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In order to go beyond GGA and hybrid DFT, one option is to use wave function correlation methods. Recently, second-order Møller-Plesset perturbation theory (MP2) and random phase approximation (RPA) have been added to CP2K . The implementations are aimed at condensed phase calculations, and in the case of MP2 provides energies, forces and stress, as well as electron (spin) densities at O(N^5) cost, while RPA provides energies at O(N^4) or O(N^3) cost.

However, significant computational resources are needed for the condensed phase calculations (for the overview see the corresponding lecture). Therefore, we will perform the gas phase calculations in this tutorial, even though RI-GPW is not very efficient in this case.

Some references:

MP2 and RPA energies implementations in CP2K:10.1016/j.cpc.2014.10.021, 10.1021/ct4002202, 10.1021/ct300531w

MP2 forces: 10.1063/1.4919238 and 10.1021/acs.jctc.6b00015

Cubic scaling RPA implementation: 10.1021/acs.jctc.6b00840

Since the correlated wave function calculations are expensive, please use 4 cores for execution (with one OMP thread):

mpirun -np 4 -x OMP_NUM_THREADS=1  cp2k.psmp -i input -o output

1. Task: Benzene dimer MP2 binding energy

Employ the provided input file to compute the benzene dimer binding energy. The provided dimer geometry is optimized already. To obtain the energy of the monomer, geometry optimization is in principle necessary, although the geometry of a benzene monomer must be close to the one in the dimer. This can be done by removing one of the benzene molecules editing the coordinates in the input. Perform a few optimization steps to see how much the energy changes and to calculate electron density (saved as .cube files) at the MP2 level. If energy does not change much, this means that the monomer structure is already close to the optimal. Compare the times needed for one geometry optimization step (energy+forces) for a monomer and energy for a dimer, as well as the relative timing for energy and forces evaluation for a monomer.

During the optimization of benzene, one will calculate gradient which, in turn, requires density matrices. Hence, one can calculate electronic densities. Add the following to the &GLOBAL section:


and the following lines to the &DFT section


Importantly, during the force calculations one will have to solve the coupled-perturbed equations invoking exact exchange calculations. If there is enough memory we can reuse the integrals from the HF calculation by setting the following keyword in the &RI_MP2 section:


Perform two optimizations setting FREE_HFX_BUFFER to .TRUE. and .FALSE. Compare the overall timings and especially the times for performing Hartree-Fock exchange calculations:

integrate_four_center               71 12.3    2.261    4.012  108.805  109.179

The last number in the line is the real time of execution. The memory distribution between the RI-MP2 integrals and HFX integrals are tuned by the MEMORY keyword in the &WF_CORRELATION section and the &MEMORY section in the &HF section:

  MAX_MEMORY  1800

At the optimized (or the most optimized) geometry of benzene monomer perform a Hartree-Fock calculation to compare electron densities. Visualize them with VMD. Density differences can be computed with cubecruncher available in the executable directory.

2. Task: Benzene monomer RPA energy: frequency integration

RPA is HFX+RPA correlation. It can be performed with HFX orbitals and eigenvalues, but also based on e.g. GGA or hybrid orbitals. Two advantages over MP2 are : 1) scaling is O(N^4) (for RI-dRPA with frequency integration). 2) it is applicable for systems with a small gap. A O(N^3)implementation is also available in CP2K, although with a very large prefactor, which makes it worth using for very large systems.

Here, we look at the convergence of the RPA energy as a function of the number of integration points (RPA_NUM_QUAD_POINTS). Change this parameter in the range 4-16. For large systems the time needed for the calculations is proportional to the number of integration points.

Use the following section to change to RPA:

      ! with WF correlation
        ! use the RI-GPW approach
          ! number of quadrature points, essential for accurate energies.
          ! small gap systems need more points
          RPA_NUM_QUAD_POINTS  8
          ! essentially always use minimax
            FRACTION 1.0000000
              EPS_SCHWARZ 1.0E-8
            &END SCREENING
          &END HF
        &END RI_RPA
        MEMORY    1800
        NUMBER_PROC  1

It is strongly recommended to use the MINIMAX integration scheme.

