Complete Minimization of
Ne
5

Here we are doing a diatom-triatom minimization from scratch of the Neon pentamer. We would like this to be a function of the distance between the center of mass of the diatom and the center of mass of the triatom.

The Pairwise Potential Definition

We start by making the potential a series of coupled Leonard-Jones potentials for each Ne to Ne bond.
The pairwise Leonard-Jones potential for a given Ne pair is defined as:

Source


Constant Definitions


Actual Pairwise Leonard-Jones Potential


Tetrahedron dimensions and lengths:

Some of these general dimensions for a tetrahedral solid may come in handy later.
In[]:=
A=rm;​​Edge=A;​​Height=(Sqrt[6]/3.0)*A;​​CentroidToVertex=Sqrt[3]/(2.0*Sqrt[2])*A;​​CentroidToMidEdge=(1.0/(2*Sqrt[2]))*A;​​CentroidToCenterFace=.25*Height;​​FaceHeight=(Sqrt[3]/2.0)*A;​​CenterFaceToVertex=(1.0/Sqrt[3])*A;​​CenterFaceToCenterEdge=(1.0/(2*Sqrt[3]))*A;​​AngleEdgeToFace=54.735600000000005`;​​AngleFaceToFace=70.5288;​​AngleTetrahedral=109.47122064;
In[]:=
​

The Total Potential

This is the potential of the total
Ne
5
molecule, which is defined her as an expansion of the Yang, Poirier
Ne
3
paper, i.e.:​
V=
V
12
+
V
13
+
V
14
+
V
15
+
V
23
+
V
24
+
V
25
+
V
34
+
V
35
+
V
45
​ with:​
V
ij
=4ϵ
12
σ

r
ij
-
6
σ

r
ij
and

r
ij
=

r
i
-

r
j

​​The mathematica version of this is:
In[]:=
VTot[r12_,r13_,r14_,r15_,r23_,r24_,r25_,r34_,r35_,r45_]:=​​VLJ[r12]+VLJ[r13]+VLJ[r14]+VLJ[r15]+​​VLJ[r23]+VLJ[r24]+VLJ[r25]+​​VLJ[r34]+VLJ[r35]+​​VLJ[r45];
Let’s check what this is at all equilibrium values, to get a sense of what values we are dealing with.
In[]:=
VTot[rm,rm,rm,rm,rm,rm,rm,rm,rm,rm]
Out[]=
-10.
Which is what it should be when everything is normalized to 1, good.

The Coordinate System


Initial Configuration of Atoms


Constraint


Saving to Arrays


Combining all into a single minimum energy path

Minimum points up to first kink

In[]:=
FirstArrs=Table[dummyBack2[[i]],{i,1,41}];​​FirstPots=Table[VTotalsCutBack2[[i]],{i,1,41}];​​SectionOne=Table[{FirstArrs[[i]],FirstPots[[i]]},{i,Length[FirstArrs]}]
Out[]=
{{0.,-9.10378},{0.001,-9.07615},{0.002,-9.05067},{0.003,-9.0265},{0.004,-9.00343},{0.005,-8.98129},{0.006,-8.95998},{0.007,-8.93938},{0.008,-8.91946},{0.009,-8.90013},{0.01,-8.88137},{0.011,-8.86314},{0.012,-8.84539},{0.013,-8.82809},{0.014,-8.81123},{0.015,-8.79478},{0.016,-8.77871},{0.017,-8.76301},{0.018,-8.74767},{0.019,-8.73265},{0.02,-8.71796},{0.021,-8.70358},{0.022,-8.68948},{0.023,-8.67564},{0.024,-8.66214},{0.025,-8.64887},{0.026,-8.63585},{0.027,-8.62308},{0.028,-8.61052},{0.029,-8.59824},{0.03,-8.58732},{0.031,-8.57664},{0.032,-8.56647},{0.033,-8.55678},{0.034,-8.54752},{0.035,-8.53866},{0.036,-8.53016},{0.037,-8.522},{0.038,-8.51416},{0.039,-8.50662},{0.04,-8.49936}}

