DFT with Hybrid Functionals

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DFT with Hybrid Functionals Sanliang Ling University College London 4th CP2K Tutorial, 31st August 4th September 2015, Zurich

What are hybrid functionals? Hybrid functionals: mixing non-local Hartree-Fock exchange with local/semi-local DFT/GGA exchange 2

3 Why do we need hybrid DFT? Improved description of the thermochemistry (e.g. atomisation energy, heats of formation, etc) of molecular systems Improved description of the lattice constants, surface energies, ionisation potentials and band gaps of semiconductors and insulators

Why do we need hybrid DFT? /PBE0 Mean absolute errors (Mae) for atomisation energies (kcal/mol) Scuseria et al., J. Chem. Phys., 110, 5029 (1999) 4

Why do we need hybrid DFT? Band gap error (ev) statistics for the SC/40 test set Scuseria et al., J. Chem. Phys., 123, 174101 (2005) 5

Hybrid DFT Calculations with CP2K Total energy as a functional of the electron density Exchange-correlation energy with a hybrid functional Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 6

Hybrid DFT Calculations with CP2K Hartree-Fock exchange energy Four-centre two-electron integrals (ERIs): O(N 4 ) scaling Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 7

Hybrid DFT Calculations with CP2K Permutational symmetry of the ERIs O(N 4 ) Guidon et al., J. Chem. Phys., 128, 214104 (2008) 8

Hybrid DFT Calculations with CP2K Integral screening: Schwarz inequality upper bound for ERIs Ahlrichs et al., J. Comput. Chem., 10, 104 (1989) O(N 2 ) (see EPS_SCHWARZ keyword in &SCREENING subsection) Guidon et al., J. Chem. Phys., 128, 214104 (2008) 9

Hybrid DFT Calculations with CP2K Density matrix screening O(N 2 ) O(N) (see SCREEN_ON_INITIAL_P keyword in &SCREENING subsection) Guidon et al., J. Chem. Phys., 128, 214104 (2008) 10

Interaction potential Standard Coulomb potential (used in B3LYP, PBE0, etc) (see POTENTIAL_TYPE keyword in &INTERACTION_POTENTIAL subsection) Guidon et al., J. Chem. Phys., 128, 214104 (2008) 11

Interaction potential Range-separated Coulomb potential parameter short-range (used in HSE06, etc) long-range (see POTENTIAL_TYPE keyword in &INTERACTION_POTENTIAL subsection) Guidon et al., J. Chem. Phys., 128, 214104 (2008) Heyd & Scuseria, J. Chem. Phys., 118, 8207 (2003) 12

Interaction potential Truncated Coulomb potential parameter (used in PBE0-TC-LRC, etc) (see POTENTIAL_TYPE keyword in &INTERACTION_POTENTIAL subsection) Guidon et al., J. Chem. Theory Comput., 5, 3010 (2008) Spencer & Alavi, Phys. Rev. B, 77, 193110 (2008) 13

Auxiliary Density Matrix Methods Introducing auxiliary density matrix approximation original density matrix auxiliary density matrix Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 14

Auxiliary Density Matrix Methods The total energy with ADMM Kohn-Sham Matrix with ADMM The Kohn-Sham equation K C total SC Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 15

Auxiliary Density Matrix Methods Construction of auxiliary density matrix (see ADMM_PURIFICATION_METHOD keyword in &AUXILIARY_DENSITY_MATRIX_METHOD subsection) Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 16

Auxiliary Density Matrix Methods Construction of auxiliary density matrix: enforcing orthonormality Lagrangian multipliers (see ADMM_PURIFICATION_METHOD keyword in &AUXILIARY_DENSITY_MATRIX_METHOD subsection) Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 17

Auxiliary Density Matrix Methods Construction of auxiliary density matrix: constraint on total charge (see EXCH_SCALING_MODEL keyword in &AUXILIARY_DENSITY_MATRIX_METHOD subsection) Merlot et al., J. Chem. Phys., 141, 094101 (2014) 18

Auxiliary Density Matrix Methods GGA exchange corrections using different functions DFT = B88, PBE, OPTX, KT3X (see EXCH_CORRECTION_FUNC keyword in &AUXILIARY_DENSITY_MATRIX_METHOD subsection) Merlot et al., J. Chem. Phys., 141, 094101 (2014) 19

ADMM basis sets (see $CP2K/cp2k/data/BASIS_ADMM) Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 20