Required files

Input files and a wave function restart file can be found in the archive: mp2_rpa.tar.gz input file for an RI-MP2 calculation on a benzene dimer

  PROJECT      benzene_dimer

  METHOD         Quickstep
    ! specification of basis and potential files (cp2k/data)
    ! Wave function restart file to accelerate HF convergence
    WFN_RESTART_FILE_NAME  benzene_dimer.wfn
      CUTOFF    400

      EPS_DEFAULT  1.0E-10
      EPS_PGF_ORB  1.0E-8
    &END QS

    ! standard OT
      EPS_SCF 1.0E-6
      MAX_SCF 40
        MINIMIZER       CG
        EPS_SCF 1.0E-6
        MAX_SCF 10
    &END SCF
    ! Non periodic calculation needs Poisson solver: use wavelet solver

      ! no XC functional
      &XC_FUNCTIONAL NONE                       
      ! and 100%HFX
        FRACTION    1.0 
          EPS_SCHWARZ          1.0E-9
          MAX_MEMORY  1800  
      &END HF
      ! with WF correlation
        ! use the RI-GPW approach
        METHOD  RI_MP2_GPW
           CUTOFF      200
           REL_CUTOFF  50
        ! block sizes and memory affect parallel efficiency
          BLOCK_SIZE       1       
        MEMORY    1800
        NUMBER_PROC  1
    &END XC
    ! sufficiently large non-periodic unit cell
      ABC [angstrom]  12 12 12
      PERIODIC NONE           ! Non periodic calculation.
! specification of the system with coordinates 
  H         8.5709951714        6.1617188657        9.1769626364
  C         8.0387778483        5.9757981379        8.2528397815
  C         6.6427474511        5.9561428224        8.2430496957
  H         6.0937543445        6.1249650636        9.1610309894
  C         5.9546039990        5.7314555716        7.0475649921
  H         4.8717398728        5.7196290593        7.0404902271
  C         6.6635660065        5.5279574532        5.8611612413
  H         6.1300650069        5.3536860275        4.9350551393
  C         8.0615723757        5.5446390494        5.8709776417
  H         8.6121087088        5.3836499080        4.9523278088
  C         8.7479484762        5.7683467907        7.0666774749
  H         9.8306982448        5.7893185194        7.0736980974
  H         6.4284687494        8.8405192314        5.8228620343
  H         8.9057373702        8.8752743844        5.8387264727
  C         6.9607954533        9.0255100555        6.7471184900
  C         8.3568440363        9.0439429763        6.7567676586
  C         6.2520232187        9.2329626290        7.9335080936
  C         9.0454365692        9.2673522387        7.9522880615
  H         5.1692478273        9.2131682112        7.9263504668
  H        10.1283042427        9.2780238064        7.9594286415
  C         6.9387312641        9.4553058890        9.1292884396
  C         8.3367433685        9.4704785560        9.1388664843
  H         6.3883177411        9.6163935534       10.0479844468
  H         8.8703567259        9.6435467092       10.0651664724

    ! keep atoms away from box borders, 
    ! a requirement for the wavelet Poisson solver
    ! MP2 needs correlation consistent basis set
    ! RI-MP2 needs an auxiliary basis set
    ! We employ HF pseudo potentials
    &KIND H
      BASIS_SET         cc-TZV2P-GTH
      RI_AUX_BASIS_SET  TZ_fiodena_opt
      POTENTIAL         GTH-HF-q1      
    &KIND C
      BASIS_SET         cc-TZV2P-GTH
      RI_AUX_BASIS_SET  TZ_fiodena_opt
      POTENTIAL         GTH-HF-q4

! how to propagate the system, selection via RUN_TYPE in the &GLOBAL section
   OPTIMIZER BFGS ! Good choice for 'small' systems like benzene
   MAX_DR    [bohr] 0.003 ! adjust target as needed
exercises/2017_uzh_cp2k-tutorial/wfc.txt · Last modified: 2020/08/21 10:15 by