# How to run 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\(_2\)O) molecule as an example.

The example files are contained in geometry_optimisation.tgz that comes with this tutorial. 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 that 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 BASIS_SET DZVP-GTH-PADE POTENTIAL GTH-PADE-q1 &END KIND &KIND O BASIS_SET DZVP-GTH-PADE POTENTIAL GTH-PADE-q6 &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 &XC_FUNCTIONAL PADE &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 Max. gradient = 0.0182785685 Conv. limit for gradients = 0.0010000000 Conv. for gradients = NO RMS gradient = 0.0091312361 Conv. limit for RMS grad. = 0.0010000000 Conv. for gradients = NO ---------------------------------------------------

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 Max. gradient = 0.0001787448 Conv. limit for gradients = 0.0010000000 Conv. in gradients = YES RMS gradient = 0.0000786642 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