Table of Contents
How to Calculate Energy and Forces
Introduction
In this tutorial, we are going to show the reader how to perform a
simple static self-consistent Kohn-Sham Density Functional Theory
energy and force calculation on a system using QUICKSTEP.
We will use face centred cubic bulk Si, with 8 atoms in a cubic unit
cell as an example. The example files are contained in static_calculation.tgz. The calculations were carried out with
CP2K version 2.4.
The reader should note that the integration grid settings used in the example calculations have already chosen to be sufficient for the given accuracy. A separate tutorial is available to show how to achieve this.
Input Files
We first look at the input files required for this calculation. The necessary input files are:
Si_bulk8.inp: the main input file for the calculation, which defines the system and the job parametersBASIS_SET: file containing parameters for the basis sets that can be used byCP2Kfor this calculationGTH_POTENTIALS: file containing parameters for the pseudopotentials that can be used byCP2Kfor this calculation
A list of basis set and pseudopotential files may be found in
cp2k/data/ that comes with a cp2k source release. These
should cover most of the commonly used elements. The user will,
however, need to produce their own main input file for a given
calculation.
Let us look at the main input: Si_bulk8.inp in detail. The name of
this file can be arbitrary, so is the file extension. The input file
is structured into ordered blocks and keywords, the order of which
are unimportant. Each input block is referred to as a “section” in
this tutorial, and some sections are “subsections” of other
sections. Full definitions of the input file format and keywords is
available via the ''CP2K'' input reference manual.
The input file is shown below:
&GLOBAL
PROJECT Si_bulk8
RUN_TYPE ENERGY_FORCE
PRINT_LEVEL LOW
&END GLOBAL
&FORCE_EVAL
METHOD Quickstep
&SUBSYS
&KIND Si
ELEMENT Si
BASIS_SET DZVP-GTH-PADE
POTENTIAL GTH-PADE-q4
&END KIND
&CELL
A 5.430697500 0.000000000 0.000000000
B 0.000000000 5.430697500 0.000000000
C 0.000000000 0.000000000 5.430697500
&END CELL
&COORD
Si 0.000000000 0.000000000 0.000000000
Si 0.000000000 2.715348700 2.715348700
Si 2.715348700 2.715348700 0.000000000
Si 2.715348700 0.000000000 2.715348700
Si 4.073023100 1.357674400 4.073023100
Si 1.357674400 1.357674400 1.357674400
Si 1.357674400 4.073023100 4.073023100
Si 4.073023100 4.073023100 1.357674400
&END COORD
&END SUBSYS
&DFT
BASIS_SET_FILE_NAME BASIS_SET
POTENTIAL_FILE_NAME GTH_POTENTIALS
&QS
EPS_DEFAULT 1.0E-10
&END QS
&MGRID
NGRIDS 4
CUTOFF 300
REL_CUTOFF 60
&END MGRID
&XC
&XC_FUNCTIONAL PADE
&END XC_FUNCTIONAL
&END XC
&SCF
SCF_GUESS ATOMIC
EPS_SCF 1.0E-7
MAX_SCF 300
&DIAGONALIZATION ON
ALGORITHM STANDARD
&END DIAGONALIZATION
&MIXING T
METHOD BROYDEN_MIXING
ALPHA 0.4
NBROYDEN 8
&END MIXING
&END SCF
&END DFT
&PRINT
&FORCES ON
&END FORCES
&END PRINT
&END FORCE_EVAL
The main sections in the input file are:
- ''GLOBAL'': contains general options for the
CP2Krun, such as the name of the job, the type of run etc. - ''FORCE_EVAL'': contains all parameters associated with the evaluation of forces on atoms, this includes the initial atomic coordinates.
We look at each section in detail. The GLOBAL section in
Si_bulk8.inp is presented below:
&GLOBAL PROJECT Si_bulk8 RUN_TYPE ENERGY_FORCE PRINT_LEVEL LOW &END GLOBAL
We will be doing a static energy and force calculation, in this
case, we must set ''RUN_TYPE'' to ENERGY_FORCE. Keyword ''PROJECT'' is an
alias for PROJECT_NAME, which sets the root-name of the
calculation, in this case Si_bulk8. Any output files automatically
generated by CP2K will have the name prefixed by
Si_bulk8. ''PRINT_LEVEL'' controls the default verbosity of the main
output of CP2K, in this example, it is set to “low”. The
verbosity of the output can be fine-tuned by overriding this setting
in each individual subsection of the input.
