PotentialExpandedOnClm

Given the onsite energies of the orbitals of all possible irreducible representations a potential expanded on renormalised Spherical Harmonics is created.

Within crystal field or ligand field theory it is common practice to expand a potential on Spherical Harmonics. A potential expanded as such can be used to create a crystal field operator with the function NewOperator(“C”, …). For more information see the documentation of the crystal field operator.

• PointGroup : String
• l

Integer

• Energy

Table of Doubles.

• Akm : Table containing k_m_coefficient_k_m_coefficient_..._example local Opp = Chop(NewOperator(“CF”,3,{0,1,2},Akm)) print(“\nAkm = ”) print(Akm) print(“\n Operator on basis of spherical Harmonics as a matrix”) print(“p_{-1},p_{0},p_{1}”) print(Chop(OperatorToMatrix(Opp))) print(“\n Operator on basis of Kubic Harmonics as a matrix”) print(“p_x, p_y, p_z”) print(Chop(OperatorToMatrix(Rotate(Opp,rotmatKp))))

end

function printAkmd(Akm)

local t=sqrt(1/2)
local u=I*sqrt(1/2)
local rotmatKd   = {{ t, 0, 0, 0, t },
                    { 0, 0, 1, 0, 0 },
                    { 0, u, 0, u, 0 },
                    { 0, t, 0,-t, 0 },
                    { u, 0, 0, 0,-u }}
local Opp = Chop(NewOperator("CF",5,{0,1,2,3,4},Akm))
print("\nAkm = ")
print(Akm)
print("\n Operator on basis of spherical Harmonics as a matrix")
print("d_{-2},d_{-1},d_{0},d_{1},d_{2}")
print(Chop(OperatorToMatrix(Opp)))
print("\n Operator on basis of Kubic Harmonics as a matrix")
print("d_{x^2-y^2},d_{z^2},d_{yz},d_{xz},d_{xy}")
print(Chop(OperatorToMatrix(Rotate(Opp,rotmatKd))))

end

function printAkmf(Akm)

local t=sqrt(1/2)
local u=I*sqrt(1/2)
local d=sqrt(3/16)
local q=sqrt(5/16)
local e=I*sqrt(3/16)
local r=I*sqrt(5/16)
local rotmatKf   = {{ 0, u, 0, 0, 0,-u, 0 },
                    { q, 0,-d, 0, d, 0,-q },
                    {-r, 0,-e, 0,-e, 0,-r },
                    { 0, 0, 0, 1, 0, 0, 0 },
                    {-d, 0,-q, 0, q, 0, d },
                    {-e, 0, r, 0, r, 0,-e },
                    { 0, t, 0, 0, 0, t, 0 }}
local Opp = Chop(NewOperator("CF",7,{0,1,2,3,4,5,6},Akm))
print("\nAkm = ")
print(Akm)
print("\n Operator on basis of spherical Harmonics as a matrix")
print("f_{-3},f_{-2},f_{-1},f_{0},f_{1},f_{2},f_{3}")
print(Chop(OperatorToMatrix(Opp)))
print("\n Operator on basis of Kubic Harmonics as a matrix")
print("f_{xyz},f_{5x^3-3x},f_{5y^3-3y},f_{5z^3-3z},f_{x(y^2-z^2)},f_{y(z^2-x^2)},f_{z(x^2-y^2)}")
print(Chop(OperatorToMatrix(Rotate(Opp,rotmatKf))))

end

print(“\nl=1 Oh”) Et1u=1 Akm = PotentialExpandedOnClm(“Oh”,1,{Et1u}) printAkmp(Akm)

print(“\nl=2 Oh”) Eeg=1 Et2g=2 Akm = PotentialExpandedOnClm(“Oh”,2,{Eeg,Et2g}) printAkmd(Akm)

print(“\nl=3 Oh”) Ea2u=1 Et1u=2 Et2u=3 Akm = PotentialExpandedOnClm(“Oh”,3,{Ea2u,Et1u,Et2u}) printAkmf(Akm) </code>

l=1 Oh
 
Akm = 
{ { 0 , 0 , 1 } }
 
 Operator on basis of spherical Harmonics as a matrix
p_{-1},p_{0},p_{1}
{ { 1 , 0 , 0 } , 
  { 0 , 1 , 0 } , 
  { 0 , 0 , 1 } }
 
 Operator on basis of Kubic Harmonics as a matrix
p_x, p_y, p_z
{ { 1 , 0 , 0 } , 
  { 0 , 1 , 0 } , 
  { 0 , 0 , 1 } }
 
l=2 Oh
 
Akm = 
{ { 0 , 0 , 1.6 } , 
  { 4 , 0 , -2.1 } , 
  { 4 , -4 , -1.2549900398011 } , 
  { 4 , 4 , -1.2549900398011 } }
 
 Operator on basis of spherical Harmonics as a matrix
d_{-2},d_{-1},d_{0},d_{1},d_{2}
{ { 1.5 , 0 , 0 , 0 , -0.5 } , 
  { 0 , 2 , 0 , 0 , 0 } , 
  { 0 , 0 , 1 , 0 , 0 } , 
  { 0 , 0 , 0 , 2 , 0 } , 
  { -0.5 , 0 , 0 , 0 , 1.5 } }
 
 Operator on basis of Kubic Harmonics as a matrix
d_{x^2-y^2},d_{z^2},d_{yz},d_{xz},d_{xy}
{ { 1 , 0 , 0 , 0 , 0 } , 
  { 0 , 1 , 0 , 0 , 0 } , 
  { 0 , 0 , 2 , 0 , 0 } , 
  { 0 , 0 , 0 , 2 , 0 } , 
  { 0 , 0 , 0 , 0 , 2 } }
 
l=3 Oh
 
Akm = 
{ { 0 , 0 , 2.2857142857143 } , 
  { 4 , 0 , 0.75 } , 
  { 4 , -4 , 0.4482107285004 } , 
  { 4 , 4 , 0.4482107285004 } , 
  { 6 , 0 , -1.8107142857143 } , 
  { 6 , -4 , 3.38753624124 } , 
  { 6 , 4 , 3.38753624124 } }
 
 Operator on basis of spherical Harmonics as a matrix
f_{-3},f_{-2},f_{-1},f_{0},f_{1},f_{2},f_{3}
{ { 2.375 , 0 , 0 , 0 , -0.48412291827593 , 0 , 0 } , 
  { 0 , 2 , 0 , 0 , 0 , 1 , 0 } , 
  { 0 , 0 , 2.625 , 0 , 0 , 0 , -0.48412291827593 } , 
  { 0 , 0 , 0 , 2 , 0 , 0 , 0 } , 
  { -0.48412291827593 , 0 , 0 , 0 , 2.625 , 0 , 0 } , 
  { 0 , 1 , 0 , 0 , 0 , 2 , 0 } , 
  { 0 , 0 , -0.48412291827593 , 0 , 0 , 0 , 2.375 } }
 
 Operator on basis of Kubic Harmonics as a matrix
f_{xyz},f_{5x^3-3x},f_{5y^3-3y},f_{5z^3-3z},f_{x(y^2-z^2)},f_{y(z^2-x^2)},f_{z(x^2-y^2)}
{ { 1 , 0 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 2 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 2 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 2 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 3 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 3 , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 0 , 3 } }