In this exercise, you will simulate a fluid of monoatomic particles that interact with a Lennard-Jones potential. The method to be used is molecular dynamics (MD) with periodic boundary conditions using CP2K. The aim is to explore the method, calculate the radial distribution function $g(r)$ and investigate a variety of ensembles.
You are expected to carry out an MD simulation of Lennard-Jones (L-J) fluid containing mono-atomic particles. The L-J potential energy expression in use is
$U(x)=4\epsilon \left [\left ( \frac{\sigma }{r_{ij}} \right )^{12}- \left ( \frac{\sigma }{r_{ij}} \right )^{6} \right ]$
where $\epsilon$ is the well depth, $\sigma$ is related to the minimum of the Lennard-Jones potential, and $r_{ij}$ is the distance between atoms i and j.
Periodic boundary conditions (PBC) should be used in this simulation. The atom near the ”edge” of the simulation box interacts with atoms contained in the neighboring image of the box. In computer simulations, one of these is the original simulation box, and others are copies called images. During the simulation, only the properties of the original simulation box need to be recorded and propagated. The minimum-image convention is a common form of PBC particle bookkeeping in which each individual particle in the simulation interacts with the closest image of the remaining particles in the system. To prevent artifacts, it requires a cut-off value for $r_{ij}$ of the L-J potential. For realistic results, the cut-off should be less than half of the simulation box size and larger than $\sigma$. The radial distribution function, (or pair correlation function) $g(r)$, in a system of particles (atoms, molecules, colloids, etc.), describes how the density varies as a function of distance from a reference particle.
In this section, we provide you with an example CP2K input for an MD calculation. Extensive comments have been added to the file, which starts with the symbol '#'.
Load the CP2K module as explained in Exercise 0, create a directory ex1 and change to it:
mkdir ex1 cd ex1
Save the following input to a file named argon.inp (for example using $ vim argon.inp):
## It's highly recommended to go to
## https://manual.cp2k.org/
## and learn how to set up a CP2K
## calculation correctly, using the manual.
&GLOBAL
PROJECT ar108 #Project Name
RUN_TYPE md #Calculation Type: MD (molecular dynamics), GEO_OPT (Geometry Optimization), Energy (Energy Calculation)
&END GLOBAL
&MOTION
&MD
ENSEMBLE NVE #The ensemble for MD propagation, NVE (microcanonical), NVT (canonical), NPT_I (NPT with isotropic cell)
STEPS 100 #The number of MD steps to perform
TIMESTEP 5. #The length of an integration step (fs)
TEMPERATURE 85.0 #The temperature in K used to initialize the velocities with init and pos restart, and in the NPT/NVT simulations
&END MD
&END MOTION
&FORCE_EVAL
METHOD FIST #Method to calculate force: Fist (Molecular Mechanics), QS or QUICKSTEP (Electronic structure methods, like DFT)
&MM
&FORCEFIELD
&CHARGE #charge of the MM atoms
ATOM Ar #Defines the atomic kind of the charge
CHARGE 0.0 #Defines the charge of the MM atom in an electron charge unit
&END
&NONBONDED
&LENNARD-JONES #LENNARD-JONES potential type.Functional form: V(r) = 4.0 * EPSILON * [(SIGMA/r)^12-(SIGMA/r)^6]
atoms Ar Ar #Defines the atomic kind involved in the nonbonded potential
EPSILON 119.8 #Defines the EPSILON parameter of the LJ potential (K_e)
SIGMA 3.405 #Defines the SIGMA parameter of the LJ potential (Angstrom)
RCUT 8.4 #Defines the cutoff parameter of the LJ potential
&END LENNARD-JONES
&END NONBONDED
&END FORCEFIELD
&POISSON # Poisson solver
&EWALD
EWALD_TYPE none
&END EWALD
&END POISSON
