constructed space is different. Transforming the Hamiltonian and all operators consistently must preserve spectra and physical observables, including applicable principal g values. Matrix elements and eigenvectors need not be identical. Mixing retained and discarded states changes the projection and is not this passive check.
Likewise, comparing eigensolver backends on the same matrix remains a numerical equivalence test. MPPT non-invariance does not excuse a failed residual, non-Hermitian matrix, broken data transfer or failed passive-covariance check.
Completeness In S, L And J#
S: a selected determinant list with a fixed spin projection is not by itself evidence of spin completeness. For a scalar spin-free Hamiltonian, check that the represented space supports the intended total-spin sectors; closure under S² and spin diagnostics help distinguish incomplete sectors. SOC construction also requires the relevant spin components and scalar states. With SOC, S need not remain a good quantum number.
L and J: assess completeness when the system has the corresponding rotational symmetry. L and S characterize scalar free-atom terms; J is relevant to a rotationally symmetric atom with SOC. General molecular or crystal-field Hamiltonians do not have full atomic L/J symmetry. Physical spatial-symmetry breaking can split multiplets even with a complete electronic space.
Space closure and symmetry of the approximate Hamiltonian are separate checks. Angular-momentum expectation values alone do not certify a complete multiplet; an orbital-shell label or a large determinant count does not certify it either. Assess both the selected model space and the states retained downstream.
Selection Does Not Guarantee Monotonic Restoration#
Changing selection thresholds, determinant limits, targeted roots or root weights changes the balance of correlation and symmetry components. Increasing the determinant count can decrease or increase splittings before convergence. Converged selected-CI energy alone does not certify converged MPPT2 gaps or magnetic response. Spin completion alone does not establish L/J completeness.
Record the actual selection settings, model-space and retained-state sizes, angular diagnostics, multiplet widths and observables across a convergence sequence. Attribute a splitting quantitatively only after controlled comparisons separate selection, angular completeness, input functions and partitioning from other approximations. Do not erase gaps or enlarge numerical tolerances merely to recover an expected multiplet.
Kramers Protection Has Its Own Conditions#
At zero magnetic field, an odd-electron system has Kramers degeneracy if its represented space is closed under time reversal and its approximate Hamiltonian preserves that symmetry. Verify partner completeness and time-reversal residuals independently of S/L/J completeness. Non-invariance of MPPT does not remove this conditional protection.
Distinguish splitting within a Kramers pair from energy differences between different Kramers pairs in a larger multiplet. Even-electron systems have no general Kramers guarantee, but other verified symmetries may protect degeneracy.
Acceptance And Reporting#
Identify whether a comparison holds matrices fixed or rebuilds MPPT.
State the physical symmetry, angular completeness and time-reversal conditions that justify each expected degeneracy.
Use exact angular algebra or a matched finite-space oracle only where its assumptions hold; a complete small CASCI control does not impose the same multiplet degeneracy on general MPPT2 calculations.
Report raw gaps, residuals and convergence trends. A single run cannot establish the quantitative source of a non-Kramers splitting.
Preserve sign/phase and degenerate-subspace freedom when comparing eigenvectors; compare invariant quantities for the same operator and space.
These are interpretation and validation rules, not a claim that every run automatically computes a full S/L/J completeness certificate.