ADMM basis sets Choice of auxiliary basis set for ADMM FIT3: three Gaussian exponents for each valence orbital cfit3: a contraction of FIT3 pfit3: FIT3 + polarization functions cpfit3: cfit3 + polarization functions aug-fit3, aug-cfit3, aug-pfit3, augcpfit3: augmented with a diffuse function (see $CP2K/cp2k/data/BASIS_ADMM) Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 21

22 Basis optimisation with OPTIMIZE_BASIS Choosing a reference (complete) basis Performing accurate molecular calculations with ref. basis Choosing a form of the basis to be fitted Minimizing the objective function

ADMM basis sets (see $CP2K/cp2k/data/BASIS_ADMM_MOLOPT) 23

Uncontracted basis sets FIT10: 4s + 3p + 3d ADMM basis sets FIT11: 4s + 3p + 3d + 1f FIT12: 4s + 3p + 4d + 1f FIT13: 4s + 4p + 4d + 1f Contracted basis sets (double-z quality) cfit10 / cfit11 / cfit12 / cfit13 recommended for solids Names of the ADMM basis sets for main group elements will differ slightly, and usually the first ADMM basis set will not contain polarisation function. (see $CP2K/cp2k/data/BASIS_ADMM_MOLOPT) 24

Some general suggestions Always check the convergence of CUTOFF (see http://www.cp2k.org/howto:converging_cutoff) Always check the convergence of properties (e.g. lattice parameters, band gaps) with respect to supercell sizes Always start from pre-converged GGA (e.g. PBE) wavefunction and geometry Always check the convergence of primary and ADMM basis sets (start from a small basis and gradually increase the size) ADMM has only been implemented for use with GPW 25

Work Flow 1. Convergence test (primary basis, CUTOFF, supercell, etc) 2. GGA optimisation with selected primary basis 3. Name GGA wave function file for use with ADMM 4. Construct ADMM input with auxiliary basis (e.g. FIT3) 5. Run calculations and check convergence of ADMM basis 26

Input Structure: ADMM &DFT BASIS_SET_FILE_NAME./BASIS_MOLOPT (files can be found in $CP2K/cp2k/data) BASIS_SET_FILE_NAME./BASIS_ADMM WFN_RESTART_FILE_NAME ${project}-restart.wfn &SCF SCF_GUESS RESTART &END SCF &AUXILIARY_DENSITY_MATRIX_METHOD METHOD BASIS_PROJECTION ADMM_PURIFICATION_METHOD MO_DIAG &END AUXILIARY_DENSITY_MATRIX_METHOD &XC &END XC &END DFT &SUBSYS &KIND Si BASIS_SET DZVP-MOLOPT-SR-GTH AUX_FIT_BASIS_SET FIT3 POTENTIAL GTH-PBE-q4 &END KIND &END SUBSYS (see examples in $CP2K/cp2k/tests/QS/regtest-admm-1/2/3/4) 27

28 Which functional to use? PBE0-TC-LRC HSE06 Guidon et al., J. Chem. Theory Comput., 5, 3010 (2008) Spencer & Alavi, Phys. Rev. B, 77, 193110 (2008) Scuceria et al., J. Chem. Phys., 125, 224106 (2006)

Input Structure: PBE0 vs. HSE06 &XC &XC_FUNCTIONAL &PBE SCALE_X 0.75 SCALE_C 1.0 &END PBE &PBE_HOLE_T_C_LR CUTOFF_RADIUS 2.0 SCALE_X 0.25 &END PBE_HOLE_T_C_LR &END XC_FUNCTIONAL &HF &SCREENING EPS_SCHWARZ 1.0E-6 SCREEN_ON_INITIAL_P FALSE &END SCREENING &INTERACTION_POTENTIAL POTENTIAL_TYPE TRUNCATED CUTOFF_RADIUS 2.0 T_C_G_DATA./t_c_g.dat &END INTERACTION_POTENTIAL &MEMORY MAX_MEMORY 2400 EPS_STORAGE_SCALING 0.1 &END MEMORY FRACTION 0.25 &END HF &END XC PBE0-TC-LRC (t_c_g.dat can be found in $CP2K/cp2k/data) &XC &XC_FUNCTIONAL &PBE SCALE_X 0.0 SCALE_C 1.0 &END PBE &XWPBE SCALE_X -0.25 SCALE_X0 1.0 OMEGA 0.11 &END XWPBE &END XC_FUNCTIONAL &HF &SCREENING EPS_SCHWARZ 1.0E-6 SCREEN_ON_INITIAL_P FALSE &END SCREENING &INTERACTION_POTENTIAL POTENTIAL_TYPE SHORTRANGE OMEGA 0.11 &END INTERACTION_POTENTIAL &MEMORY MAX_MEMORY 2400 EPS_STORAGE_SCALING 0.1 &END MEMORY FRACTION 0.25 &END HF &END XC HSE06 (see examples in $CP2K/cp2k/tests/QS/regtest-admm-1/2/3/4) 29

