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WOLFRAM NOTEBOOK
Complete Minimization of Ne5
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
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:
The pairwise Leonard-Jones potential for a given Ne pair is defined as:
Source
Source
Constant Definitions
Constant Definitions
Actual Pairwise Leonard-Jones Potential
Actual Pairwise Leonard-Jones Potential
Tetrahedron dimensions and lengths:
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
The Total Potential
This is the potential of the total molecule, which is defined her as an expansion of the Yang, Poirier paper, i.e.: with: =4ϵ- and =-The mathematica version of this is:
Ne
5
Ne
3
V=+++++++++
V
12
V
13
V
14
V
15
V
23
V
24
V
25
V
34
V
35
V
45
V
ij
12
σ
r
ij
6
σ
r
ij
r
ij
r
i
r
j
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
The Coordinate System
Initial Configuration of Atoms
Initial Configuration of Atoms
Constraint
Constraint
Saving to Arrays
Saving to Arrays
Combining all into a single minimum energy path
Combining all into a single minimum energy path
Minimum points up to first kink
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
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
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]}]
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
Switch to lower energy curve after R = 0.75 to follow right path to z1init
Put all together
Put all together
Curve Fitting (EXECUTE EVERYTHING UP TO HERE)
Curve Fitting (EXECUTE EVERYTHING UP TO HERE)
Polynomial
Polynomial
Plot showing the difference from the points
Plot showing the difference from the points
3d Plots
3d Plots
R = z1init to 0
R = z1init to 0