YtoZMatrix

YtoZMatrix($orb$) takes the angular momentum $l$ or the name of a non-relativistic atomic orbital and returns the corresponding matrix to rotate from a basis of spherical harmonics to tesseral harmonics. It is also possible to give a list of $l$ numbers or orbital names, in which case the output is a block diagonal matrix with the rotation matrices as entries.

• $orb$ : An integer number, or a list of integer numbers, or a string that can be interpreted as a non-relativistic atomic orbital, or a list or strings.
• Options:

  • “addSpin” : bool determining if spin-space is considered in the matrix (resulting in matrices double in size). (Default true)

• $R$ : Rotation Matrix from a basis of spherical to tesseral harmonics.

Example.Quanty

print("")
print("YtoZMatrix(0)")
print(YtoZMatrix(0))
 
print("")
print("YtoZMatrix(\"s\", {{\"addSpin\",false}})")
print(YtoZMatrix("s", {{"addSpin",false}}))
 
print("")
print("YtoZMatrix(1)")
print(YtoZMatrix(1))
 
print("")
print("YtoZMatrix(\"p\")")
print(YtoZMatrix("p"))
 
print("")
print("YtoZMatrix(2)")
print(YtoZMatrix(2))
 
print("")
print("YtoZMatrix(\"d\")")
print(YtoZMatrix("d"))
 
print("")
print("YtoZMatrix({0,1,2}, {{\"addSpin\",false}})")
print(YtoZMatrix({0,1,2}, {{"addSpin",false}}))
 
print("")
print("YtoZMatrix({\"s\",\"p\",\"d\"}, {{\"addSpin\",false}})")
print(YtoZMatrix({"s","p","d"}, {{"addSpin",false}}))
 
print("\n\n")
print("A more realistic example")
Orbitals = {"1s","2s","2p"}
Indices, NF = CreateAtomicIndicesDict(Orbitals)
 
--Some Operator definition on spherical harmonics
op = NewOperator("U", NF, Indices["2p_Up"], Indices["2p_Dn"],{0,1})
print("Operator on a basis of spherical harmonics")
print(op)
 
opY = Rotate(op, YtoZMatrix(Orbitals))
print("Operator on a basis of tesseral harmonics")
print(opY)

YtoZMatrix(0)
{ { 1 , 0 } , 
  { 0 , 1 } }
 
YtoZMatrix("s", {{"addSpin",false}})
{ { 1 } }
 
YtoZMatrix(1)
{ { (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 } , 
  { 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) } , 
  { 0 , 0 , 1 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 1 , 0 , 0 } , 
  { 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 } , 
  { 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 } }
 
YtoZMatrix("p")
{ { (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 } , 
  { 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) } , 
  { 0 , 0 , 1 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 1 , 0 , 0 } , 
  { 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 } , 
  { 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 } }
 
YtoZMatrix(2)
{ { (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) , 0 } , 
  { 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) } , 
  { 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 } , 
  { 0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 } , 
  { 0 , 0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 } }
 
YtoZMatrix("d")
{ { (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) , 0 } , 
  { 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) } , 
  { 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 } , 
  { 0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 } , 
  { 0 , 0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 } }
 
YtoZMatrix({0,1,2}, {{"addSpin",false}})
{ { 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , (0 + 0.70710678118655 I) , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0.70710678118655 , 0 , -0.70710678118655 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 - 0.70710678118655 I) } , 
  { 0 , 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , (0 + 0.70710678118655 I) , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , -0.70710678118655 , 0 } , 
  { 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , 0.70710678118655 } }
 
YtoZMatrix({"s","p","d"}, {{"addSpin",false}})
{ { 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , (0 + 0.70710678118655 I) , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0.70710678118655 , 0 , -0.70710678118655 , 0 , 0 , 0 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 - 0.70710678118655 I) } , 
  { 0 , 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , (0 + 0.70710678118655 I) , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 } , 
  { 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , -0.70710678118655 , 0 } , 
  { 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , 0.70710678118655 } }
 
 
 
A more realistic example
Operator on a basis of spherical harmonics
 
Operator: Coulomb Operator
QComplex         =          0 (Real==0 or Complex==1 or Mixed==2)
MaxLength        =          4 (largest number of product of lader operators)
NFermionic modes =         10 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis)
NBosonic modes   =          0 (Number of bosonic modes (phonon modes, ...) in the one particle basis)
 