Minimum after first kink

In[]:=
test=Table[ArrsCut2[[Length[ArrsCut2]+1-i]],{i,Length[ArrsCut2]}];​​test2=Table[VTotalsCut2[[Length[ArrsCut2]+1-i]],{i,Length[ArrsCut2]}];
In[]:=
SecondArrs=Table[test[[i]],{i,42,Length[test]}];​​SecondPots=Table[test2[[i]],{i,42,Length[test]}];​​SectionTwo=Table[{SecondArrs[[i]],SecondPots[[i]]},{i,Length[SecondPots]}]
Out[]=
{{0.041,-8.49734},{0.042,-8.49498},{0.043,-8.49285},{0.044,-8.49093},{0.045,-8.48921},{0.046,-8.48767},{0.047,-8.4863},{0.048,-8.48511},{0.049,-8.48407},{0.05,-8.48318},{0.051,-8.48243},{0.052,-8.48182},{0.053,-8.48134},{0.054,-8.48098},{0.055,-8.48075},{0.056,-8.48064},{0.057,-8.48063},{0.058,-8.48074},{0.059,-8.48095},{0.06,-8.48127},{0.061,-8.48168},{0.062,-8.4822},{0.063,-8.48281},{0.064,-8.48351},{0.065,-8.48431},{0.066,-8.48519},{0.067,-8.48617},{0.068,-8.48723},{0.069,-8.48837},{0.07,-8.4896},{0.071,-8.49092},{0.072,-8.49231},{0.073,-8.49379},{0.074,-8.49535},{0.075,-8.49699},{0.076,-8.49872},{0.077,-8.50052},{0.078,-8.5024},{0.079,-8.50435},{0.08,-8.50639},{0.081,-8.50851},{0.082,-8.51071},{0.083,-8.51298},{0.084,-8.51534},{0.085,-8.51777},{0.086,-8.52029},{0.087,-8.52288},{0.088,-8.52556},{0.089,-8.52832},{0.09,-8.53116},{0.091,-8.53408},{0.092,-8.53709},{0.093,-8.54018},{0.094,-8.54336},{0.095,-8.54662},{0.096,-8.54997},{0.097,-8.55341},{0.098,-8.55694},{0.099,-8.56056},{0.1,-8.56428},{0.101,-8.56808},{0.102,-8.57199},{0.103,-8.57599},{0.104,-8.58008},{0.105,-8.58428},{0.106,-8.58858},{0.107,-8.59299},{0.108,-8.5975},{0.109,-8.60212},{0.11,-8.60684},{0.111,-8.61167},{0.112,-8.61662},{0.113,-8.62168},{0.114,-8.62685},{0.115,-8.63213},{0.116,-8.63752},{0.117,-8.64302},{0.118,-8.64864},{0.119,-8.65436},{0.12,-8.66019},{0.121,-8.66611},{0.122,-8.67213},{0.123,-8.67823},{0.124,-8.68441},{0.125,-8.69066},{0.126,-8.69696},{0.127,-8.70332},{0.128,-8.70971},{0.129,-8.71614},{0.13,-8.72259},{0.131,-8.72906},{0.132,-8.73554},{0.133,-8.74202},{0.134,-8.74849},{0.135,-8.75496},{0.136,-8.76141},{0.137,-8.76785},{0.138,-8.77426},{0.139,-8.78065},{0.14,-8.78701},{0.141,-8.79333},{0.142,-8.79963},{0.143,-8.80588},{0.144,-8.81209},{0.145,-8.81827},{0.146,-8.8244},{0.147,-8.83049},{0.148,-8.83653},{0.149,-8.84252},{0.15,-8.84846},{0.151,-8.85435},{0.152,-8.86019},{0.153,-8.86598},{0.154,-8.87171},{0.155,-8.87739},{0.156,-8.88301},{0.157,-8.88857},{0.158,-8.89407},{0.159,-8.89952},{0.16,-8.9049},{0.161,-8.91023},{0.162,-8.91549},{0.163,-8.92069},{0.164,-8.92582},{0.165,-8.9309},{0.166,-8.9359},{0.167,-8.94084},{0.168,-8.94572},{0.169,-8.95053},{0.17,-8.95527},{0.171,-8.95995},{0.172,-8.96455},{0.173,-8.96909},{0.174,-8.97355},{0.175,-8.97795},{0.176,-8.98228},{0.177,-8.98653},{0.178,-8.99072},{0.179,-8.99483},{0.18,-8.99887},{0.181,-9.00284},{0.182,-9.00674},{0.183,-9.01056},{0.184,-9.0143},{0.185,-9.01798},{0.186,-9.02158},{0.187,-9.0251},{0.188,-9.02855},{0.189,-9.03193},{0.19,-9.03523},{0.191,-9.03845},{0.192,-9.0416},{0.193,-9.04467},{0.194,-9.04766},{0.195,-9.05058},{0.196,-9.05342},{0.197,-9.05618},{0.198,-9.05887},{0.199,-9.06147},{0.2,-9.064}}

