### Table of Contents

# Hartree-Fock exchange

The purpose of this section is to explain how to compute hybrid functionals (or Hartree-Fock exchange, HFX) with CP2K in condensed phase systems. It is based on the developments described in 10.1021/ct900494g and 10.1063/1.2931945, and its efficient extension (ADMM) described in 10.1021/ct1002225.

Hartree-Fock exchange in CP2K is based on four center integrals, these are computed with an external library (libint). Do these exercises, CP2K must be linked to this library.

This approach has a computational cost that depends strongly on the nature of the basis, unless combined with ADMM (see below), do not use MOLOPT basis sets with HFX. We use basis sets from `HFX_BASIS`

, which are suitable.

## Truncated Coulomb operator

To enable HFX in the condensed phase (described at the Gamma point only), CP2K employs a truncated Coulomb operator for the exchange part. The physical picture is that we do not want to have 'self-exchange interactions' of an electron with its image in neighboring unit cells. As a rule of thumb, the maximum range (truncation radius) is L/2 where L is the smallest edge of the unit cell. The convergence of the exchange energy is exponential wrt. this radius. Typically, 5-6A provides good results, but this depends on the nature of the system, i.e. the band gap or the range of the maximally localized Wannier orbitals.

### 1st task : GGA restart wfn

using the water input from the previous exercise, we will perform a single point GGA calculation to generate an initial wavefunction (wfn) restart. HFX calculations benefit from this.

Change the input to:

`RUN_TYPE ENERGY`

`IOLEVEL MEDIUM`

`RESTART ON`

- comment section
`&EXT_RESTART`

Run the input and rename the generated wfn file (`WATER-RESTART.wfn`

) to `WATER-RESTART-GGA.wfn`

.
Also make a note of the `HOMO - LUMO gap [eV]`

### 2nd task: PBE0-D3 water

To do a hybrid calculation, we just change the &XC section. Various examples can be found in the regtests, but here we employ a section equivalent to PBE0-D3.

Change the input to:

`SCF_GUESS RESTART`

`WFN_RESTART_FILE_NAME WATER-RESTART-GGA.wfn`

And employ the following `&XC`

section:

! specify the exchange and correlation treatment &XC ! use a PBE0 functional &XC_FUNCTIONAL &PBE ! 75% GGA exchange SCALE_X 0.75 ! 100% GGA correlation SCALE_C 1.0 &END PBE &END XC_FUNCTIONAL &HF ! 25 % HFX exchange FRACTION 0.25 &SCREENING ! important parameter to get stable HFX calcs EPS_SCHWARZ 1.0E-6 ! needs a good (GGA) initial guess SCREEN_ON_INITIAL_P TRUE &END &INTERACTION_POTENTIAL ! for condensed phase systems POTENTIAL_TYPE TRUNCATED ! should be less than halve the cell CUTOFF_RADIUS 6.0 ! data file needed with the truncated operator T_C_G_DATA ./t_c_g.dat &END &MEMORY ! In MB per MPI rank.. use as much as need to get in-core operation MAX_MEMORY 4000 EPS_STORAGE_SCALING 0.1 &END &END ! adding Grimme's D3 correction (by default without C9 terms) &VDW_POTENTIAL POTENTIAL_TYPE PAIR_POTENTIAL &PAIR_POTENTIAL PARAMETER_FILE_NAME dftd3.dat TYPE DFTD3 REFERENCE_FUNCTIONAL PBE0 R_CUTOFF [angstrom] 16 &END &END VDW_POTENTIAL &END XC

Topics:

`EPS_SCHWARZ`

,`EPS_PGF_ORB`

,`SCREEN_ON_INITIAL_P`

: parameters to guarantee stable SCF.- Fraction of exchange (
`SCALE_X`

,`FRACTION`

).