A few more remarks Libint library is used to calculate ERIs and their analytic derivatives, see $CP2K/cp2k/INSTALL for more details on installation and linking with CP2K Hybrid DFT calculations of large systems are memory demanding, try to increase MAX_MEMORY or run the job with more MPI processes For extremely large hybrid DFT calculations, try hybrid MPI/OpenMP binary (i.e. cp2k.psmp) 30

Example: Diamond Band Gap 3x3x3 supercell Guidon, Hutter and VandeVondele, J. Chem. Theory Comput., 6, 2348 (2010) 31

Spreafico & VandeVondele, PCCP, 16, 26144 (2014) 32 Example: excess electrons in TiO 2 E pol = E loc(n+1) E deloc(n+1)

GGA with on-site Coulomb interaction: GGA+U Input Structure: GGA+U &DFT PLUS_U_METHOD &END DFT &KIND Ti BASIS_SET DZVP-MOLOPT-SR-GTH POTENTIAL GTH-PBE-q12 &DFT_PLUS_U T L 2 U_MINUS_J [ev] 3.9 &END DFT_PLUS_U &END KIND MULLIKEN ###LOWDIN specify which orbital to add GGA+U Dudarev et al., Phys. Rev. B, 57, 1505 (1998) specify effective on-site Coulomb interaction parameter (see examples in $CP2K/cp2k/tests/QS/regtest-plus_u) 33

Magnetic systems Fe1 Fe2 Fe1 Fe2 Fe1 O: 2s 2 2p 4 O 2- : 2s 2 2p 6 &KIND O BASIS_SET DZVP-MOLOPT-SR-GTH POTENTIAL GTH-PBE-q6 &BS &ALPHA NEL +2 L 1 N 2 &END ALPHA &BETA NEL +2 L 1 N 2 &END BETA &END BS &END KIND spin channel orbital occupation change angular momentum quantum number principal quantum number Fe2 Hematite (Fe 2 O 3 ) antiferromagnetic (see examples in $CP2K/cp2k/tests/QS/regtest-bs) 34

Magnetic systems Fe: 3d 6 4s 2 Fe 3+ : 3d 5 Fe1 Fe2 Fe1 Fe2 Fe1 Fe2 Hematite (Fe 2 O 3 ) antiferromagnetic &KIND Fe1 ELEMENT Fe BASIS_SET DZVP-MOLOPT-SR-GTH POTENTIAL GTH-PBE-q16 &DFT_PLUS_U L 2 U_MINUS_J [ev] 5.0 &END DFT_PLUS_U &BS &ALPHA NEL +4-2 L 2 0 N 3 4 &END ALPHA &BETA NEL -6-2 L 2 0 N 3 4 &END BETA &END BS &END KIND (see examples in $CP2K/cp2k/tests/QS/regtest-bs) 35

Magnetic systems Fe: 3d 6 4s 2 Fe 3+ : 3d 5 Fe1 Fe2 Fe1 Fe2 Fe1 Fe2 Hematite (Fe 2 O 3 ) antiferromagnetic &KIND Fe2 ELEMENT Fe BASIS_SET DZVP-MOLOPT-SR-GTH POTENTIAL GTH-PBE-q16 &DFT_PLUS_U L 2 U_MINUS_J [ev] 5.0 &END DFT_PLUS_U &BS &ALPHA NEL -6-2 L 2 0 N 3 4 &END ALPHA &BETA NEL +4-2 L 2 0 N 3 4 &END BETA &END BS &END KIND (see examples in $CP2K/cp2k/tests/QS/regtest-bs) 36

37 Some general suggestions Always check the convergence of CUTOFF (see http://www.cp2k.org/howto:converging_cutoff) Always start from a pre-converged GGA (e.g. PBE) wavefunction and geometry For GGA+U calculations, do not use U_MINUS_J values derived from other codes directly

Further Reading Hybrid Functionals (by Prof Joost VandeVondele) http://www.cecam.org/upload/talk/presentation_5766.pdf Self-Interaction Energy and Dispersion (by Prof Juerg Hutter) http://www.cecam.org/upload/talk/presentation_2988.pdf Exchange-Correlation Functionals (by Dr Manuel Guidon) http://www.cecam.org/upload/talk/presentation_2987.pdf 38

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