Processing math: 34%

+Table of Contents

Orientation Y

Symmetry Operations

In the Cs Point Group, with orientation Y there are the following symmetry operations

Operator Orientation
E {0,0,0} ,
σh {0,1,0} ,

Different Settings

Character Table

E(1) σh(1)
A' 1 1
A'' 1 −1

Product Table

A' A''
A' A' A''
A'' A'' A'

Sub Groups with compatible settings

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Any potential (function) can be written as a sum over spherical harmonics. V(r,θ,ϕ)=∞∑k=0k∑m=−kAk,m(r)C(m)k(θ,ϕ) Here Ak,m(r) is a radial function and C(m)k(θ,ϕ) a renormalised spherical harmonics. C(m)k(θ,ϕ)=√4π2k+1Y(m)k(θ,ϕ) The presence of symmetry induces relations between the expansion coefficients such that V(r,θ,ϕ) is invariant under all symmetry operations. For the Cs Point group with orientation Y the form of the expansion coefficients is:

Expansion

Ak,m={A(0,0)k=0∧m=0−A(1,1)k=1∧m=−1A(1,0)k=1∧m=0A(1,1)k=1∧m=1A(2,2)k=2∧(m=−2∨m=2)−A(2,1)k=2∧m=−1A(2,0)k=2∧m=0A(2,1)k=2∧m=1−A(3,3)k=3∧m=−3A(3,2)k=3∧(m=−2∨m=2)−A(3,1)k=3∧m=−1A(3,0)k=3∧m=0A(3,1)k=3∧m=1A(3,3)k=3∧m=3A(4,4)k=4∧(m=−4∨m=4)−A(4,3)k=4∧m=−3A(4,2)k=4∧(m=−2∨m=2)−A(4,1)k=4∧m=−1A(4,0)k=4∧m=0A(4,1)k=4∧m=1A(4,3)k=4∧m=3−A(5,5)k=5∧m=−5A(5,4)k=5∧(m=−4∨m=4)−A(5,3)k=5∧m=−3A(5,2)k=5∧(m=−2∨m=2)−A(5,1)k=5∧m=−1A(5,0)k=5∧m=0A(5,1)k=5∧m=1A(5,3)k=5∧m=3A(5,5)k=5∧m=5A(6,6)k=6∧(m=−6∨m=6)−A(6,5)k=6∧m=−5A(6,4)k=6∧(m=−4∨m=4)−A(6,3)k=6∧m=−3A(6,2)k=6∧(m=−2∨m=2)−A(6,1)k=6∧m=−1A(6,0)k=6∧m=0A(6,1)k=6∧m=1A(6,3)k=6∧m=3A(6,5)k=6∧m=5

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}, {A[1, 1], k == 1 && m == 1}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {-A[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1], k == 2 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}, {A[3, 3], k == 3 && m == 3}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {-A[4, 3], k == 4 && m == -3}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {-A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1], k == 4 && m == 1}, {A[4, 3], k == 4 && m == 3}, {-A[5, 5], k == 5 && m == -5}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {-A[5, 3], k == 5 && m == -3}, {A[5, 2], k == 5 && (m == -2 || m == 2)}, {-A[5, 1], k == 5 && m == -1}, {A[5, 0], k == 5 && m == 0}, {A[5, 1], k == 5 && m == 1}, {A[5, 3], k == 5 && m == 3}, {A[5, 5], k == 5 && m == 5}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {-A[6, 5], k == 6 && m == -5}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {-A[6, 3], k == 6 && m == -3}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {-A[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {A[6, 1], k == 6 && m == 1}, {A[6, 3], k == 6 && m == 3}, {A[6, 5], k == 6 && m == 5}}, 0]

