The Slater atomic model

Reference : J. C. Slater, Phys. Rev. 36 (1930) 57
The basic idea is to replace the electron-electron interaction by effective nuclear attractions due to screening parameters, and to use an independent-electron Bohr model for each electron separately. The total energy is the sum of the individual orbital energies, and the size of an atom is determined by the according radiuus of the Bohr model. Only 3 screening parameters are needed, and effective principal quantum numbers for the shells 4 to 7. The model results in fairly good radii and ionization energies, and "explains" that orbital energies are dependent on orbital occupation. All 3rd-row transition metals are calculated having a 4(s^2) 3(d^(n-2)) ground state.

Constants and parameters

Natural constants

autoeV=27.21;
Bohr=0.529177;

Parameters of the model

S31=0.31;S35=0.35;S85=0.85; Neff4=3.7;Neff5=4.0;Neff6=4.2; Neff7=4.3;

Initialization of the data

n1s=0;n2sp=0;n3sp=0;n4sp=0;n3d=0;n4f=0;n5sp=0;n4d=0;n5f=0;n5d=0;n6sp=0;n7s=0;

Input defining the atom

Z=22;
n1s=2;
n2sp=6;
n3sp=6;
n4sp=2;
n3d=2;
n4f=0;
n5sp=0;
n4d=0;
n5f=0;
n5d=0;
n6sp=0;
n7s=0;

The calculation

The effective nuclear charges

Zeff1s=Z-(n1s-1)*S31;
Zeff2sp=Z-S85*n1s-S35*(n2sp-1);
Zeff3sp=Z-n1s-S85*n2sp-S35*(n3sp-1);
Zeff3d=Z-n1s-n2sp-n3sp-S35*(n3d-1);
Zeff4sp =Z-n1s-n2sp-n3sp*S85-S85*n3d -S35*(n4sp-1);
Zeff4d =Z-n1s-n2sp-n3sp-n3d -n4sp -(n4d-1)*S35 ;
Zeff4f =Z-n1s-n2sp-n3sp-n3d -n4sp -n4d-S35*(n4f-1);
Zeff5sp =Z-n1s-n2sp-n3sp-n3d-S85*(n4sp+n4d+n4f)-S35*(n5sp-1);
Zeff5d =Z-n1s-n2sp-n3sp-n3d-n4sp-n4d-n4f-n5sp-(n5d-1)*S35;
Zeff5f =Z-n1s-n2sp-n3sp-n3d-n4sp-n4d-n4f-n5sp-n5d-(n5f-1)*S35;
Zeff6sp =Z-n1s-n2sp-n3sp-n3d-n4sp-n4d-n4f-S85*(n5sp+n5d+n5f)-(n6sp-1)*S35;
Zeff7s =Z-n1s-n2sp-n3sp-n3d-n4sp-n4d-n4f-n5sp-n5d-n5f- n6sp*S85-(n7s-1)*S35;

The orbital energies

E1s=-0.5*Zeff1s^2;
E2sp=-0.5/4*Zeff2sp^2;
E3sp=-0.5/9*Zeff3sp^2;
E3d=-0.5/9*Zeff3d^2;
E4sp=-0.5*(Zeff4sp/Neff4)^2;
E4d=-0.5*(Zeff4d/Neff4)^2;
E4f=-0.5*(Zeff4f/Neff4)^2;
E5sp=-0.5*(Zeff5sp/Neff5)^2;
E5d=-0.5*(Zeff5d/Neff5)^2;
E5f=-0.5*(Zeff5f/Neff5)^2;
E6sp=-0.5*(Zeff6sp/Neff6)^2;
E7s=-0.5*(Zeff7s/Neff7)^2;

The orbital radii

R1s=1./Zeff1s;
R2sp=4./Zeff2sp;
R3sp=9./Zeff3sp;
R3d=9./Zeff3d;
R4sp=Neff4*Neff4/Zeff4sp;
R4d=Neff4*Neff4/Zeff4d;
R4f=Neff4*Neff4/Zeff4f;
R5sp=Neff5*Neff5/Zeff5sp;
R5d=Neff5*Neff5/Zeff5d;
R5f=Neff5*Neff5/Zeff5f;
R6sp=Neff6*Neff6/Zeff6sp;
R7s=Neff7*Neff7/Zeff7s;

The total energy

Etot=n1s*E1s+n2sp*E2sp+n3sp*E3sp+n4sp*E4sp+n5sp*E5sp+n6sp*E6sp+n7s*E7s+n3d*E3d+n4d*E4d+n5d*E5d+n4f*E4f+n5f*E5f

The results

Print["Nuclear charge Z = ",Z]
 