Run this input, it should lead to an output like:

HFX_MEM_INFO| Number of cart. primitive ERI's calculated: 20780449251 HFX_MEM_INFO| Number of sph. ERI's calculated: 4626861713 HFX_MEM_INFO| Number of sph. ERI's stored in-core: 1440639962 HFX_MEM_INFO| Number of sph. ERI's stored on disk: 0 HFX_MEM_INFO| Number of sph. ERI's calculated on the fly: 0 HFX_MEM_INFO| Total memory consumption ERI's RAM [MB's]: 1368 HFX_MEM_INFO| Whereof max-vals [MB's]: 78 HFX_MEM_INFO| Total compression factor ERI's RAM: 8.03 HFX_MEM_INFO| Total memory consumption ERI's disk [MB's]: 0 HFX_MEM_INFO| Total compression factor ERI's disk: 0.00 HFX_MEM_INFO| Size of density/Fock matrix [MB's]: 14 HFX_MEM_INFO| Size of buffers [MB's]: 2 HFX_MEM_INFO| Number of periodic image cells considered: 27 HFX_MEM_INFO| Est. max. program size after HFX [MB's]: 243 1 OT DIIS 0.80E-01 68.5 0.00096341 -1102.1375916328 -1.10E+03 Trace(PS): 512.0000000000 Electronic density on regular grids: -511.9999999886 0.0000000114 Core density on regular grids: 511.9999999836 -0.0000000164 Total charge density on r-space grids: -0.0000000051 Total charge density g-space grids: -0.0000000051 2 OT DIIS 0.80E-01 4.1 0.00064929 -1102.1607534171 -2.32E-02

Topics:

- in-core operation.
- How to detect instabilities due to too aggressive screening.

Question: What is the HOMO-LUMO gap for this configuration ? How does this compare to the GGA result ? Adjust the fraction of exchange (modify the input in two places!) to 20% and/or 30%, how does this influence the gap ?

## Truncated Coulomb operator with long range correction

Like in the HSE functional, the difference between the operator used for exchange and 1/r, can be accounted for by a special GGA exchange functional. Also for the truncated coulomb operator this is possible, and allows for xc functionals that embed very short range exchange operators only. This can be used to speedup the calculation, while retaining the benefits of HFX. The functional employed in this way smoothly goes from PBE to PBE0 as the range goes from 0 to Infinity.

### 3rd task

Add to the &XC_FUNCTIONAL section (i.e. in addition to &PBE) the following section:

&PBE_HOLE_T_C_LR CUTOFF_RADIUS 2.5 SCALE_X 0.25 &END

and employ the same `CUTOFF_RADIUS`

for the `INTERACTION_POTENTIAL`

.

Rerun the single point energy calculation and note the band gap.

- Is such a short range sufficient to have a sizable effect on the band gap ?
- is
`HFX_MEM_INFO| Number of cart. primitive ERI's calculated`

very different for calculations with 2.5 and 6.0A truncation radius ?

## Auxiliary Density Matrix Methods (ADMM)

ADMM is an approach to mitigate the cost of HFX for large basis sets. In particular, if MOLOPT basis sets are used, standard HFX becomes too expensive (CP2K can not deal efficiently with highly contracted AOs). In ADMM, an `AUX_FIT_BASIS_SET`

is introduced, which is used to create an auxiliary density matrix (ADM) by projection. HFX is evaluated for this ADM, while the error introduced by using an ADM is corrected for with a GGA exchange functional.

### 4rd task : introduce ADMM

Make the following changes:

- insert and additional line
`BASIS_SET_FILE_NAME BASIS_ADMM`

(and copy that file from cp2k/data as needed). - insert for each
`&KIND`

a line`AUX_FIT_BASIS_SET cFIT3`

- insert a secion
`&AUXILIARY_DENSITY_MATRIX_METHOD`

! use ADMM &AUXILIARY_DENSITY_MATRIX_METHOD ! recommended, i.e. use a smaller basis for HFX ! each kind will need an AUX_FIT_BASIS_SET. METHOD BASIS_PROJECTION ! recommended, this method is stable and allows for MD. ! can be expensive for large systems ADMM_PURIFICATION_METHOD MO_DIAG &END

Run the input, what's the `HOMO - LUMO gap`

?