&END MM
&SUBSYS
&CELL
ABC 17.1580 17.158 17.158 #Simulation box size
&END CELL
&COORD
Ar -8.53869012951987116 -15.5816257770688615 2.85663672298278293
Ar 1.53007304829383051 9.28528179040142554 11.1777824543317941
Ar 11.9910225119590699 -7.48825329565798015 -9.96545306345559823
Ar -12.6782400030290496 -3.34105872014234606 4.07471097818485806
Ar -1.77046254278594462 -0.232459464264201887 13.2012946017273016
Ar 8.01761371186688443 -2.57249587730733298 -4.12720554747711432
Ar 8.57849517232300052 4.01396664624232002 5.57368821983998419
Ar -3.89200679277030925 -10.2930917801117356 -6.98640232289045482
Ar -3.35457160564444568 -16.1119619276890056 16.1358515626317427
Ar 9.78957155103081966 -16.2628264194939263 -5.69790857071688350
Ar 0.505143495414835719 -4.22978415759568183 12.4854171634357307
Ar 15.5632243939617503 -7.98048905093276240 2.20994708545912832
Ar -5.40741643995084953 -2.64764457113743079 -0.681485212640798199
Ar -0.983719068448489081E-01 -1.73674004862212694 -7.11915545117132265
Ar 7.52655781331927187 -5.52969969672439632 -12.8886150439489313
Ar -5.45655410995716128 0.564445754429787061 2.03902510096247536
Ar -11.8590998267164665 3.40407446386207724 3.72687933934436399
Ar 16.7175362589401821 -7.47132377347522780 -1.02274476672697889
Ar -20.4572129717055340 -5.73700807719791683 4.81845086375497811
Ar 14.8485522289272627 -1.41608633045414667 -16.0839111490847451
Ar 8.04379470511429595 -8.14033814842439263 -4.75543123809189261
Ar 12.2738439612049568 -1.70589834674486429 12.9622486199573572
Ar -0.421851806372696092 -11.1177490353157999 20.4545363332536283
Ar 2.28194341698637571 5.92083917539752136 -11.1732449877738436
Ar -13.9648466918215064 8.77923885764231926 8.07373370482465091
Ar -10.3147439499058429 6.38529561240966004 -15.3411964215061527
Ar -2.71899964647918457 -21.4890074469143855 10.8678096818980006
Ar -17.7923879123397271 -10.7840901151121251 -4.83954996524571968
Ar 5.23494138507746420 -6.79222906792632841 -6.07187690814296133
Ar 3.52448750638480446 -10.1225951872349782 2.96829048662758721
Ar -16.1586602901979361 -5.18274316385346445 8.57072694078649455
Ar -5.80982824422251287 4.32640193501643733 2.55599101868223322
Ar 6.29160109084684382 27.5741337288405717 15.0246410590392632
Ar -3.18741711710350684 23.2996469099840624 -16.8034854143018748
Ar -4.20225755039435622 9.36037725943080190 16.5891306154890081
Ar -7.64392908749747946 -9.52432384411045341 -29.8228731471089645
Ar 0.545352525792712428 13.9240554617015260 -0.383786780333776500
Ar -5.27432886808646906 -5.53813781787395865 -20.3014703747109415
Ar 22.9921850152838871 6.78619371666398941 -1.98289905290632484
Ar 19.7720034229251880 -10.2373337687313679 -3.33081818566269172
Ar 0.156776902886395425 6.59630118110908725 8.90749062505743083
Ar 5.57937381862174053 0.233106223140015806 1.02752287819280941
Ar -3.64343561800208793 3.96448881012491006 25.8752124557059595
Ar -0.248491698112870391 20.4489725648023182 -2.51220445353457666
Ar 2.93626708600658270 0.859812213376437984 9.96743307236779508
Ar 3.30384315693043895 -2.92421266591109408 -6.34927042371499883
Ar -6.15490235244551265 -6.84961480075890883 -6.46204144605644260
Ar -23.2388291761596619 -28.1213094673208666 7.13721047187827917
Ar 4.11526291325474780 2.71564143367947342 -0.852030043744060328
Ar 14.6194148692240713 2.80815182256426210 1.93601975975151541
Ar 18.9667954753247869 16.5700888519293095 13.3423444868082761
Ar -28.6124161416877705 2.84353637083477562 -9.23601973326721648
Ar -5.97004594556101331 -16.2230172568109978 -9.22928061840017477
Ar 10.0481077882725955 16.3854819569745231 5.12578711346205651
Ar -7.22508507825336643 6.34615422233080650 -0.680757463730119028