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = { 0, A(0,0)} , 
       {1, 0, A(1,0)} , 
       {1,-1, (-1)*(A(1,1))} , 
       {1, 1, A(1,1)} , 
       {2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1))} , 
       {2, 1, A(2,1)} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} , 
       {3, 0, A(3,0)} , 
       {3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, A(3,2)} , 
       {3, 2, A(3,2)} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1)*(A(4,1))} , 
       {4, 1, A(4,1)} , 
       {4,-2, A(4,2)} , 
       {4, 2, A(4,2)} , 
       {4,-3, (-1)*(A(4,3))} , 
       {4, 3, A(4,3)} , 
       {4,-4, A(4,4)} , 
       {4, 4, A(4,4)} , 
       {5, 0, A(5,0)} , 
       {5,-1, (-1)*(A(5,1))} , 
       {5, 1, A(5,1)} , 
       {5,-2, A(5,2)} , 
       {5, 2, A(5,2)} , 
       {5,-3, (-1)*(A(5,3))} , 
       {5, 3, A(5,3)} , 
       {5,-4, A(5,4)} , 
       {5, 4, A(5,4)} , 
       {5,-5, (-1)*(A(5,5))} , 
       {5, 5, A(5,5)} , 
       {6, 0, A(6,0)} , 
       {6,-1, (-1)*(A(6,1))} , 
       {6, 1, A(6,1)} , 
       {6,-2, A(6,2)} , 
       {6, 2, A(6,2)} , 
       {6,-3, (-1)*(A(6,3))} , 
       {6, 3, A(6,3)} , 
       {6,-4, A(6,4)} , 
       {6, 4, A(6,4)} , 
       {6,-5, (-1)*(A(6,5))} , 
       {6, 5, A(6,5)} , 
       {6,-6, A(6,6)} , 
       {6, 6, A(6,6)} }

One particle coupling on a basis of spherical harmonics

The operator representing the potential in second quantisation is given as: O=∑n″,l″,m″,n′,l′,m′⟨ψn″,l″,m″(r,θ,ϕ)|V(r,θ,ϕ)|ψn′,l′,m′(r,θ,ϕ)⟩a†n″,l″,m″a†n′,l′,m′ For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. ψn,l,m(r,θ,ϕ)=Rn,l(r)Y(l)m(θ,ϕ). With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter. An″l″,n′l′(k,m)=⟨Rn″,l″|Ak,m(r)|Rn′,l′⟩ Note the difference between the function Ak,m and the parameter An″l″,n′l′(k,m)

we can express the operator as O=∑n″,l″,m″,n′,l′,m′,k,mAn″l″,n′l′(k,m)⟨Y(m″)l″(θ,ϕ)|C(m)k(θ,ϕ)|Y(m′)l′(θ,ϕ)⟩a†n″,l″,m″a†n′,l′,m′

The table below shows the expectation value of O on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle Al″,l′(k,m) can be complex. Instead of allowing complex parameters we took Al″,l′(k,m)+IBl″,l′(k,m) (with both A and B real) as the expansion parameter.