## Chasing charge localization in liquid water

The combination of truncated exchange and ADMM results in the most effective way to run AIMD with hybrid functionals. In some systems the difference between GGA DFT and hybrids is very large. One such systems is liquid water after ionization (i.e. charge +1), where only with hybrids the expected species (OH radicals) are formed. See 10.1063/1.3664746.

### 5th task : ionized water

adapt the admm input for water to reflect the ionized state:

! Charge and multiplicity LSD CHARGE 1 MULTIPLICITY 2

because the system is electronically very difficult initially, we'll reduce the convergence threshold `EPS_SCF 1.0E-5`

(twice).

Note that the `WFN_RESTART_FILE_NAME`

must point to a GGA calculation of the same charge and multiplicity (do this calculations first).

Run single point energy calculations varying the fraction of exchange from 0.25 to 0.50, does the mulliken spin population reproduce Fig. 2 in 10.1021/ct1002225 ?

For the fraction 0.5, run AIMD for about 50-100fs (if time permits), what happens with the water molecule on which the spin was localized ? Do you results agree with 10.1063/1.3664746 ?

## Required files

No new files are required for this exercise. If you're stuck, you can use the following worked out examples.

- water_pbe0_cheating.inp
&GLOBAL ! the project name is made part of most output files... useful to keep order PROJECT WATER ! various runtypes (energy, geo_opt, etc.) available. RUN_TYPE ENERGY ! limit the runs to 30min WALLTIME 1800 ! reduce the amount of IO IOLEVEL MEDIUM &END GLOBAL &FORCE_EVAL ! the electronic structure part of CP2K is named Quickstep METHOD Quickstep &DFT ! basis sets and pseudopotential files can be found in cp2k/data BASIS_SET_FILE_NAME HFX_BASIS POTENTIAL_FILE_NAME GTH_POTENTIALS ! GGA restart to provide a good initial density matrix WFN_RESTART_FILE_NAME WATER-RESTART-GGA.wfn ! Charge and multiplicity CHARGE 0 MULTIPLICITY 1 &MGRID ! PW cutoff ... depends on the element (basis) too small cutoffs lead to the eggbox effect. ! certain calculations (e.g. geometry optimization, vibrational frequencies, ! NPT and cell optimizations, need higher cutoffs) CUTOFF [Ry] 400 &END &QS ! use the GPW method (i.e. pseudopotential based calculations with the Gaussian and Plane Waves scheme). METHOD GPW ! default threshold for numerics ~ roughly numerical accuracy of the total energy per electron, ! sets reasonable values for all other thresholds. EPS_DEFAULT 1.0E-10 ! used for MD, the method used to generate the initial guess. EXTRAPOLATION ASPC &END &POISSON PERIODIC XYZ ! the default, gas phase systems should have 'NONE' and a wavelet solver &END &PRINT ! at the end of the SCF procedure generate cube files of the density &E_DENSITY_CUBE OFF &END E_DENSITY_CUBE ! compute eigenvalues and homo-lumo gap each 10nd MD step &MO_CUBES NLUMO 4 NHOMO 4 WRITE_CUBE .FALSE. &EACH MD 10 &END &END &END ! use the OT METHOD for robust and efficient SCF, suitable for all non-metallic systems. &SCF SCF_GUESS RESTART ! can be used to RESTART an interrupted calculation MAX_SCF 30 EPS_SCF 1.0E-6 ! accuracy of the SCF procedure typically 1.0E-6 - 1.0E-7 &OT ! an accurate preconditioner suitable also for larger systems PRECONDITIONER FULL_SINGLE_INVERSE ! the most robust choice (DIIS might sometimes be faster, but not as stable). MINIMIZER DIIS &END OT &OUTER_SCF ! repeat the inner SCF cycle 10 times MAX_SCF 10 EPS_SCF 1.0E-6 ! must match the above &END ! do not store the wfn during MD &PRINT &RESTART ON &END &END &END SCF ! specify the exchange and correlation treatment &XC ! use a PBE0 functional &XC_FUNCTIONAL &PBE ! 