====== 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''
     
<code - 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
</code>

=== 2. Step ===
Run CP2K with the following arguments:
<code>
$ cp2k.popt -i energy.inp -o energy.out
</code>

=== 3. Step ===
Afterwards the file energy.out will look like this:
<code>
 **** **** ******  **  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
</code>

If you get the closing Banner you know that cp2k worked. The following line tells you the result:
<code>
ENERGY| Total FORCE_EVAL ( FIST ) energy (a.u.):              0.003617048870059
</code>

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.

<note important>
The output file will be rewritten every time you run a calculation, unless you change its name.
</note>

To do so: 
<code>
$ mv energy.out energy_dist3A.out
</code>

<note tip>
If you run multiple calculations, it is always good to keep track of what you have done by producing an input and an output for every distance you are planning to run. 
</note>

For doing so:
<code>
$ cp energy.inp energy_dist2A.inp 
</code>
then edit the input file with the new coordinates (e.g. 2 Å). 
you can now run CP2K and produce the output file:
<code>
$ cp2k.popt -i energy_dist2A.inp -o energy_dist2A.out
</code>

=== 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.

<code>
         &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
</code>

=== 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. \\

$$ \sigma_{ij}= \sqrt{\sigma_i\sigma_j}$$ \\
$$ \epsilon_{ij}= \sqrt{\epsilon_i\epsilon_j}$$

<note tip>
Remember that you are running  with two different atom types. For this reason some of the input sections must be duplicated for the two kinds of atoms present
</note>

  * The " LENNARD-JONES " section must be present for all the three possible couples: Ar-Ar, Xe-Xe and Xe-Ar.  Example: 

<code>
      &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 
</code>

  * The " CHARGE " section must be also duplicated: 

<code>
         &CHARGE
          ATOM Xe
          CHARGE 0.0
         &END CHARGE
         &CHARGE
          ATOM Ar
          CHARGE 0.0
         &END CHARGE
         
</code>

===== 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?