We now explain the section FORCE_EVAL line-by-line.
METHOD Quickstep
The keyword ''METHOD'' chooses the method for evaluating the forces on
atoms to QUICKSTEP, i.e. Density Functional Theory using the
Gaussian and Planewaves (GPW) method.
&SUBSYS
&KIND Si
ELEMENT Si
BASIS_SET DZVP-GTH-PADE
POTENTIAL GTH-PADE-q4
&END KIND
&CELL
A 5.430697500 0.000000000 0.000000000
B 0.000000000 5.430697500 0.000000000
C 0.000000000 0.000000000 5.430697500
&END CELL
&COORD
Si 0.000000000 0.000000000 0.000000000
Si 0.000000000 2.715348700 2.715348700
Si 2.715348700 2.715348700 0.000000000
Si 2.715348700 0.000000000 2.715348700
Si 4.073023100 1.357674400 4.073023100
Si 1.357674400 1.357674400 1.357674400
Si 1.357674400 4.073023100 4.073023100
Si 4.073023100 4.073023100 1.357674400
&END COORD
&END SUBSYS
The subsection ''SUBSYS'' defines the simulation unit cell and the initial coordinates of atoms in the calculation.
The subsection ''KIND'' gives definitions of elements in the
calculation. There must be one KIND subsection per element. In
this example, for Si, we have defined the basis set to be used:
DZVP-GTH-PADE (double-\(\zeta\) with polarisation basis optimised
for Geodecker-Teter-Hutter PADE LDA pseudopotential); and the
pseudopotential: GTH-PADE-q4 (Geodecker-Teter-Hutter PADE LDA
pseudopotential with 4 valence electrons).
The basis set and pseudopotential names must correspond to an
existing entry in the corresponding basis set and pseudopotential
files defined by ''BASIS_SET_FILE_NAME'' and ''POTENTIAL_FILE_NAME''
keywords in ''DFT'' subsection, in FORCE_EVAL section. The chosen
basis for Si corresponds to parameters:
Si DZVP-GTH-PADE
2
3 0 1 4 2 2
1.2032422345 0.3290350445 0.0000000000 0.0474539126 0.0000000000
0.4688409786 -0.2533118323 0.0000000000 -0.2594473573 0.0000000000
0.1679863234 -0.7870946277 0.0000000000 -0.5440929303 0.0000000000
0.0575619526 -0.1909898479 1.0000000000 -0.3624010364 1.0000000000
3 2 2 1 1
0.4500000000 1.0000000000
in file BASIS_SET; and the chosen pseudopotential corresponds to
parameters:
Si GTH-PADE-q4 GTH-LDA-q4
2 2
0.44000000 1 -7.33610297
2
0.42273813 2 5.90692831 -1.26189397
3.25819622
0.48427842 1 2.72701346
in file GTH_POTENTIALS.
The subsection ''CELL'' defines the simulation unit cell used in a calculation. In this example, we define the unit cell as cubic, with lattice constant equal to 5.4306975 Angstroms. “Angstrom” is the default unit for cell vectors. ''A'', ''B'' and ''C'' are the first, second and third lattice (cell) vectors. There are many ways to define the cell, see ''CP2K'' input reference manual for more details.
The initial atomic coordinates are specified in the ''COORD''
subsection. The default input format for atomic coordinates in
CP2K is:
<ATOM_KIND> X Y Z
where X, Y and Z are Cartesian coordinates in Angstroms. This
can be changed by configuring keyword ''SCALED'' to .TRUE., in the
COORD subsection, which makes the coordinate input X Y Z to
be fractional with respect to the lattice vectors. One can also
change the unit for the Cartesian coordinates by setting the keyword
''UNIT'' with in the subsection. <ATOM_KIND> should be a label that
corresponds to the definition of the elements in the ''KIND''
subsections.
After the SUBSYS section in the input file Si_bulk8.inp follows
the ''DFT'' subsection, which controls all aspects of the
self-consistent Kohn-Sham Density Functional Theory
calculation. This subsection is only relevant if and only if the
METHOD keyword in FORCE_EVAL is set to QUICKSTEP.
