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…Lianheng Tong)
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# Geometry Optimisation | ||
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## Introduction | ||
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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. | ||
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The example files are contained in | ||
[geometry_optimisation.tgz](https://www.cp2k.org/_media/geometry_optimisation.tgz) that comes with | ||
this tutorial. The calculation was carried out with CP2K version 2.4. | ||
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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](https://www.cp2k.org/howto:static_calculation)), 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](https://www.cp2k.org/howto:converging_cutoff)). | ||
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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. | ||
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```{note} | ||
The [ALPHA](#CP2K_INPUT.FORCE_EVAL.DFT.SCF.MIXING.ALPHA) and | ||
[NBUFFER](#CP2K_INPUT.FORCE_EVAL.DFT.SCF.MIXING.NBUFFER) parameters might have to be reduced to | ||
avoid a [bad condition number](https://github.com/cp2k/cp2k/issues/2360). | ||
``` | ||
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## Input Files | ||
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The input file for a geometry calculation is shown below: | ||
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```none | ||
&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 | ||
&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 | ||
``` | ||
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The reader should already be familiar with the [GLOBAL](#CP2K_INPUT.GLOBAL) and | ||
[FORCE_EVAL](#CP2K_INPUT.FORCE_EVAL) sections. For geometry optimisation calculations, we must set | ||
[RUN_TYPE](#CP2K_INPUT.GLOBAL.RUN_TYPE) in [GLOBAL](#CP2K_INPUT.GLOBAL) section to `GEO_OPT`: | ||
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```none | ||
RUN_TYPE GEO_OPT | ||
``` | ||
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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. | ||
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The important section for geometry optimisation settings are contained in subsection | ||
[GEO_OPT](#CP2K_INPUT.MOTION.GEO_OPT) of [MOTION](#CP2K_INPUT.MOTION) section. Note that | ||
[GEO_OPT](#CP2K_INPUT.MOTION.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. | ||
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```none | ||
&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 | ||
``` | ||
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The [TYPE](#CP2K_INPUT.MOTION.GEO_OPT.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](#CP2K_INPUT.MOTION.GEO_OPT.MAX_DR), | ||
[MAX_FORCE](#CP2K_INPUT.MOTION.GEO_OPT.MAX_FORCE), [RMS_DR](#CP2K_INPUT.MOTION.GEO_OPT.RMS_DR) and | ||
[RMS_FORCE](#CP2K_INPUT.MOTION.GEO_OPT.RMS_FORCE) set the criteria of whether an optimised geometry | ||
is reached. [MAX_DR](#CP2K_INPUT.MOTION.GEO_OPT.MAX_DR) and | ||
[RMS_DR](#CP2K_INPUT.MOTION.GEO_OPT.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](#CP2K_INPUT.MOTION.GEO_OPT.MAX_FORCE) and | ||
[RMS_FORCE](#CP2K_INPUT.MOTION.GEO_OPT.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](#CP2K_INPUT.MOTION.GEO_OPT.MAX_ITER) sets the | ||
maximum number of geometry optimisation iterations. | ||
[OPTIMIZER](#CP2K_INPUT.MOTION.GEO_OPT.OPTIMIZER) sets the algorithm for finding the stationary | ||
points; in this example we have chosen the conjugate gradients (`CG`) method. | ||
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The [CG](#CP2K_INPUT.MOTION.GEO_OPT.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. | ||
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```none | ||
&CONSTRAINT | ||
&FIXED_ATOMS | ||
COMPONENTS_TO_FIX XYZ | ||
LIST 1 | ||
&END FIXED_ATOMS | ||
&END CONSTRAINT | ||
``` | ||
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We can add constraints to atomic movements by using the [CONSTRAINT](#CP2K_INPUT.MOTION.CONSTRAINT) | ||
subsection in [MOTION](#CP2K_INPUT.MOTION) section. In this example, we choose to fix particular | ||
atoms using the [FIXED_ATOMS](#CP2K_INPUT.MOTION.CONSTRAINT.FIXED_ATOMS) subsection. The keyword | ||
[COMPONENTS_TO_FIX](#CP2K_INPUT.MOTION.CONSTRAINT.FIXED_ATOMS.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](#CP2K_INPUT.MOTION.CONSTRAINT.FIXED_ATOMS.LIST) keyword: | ||
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```none | ||
LIST 1 2 3 ... N | ||
``` | ||
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The numbers to the right of [LIST](#CP2K_INPUT.MOTION.CONSTRAINT.FIXED_ATOMS.LIST) are the list of | ||
atomic indices, and correspond to the order (from top to bottom) of the atoms given in the | ||
[COORD](#CP2K_INPUT.FORCE_EVAL.SUBSYS.COORD) subsection of [SUBSYS](#CP2K_INPUT.FORCE_EVAL.SUBSYS) | ||
(of [FORCE_EVAL](#CP2K_INPUT.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. | ||
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## Results | ||
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The example is run using the serial version of the CP2K binaries: | ||
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```none | ||
cp2k.sopt -o H2O.out H2O.inp & | ||
``` | ||
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After the job has finished, you should obtain the following files: | ||
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- `H2O.out` | ||
- `H2O-pos-1.xyz` | ||
- `H2O-1.restart` | ||
- `H2O-1.restart.bak-1` | ||
- `H2O-1.restart.bak-2` | ||
- `H2O-1.restart.bak-3` | ||
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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: | ||
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```none | ||
cp2k.sopt -o H2O.out H2O-1.restart & | ||
``` | ||
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You can of course also use `H2O-1.restart` as a template for writing an input for further | ||
calculations using the relaxed atomic structures. | ||
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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`. | ||
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In the main output file `H2O.out`, at the end of each geometry optimisation step, we will have the | ||
following information: | ||
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```none | ||
-------- 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 | ||
--------------------------------------------------- | ||
``` | ||
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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`. | ||
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At the end of geometry optimisation, one should obtain something like: | ||
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```none | ||
-------- 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 | ||
--------------------------------------------------- | ||
``` | ||
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which clearly shows all criteria have been satisfied. | ||
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The final Kohn-Sham energies can be obtained at the end of the output: | ||
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```none | ||
******************************************************************************* | ||
*** 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 | ||
``` |
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