Ar -12.0138912984383506 -10.4653110276797570 -6.43434787584580103
Ar -8.53169926903037457 12.8976589212818862 -0.890361252446473683
Ar -25.3700692950848676 3.33119906434656077 -7.93917685683272722
Ar 2.90163480643285920 -12.9668181360039672 -7.94907759259854707
Ar -13.5963940986222074 11.9896580951935974 -3.55068754869933789
Ar 13.1416029517342476 4.97143783446568488 -3.50841252726170705
Ar -13.3295460955805254 16.0410015777677764 7.05282797577515375
Ar -4.15068335494176122 -19.5111913798076593 21.0255971827539376
Ar -8.42944270819351793 16.3065160593537009 -18.2887817284733885
Ar 0.788636333898691255 9.59016836817029095 22.1772606194495872
Ar -2.92606778628861974 -7.97408054890791007 -21.3519900334304964
Ar -6.39959865978756426 -4.56280461803643256 2.75533571094951402
Ar -4.04423878093174860 -14.9275965394452506 -5.58561473738824965
Ar -27.8524912514281482 0.802052180719123098 -3.02663789713126441
Ar -2.83966529645897792 7.11627121253196915 6.18547332762273783
Ar -8.68327887612401739 -6.67088300493855879 -9.15815450219801264
Ar -11.9620847111198501 -2.20956249614563038 -1.83979975374852245
Ar 22.6848553724304907 12.2047209420099971 1.01238797839832362
Ar 6.29501012040417507 -0.769712471349173866 -6.91454332254278281
Ar 3.49995546789933476 -8.00704920137973453 -0.426526631939732892
Ar 0.385154812289867643 17.8769740351009112 -17.4065226240143041
Ar 21.2288869131365736 10.2327102035561044 -13.0872200859088803
Ar 1.22082587001210396 5.83597435065779457 16.8450099266840283
Ar -7.08754036219628425 6.03412971863339109 -22.3251445579668015
Ar -0.244265849036998037E-01 17.4693605251376454 7.37116730966604194
Ar 15.0981822679441553 9.88940516251130397 -8.49382740142986670
Ar -6.57877688336587152 -15.0484532074656290 14.7230359830473887
Ar -2.22666666633409394 -4.18421900331013674 -2.47007887105670587
Ar 5.20621069851729867 -22.6565181989138011 7.39475674805799521
Ar -8.85828800414884299 -2.47510661993999781 2.35441398531938617
Ar 6.75202354538700167 0.430391383628436597 5.43492495261394382
Ar 11.9263127546080856 8.13267254152258445 2.40081132956567966
Ar -14.5507562394484040 -0.471540677239574602 -13.7058431104765983
Ar 14.1157692422228553 -2.98968593175088149 24.6842798176059546
Ar -3.35107336204723527 -0.681362546744063047 -7.37039916831594510
Ar 7.79269876443546838 3.30687615091469800 -0.732378021069576002E-01
Ar -1.13289059102623746 -17.1672835835708497 -12.9126466371968966
Ar -9.21054349522787241 -10.4846510042527843 -8.38485797788161591
Ar -6.47848777956778044 -3.90736653076878993 -10.6499668409808841
Ar 0.987874979233200667 13.7363585340729077 5.07209659800543733
Ar 8.86097814789463278 9.96103887786039799 1.09373795795780060
Ar -6.58068766844202013 -20.4019345282015756 -7.28935608176262662
Ar -0.448977062720621045 19.5862520159664086 11.0351198968750293
Ar 7.36056937465398153 -2.69594281683156067 7.26081874603436361
Ar 13.8791344546872004 12.1903465249438128 1.24889885444881155
Ar -3.65782753722175302 19.7829061761924159 -11.8161510229542408
Ar 1.49729450944005649 -5.39289977250827679 1.92445849672255198
Ar 18.5861605633917577 3.00868366398259690 2.06440131010935168
Ar 6.20730767975507014 -9.47418398815358032 5.54930507752316249
Ar 3.65054837888884753 3.43181054126032858 -4.31160813615129435
Ar 2.67862616463048520 2.29300605146530545 5.98502962150055051
Ar 24.5113122275150914 4.00733170976478448 13.1412501215423774
Ar -0.600233262008137092 3.62825631372324597 6.38411284716526772
&END COORD
&TOPOLOGY
&END TOPOLOGY
&END SUBSYS
&END FORCE_EVAL
& start a section and must be terminated with an &END SECTION-NAME. Other instructions are called keywords. The indentation is ignored but recommended for readability.
