Table of Contents
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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.
Input
-
$orb$ : 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.
Output
-
$R$ : Rotation Matrix from a basis of spherical to tesseral harmonics.
Example
Input
- Example.Quanty
print("") print("YtoZMatrix(0)") print(YtoZMatrix(0)) print("") print("YtoZMatrix(\"s\")") print(YtoZMatrix("s")) 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})") print(YtoZMatrix({0,1,2})) print("") print("YtoZMatrix({\"s\",\"p\",\"d\"})") print(YtoZMatrix({"s","p","d"})) 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)
Result
YtoZMatrix(0)
{ { 1 , 0 } ,
{ 0 , 1 } }
YtoZMatrix("s")
{ { 1 , 0 } ,
{ 0 , 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})
{ { 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 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 , 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 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) } ,
{ 0 , 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 , 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 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 } }
YtoZMatrix({"s","p","d"})
{ { 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 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 , 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 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , (0 - 0.70710678118655 I) } ,
{ 0 , 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 , 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 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 } ,
{ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0.70710678118655 , 0 , 0 , 0 , 0 , 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
