Computation of the Lennard Jones curve
In this exercise you will compute the Lennard-Jones energy curve for a system of two Ar atoms.
In Part I you find the instructions for computing the energy of two Ar atoms at a distance $r=3.00 Å$.
In Part II you find the instructions for getting the energy profile as a function of $r$.
Additonal parameters for Xe and combination rules to obtain new parameters are provided in Part III and IV.
Part I: Single Point (Energy) calculation
In this section a commented CP2K input example for a single point calculation is provided. Comments are added and signaled with '!'.
1. Step
Save the following input to a file named energy.inp
- energy.inp
&GLOBAL ! section to select the kind of calculation RUN_TYPE ENERGY ! select type of calculation. In this case: ENERGY (=Single point calculation) &END GLOBAL &FORCE_EVAL ! section with parameters and system description METHOD FIST ! Molecular Mechanics method &MM ! specification of MM parameters &FORCEFIELD ! parameters needed to describe the potential &SPLINE EMAX_SPLINE 10000 ! numeric parameter to ensure calculation stability. Should not be changed &END &NONBONDED ! parameters for the non bonded interactions &LENNARD-JONES ! Lennard-Jones parameters atoms Ar Ar EPSILON [K_e] 119.8 SIGMA [angstrom] 3.401 RCUT [angstrom] 25.0 &END LENNARD-JONES &END NONBONDED &CHARGE ATOM Ar CHARGE 0.0 &END CHARGE &END FORCEFIELD &POISSON ! solver for non periodic calculations PERIODIC NONE &EWALD EWALD_TYPE none &END EWALD &END POISSON &END MM &SUBSYS ! system description &CELL ABC [angstrom] 10 10 10 PERIODIC NONE &END CELL &COORD UNIT angstrom Ar 0 0 0 Ar 3 0 0 &END COORD &END SUBSYS &END FORCE_EVAL
2. Step
Run CP2K with the following arguments:
$ cp2k.popt -i energy.inp -o energy.out
3. Step
Afterwards the file energy.out will look like this:
**** **** ****** ** PROGRAM STARTED AT 2014-01-20 11:32:08.142 ***** ** *** *** ** PROGRAM STARTED ON some_server.ethz.ch ** **** ****** PROGRAM STARTED BY you ***** ** ** ** ** PROGRAM PROCESS ID 24183 **** ** ******* ** PROGRAM STARTED IN /home/you/XERCISES ... ENERGY| Total FORCE_EVAL ( FIST ) energy (a.u.): 0.003617048870059 ... **** **** ****** ** PROGRAM ENDED AT 2014-01-20 12:24:18.154 ***** ** *** *** ** PROGRAM RAN ON some_server.ethz.ch ** **** ****** PROGRAM RAN BY you ***** ** ** ** ** PROGRAM PROCESS ID 24993 **** ** ******* ** PROGRAM STOPPED IN /home/you/EXERCISES
If you get the closing Banner you know that cp2k worked. The following line tells you the result:
ENERGY| Total FORCE_EVAL ( FIST ) energy (a.u.): 0.003617048870059
This is the energy (in Hartree) for a system of 2 Ar atoms at distance $ r=3.00 Å$
Note, that in the input-file EPSILON is given in units of Kelvin, whereas in the output the energy is printed in Hartree, which is the unit of energy in the system of atomic units (a.u.). To convert from Kelvin to Hartree you have to multiply with the Boltzmann constant $ k_\text{b} = 3.1668154 \cdot 10^{-6} \frac{E_\text{H}}{\text{K}} $ .
Part II: Computation of the LJ energy curve
In this section a few scripts to get the LJ energy profiles are presented.
1. Step
In order to get a good profile, a set of energy values as a function of the interatomic distance is needed. You can use the energy.inp input file and change the Ar coordinates in order to get different starting distances.
To do so:
$ mv energy.out energy_dist3A.out
For doing so:
$ cp energy.inp energy_dist2A.inp
then edit the input file with the new coordinates (e.g. 2 Å). you can now run CP2K and produce the output file:
$ cp2k.popt -i energy_dist2A.inp -o energy_dist2A.out
2. Step
When you have tested a few distances, you can produce a table looking like:
| Input file | Distance (Å) | Energy (Eh) |
|---|---|---|
| energy_dist1A.inp | 1 | … |
| energy_dist2A.inp | 2 | … |
| energy_dist3A.inp | 3 | … |
| … | … | … |
This is the Lennard Jones energy curve for two Ar atoms. By using any plotting program you can now get a representation of the energy profile.
3. Step
Here are reported the LJ parameters for Xe atoms. Those are to replace the Ar parameters in the input file, along with your Xe coordinates that have to replace the Ar coordinates. A new LJ curve for Xe atoms can be now generated.
&NONBONDED
&LENNARD-JONES ! Lennard-Jones Xe parameters
atoms Xe Xe
EPSILON [K_e] 232
SIGMA [angstrom] 3.98
RCUT [angstrom] 25.0
&END LENNARD-JONES
&END NONBONDED
&CHARGE
ATOM Xe
CHARGE 0.0
&END CHARGE
4. Step
Here are reported the combination rules for pairs unlike pairs, i.e. for pairs of non identical atoms.
Once generated the ε and σ parameters for the couple Ar/Xe, generate once more the LJ dissociation curve.
Compare the “mixed” curve to the two “pure” curves and report the position and depth of the minimum.
$$ \sigma_{ij}= \sqrt{\sigma_i\sigma_j}$$
$$ \epsilon_{ij}= \sqrt{\epsilon_i\epsilon_j}$$
- The “ LENNARD-JONES ” section must be present for all the three possible couples: Ar-Ar, Xe-Xe and Xe-Ar. Example:
&LENNARD-JONES ! Lennard-Jones parameters for Ar-Ar interaction
atoms Ar Ar
EPSILON [K_e] 119.8
SIGMA [angstrom] 3.401
RCUT [angstrom] 25.0
&END LENNARD-JONES
&LENNARD-JONES ! Lennard-Jones Xe-Xe parameters
atoms Xe Xe
EPSILON [K_e] 232
SIGMA [angstrom] 3.98
RCUT [angstrom] 25.0
&END LENNARD-JONES
&LENNARD-JONES ! Lennard-Jones parameters for Ar-Xe interaction
atoms Ar Xe
EPSILON [K_e] YOUR EPSILON
SIGMA [angstrom] YOUR SIGMA
RCUT [angstrom] 25.0
&END LENNARD-JONES
- The “ CHARGE ” section must be also duplicated:
&CHARGE
ATOM Xe
CHARGE 0.0
&END CHARGE
&CHARGE
ATOM Ar
CHARGE 0.0
&END CHARGE
Questions
- Sketch the LJ energy curve for the two set of parameters ($\sigma$ and $\epsilon$) provided.
- Report, for both curves, the minimum energy distance and the depth of the minimum.
- What are the major differences between the curves? How do they relate to the sets of parameters provided?