75% GGA exchange SCALE_X 0.75 ! 100% GGA correlation SCALE_C 1.0 &END PBE &END XC_FUNCTIONAL &HF ! 25 % HFX exchange FRACTION 0.25 &SCREENING ! important parameter to get stable HFX calcs EPS_SCHWARZ 1.0E-6 ! needs a good (GGA) initial guess SCREEN_ON_INITIAL_P TRUE &END &INTERACTION_POTENTIAL ! for condensed phase systems POTENTIAL_TYPE TRUNCATED ! should be less than halve the cell CUTOFF_RADIUS 6.0 ! data file needed with the truncated operator T_C_G_DATA ./t_c_g.dat &END &MEMORY ! In MB per MPI rank.. use as much as need to get in-core operation MAX_MEMORY 4000 ! additional accuracy for storing compressed results EPS_STORAGE_SCALING 0.1 &END &END ! adding Grimme's D3 correction (by default without C9 terms) &VDW_POTENTIAL POTENTIAL_TYPE PAIR_POTENTIAL &PAIR_POTENTIAL PARAMETER_FILE_NAME dftd3.dat TYPE DFTD3 REFERENCE_FUNCTIONAL PBE0 R_CUTOFF [angstrom] 16 &END &END VDW_POTENTIAL &END XC &END DFT ! description of the system &SUBSYS &CELL ! unit cells that are orthorhombic are more efficient with CP2K ABC [angstrom] 12.42 12.42 12.42 &END CELL ! atom coordinates can be in the &COORD section, ! or provided as an external file. &TOPOLOGY COORD_FILE_NAME water.xyz COORD_FILE_FORMAT XYZ &END ! MOLOPT basis sets are fairly costly, ! but in the 'DZVP-MOLOPT-SR-GTH' available for all elements ! their contracted nature makes them suitable ! for condensed and gas phase systems alike. &KIND H BASIS_SET DZVP-GTH POTENTIAL GTH-PBE-q1 &END KIND &KIND O BASIS_SET DZVP-GTH POTENTIAL GTH-PBE-q6 &END KIND &END SUBSYS &END FORCE_EVAL ! how to propagate the system, selection via RUN_TYPE in the &GLOBAL section &MOTION &GEO_OPT OPTIMIZER BFGS ! Good choice for 'small' systems (use LBFGS for large systems) MAX_ITER 100 MAX_DR [bohr] 0.003 ! adjust target as needed &BFGS &END &END &MD ENSEMBLE NVT ! sampling the canonical ensemble, accurate properties might need NVE TEMPERATURE [K] 300 TIMESTEP [fs] 0.5 STEPS 1000 # GLE thermostat as generated at http://epfl-cosmo.github.io/gle4md # GLE provides an effective NVT sampling. &THERMOSTAT REGION MASSIVE TYPE GLE &GLE NDIM 5 A_SCALE [ps^-1] 1.00 A_LIST 1.859575861256e+2 2.726385349840e-1 1.152610045461e+1 -3.641457826260e+1 2.317337581602e+2 A_LIST -2.780952471206e-1 8.595159180871e-5 7.218904801765e-1 -1.984453934386e-1 4.240925758342e-1 A_LIST -1.482580813121e+1 -7.218904801765e-1 1.359090212128e+0 5.149889628035e+0 -9.994926845099e+0 A_LIST -1.037218912688e+1 1.984453934386e-1 -5.149889628035e+0 2.666191089117e+1 1.150771549531e+1 A_LIST 2.180134636042e+2 -4.240925758342e-1 9.994926845099e+0 -1.150771549531e+1 3.095839456559e+2 &END GLE &END THERMOSTAT &END &PRINT &TRAJECTORY &EACH MD 1 &END EACH &END TRAJECTORY &VELOCITIES OFF &END VELOCITIES &FORCES OFF &END FORCES &RESTART_HISTORY &EACH MD 500 &END EACH &END RESTART_HISTORY &RESTART BACKUP_COPIES 3 &EACH MD 1 &END EACH &END RESTART &END PRINT &END