If [n1s>0, Print[" 1s  : ",n1s," electrons; Zeff = ",Zeff1s," ; E1s  = ",E1s," a.u. = ",E1s*autoeV," eV \n     Orbital radius = ",R1s," bohr = ",R1s*Bohr," Angstrom" ]] ;
If [n2sp>0,Print[" 2sp : " ,n2sp," electrons; Zeff = " ,Zeff2sp," ; E2sp = ",E2sp," a.u. = ",E2sp*autoeV," eV \n     Orbital radius = ",R2sp," bohr = ",R2sp*Bohr," Angstrom"]] ;
If [n3sp>0,Print[" 3sp : " ,n3sp," electrons; Zeff = " ,Zeff3sp," ; E3sp = ",E3sp," a.u. = ",E3sp*autoeV," eV  \n     Orbital radius = ",R3sp," bohr = ",R3sp*Bohr," Angstrom"]] ;
If [n3d>0, Print[" 3d  : ",n3d," electrons; Zeff = ",Zeff3d," ; E3d  = ",E3d," a.u. = ",E3d*autoeV," eV  \n     Orbital radius = ",R3d," bohr = ",R3d*Bohr," Angstrom"]] ;
If [n4sp>0,Print[" 4sp : " ,n4sp," electrons; Zeff = " ,Zeff4sp," ; E4sp = ",E4sp," a.u. = ",E4sp*autoeV," eV  \n     Orbital radius = ",R4sp," bohr = ",R4sp*Bohr," Angstrom"]] ;
If [n4d>0, Print[" 4d  : ",n4d," electrons; Zeff = ",Zeff4d," ; E4d  = ",E4d," a.u. = ",E4d*autoeV," eV  \n     Orbital radius = ",R4d," bohr = ",R4d*Bohr," Angstrom"]] ;
If [n4f>0, Print[" 4f  : ",n4f," electrons; Zeff = ",Zeff4f," ; E4f  = ",E4f," a.u. = ",E4f*autoeV," eV  \n     Orbital radius = ",R4f," bohr = ",R4f*Bohr," Angstrom"]] ;
If [n5sp>0,Print[" 5sp : " ,n5sp," electrons; Zeff = " ,Zeff5sp," ; E5sp = ",E5sp," a.u. = ",E5sp*autoeV," eV  \n     Orbital radius = ",R5sp," bohr = ",R5sp*Bohr," Angstrom"]] ;
If [n5f>0, Print[" 5f  : ",n5f," electrons; Zeff = ",Zeff5f," ; E5f  = ",E5f," a.u. = ",E5f*autoeV," eV  \n     Orbital radius = ",R5f," bohr = ",R5f*Bohr," Angstrom"]] ;
If [n5d>0, Print[" 5d  : ",n5d," electrons; Zeff = ",Zeff5d," ; E5d  = ",E5d," a.u. = ",E5d*autoeV," eV  \n     Orbital radius = ",R5d," bohr = ",R5d*Bohr," Angstrom"]] ;
If [n6sp>0,Print[" 6sp : " ,n6sp," electrons; Zeff = " ,Zeff6sp," ; E6sp = ",E6sp," a.u. = ",E6sp*autoeV," eV  \n     Orbital radius = ",R6sp," bohr = ",R6sp*Bohr," Angstrom"]] ;
If [n7s>0, Print[" 7s  : ",n7s," electrons; Zeff = ",Zeff7s," ; E7s  = ",E7s," a.u. = ",E7s*autoeV," eV "]] ;Print[" Total energy = ",Etot," a.u. = ",Etot*autoeV," eV  "];
Graphics[{Thick,If [n1s>0,{Black,Circle[{0,0},{R1s,R1s}]}],If [n2sp>0,{Black,Circle[{0,0},{R2sp,R2sp}]}],If [n3sp>0,{Black,Circle[{0,0},{R3sp,R3sp}]}],
If [n4sp>0,{Black,Circle[{0,0},{R4sp,R4sp}]}],
If [n5sp>0,{Black,Circle[{0,0},{R5sp,R5sp}]}],
If [n6sp>0,{Black,Circle[{0,0},{R6sp,R6sp}]}],
If [n7s>0,{Black,Circle[{0,0},{R7s,R7s}]}],
If [n3d>0,{Red,Circle[{0,0},{R3d,R3d}]}],
If [n4d>0,{Red,Circle[{0,0},{R4d,R4d}]}],
If [n5d>0,{Red,Circle[{0,0},{R5d,R5d}]}],
If [n4f>0,{Blue,Circle[{0,0},{R4f,R4f}]}],
If [n5f>0,{Blue,Circle[{0,0},{R5f,R5f}]}]}]

Credits

Author : Peter Reinhardt (Paris) 2012,       peter.reinhardt@upmc.fr