7.4 PBE0 with ADMM: separate model change from acceleration

These are unexecuted teaching inputs and starting models. Original diagrams are schematics, not calculated results. Validate version-specific syntax, licensed or authorized data, numerical convergence and the scientific model before using this workflow.

7.4.1 Model, units and provenance

Keep basis/potential files and executable versions traceable. Grid controls use Ry; common energy/force outputs use hartree and hartree/bohr. Read the unit in each output heading.

Shared inputs, conventions and evidence

Original schematic: PBE0 with ADMM: separate model change from acceleration. No numerical results are claimed.

7.4.2 Unexecuted inputs and explicit deltas

Use the accompanying instructions to identify the parent calculation and placement of every delta; a snippet is not automatically a standalone input. Preserve all blank-line and file-provenance requirements.

7.4.2.1 Input block 1

&XC
  &XC_FUNCTIONAL
    &PBE
      SCALE_X 0.75
      SCALE_C 1.00
    &END PBE
  &END XC_FUNCTIONAL
  &HF
    FRACTION 0.25
    &INTERACTION_POTENTIAL
      POTENTIAL_TYPE COULOMB
    &END INTERACTION_POTENTIAL
    &SCREENING
      EPS_SCHWARZ 1.0E-7
    &END SCREENING
  &END HF
&END XC

7.4.2.2 Input block 2

! Inside FORCE_EVAL/DFT
BASIS_SET_FILE_NAME BASIS_ADMM
&AUXILIARY_DENSITY_MATRIX_METHOD
  METHOD BASIS_PROJECTION
  ADMM_PURIFICATION_METHOD MO_DIAG
  EXCH_CORRECTION_FUNC PBEX
&END AUXILIARY_DENSITY_MATRIX_METHOD

! Inside each existing H and O KIND
BASIS_SET AUX_FIT cFIT3

7.4.3 Worked investigation

Question and intuition. Moving from PBE to PBE0 changes the electronic model. Introducing ADMM then approximates the costliest part of that hybrid calculation. These are two different comparisons. PBE0 combines 25% Hartree–Fock exchange, 75% PBE exchange, and full PBE correlation. ADMM projects the density matrix to a smaller auxiliary basis for exchange and applies a density-functional correction. A fast ADMM result should be judged against a suitably converged hybrid reference, not merely against the original PBE energy. This case uses an isolated water dimer with fixed geometry, so interaction-energy changes can be examined before geometry relaxation obscures the comparison.

Prerequisites. Have monomer and dimer coordinates with an explicit definition of frozen versus relaxed monomers. Converge primary basis, grid, isolated electrostatics, and SCF. Use data files shipped with the selected CP2K release; confirm that every element has the specified primary and auxiliary basis entries. A small system makes an unapproximated HFX comparison feasible in principle, although cost depends strongly on basis contraction. The UZH 2017 hybrid tutorial introduces this cost/accuracy distinction and condensed-phase truncation issues.

Original input deltas: valid section syntax. Replace FORCE_EVAL/DFT/XC with the following explicit PBE0 construction. It uses the untruncated Coulomb exchange operator for an isolated molecule. It must not be copied unchanged as a periodic condensed-phase hybrid recipe.

See input block 1 above.

For the ADMM branch only, append the auxiliary data file and section inside FORCE_EVAL/DFT, and add the AUX_FIT line to each existing H and O KIND in FORCE_EVAL/SUBSYS. Keep its orbital BASIS_SET and POTENTIAL lines.

See input block 2 above.

Fractions and exchange scaling are dimensionless. EPS_SCHWARZ is a numerical screening threshold, not a physical mixing coefficient. The 2025.2 XC definition, exchange operator, and ADMM reference define these controls. MO_DIAG is the documented purification choice for MD; this introductory branch assumes compatible closed-shell OT settings rather than arbitrary smeared occupations.

Numbered procedure. 1. Prepare three branches with identical geometry and primary basis: PBE, PBE0 without ADMM, and PBE0 with ADMM. Give each a separate directory and clear method label. 2. Obtain compatible starting orbitals; reconverge under each branch's Hamiltonian. Restarting from PBE does not mean the final result remains PBE. 3. Compute dimer and matching monomer energies consistently. Form Eint = Edimer − Emonomer A − Emonomer B, documenting ghost-basis/counterpoise treatment if used. 4. Compare PBE0−PBE as a method change, and PBE0+ADMM−PBE0 as an approximation error for the chosen observable. Also compare forces at selected distorted geometries if MD is planned. 5. Tighten EPS_SCHWARZ and SCF, then test a better auxiliary basis available for both elements. Keep these tests separate from primary-basis improvements. 6. Record wall time, SCF iteration count, memory, and identical hardware allocation. A smaller calculation can show little acceleration because projection and setup costs remain.

Interpretation and pitfalls. Total-energy offsets between ADMM variants can be misleading; benchmark consistent energy differences and forces. Do not combine a dimer from one correction scheme with monomers from another. Basis-set superposition error remains a distinct issue. ADMM does not add missing dispersion, and PBE0 is not a universal cure for strong correlation. For periodic HFX, exchange range, cell geometry, and operator treatment require their own convergence study; a radius shorter than all chemically relevant exchange contributions changes the calculation. Do not vary HF/FRACTION alone while leaving the semilocal exchange scale inconsistent with the intended hybrid.

Exercise. Design an error budget for a hydrogen-bond energy: primary basis, grid, SCF, exchange screening, auxiliary fit, and geometry. Choose an application-specific tolerance before looking at results. Explain why a small absolute total-energy difference does not prove accurate forces, and why comparing two differently relaxed geometries cannot isolate ADMM error.