Examples from Li to Cs: in preparation

Explanation of how we have rendered Kimballs atoms spherical, using the Xenon atom. Note, this is the method best studied up to now, but there are other possibilities to explore, e.g. full construction of core pseudopotentials or modelling the inner shells s-p-d- with spherical clouds, using n,l,m quantum numbers, e.g. Ne atom and isoelectronic ions up to Ar⁸⁺.
Below, Xe is coded in Mathematica. You can see that from every inner shell a spherical transform S2...6 is inserted into the next shell optimization, all the way up to the topmost Xe-shell. This cascade produces an all spherical atom.- The examples on this page are not yet energy-optimized with the help of FDA.

(* Xe atom Kimball, Ne,Ar,Ni,Kr,Pd-centers, 4 5sp-spheres tetrahedral 26.12.2011 *)
Clear[k1,k2,k3,k4,k5,k6,k7,sig1,sig2,sig3,sig4,sig5,sig6,sig7,c,z,R1,R2,R3,R4,R5,
R6,R7,S2,S3,S4,S5,S6,S7];
c = {k1 -> 1.0, k2 -> 1.0, k3 -> 1.0, k4 -> 1.0, k5 -> 1.0, k6 -> 1.0, k7 -> 1.0, 
sig1 -> 0.3, sig2 -> 0.3, sig3 -> 0.3, sig4 -> 0.3, sig5 -> 0.3, sig6 -> 0.3, sig7 -> 0.3};
z=54.;


(* He+Ne shell *)
T = (2.*9./8.)*k1/R1^2+(8.*9./8.)*k2/R2^2 /. c;
ad = Sqrt[3./8.]; 
Vee=3.0*sig1/R1+12.*sig2/R2+16/(R1+R2)+24*ad/(R1+R2) /. c;
Vne=-3.0*z/R1-8.0*z/(R1+R2);
S2 = R2*4^(1/3);          transform Ne shell tetrahedron to equivalent sphere


(* Ar shell *)
T = T + (8.*9./8.)*k3/R3^2 /. c;
Vee = Vee+12.*sig3/R3+80./(S2+R3)+24.*ad/(S2+R3) /. c;
Vne = Vne-8.0*z/(S2+R3);
S3 = R3*4^(1/3);          transform Ar shell tetrahedron to equivalent sphere


(* Ni shell trigbipyr, d10 with 5*2 charges; *)
T = T + (10.*9./8.)*k4/R4^2 /. c;
Vee = Vee+5.*3.*sig4/R4+(180.+25.898766)/(S3+R4) /. c;
Vne = Vne - 10.0*z/(S3+R4);
S4 = R4*5^(1/3);          transform Ni shell trigpyr to equivalent sphere


(* Kr shell *)
T = T + (8.*9./8.)*k5/R5^2 /. c;
Vee = Vee+12.*sig5/R5+224./(S4+R5)+24.*ad/(S4+R5) /. c;
Vne = Vne-8.0*z/(S4+R5);
S5 = R5*4^(1/3);          transform Kr shell tetrahedron to equivalent sphere


(* Pd shell trigbipyr, d10 with 5*2 charges *)

T = T + (10.*9./8)*k6/R6^2 /. c;
Vee = Vee+5.*3.*sig6/R6+(360.+25.898766)/(S5+R6) /. c;
Vne = Vne - 10.0*z/(S5+R6);
S6 = R6*5^(1/3);          transform Pd shell trigpyr to equivalent sphere


(* Xe shell *)
T = T + (8.*9./8.)*k7/R7^2 /. c;
Vee = Vee+12.*sig7/R7+(368+24.*ad)/(S6+R7) /. c;
Vne = Vne-8.0*z/(S6+R7);
S7 = R7*4^(1/3);          transform Xe shell tetrahedron to equivalent sphere


func = T + Vee + Vne;


Minimization of energy routine. Startvalues have been pasted back
three times from results to improve minimum location:

t = FindMinimum[func, {R1,0.0239334}, {R2,0.0684705}, {R3,0.1245481},
{R4,0.2002774},{R5,0.2988439},{R6,0.5253633},{R7,1.3095930},
{Method -> Automatic}, {MaxIterations -> 500}]


N[Vne /. c /. t[[2]],10]
N[Vee /. c /. t[[2]],10]
N[-(Vee+Vne)/T /. c /. t[[2]],10]
N[(S2) /. c /. t[[2]],10]
N[(S3) /. c /. t[[2]],10]
N[(S4) /. c /. t[[2]],10]
N[(S5) /. c /. t[[2]],10]
N[(S6) /. c /. t[[2]],10]
N[S7 /. c /. t[[2]],10]

Result of minimization of energy: First number Etot

Vne:

Vee:

Virial ratio:

Radii S2 to S7 of equivalent spheres:

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Converted by Mathematica      September 16, 2012