BASIS_SET_FILE_NAME BASIS_SET POTENTIAL_FILE_NAME GTH_POTENTIALS
As already mentioned above, the keywords ''BASIS_SET_FILE_NAME'' and ''POTENTIAL_FILE_NAME'' set the files that contains basis set and pseudopotential parameters.
&QS EPS_DEFAULT 1.0E-10 &END QS
The ''QS'' subsection contains general control parameters used by
QUICKSTEP. ''EPS_DEFAULT'' sets the default value for all tolerances
used within QUICKSTEP. The individual tolerances (EPS_*) can be
set, and they will override the EPS_DEFAULT value.
&MGRID NGRIDS 4 CUTOFF 300 REL_CUTOFF 60 &END MGRID
The ''MGRID'' subsection defines how the integration grid used in
QUICKSTEP calculations should be setup. QUICKSTEP uses a
multi-grid method for representing Gaussian functions numerically on
the grid. Narrow and sharp Gaussians are mapped onto a finer grid
than wider and smoother Gaussians. In this case, we are telling the
code to set up 4 levels of multi-grids, with the planewave cutoff of
the finest grid set to be 300 Ry, and with the grid spacing
underneath any Gaussian functions to be finer than the equivalent
planewave cutoff of 60 Ry. The users should read the tutorial
“Converging the CUTOFF and REL_CUTOFF” for details on how these
parameters affect the grid constructed, and how to define a
sufficient grid for their calculation. In this example, the grid
defined has already been found to be sufficient for the energy and
force calculation.
The ''XC'' subsection follows:
&XC &XC_FUNCTIONAL PADE &END XC_FUNCTIONAL &END XC
This defines which exchange-correlation density functional we want to use. In this we choose PADE LDA functional, which is consistent with the basis set and pseudopotential we have chosen.
&SCF
SCF_GUESS ATOMIC
EPS_SCF 1.0E-7
MAX_SCF 300
&DIAGONALIZATION
ALGORITHM STANDARD
&END DIAGONALIZATION
&MIXING
METHOD BROYDEN_MIXING
ALPHA 0.4
NBROYDEN 8
&END MIXING
&END SCF
The ''SCF'' subsection defines all the settings related to methods used to find a self-consistent solution of the Kohn-Sham DFT formalism.
''SCF_GUESS'' sets how the initial trial electron density function
\(\rho(\vec{r})\) is to be generated. In this example (ATOMIC), the
initial density is to be generated using overlapping of atomic
charge densities. A good starting point for the electron density in
the self-consistency loop is important in obtaining a convergent
result quickly. ''EPS_SCF'' sets the tolerance of the charge density
residual. This overrides the EPS_DEFAULT value set in QS
subsection. ''MAX_SCF'' sets the maximum number of self-consistency
loops QUICKSTEP is allowed to perform for each ground-state energy
calculation.
&DIAGONALIZATION ON ALGORITHM STANDARD &END DIAGONALIZATION
The ''DIAGONALIZATION '' subsection tells the code to use the
traditional diagonalisation method for finding the ground state
Kohn-Sham energy and electron density. The subsection heading also
takes an argument, and in this case is set to “ON”, which
equivalent to “.TRUE.” or “T”, and indicates that the
diagonalisation method is turned on. One can also omit the value of
the subsection heading, which defaults to “.TRUE.”. The
alternative to diagonalisation is to use the Orbital Transform (OT)
method, in which case, the user should either delete the
DIAGONALIZATION block or change “ON” to “OFF” (or
“.FALSE.”), and add the ''OT'' subsection instead. The ''ALGORITHM''
keyword sets the algorithm to use for diagonalisation of the
Kohn-Sham Hamiltonian. “STANDARD” means the standard
LAPACK/SCALAPACK subroutines are to be used for diagonalisation.
&MIXING T METHOD BROYDEN_MIXING ALPHA 0.4 NBROYDEN 8 &END MIXING
The ''MIXING'' subsection contains all the parameters associated with
charge mixing in a self-consistency calculation. The subsection also
admits a value, which can be either .TRUE. (T) or .FALSE.
(F), which switches charge mixing on or off. The default is
.TRUE.. Note that this subsection only applies to the traditional
diagonalisation method. The OT method uses a different approach for
charge mixing, and is explained in other tutorials. The keyword
''ALPHA'' sets the mixing parameter; in this example 0.4 of the output
density will be mixed with 0.6 of the input density to form the new
input density in the next SCF iteration. The keyword ''METHOD'' sets
the mixing method; in this case, we will use Broyden mixing. The
keyword ''NBROYDEN'' is an alias to the parameter NBUFFER, and it
sets the number of histories to be used in the Broyden mixing
algorithm.