Y(0)0 Y(1)−1 Y(1)0 Y(1)1 Y(2)−2 Y(2)−1 Y(2)0 Y(2)1 Y(2)2 Y(3)−3 Y(3)−2 Y(3)−1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
Y(0)0Ass(0,0)−Asp(1,1)√3Asp(1,0)√3Asp(1,1)√3Asd(2,2)√5−Asd(2,1)√5Asd(2,0)√5Asd(2,1)√5Asd(2,2)√5−Asf(3,3)√7Asf(3,2)√7−Asf(3,1)√7Asf(3,0)√7Asf(3,1)√7Asf(3,2)√7Asf(3,3)√7
Y(1)−1−Asp(1,1)√3App(0,0)−15App(2,0)−15√3App(2,1)−15√6App(2,2)17√35Apd(3,1)−√25Apd(1,1)Apd(1,0)√5−3Apd(3,0)7√5Apd(1,1)√15−37√25Apd(3,1)−17√6Apd(3,2)−37Apd(3,3)3Apf(2,2)√35−Apf(4,2)3√21Apf(4,1)3√7−√635Apf(2,1)35√27Apf(2,0)−13√27Apf(4,0)3Apf(2,1)5√7−13√1021Apf(4,1)15√37Apf(2,2)−13√57Apf(4,2)−13Apf(4,3)−2Apf(4,4)3√3
Y(1)0Asp(1,0)√3−15√3App(2,1)App(0,0)+25App(2,0)15√3App(2,1)17√3Apd(3,2)−Apd(1,1)√5−27√65Apd(3,1)2Apd(1,0)√15+37√35Apd(3,0)Apd(1,1)√5+27√65Apd(3,1)17√3Apd(3,2)−Apf(4,3)3√3√335Apf(2,2)+2Apf(4,2)3√7−25√67Apf(2,1)−13√57Apf(4,1)35√37Apf(2,0)+4Apf(4,0)3√2125√67Apf(2,1)+13√57Apf(4,1)√335Apf(2,2)+2Apf(4,2)3√7Apf(4,3)3√3
Y(1)1Asp(1,1)√3−15√6App(2,2)15√3App(2,1)App(0,0)−15App(2,0)37Apd(3,3)−17√6Apd(3,2)37√25Apd(3,1)−Apd(1,1)√15Apd(1,0)√5−3Apd(3,0)7√5√25Apd(1,1)−17√35Apd(3,1)−2Apf(4,4)3√313Apf(4,3)15√37Apf(2,2)−13√57Apf(4,2)13√1021Apf(4,1)−3Apf(2,1)5√735√27Apf(2,0)−13√27Apf(4,0)√635Apf(2,1)−Apf(4,1)3√73Apf(2,2)√35−Apf(4,2)3√21
Y(2)−2Asd(2,2)√517√35Apd(3,1)−√25Apd(1,1)17√3Apd(3,2)37Apd(3,3)Add(0,0)−27Add(2,0)+121Add(4,0)121√5Add(4,1)−17√6Add(2,1)17√53Add(4,2)−27Add(2,2)13√57Add(4,3)13√107Add(4,4)−√37Adf(1,1)+13√27Adf(3,1)−133√57Adf(5,1)Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7Adf(1,1)√35−2√2105Adf(3,1)+5Adf(5,1)11√21533Adf(5,2)−2Adf(3,2)3√7533√2Adf(5,3)−13√27Adf(3,3)111√10Adf(5,4)511√23Adf(5,5)
Y(2)−1−Asd(2,1)√5Apd(1,0)√5−3Apd(3,0)7√5−Apd(1,1)√5−27√65Apd(3,1)−17√6Apd(3,2)121√5Add(4,1)−17√6Add(2,1)Add(0,0)+17Add(2,0)−421Add(4,0)−17Add(2,1)−17√103Add(4,1)−17√6Add(2,2)−221√10Add(4,2)−13√57Add(4,3)13√57Adf(3,2)−133√5Adf(5,2)−√27Adf(1,1)−Adf(3,1)√21+211√1021Adf(5,1)2√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0)√335Adf(1,1)−13√235Adf(3,1)−20Adf(5,1)33√7−Adf(3,2)√21−5Adf(5,2)11√3−13√57Adf(3,3)−433√5Adf(5,3)−211√53Adf(5,4)
Y(2)0Asd(2,0)√5Apd(1,1)√15−37√25Apd(3,1)2Apd(1,0)√15+37√35Apd(3,0)37√25Apd(3,1)−Apd(1,1)√1517√53Add(4,2)−27Add(2,2)−17Add(2,1)−17√103Add(4,1)Add(0,0)+27Add(2,0)+27Add(4,0)17Add(2,1)+17√103Add(4,1)17√53Add(4,2)−27Add(2,2)13√57Adf(3,3)−233√5Adf(5,3)111√5Adf(5,2)−√635Adf(1,1)−Adf(3,1)√35−511√27Adf(5,1)3Adf(1,0)√35+4Adf(3,0)3√35+1033√57Adf(5,0)√635Adf(1,1)+Adf(3,1)√35+511√27Adf(5,1)111√5Adf(5,2)233√5Adf(5,3)−13√57Adf(3,3)
Y(2)1Asd(2,1)√5−17√6Apd(3,2)Apd(1,1)√5+27√65Apd(3,1)Apd(1,0)√5−3Apd(3,0)7√513√57Add(4,3)−17√6Add(2,2)−221√10Add(4,2)17Add(2,1)+17√103Add(4,1)Add(0,0)+17Add(2,0)−421Add(4,0)17√6Add(2,1)−121√5Add(4,1)−211√53Adf(5,4)13√57Adf(3,3)+433√5Adf(5,3)−Adf(3,2)√21−5Adf(5,2)11√3−√335Adf(1,1)+13√235Adf(3,1)+20Adf(5,1)33√72√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0)√27Adf(1,1)+Adf(3,1)√21−211√1021Adf(5,1)13√57Adf(3,2)−133√5Adf(5,2)
Y(2)2Asd(2,2)√5−37Apd(3,3)17√3Apd(3,2)√25Apd(1,1)−17√35Apd(3,1)13√107Add(4,4)−13√57Add(4,3)17√53Add(4,2)−27Add(2,2)17√6Add(2,1)−121√5Add(4,1)Add(0,0)−27Add(2,0)+121Add(4,0)−511√23Adf(5,5)111√10Adf(5,4)13√27Adf(3,3)−533√2Adf(5,3)533Adf(5,2)−2Adf(3,2)3√7−Adf(1,1)√35+2√2105Adf(3,1)−5Adf(5,1)11√21Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7√37Adf(1,1)−13√27Adf(3,1)+133√57Adf(5,1)