Run a CP2K calculation with the following command:
$ cp2k.popt -i argon.inp -o argon.out
$ cp2k.popt -i argon.inp | tee argon.out
which will write the output simultaneously to a file argon.out and show it in the terminal.
In the previous sections, you have already run NVE ensemble molecular dynamics calculations for liquid Ar. In this section, we will focus on the NVT ensembles.
Although the most popular Nose-Hoover thermostat is commonly-used in other MD codes, the original Nose-Hoover thermostat has an ergodic issue. This has been solved by Mark Tuckerman et al. See Nose-Hoover Chain Thermostat In CP2K, the default length of the Nose-Hoover chain is set to 3. (See Manual %MOTION%MD%THERMOSTAT%NOSE)
To set up an NVT calculation, change the settings in the &MD section as shown below:
&MD
ENSEMBLE NVT
STEPS 3000
TIMESTEP 5
TEMPERATURE 298
&THERMOSTAT
REGION MASSIVE
&NOSE #Uses the Nose-Hoover thermostat
TIMECON 100 #timeconstant of the thermostat chain, how often does the thermostat adjust your system
&END NOSE
&END
&END MD
Alternatively, one can also use Canonical sampling through velocity rescaling (CSVR) as developed by Giovanni Bussi et al.
&MD
ENSEMBLE NVT
STEPS 3000
TIMESTEP 5
TEMPERATURE 298
&THERMOSTAT
&CSVR
TIMECON 100 #timeconstant of the CSVR, how often does the thermostat adjust your system
&END CSVR
&END
&END MD
To set up an NPT calculation, change the settings in the &MD section as shown below:
&FORCE_EVAL
...
STRESS_TENSOR ANALYTICAL
...
&END FORCE_EVAL
...
...
&MD
ENSEMBLE NPT_I #constant temperature and pressure using an isotropic cell
STEPS 3000
TIMESTEP 5.
TEMPERATURE 85.0
&BAROSTAT
PRESSURE 0. # PRESSURE, unit[bar]
TIMECON 1000
&END BAROSTAT
&THERMOSTAT
&NOSE
TIMECON 1000
&END NOSE
&END THERMOSTAT
&END MD
For the purposes of this example, you should see a file called ar108-1.restart (at least for Part 1 of this exercise). Note: there will also be multiple backup files (.bak-# suffixes) which you do not need to care about.
These files are nothing but another input file. However, their parameters are set such that they continue a CP2K calculation from the last step of the simulation which generated the RESTART file.
Here are two options for how you can use these RESTART-files:
1. Directly using the RESTART as an input.
cp ar108-1.restart argon_follow_up.inp
cp2k -i argon_followup.inp -o argon_followup.out
2. You can also tell CP2K to load a specific RESTART-file.
argon_followup.inp
&EXT_RESTART RESTART_FILE_NAME ar108-1.restart &END EXT_RESTART
cp2k -i argon_followup.inp -o argon_followup.out
In this section we analyze the dependence of the radial distribution function (rdf), $g(r)$, on the temperature of the system. To do so, you should plot $g(r)$ against various temperatures and examine the effects. You can use VMD (as explained below) or write your own program (Fortran, C, C++, Python etc.) to calculate the rdf.
