# Geometry Optimisation

## Introduction

This tutorial is designed to illustrate how to relax the structure of a system (without changing the cell dimensions) using CP2K. We use the relaxation of a water ($$H_2O$$) molecule as an example.

The example files can be downloaded from here. The calculation was carried out with CP2K version 2.4.

It should be noted that before running the geometry optimisation, the reader should have already know how to perform a simple Kohn-Sham Density Functional Theory energy and force calculation (this is covered in tutorial Calculating Energy and Forces using QUICKSTEP), and they should also know how how to find a sufficient grid cutoff for the static energy calculations (this is covered in tutorial Converging the CUTOFF and REL_CUTOFF).

DIIS (direct inversion in the iterative subspace or direct inversion of the iterative subspace), also known as Pulay mixing, is an extrapolation technique. DIIS was developed by Peter Pulay in the field of computational quantum chemistry with the intent to accelerate and stabilize the convergence of the Hartree–Fock self-consistent field method.

Note

The ALPHA and NBUFFER parameters might have to be reduced to avoid a bad condition number.

## Input Files

The input file for a geometry calculation is shown below:

&GLOBAL
PROJECT H2O
RUN_TYPE GEO_OPT
PRINT_LEVEL LOW
&END GLOBAL
&FORCE_EVAL
METHOD QS
&SUBSYS
&CELL
ABC 12.4138 12.4138 12.4138
&END CELL
&COORD
O      12.235322       1.376642      10.869880
H      12.415139       2.233125      11.257611
H      11.922476       1.573799       9.986994
&END COORD
&KIND H
&END KIND
&KIND O
&END KIND
&END SUBSYS
&DFT
BASIS_SET_FILE_NAME ./BASIS_SET
POTENTIAL_FILE_NAME ./POTENTIAL
&QS
EPS_DEFAULT 1.0E-7
&END QS
&MGRID
CUTOFF 200
NGRIDS 4
REL_CUTOFF 30
&END MGRID
&SCF
SCF_GUESS ATOMIC
EPS_SCF 1.0E-05
MAX_SCF 200
&DIAGONALIZATION T
ALGORITHM STANDARD
&END DIAGONALIZATION
&MIXING T
ALPHA 0.5
METHOD PULAY_MIXING
NPULAY 5
&END MIXING
&PRINT
&RESTART OFF
&END RESTART
&END PRINT
&END SCF
&XC
&END XC_FUNCTIONAL
&END XC
&END DFT
&END FORCE_EVAL
&MOTION
&GEO_OPT
TYPE MINIMIZATION
MAX_DR    1.0E-03
MAX_FORCE 1.0E-03
RMS_DR    1.0E-03
RMS_FORCE 1.0E-03
MAX_ITER 200
OPTIMIZER CG
&CG
MAX_STEEP_STEPS  0
RESTART_LIMIT 9.0E-01
&END CG
&END GEO_OPT
&CONSTRAINT
&FIXED_ATOMS
COMPONENTS_TO_FIX XYZ
LIST 1
&END FIXED_ATOMS
&END CONSTRAINT
&END MOTION


The reader should already be familiar with the GLOBAL and FORCE_EVAL sections. For geometry optimisation calculations, we must set RUN_TYPE in GLOBAL section to GEO_OPT:

RUN_TYPE GEO_OPT


In this example, we note that we have chosen diagonalisation of the Kohn-Sham Hamiltonian for the evaluation of wavefunctions, and used Pulay mixing for the self-consistency loops. 5 histories are used for Pulay mixing.

The important section for geometry optimisation settings are contained in subsection GEO_OPT of MOTION section. Note that GEO_OPT subsection only applies to the calculation where the cell dimensions do not change. Calculations which allows the relaxation of the cell are covered in a separate tutorial.

&GEO_OPT
TYPE MINIMIZATION
MAX_DR    1.0E-03
MAX_FORCE 1.0E-03
RMS_DR    1.0E-03
RMS_FORCE 1.0E-03
MAX_ITER 200
OPTIMIZER CG
&CG
MAX_STEEP_STEPS  0
RESTART_LIMIT 9.0E-01
&END CG
&END GEO_OPT


The TYPE keyword sets whether the geometry optimisation is for finding the local minima (MINIMIZATION) or for finding the saddle point transition state (TRANSITION_STATE). The keywords MAX_DR, MAX_FORCE, RMS_DR and RMS_FORCE set the criteria of whether an optimised geometry is reached. MAX_DR and RMS_DR (in Bohr) are the tolerance on the maximum and root-mean-square of atomic displacements from the previous geometry optimisation iteration; MAX_FORCE and RMS_FORCE (in Bohr/Hartree) are the tolerance on the maximum and root-mean-square of atomic forces. The geometry is considered to be optimised only when all four criteria are satisfied. The keyword MAX_ITER sets the maximum number of geometry optimisation iterations. OPTIMIZER sets the algorithm for finding the stationary points; in this example we have chosen the conjugate gradients (CG) method.

The CG subsection sets options for the conjugate gradients algorithm. In this case, we have configured it so that no steepest descent steps are to be performed before the start of the conjugate gradients algorithm; and the CG algorithm should be reset (and one steepest descent step is performed) if the cosine of the angles between two consecutive searching directions is less than 0.9.