Y(3)−3−Asf(3,3)√73Apf(2,2)√35−Apf(4,2)3√21−Apf(4,3)3√3−2Apf(4,4)3√3−√37Adf(1,1)+13√27Adf(3,1)−133√57Adf(5,1)13√57Adf(3,2)−133√5Adf(5,2)13√57Adf(3,3)−233√5Adf(5,3)−211√53Adf(5,4)−511√23Adf(5,5)Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0)−13Aff(2,1)+111√103Aff(4,1)−5429√7Aff(6,1)−13√25Aff(2,2)+111√6Aff(4,2)−10429√7Aff(6,2)111√7Aff(4,3)−10143√73Aff(6,3)111√143Aff(4,4)−5143√703Aff(6,4)−513√1433Aff(6,5)−1013√733Aff(6,6)
Y(3)−2Asf(3,2)√7Apf(4,1)3√7−√635Apf(2,1)√335Apf(2,2)+2Apf(4,2)3√713Apf(4,3)Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7−√27Adf(1,1)−Adf(3,1)√21+211√1021Adf(5,1)111√5Adf(5,2)13√57Adf(3,3)+433√5Adf(5,3)111√10Adf(5,4)−13Aff(2,1)+111√103Aff(4,1)−5429√7Aff(6,1)Aff(0,0)−733Aff(4,0)+10143Aff(6,0)−Aff(2,1)√15−433√2Aff(4,1)+5143√353Aff(6,1)−2Aff(2,2)3√5−Aff(4,2)11√3+20429√14Aff(6,2)133√14Aff(4,3)+5143√42Aff(6,3)133√70Aff(4,4)+10143√14Aff(6,4)513√1433Aff(6,5)
Y(3)−1−Asf(3,1)√735√27Apf(2,0)−13√27Apf(4,0)−25√67Apf(2,1)−13√57Apf(4,1)15√37Apf(2,2)−13√57Apf(4,2)Adf(1,1)√35−2√2105Adf(3,1)+5Adf(5,1)11√212√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0)−√635Adf(1,1)−Adf(3,1)√35−511√27Adf(5,1)−Adf(3,2)√21−5Adf(5,2)11√313√27Adf(3,3)−533√2Adf(5,3)−13√25Aff(2,2)+111√6Aff(4,2)−10429√7Aff(6,2)−Aff(2,1)√15−433√2Aff(4,1)+5143√353Aff(6,1)Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0)−115√2Aff(2,1)−111√53Aff(4,1)−25429√14Aff(6,1)−25√23Aff(2,2)−233√10Aff(4,2)−10143√353Aff(6,2)−133√14Aff(4,3)−5143√42Aff(6,3)111√143Aff(4,4)−5143√703Aff(6,4)
Y(3)0Asf(3,0)√73Apf(2,1)5√7−13√1021Apf(4,1)35√37Apf(2,0)+4Apf(4,0)3√2113√1021Apf(4,1)−3Apf(2,1)5√7533Adf(5,2)−2Adf(3,2)3√7√335Adf(1,1)−13√235Adf(3,1)−20Adf(5,1)33√73Adf(1,0)√35+4Adf(3,0)3√35+1033√57Adf(5,0)−√335Adf(1,1)+13√235Adf(3,1)+20Adf(5,1)33√7533Adf(5,2)−2Adf(3,2)3√7111√7Aff(4,3)−10143√73Aff(6,3)−2Aff(2,2)3√5−Aff(4,2)11√3+20429√14Aff(6,2)−115√2Aff(2,1)−111√53Aff(4,1)−25429√14Aff(6,1)Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)115√2Aff(2,1)+111√53Aff(4,1)+25429√14Aff(6,1)−2Aff(2,2)3√5−Aff(4,2)11√3+20429√14Aff(6,2)10143√73Aff(6,3)−111√7Aff(4,3)
Y(3)1Asf(3,1)√715√37Apf(2,2)−13√57Apf(4,2)25√67Apf(2,1)+13√57Apf(4,1)35√27Apf(2,0)−13√27Apf(4,0)533√2Adf(5,3)−13√27Adf(3,3)−Adf(3,2)√21−5Adf(5,2)11√3√635Adf(1,1)+Adf(3,1)√35+511√27Adf(5,1)2√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0)−Adf(1,1)√35+2√2105Adf(3,1)−5Adf(5,1)11√21111√143Aff(4,4)−5143√703Aff(6,4)133√14Aff(4,3)+5143√42Aff(6,3)−25√23Aff(2,2)−233√10Aff(4,2)−10143√353Aff(6,2)115√2Aff(2,1)+111√53Aff(4,1)+25429√14Aff(6,1)Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0)Aff(2,1)√15+433√2Aff(4,1)−5143√353Aff(6,1)−13√25Aff(2,2)+111√6Aff(4,2)−10429√7Aff(6,2)
Y(3)2Asf(3,2)√7−13Apf(4,3)√335Apf(2,2)+2Apf(4,2)3√7√635Apf(2,1)−Apf(4,1)3√7111√10Adf(5,4)−13√57Adf(3,3)−433√5Adf(5,3)111√5Adf(5,2)√27Adf(1,1)+Adf(3,1)√21−211√1021Adf(5,1)Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7−513√1433Aff(6,5)133√70Aff(4,4)+10143√14Aff(6,4)−133√14Aff(4,3)−5143√42Aff(6,3)−2Aff(2,2)3√5−Aff(4,2)11√3+20429√14Aff(6,2)Aff(2,1)√15+433√2Aff(4,1)−5143√353Aff(6,1)Aff(0,0)−733Aff(4,0)+10143Aff(6,0)13Aff(2,1)−111√103Aff(4,1)+5429√7Aff(6,1)
Y(3)3Asf(3,3)√7−2Apf(4,4)3√3Apf(4,3)3√33Apf(2,2)√35−Apf(4,2)3√21511√23Adf(5,5)−211√53Adf(5,4)233√5Adf(5,3)−13√57Adf(3,3)13√57Adf(3,2)−133√5Adf(5,2)√37Adf(1,1)−13√27Adf(3,1)+133√57Adf(5,1)−1013√733Aff(6,6)513√1433Aff(6,5)111√143Aff(4,4)−5143√703Aff(6,4)10143√73Aff(6,3)−111√7Aff(4,3)−13√25Aff(2,2)+111√6Aff(4,2)−10429√7Aff(6,2)13Aff(2,1)−111√103Aff(4,1)+5429√7Aff(6,1)Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0)