Switchover to coarse grain after R = 0.2 going to R = 0.75

In[]:=
test2=Table[VTotals[[Length[Arrs]+1-i]],{i,Length[Arrs]}];
In[]:=
ThirdArrs=Table[dummy[[i]],{i,22,76}];​​ThirdPots=Table[test2[[i]],{i,22,76}];​​SectionThree=Table[{ThirdArrs[[i]],ThirdPots[[i]]},{i,Length[ThirdPots]}]
Part
:Part 22 of {} does not exist.
Part
:Part 23 of {} does not exist.
Part
:Part 24 of {} does not exist.
General
:Further output of Part::partw will be suppressed during this calculation.
Out[]=
{{0.21,{}〚22〛},{0.22,{}〚23〛},{0.23,{}〚24〛},{0.24,{}〚25〛},{0.25,{}〚26〛},{0.26,{}〚27〛},{0.27,{}〚28〛},{0.28,{}〚29〛},{0.29,{}〚30〛},{0.3,{}〚31〛},{0.31,{}〚32〛},{0.32,{}〚33〛},{0.33,{}〚34〛},{0.34,{}〚35〛},{0.35,{}〚36〛},{0.36,{}〚37〛},{0.37,{}〚38〛},{0.38,{}〚39〛},{0.39,{}〚40〛},{0.4,{}〚41〛},{0.41,{}〚42〛},{0.42,{}〚43〛},{0.43,{}〚44〛},{0.44,{}〚45〛},{0.45,{}〚46〛},{0.46,{}〚47〛},{0.47,{}〚48〛},{0.48,{}〚49〛},{0.49,{}〚50〛},{0.5,{}〚51〛},{0.51,{}〚52〛},{0.52,{}〚53〛},{0.53,{}〚54〛},{0.54,{}〚55〛},{0.55,{}〚56〛},{0.56,{}〚57〛},{0.57,{}〚58〛},{0.58,{}〚59〛},{0.59,{}〚60〛},{0.6,{}〚61〛},{0.61,{}〚62〛},{0.62,{}〚63〛},{0.63,{}〚64〛},{0.64,{}〚65〛},{0.65,{}〚66〛},{0.66,{}〚67〛},{0.67,{}〚68〛},{0.68,{}〚69〛},{0.69,{}〚70〛},{0.7,{}〚71〛},{0.71,{}〚72〛},{0.72,{}〚73〛},{0.73,{}〚74〛},{0.74,{}〚75〛},{0.75,{}〚76〛}}

Switch to lower energy curve after R = 0.75 to follow right path to z1init


Put all together

Curve Fitting (EXECUTE EVERYTHING UP TO HERE)

Polynomial

Plot showing the difference from the points

3d Plots

R = z1init to 0