Orbital Transformation

Orbital transformation (OT) minimizes the electronic energy while maintaining an orthonormal orbital subspace. It avoids a full diagonalization and is often efficient for large insulating systems. CP2K also supports complex k-point orbitals, fractional occupations, and finite-temperature free-energy minimization with OT.

Two input sections expose related but distinct algorithms:

  • &SCF%OT is a direct SCF method. The orbitals and, when requested, their rotations and auxiliary energies are optimization variables.

  • &SCF%DIAGONALIZATION%OT is an iterative eigensolver for a fixed Kohn–Sham matrix. The surrounding diagonalization SCF path assigns occupations, constructs the density, and performs density mixing.

The distinction is important for metallic calculations because only direct OT needs explicit rotation and auxiliary-energy variables.

Fixed occupations

The conventional direct OT setup is appropriate when the occupied subspace is separated from the virtual space and the occupations remain fixed:

&SCF
  &OT
    ALGORITHM STRICT
    MINIMIZER CG
    PRECONDITIONER FULL_SINGLE_INVERSE
  &END OT
&END SCF

STRICT is the default algorithm and CG is the default, generally robust minimizer. The preconditioner controls the orbital part of the optimization. Its suitability and cost depend on the system, basis, and electronic-structure method; it should therefore be tested for the target calculation rather than selected solely from a nominal hierarchy.

The same direct OT formulation supports complex, weighted k-point sets, including symmetry-reduced meshes. Converge the k-point mesh for every target property and validate experimental atomic symmetry reduction against an equivalent full mesh; see K-Points.

Fractional occupations and metals

At finite electronic temperature, direct OT minimizes a fixed-electron-number free-energy functional. It optimizes the orbital subspace together with rotations inside that subspace and auxiliary orbital energies that determine the occupations. A representative input is:

&SCF
  ADDED_MOS AUTO
  &SMEAR ON
    METHOD FERMI_DIRAC
    ELECTRONIC_TEMPERATURE 500
  &END SMEAR
  &OT
    ALGORITHM STRICT
    MINIMIZER CG
    PRECONDITIONER FULL_ALL
    ROTATION
    ENERGIES
    OCCUPATION_PRECONDITIONER
  &END OT
&END SCF

ROTATION makes occupied-column rotations variational. This is required when the energy is not invariant under such rotations, including fractional occupations. ENERGIES adds the auxiliary-energy variables needed to optimize occupations consistently with the free-energy functional. For direct OT with smearing, use both options together.

OCCUPATION_PRECONDITIONER augments the orbital metric with the coupled, fixed-electron-number occupation response. It is optional and is combined with, rather than substituted for, the independently selected PRECONDITIONER. It can improve difficult metallic response modes, but its effect on convergence remains system dependent.

Virtual-state buffer

Smearing requires enough virtual states to contain the partially occupied tail. A fixed positive ADDED_MOS value adds that many states per spin channel; ADDED_MOS -1 requests all states available in the atomic-orbital basis. ADDED_MOS AUTO selects an initial buffer for finite-temperature k-point OT and grows it when the highest available band is still appreciably occupied. It stops with a diagnostic if the atomic-orbital basis is exhausted.

AUTO removes most manual trial and error, but it cannot create states beyond the basis. A warning about occupation of the highest band therefore calls for checking the basis, electronic temperature or smearing width, and the physical electron count, not merely increasing an arbitrary integer indefinitely.

Smearing distributions

Direct finite-temperature OT supports FERMI_DIRAC, GAUSSIAN, METHFESSEL_PAXTON, and MARZARI_VANDERBILT. FERMI_DIRAC has the direct thermodynamic interpretation: the optimized quantity is the Helmholtz free energy at fixed electron number, with a common chemical potential chosen to enforce that number across spin and k-point channels. For the other distributions, CP2K uses their corresponding generalized free-energy corrections. Follow the interpretation and extrapolation guidance in &SMEAR, especially for forces and stress.

Omitting &KPOINTS uses the natural Gamma-only implementation. General k-point meshes use complex orbitals and normalized irreducible-point weights. In both cases the reported free-energy objective, rather than the uncorrected potential energy alone, determines convergence under smearing.

OT as an iterative eigensolver

To use OT inside the conventional diagonalization SCF workflow, select it below DIAGONALIZATION:

&SCF
  ADDED_MOS AUTO
  &SMEAR ON
    METHOD FERMI_DIRAC
    ELECTRONIC_TEMPERATURE 500
  &END SMEAR
  &DIAGONALIZATION ON
    ALGORITHM OT
    &OT
      ALGORITHM STRICT
      MINIMIZER CG
      PRECONDITIONER FULL_ALL
    &END OT
  &END DIAGONALIZATION
  &MIXING
    METHOD BROYDEN_MIXING
  &END MIXING
&END SCF

Here OT solves only the fixed-Hamiltonian eigenspace problem. The parent SCF driver canonicalizes that eigenspace, assigns occupations, constructs the density, and applies the selected density mixing. Consequently, DIAGONALIZATION%OT does not use ROTATION, ENERGIES, OCCUPATION_PRECONDITIONER, or NONDIAG_ENERGY; setting them does not turn them into additional eigensolver variables.

This path can be useful when robust density mixing is more important than avoiding all diagonalization-style outer iterations. It is not mathematically identical to simultaneous direct OT, even when both calculations use the same inner OT algorithm, minimizer, and preconditioner.

Practical checks

For a new metallic calculation:

  1. Converge the basis, cutoff, k-point mesh, and smearing parameter for the target property.

  2. Confirm that the first and highest available bands do not carry unintended partial occupation.

  3. Compare the free energy, electron count, chemical potential, and spin moment between MPI layouts.

  4. Compare at least one representative result with conventional diagonalization and density mixing.

  5. Treat step counts as a performance indicator, not as evidence that two methods found the same electronic state.

The free energy, entropy contribution, and extrapolated energy serve different purposes. Forces and stress under smearing are derivatives of the documented free-energy functional; use the matching quantity when validating finite differences or comparing structures.