The final ''PRINT'' subsection in FORCE_EVAL section:
&PRINT &FORCES ON &END FORCES &END PRINT
tells CP2K, in this case, to print out atomic forces in the main
output of the calculation.
Running the Calculation
To run the calculation, the reader needs to have a working CP2K
package compiled, and with the path to its binaries in the system
PATH. The files Si_bulk8.inp, BASIS_SET and GTH_POTENTIAL
should be in the same working directory. In this example, we will
use the MPI version of CP2K. Type command:
mpirun -n 2 cp2k.popt -o Si_bulk8.out Si_bulk8.inp &
in the working directory to run CP2K in parallel with 2 MPI
processes in the background. The -o option redirects the CP2K
output to file Si_bulk8.out. Note that the -o option appends
output of successive runs to Si_bulk8.out, so if the reader wants
to start completely from afresh, they must delete Si_bulk8.out
before running a new calculation.
Obtaining the Results
After the job has finished, we should obtain the following files:
Si_bulk8.outSi_bulk8-RESTART.wfnSi_bulk8-RESTART.wfn.bak-1
The file Si_bulk8.out contains the main output of the
job. Si_bulk8-RESTART.wfn is the final Kohn-Sham wavefunctions
from the calculation. Si_bulk8-RESTART.wfn.bak-<n> records the
Kohn-Sham wavefunctions obtained from the <n>-th previous SCF
step; in this case, Si_bulk8-RESTART.wfn.bak-1 contains the
wavefunctions obtained from the last SCF step. Should the reader
want to start a new calculation from the previous calculated
wavefunctions, they need to change the SCF_GUESS keyword in SCF
subsection of FORCE_EVAL section to RESTART:
SCF_GUESS RESTART
provided that the new calculation shares the same PROJECT_NAME as
the one that generated the wavefunctions. Otherwise, we would need
to rename the restart wavefunction files to correspond to the
project name of the new calculation.
We now look at the main CP2K output in more detail.
Number of electrons: 32
Number of occupied orbitals: 16
Number of molecular orbitals: 16
Number of orbital functions: 104
Number of independent orbital functions: 104
Extrapolation method: initial_guess
SCF WAVEFUNCTION OPTIMIZATION
Step Update method Time Convergence Total energy Change
------------------------------------------------------------------------------
1 NoMix/Diag. 0.40E+00 0.6 0.75558724 -32.2320848878 -3.22E+01
2 Broy./Diag. 0.40E+00 1.1 0.05667976 -31.1418135481 1.09E+00
3 Broy./Diag. 0.40E+00 1.1 0.09691469 -31.1974003416 -5.56E-02
4 Broy./Diag. 0.40E+00 1.1 0.00245608 -31.3378474040 -1.40E-01
5 Broy./Diag. 0.40E+00 1.1 0.00235460 -31.3009654398 3.69E-02
6 Broy./Diag. 0.40E+00 1.1 0.00007565 -31.2972158934 3.75E-03
7 Broy./Diag. 0.40E+00 1.1 0.00009004 -31.2977293749 -5.13E-04
8 Broy./Diag. 0.40E+00 1.1 0.00000186 -31.2978454163 -1.16E-04
9 Broy./Diag. 0.40E+00 1.1 0.00000252 -31.2978835492 -3.81E-05
10 Broy./Diag. 0.40E+00 1.1 5.6405E-09 -31.2978852054 -1.66E-06
*** SCF run converged in 10 steps ***
The above shows a typical output from a self-consistent Kohn-Sham ground state calculation. It states that we are using diagonalisation method with Broyden charge mixing, and it took 10 Broyden mixing steps (each containing a diagonalisation process to obtain the wavefunctions) to reach the required tolerance for self-consistency.
Electronic density on regular grids: -31.9999999889 0.0000000111
Core density on regular grids: 31.9999999939 -0.0000000061
Total charge density on r-space grids: 0.0000000051
Total charge density g-space grids: 0.0000000051
Overlap energy of the core charge distribution: 0.00000000005320
Self energy of the core charge distribution: -82.06393942512820
Core Hamiltonian energy: 18.06858429706010
Hartree energy: 42.41172824581682
Exchange-correlation energy: -9.71425832315952
Total energy: -31.29788520535761
ENERGY| Total FORCE_EVAL ( QS ) energy (a.u.): -31.297885372811002
ATOMIC FORCES in [a.u.]
