HFX with ADMM

The auxiliary density matrix method (ADMM) reduces the cost of Hartree-Fock exchange in hybrid DFT calculations by projecting the density matrix from the primary orbital basis onto a smaller auxiliary basis. CP2K evaluates exact exchange in the auxiliary basis and adds a correction term for the difference between the primary and auxiliary exchange descriptions.

ADMM is most useful when exact exchange is the bottleneck, especially with larger or more diffuse Gaussian basis sets. It is commonly used for hybrid DFT, and the ADMM2 variant is also supported by several post-SCF methods that reuse exact-exchange machinery.

Basic Setup

An ADMM calculation needs three pieces of input:

  • a hybrid functional or another setup that evaluates Hartree-Fock exchange,

  • an auxiliary basis set for each atomic kind, specified with BASIS_SET AUX_FIT,

  • an AUXILIARY_DENSITY_MATRIX_METHOD section that selects the ADMM variant and correction functional.

For example:

&DFT
  BASIS_SET_FILE_NAME BASIS_MOLOPT_UZH
  BASIS_SET_FILE_NAME BASIS_ADMM_UZH
  POTENTIAL_FILE_NAME POTENTIAL_UZH
  &AUXILIARY_DENSITY_MATRIX_METHOD
    ADMM_TYPE ADMMS
    EXCH_CORRECTION_FUNC PBEX
  &END AUXILIARY_DENSITY_MATRIX_METHOD
  &XC
    &XC_FUNCTIONAL PBE
    &END XC_FUNCTIONAL
    &HF
      FRACTION 0.25
    &END HF
  &END XC
&END DFT

&SUBSYS
  &KIND O
    BASIS_SET ccGRB-D-q6
    BASIS_SET AUX_FIT admm-dz-q6
    POTENTIAL GTH-HYB-q6
  &END KIND
&END SUBSYS

Choosing the Auxiliary Basis

The auxiliary basis should be chosen for the primary basis family and for the intended accuracy. For MOLOPT-style calculations, the BASIS_ADMM_MOLOPT family provides compact auxiliary bases. The newer UZH basis-set collection includes BASIS_ADMM_UZH and related basis files for correlation consistent setups. All-electron calculations can use all-electron auxiliary basis sets when available.

The auxiliary basis is part of the approximation. A too small auxiliary basis can make the exchange correction large and reduce accuracy; a too large one gives back less speedup. For production work, test at least one larger auxiliary basis or compare against a smaller reference system without ADMM.

Choosing the ADMM Variant

ADMM_TYPE is a shortcut that sets the projection, purification, and scaling options consistently. ADMM1 and ADMM2 are the original variants, while ADMMS, ADMMP, and ADMMQ use additional models introduced later. ADMM2 is often the most broadly supported variant for workflows beyond ground-state hybrid DFT.

The EXCH_CORRECTION_FUNC keyword selects the exchange functional used for the ADMM correction. It should be chosen consistently with the exchange part of the main exchange-correlation setup; PBEX is a common choice for PBE-based hybrid calculations.

Practical Checks

When using ADMM:

  • keep the same primary basis and potential convergence checks that would be used without ADMM,

  • check the sensitivity to the auxiliary basis size,

  • compare total energies, forces, or target properties against a non-ADMM reference for a small representative system,

  • remember that ADMM accelerates the exchange calculation but does not replace convergence of the primary Gaussian basis, real-space grid, or SCF thresholds.

See Also