Quadratic pseudospectrum

FORCE_EVAL / PROPERTIES / QUADRATIC_PSEUDOSPECTRUM finds states approximately localized in both position and energy from a converged GPW/GAPW calculation. It reports no topological index and requires no additional library.

For Hermitian covariant AO observables \(A_j\), queries \(\lambda_j\), positive weights \(w_j\) and overlap \(S\), the generalized eigenproblem is

\[ Q c = q S c,\qquad Q=\sum_j w_j^2(A_j-\lambda_j S)S^{-1}(A_j-\lambda_j S). \]

The lowest eigenpairs describe joint approximate states. The output includes \(\sqrt{q}\), separate energy/position residuals and expectation values. States satisfy \(c^\dagger S c=1\). Residuals concern projected finite-basis operators, not exact continuum variances.

&PROPERTIES
  &QUADRATIC_PSEUDOSPECTRUM
    FORMULATION FINITE
    POSITION [bohr] 0 0 0
    ENERGY [hartree] 0.0
    KAPPA [hartree*bohr^-1] 0.01
    NSTATES 2
    &STATES
    &END
  &END
&END

Observables

  • FINITE uses analytic Cartesian moments. It requires matching CELL/POISSON PERIODIC NONE without explicit k-points.

  • PERIODIC uses sine/cosine coordinates on the full MP_GRID torus at the frozen SCF potential. Only periodic directions are localized. Nonperiodic mesh dimensions must be one.

  • TRANSLATION uses analytic translated-Gaussian overlaps for \(U(a)\psi(r)=\psi(r+a)\). Specify repeated TRANSLATION_VECTOR and WAVE_VECTOR queries and positive TRANSLATION_SCALE instead of KAPPA. The target phase is \(\exp(i k\cdot a)\) without an additional \(2\pi\). This path includes the full continuum translation norm and reports leakage outside the AO span, rather than treating compressed AO translation matrices as unitary.

The periodic paths construct a complete analysis torus, independently of SCF symmetry reduction. SOC T uses post-SCF GTH spin-orbit coupling after a restricted calculation with suitable pseudopotentials.

Output and limits

SOLVER DENSE uses a metric-aware dense reference calculation. MAX_AO bounds the complete scalar AO space, including periodic cells. EPS_METRIC rejects ill-conditioned overlap matrices, and EPS_EIGEN bounds eigenpair residuals. NSTATES selects the number of states.

The STATES print section writes complex AO coefficients. Spin-up precedes spin-down for SOC, with cell-major ordering inside each spin block on a torus. Individual vectors in a degenerate subspace are gauge dependent. Compare subspaces, residuals and observables instead of individual coefficients.

Choose energy, position and scales for the physical question and check basis and size dependence. Broader examples are available in TopologicalCP2K.