Operator of Length   4
QComplex      =          0 (Real==0 or Complex==1)
N             =         25 (number of operators of length   4)
C  5 C  4 A  5 A  4 | -3.999999999999999E-02
C  7 C  5 A  7 A  5 |  2.000000000000000E-01
C  6 C  5 A  6 A  5 |  7.999999999999999E-02
C  7 C  4 A  7 A  4 |  7.999999999999999E-02
C  6 C  4 A  6 A  4 |  2.000000000000000E-01
C  9 C  5 A  9 A  5 |  2.000000000000000E-01
C  8 C  5 A  8 A  5 | -3.999999999999999E-02
C  9 C  4 A  9 A  4 | -3.999999999999999E-02
C  8 C  4 A  8 A  4 |  2.000000000000000E-01
C  7 C  6 A  7 A  6 | -1.600000000000000E-01
C  9 C  7 A  9 A  7 |  2.000000000000000E-01
C  8 C  7 A  8 A  7 |  7.999999999999999E-02
C  9 C  6 A  9 A  6 |  7.999999999999999E-02
C  8 C  6 A  8 A  6 |  2.000000000000000E-01
C  9 C  8 A  9 A  8 | -3.999999999999999E-02
C  6 C  5 A  7 A  4 |  1.200000000000000E-01
C  7 C  4 A  6 A  5 |  1.200000000000000E-01
C  8 C  5 A  7 A  6 | -1.200000000000000E-01
C  9 C  4 A  7 A  6 |  1.200000000000000E-01
C  8 C  5 A  9 A  4 |  2.400000000000000E-01
C  9 C  4 A  8 A  5 |  2.400000000000000E-01
C  7 C  6 A  8 A  5 | -1.200000000000000E-01
C  7 C  6 A  9 A  4 |  1.200000000000000E-01
C  8 C  7 A  9 A  6 |  1.200000000000000E-01
C  9 C  6 A  8 A  7 |  1.200000000000000E-01
 
 
Operator on a basis of tesseral harmonics
 
Operator: Coulomb Operator
QComplex         =          0 (Real==0 or Complex==1 or Mixed==2)
MaxLength        =          4 (largest number of product of lader operators)
NFermionic modes =         10 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis)
NBosonic modes   =          0 (Number of bosonic modes (phonon modes, ...) in the one particle basis)
 
Operator of Length   4
QComplex      =          0 (Real==0 or Complex==1)
N             =         27 (number of operators of length   4)
C  5 C  4 A  5 A  4 | -1.599999999999999E-01
C  9 C  8 A  5 A  4 | -1.200000000000000E-01
C  9 C  4 A  9 A  4 |  7.999999999999997E-02
C  8 C  5 A  9 A  4 |  1.200000000000000E-01
C  9 C  4 A  8 A  5 |  1.200000000000000E-01
C  8 C  5 A  8 A  5 |  7.999999999999997E-02
C  5 C  4 A  9 A  8 | -1.200000000000000E-01
C  9 C  8 A  9 A  8 | -1.599999999999999E-01
C  7 C  5 A  7 A  5 |  1.999999999999999E-01
C  9 C  7 A  9 A  7 |  1.999999999999999E-01
C  6 C  5 A  6 A  5 |  7.999999999999997E-02
C  9 C  6 A  9 A  6 |  7.999999999999997E-02
C  7 C  4 A  7 A  4 |  7.999999999999997E-02
C  8 C  7 A  8 A  7 |  7.999999999999997E-02
C  6 C  4 A  6 A  4 |  1.999999999999999E-01
C  8 C  6 A  8 A  6 |  1.999999999999999E-01
C  9 C  5 A  9 A  5 |  1.999999999999999E-01
C  8 C  4 A  8 A  4 |  1.999999999999999E-01
C  7 C  6 A  7 A  6 | -1.600000000000000E-01
C  6 C  5 A  7 A  4 |  1.200000000000000E-01
C  9 C  6 A  8 A  7 |  1.200000000000000E-01
C  7 C  4 A  6 A  5 |  1.200000000000000E-01
C  8 C  7 A  9 A  6 |  1.200000000000000E-01
C  5 C  4 A  7 A  6 | -1.200000000000000E-01
C  9 C  8 A  7 A  6 | -1.200000000000000E-01
C  7 C  6 A  5 A  4 | -1.200000000000000E-01
C  7 C  6 A  9 A  8 | -1.200000000000000E-01