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.

Input

  • PointGroup : String
  • l
Integer
  • Energy
Table of Doubles.

Output

  • Akm : Table containing “$\{\{k_1,m_1,A_{k_1,m_1}\},\{k_2,m_2,A_{k_2,m_2}\},\ldots\}$” defining the angular part of the potential such that $\big\langle \varphi_{\tau_1}(\vec{r}) \big| V(\vec{r}) \big| \varphi_{\tau_2}(\vec{r}) \big\rangle = \sum_{k=0}^{\infty}\sum_{m=-k}^{k} A_{k,m} \big\langle Y_{l_1,m_1} \big| C_{k,m} \big| Y_{l_2,m_2} \big\rangle$, with $\varphi_{\tau_1}(\vec{r})=R_{n_1,l_1}(r) Y_{l_1,m_1}(\theta\phi)$

Example

The following example creates a potential expanded on renormalised spherical Harmonics for different point-groups and prints the expansion as well as the Hamiltonian for an p, d and f shell that arises from this potential. The energies of the orbitals of a given irreducible representation are set to arbitrary values.

The example firstly defines three functions (for the p, d and f shell), that given a potential expanded on spherical harmonics print this potential, create an Hamiltonian on the basis of Spherical Harmonics and creates the Hamiltonian on a basis of Kubic Harmonics. The crystal field Hamiltonians are printed as a matrix, the basis functions used for this matrix are printed above the matrix.

Input

Example.Quanty
function printAkms(Akm)
  local Opp = Chop(NewOperator("CF",1,{0},Akm))
  print("\nAkm = ")
  print(Akm)
  print("\n Operator on basis of spherical Harmonics or Kubic Harmonics (same for s orbital as a matrix")
  print("s")
  print(Chop(OperatorToMatrix(Opp)))
end
 
function printAkmp(Akm)
  local t=sqrt(1/2)
  local u=I*sqrt(1/2)
  local rotmatKp   = {{ t, 0,-t },
                      { u, 0, u },
                      { 0, 1, 0 }}
  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)

Result

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 } }

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