&CONSTRAINT
&FIXED_ATOMS
COMPONENTS_TO_FIX XYZ
LIST 1
&END FIXED_ATOMS
&END CONSTRAINT


We can add constraints to atomic movements by using the CONSTRAINT subsection in MOTION section. In this example, we choose to fix particular atoms using the FIXED_ATOMS subsection. The keyword COMPONENTS_TO_FIX sets which of the X Y Z directions are to be fixed, and in this case, the atoms will be completely pinned in all directions (XYZ). The list of atoms to be constrained are given by the LIST keyword:

LIST 1 2 3 ... N


The numbers to the right of LIST are the list of atomic indices, and correspond to the order (from top to bottom) of the atoms given in the COORD subsection of SUBSYS (of FORCE_EVAL). In our example, we have fixed the oxygen atom during geometry optimisation, so that the water molecule will not move around while its structure is being relaxed.

## Results

The example is run using the serial version of the CP2K binaries:

cp2k.sopt -o H2O.out H2O.inp &


After the job has finished, you should obtain the following files:

• H2O.out

• H2O-pos-1.xyz

• H2O-1.restart

• H2O-1.restart.bak-1

• H2O-1.restart.bak-2

• H2O-1.restart.bak-3

Again, the file H2O.out contains the main output of the job. H2O-pos-1.xyz contains the trace of atomic coordinates at each geometry optimisation step in the xyz file format. The last set of atomic coordinates corresponds to the relaxed structure. H2O-1.restart is a CP2K input file, similar to H2O.inp, which contains the latest atomic coordinates of the water molecule. Should the job die for some reason, you can continue the job using the latest atomic coordinates by using command:

cp2k.sopt -o H2O.out H2O-1.restart &


You can of course also use H2O-1.restart as a template for writing an input for further calculations using the relaxed atomic structures.

The files H2O-1.restart.bak-* are backup restart files with atomic coordinates obtained from the previous 1, 2 and 3 geometry optimisation iterations. H2O-1.restart.bak-1 should be the same as H2O-1.restart.

In the main output file H2O.out, at the end of each geometry optimisation step, we will have the following information:

--------  Informations at step =     1 ------------
Optimization Method        =                   SD
Total Energy               =       -17.1643447508
Real energy change         =        -0.0006776683
Decrease in energy         =                  YES
Used time                  =               90.837

Convergence check :
Max. step size             =         0.0336570168
Conv. limit for step size  =         0.0010000000
Convergence in step size   =                   NO
RMS step size              =         0.0168136889
Conv. limit for RMS step   =         0.0010000000
Convergence in RMS step    =                   NO
Conv. limit for gradients  =         0.0010000000
Conv. limit for RMS grad.  =         0.0010000000
---------------------------------------------------


The above output segment states that at the end of geometry optimisation step 1, the total energy of the system is -17.1643447508 (Ha) and none of the criteria for optimised geometry has been reached. The iteration therefore will carry on, until all criteria becomes YES.

At the end of geometry optimisation, one should obtain something like:

--------  Informations at step =    11 ------------
Optimization Method        =                   SD
Total Energy               =       -17.1646204766
Real energy change         =        -0.0000000529
Decrease in energy         =                  YES
Used time                  =               49.893

Convergence check :
Max. step size             =         0.0003393150
Conv. limit for step size  =         0.0010000000
Convergence in step size   =                  YES
RMS step size              =         0.0001493298
Conv. limit for RMS step   =         0.0010000000
Convergence in RMS step    =                  YES
Conv. limit for gradients  =         0.0010000000
Conv. limit for RMS grad.  =         0.0010000000
Conv. in RMS gradients     =                  YES
---------------------------------------------------


which clearly shows all criteria have been satisfied.

The final Kohn-Sham energies can be obtained at the end of the output:

*******************************************************************************
***                    GEOMETRY OPTIMIZATION COMPLETED                      ***
*******************************************************************************

Reevaluating energy at the minimum

Number of electrons:                                                          8
Number of occupied orbitals:                                                  4
Number of molecular orbitals:                                                 4

Number of orbital functions:                                                 23
Number of independent orbital functions:                                     23

Parameters for the always stable predictor-corrector (ASPC) method:

ASPC order: 3

B(1) =   3.000000
B(2) =  -3.428571
B(3) =   1.928571
B(4) =  -0.571429
B(5) =   0.071429

Extrapolation method: ASPC

SCF WAVEFUNCTION OPTIMIZATION

Step     Update method      Time    Convergence         Total energy    Change
------------------------------------------------------------------------------
1 Pulay/Diag. 0.50E+00    0.5     0.00005615       -17.1646204762 -1.72E+01
2 Pulay/Diag. 0.50E+00    1.0     0.00000563       -17.1646347711 -1.43E-05

*** SCF run converged in     2 steps ***

Electronic density on regular grids:         -8.0000016293       -0.0000016293
Core density on regular grids:                7.9999992554       -0.0000007446
Total charge density on r-space grids:       -0.0000023739
Total charge density g-space grids:          -0.0000023739

Overlap energy of the core charge distribution:               0.00000004555422
Self energy of the core charge distribution:                -43.83289054591484
Core Hamiltonian energy:                                     12.82175605770555
Hartree energy:                                              17.97395116120845
Exchange-correlation energy:                                 -4.12745148966141

Total energy:                                               -17.16463477110803

ENERGY| Total FORCE_EVAL ( QS ) energy (a.u.):              -17.164634771108034