Rotation matrix to symmetry adapted functions (choice is not unique)

Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field

Y(0)0 Y(1)−1 Y(1)0 Y(1)1 Y(2)−2 Y(2)−1 Y(2)0 Y(2)1 Y(2)2 Y(3)−3 Y(3)−2 Y(3)−1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
s1000000000000000
pz0010000000000000
px01√20−1√2000000000000
py0i√20i√2000000000000
dz2−x20000−12√20√320−12√20000000
d3y2−r20000−√3220−120−√3220000000
dxy0000i√2000−i√20000000
dyz00000i√20i√200000000
dxz000001√20−1√200000000
fxyz0000000000i√2000−i√20
fz(5z2−r2)0000000000001000
fx(5x2−r2)000000000√540−√340√340−√54
fy(5y2−r2)000000000−i√540−i√340−i√340−i√54
fz(x2−y2)0 0 0 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } 0 \frac{1}{\sqrt{2}} 0 0 0 \frac{1}{\sqrt{2}} 0
f_{x\left(y^2-z^2\right)} \color{darkred}{ 0 } 0 0 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } -\frac{\sqrt{3}}{4} 0 -\frac{\sqrt{5}}{4} 0 \frac{\sqrt{5}}{4} 0 \frac{\sqrt{3}}{4}
f_{y\left(z^2-x^2\right)} \color{darkred}{ 0 } 0 0 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } -\frac{i \sqrt{3}}{4} 0 \frac{i \sqrt{5}}{4} 0 \frac{i \sqrt{5}}{4} 0 -\frac{i \sqrt{3}}{4}