VMD comes with an extension for exactly this purpose: In the VMD Main window open “Extensions → Analysis” click on “Radial Pair Distribution function $g(r)$“. In the appearing window use “Utilities → Set unit cell dimensions” to tell VMD the size of the simulation box you used. After that use Selection 1 and 2 to define the atomic types that you want to calculate the rdf for, for example “element Ar”. In the plot window, use the “File” menu to save the plot data.
exp_gr.dat, below). What does this say about the structure of the liquid and did you expect this?.0681 -.0789 .1362 -.0545 .2043 -.0263 .2724 -.0062 .3405 -.0008 .4086 -.0088 .4767 -.0222 .5448 -.0313 .6129 -.0305 .6810 -.0206 .7491 -.0082 .8172 -.0010 .8853 -.0031 .9534 -.0123 1.0215 -.0222 1.0896 -.0259 1.1577 -.0203 1.2258 -.0085 1.2939 .0026 1.3620 .0065 1.4301 .0011 1.4982 -.0095 1.5663 -.0180 1.6344 -.0180 1.7025 -.0086 1.7706 .0053 1.8387 .0155 1.9068 .0157 1.9749 .0058 2.0430 -.0076 2.1111 -.0153 2.1792 .0111 2.2473 .0038 2.3154 .0210 2.3835 .0295 2.4516 .0232 2.5197 .0052 2.5878 -.0128 2.6559 -.0176 2.7240 .0030 2.7921 .0241 2.8602 .0455 2.9283 .0427 2.9964 .0112 3.0645 -.0279 3.1326 -.0276 3.2007 .0721 3.2688 .3212 3.3369 .7356 3.4050 1.2831 3.4731 1.8857 3.5412 2.4408 3.6093 2.8510 3.6774 3.0542 3.7455 3.0403 3.8136 2.8497 3.8817 2.5549 3.9498 2.2343 4.0179 1.9474 4.0860 1.7218 4.1541 1.5545 4.2222 1.4240 4.2903 1.3059 4.3584 1.1856 4.4265 1.0624 4.4946 .9458 4.5627 .8471 4.6308 .7730 4.6989 .7219 4.7670 .6862 4.8351 .6571 4.9032 .6293 4.9713 .6023 5.0394 .5796 5.1075 .5651 5.1756 .5604 5.2437 .5640 5.3118 .5725 5.3799 .5829 5.4480 .5945 5.5161 .6091 5.5842 .6294 5.6523 .6569 5.7204 .6911 5.7885 .7291 5.8566 .7675 5.9247 .8043 5.9928 .8395 6.0609 .8751 6.1290 .9137 6.1971 .9569 6.2652 1.0038 6.3333 1.0520 6.4014 1.0983 6.4695 1.1398 6.5376 1.1753 6.6057 1.2048 6.6738 1.2287 6.7419 1.2475 6.8100 1.2608 6.8781 1.2685 6.9462 1.2706 7.0143 1.2679 7.0824 1.2618 7.1505 1.2536 7.2186 1.2435 7.2867 1.2304 7.3548 1.2128 7.4229 1.1891 7.4910 1.1594 7.5591 1.1254 7.6272 1.0899 7.6953 1.0561 7.7634 1.0255 7.8315 .9982 7.8996 .9726 7.9677 .9470 8.0358 .9206 8.1039 .8942 8.1720 .8698 8.2401 .8499 8.3082 .8360 8.3763 .8280 8.4444 .8247 8.5125 .8244 8.5806 .8259 8.6487 .8293 8.7168 .8355 8.7849 .8452 8.8530 .8590 8.9211 .8757 8.9892 .8937 9.0573 .9113 9.1254 .9276 9.1935 .9431 9.2616 .9587 9.3297 .9757 9.3978 .9943 9.4659 1.0138 9.5340 1.0326 9.6021 1.0492 9.6702 1.0628 9.7383 1.0738 9.8064 1.0830 9.8745 1.0913 9.9426 1.0988 10.0107 1.1048 10.0788 1.1082 10.1469 1.1079 10.2150 1.1041 10.2831 1.0977 10.3512 1.0904 10.4193 1.0835 10.4874 1.0774 10.5558 1.0716 10.6236 1.0648 10.6917 1.0559 10.7598 1.0448 10.8279 1.0322 10.8960 1.0196 10.9641 1.0083