# Atom Kind Element X Y Z
1 1 Si 0.00000000 0.00000000 0.00000000
2 1 Si 0.00000000 0.00000001 0.00000001
3 1 Si 0.00000001 0.00000001 0.00000000
4 1 Si 0.00000001 0.00000000 0.00000001
5 1 Si -0.00000001 -0.00000001 -0.00000001
6 1 Si -0.00000001 -0.00000001 -0.00000001
7 1 Si -0.00000001 -0.00000001 -0.00000001
8 1 Si -0.00000001 -0.00000001 -0.00000001
SUM OF ATOMIC FORCES -0.00000000 -0.00000000 -0.00000000 0.00000000
The above shows the results on final energies and forces. One should always check if the total number of electrons calculated from the final electron density, in this case 31.9999999939, is correct.
The results show that the force on the atoms are almost zero. This means the system is more or less relaxed, and its geometry is close to its optimal at ground state.
Adding Smearing
In the example so far, we have not used any smearing on electron occupation. This is fine for system with a large band gap. However, for metals or systems with a small gap, this may cause the calculation to be unstable, and the self-consistency loop may never converge, due to the discontinuity in the electron occupation function.
To add smearing, we need to add the subsection ''SMEAR'' inside
subsection SCF:
&SMEAR ON METHOD FERMI_DIRAC ELECTRONIC_TEMPERATURE [K] 300 &END SMEAR
This tells QUICKSTEP to use a smearing function for the electron
occupation. In this example, we use the Fermi-Dirac smearing
function, with the electron temperature being set to 300 K. Note
that, in CP2K, one can explicitly define the unit of a given input
value by using a unit enclosed in square bracket, such as “[K]”,
before the value.
This is not all, since smearing may lead to occupation of molecular
orbitals in the conduction band, we must tell CP2K to include
extra, empty, molecular orbitals into the calculation, which
otherwise would be omitted (for reducing computational cost). To do
this, we need to add the keyword ''ADDED_MOS'' in the SCF
subsection:
ADDED_MOS 10
In this example, we have asked CP2K to not to omit 10 of the
lowest empty molecular orbitals in the calculation. It should be
noted that given a chosen basis set, there is a maximum number of
molecular orbitals, i.e. the number of eigenvectors of the
Hamiltonian, one can generate. In theory, the maximum should be the
rank of the Hamiltonian generated in the calculation.
In the output of the calculation using smearing, we first notice that:
Number of electrons: 32 Number of occupied orbitals: 16 Number of molecular orbitals: 26 Number of orbital functions: 104 Number of independent orbital functions: 104
unlike in the previous case with no smearing, now 26 molecular orbitals have been used during the calculation. There are a total of 104 basis functions (“atom centred orbitals” spanned by Cartesian Gaussians) used in the calculation for the given basis set, so 26 is well within the limit of the calculation.
Electronic density on regular grids: -31.9999999889 0.0000000111 Core density on regular grids: 31.9999999939 -0.0000000061 Total charge density on r-space grids: 0.0000000051 Total charge density g-space grids: 0.0000000051 Overlap energy of the core charge distribution: 0.00000000005320 Self energy of the core charge distribution: -82.06393942512820 Core Hamiltonian energy: 18.06842027191411 Hartree energy: 42.41184371469986 Exchange-correlation energy: -9.71419454555998 Electronic entropic energy: -0.00001687947145 Fermi energy: 0.20867150262130 Total energy: -31.29788686349247 ENERGY| Total FORCE_EVAL ( QS ) energy (a.u.): -31.297887031736590
In the final energy section of the output, we notice that there is an extra entropy \((TS)\) term:
Electronic entropic energy: -0.00001687947145
This should be small for the calculation to be a reliable approximation to the zero electron temperature result. The final free energy is the sum of the total DFT energy and the entropic energy. The total DFT energy is given by:
Total energy: -31.29788686349247
and the final free energy extrapolated for \(TS \to 0\) is given by:
ENERGY| Total FORCE_EVAL ( QS ) energy (a.u.): -31.297887031736590
This is the energy to be quoted as the final result.