One particle coupling on a basis of symmetry adapted functions

After rotation we find

\text{s} p_z p_x p_y d_{z^2-x^2} d_{3y^2-r^2} d_{\text{xy}} d_{\text{yz}} d_{\text{xz}} f_{\text{xyz}} f_{z\left(5z^2-r^2\right)} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(x^2-y^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)}
\text{s} \text{Ass}(0,0) \color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }\color{darkred}{ 0 } \frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} -\frac{\text{Asd}(2,0)}{2 \sqrt{5}}-\sqrt{\frac{3}{10}} \text{Asd}(2,2) 0 0 -\sqrt{\frac{2}{5}} \text{Asd}(2,1) \color{darkred}{ 0 }\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }\color{darkred}{ 0 }\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }\color{darkred}{ 0 }
p_z \color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} } \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) -\frac{1}{5} \sqrt{6} \text{App}(2,1) 0 \color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}+\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)-\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) } 0 \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) 0 \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) 0
p_x \color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) } -\frac{1}{5} \sqrt{6} \text{App}(2,1) \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2) 0 \color{darkred}{ \sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{14} \text{Apd}(3,3) }\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)+\frac{3 \text{Apd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Apd}(3,3) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) } 0 \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) 0 -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} 0
p_y \color{darkred}{ 0 } 0 0 \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }\color{darkred}{ 0 } -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) 0 0 -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) 0 0 \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}}
d_{z^2-x^2} \frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} \color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}+\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }\color{darkred}{ \sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{14} \text{Apd}(3,3) }\color{darkred}{ 0 } \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{19}{84} \text{Add}(4,0)-\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{6} \sqrt{\frac{5}{14}} \text{Add}(4,4) -\frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)-\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) 0 0 \frac{1}{6} \sqrt{\frac{5}{7}} \text{Add}(4,3)-\frac{1}{6} \sqrt{5} \text{Add}(4,1) \color{darkred}{ 0 }\color{darkred}{ \frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)+\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)+\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }\color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,3)+\frac{5 \text{Adf}(5,1)}{6 \sqrt{14}}-\frac{5}{44} \sqrt{3} \text{Adf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }\color{darkred}{ 0 }\color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{17}{22} \sqrt{\frac{5}{42}} \text{Adf}(5,1)+\frac{1}{132} \sqrt{5} \text{Adf}(5,3)-\frac{5}{44} \text{Adf}(5,5) }\color{darkred}{ 0 }
d_{3y^2-r^2} -\frac{\text{Asd}(2,0)}{2 \sqrt{5}}-\sqrt{\frac{3}{10}} \text{Asd}(2,2) \color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)-\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)+\frac{3 \text{Apd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Apd}(3,3) }\color{darkred}{ 0 } -\frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)-\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) \text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{3}{28} \text{Add}(4,0)+\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{14}} \text{Add}(4,4) 0 0 \frac{2}{7} \sqrt{2} \text{Add}(2,1)+\frac{1}{14} \sqrt{\frac{5}{3}} \text{Add}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Add}(4,3) \color{darkred}{ 0 }\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}+\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{7}{15}} \text{Adf}(3,1)-\frac{\text{Adf}(3,3)}{6 \sqrt{7}}-\frac{5 \text{Adf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Adf}(5,3)+\frac{5}{44} \sqrt{5} \text{Adf}(5,5) }\color{darkred}{ 0 }\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)+\frac{\text{Adf}(3,0)}{\sqrt{21}}-\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }\color{darkred}{ -\sqrt{\frac{3}{14}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)-\frac{3}{22} \sqrt{\frac{5}{14}} \text{Adf}(5,1)-\frac{7}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,3)-\frac{5}{44} \sqrt{3} \text{Adf}(5,5) }\color{darkred}{ 0 }
d_{\text{xy}} 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) } 0 0 \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) 0 \color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }
d_{\text{yz}} 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) } 0 0 -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) 0 \color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }
d_{\text{xz}} -\sqrt{\frac{2}{5}} \text{Asd}(2,1) \color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }\color{darkred}{ 0 } \frac{1}{6} \sqrt{\frac{5}{7}} \text{Add}(4,3)-\frac{1}{6} \sqrt{5} \text{Add}(4,1) \frac{2}{7} \sqrt{2} \text{Add}(2,1)+\frac{1}{14} \sqrt{\frac{5}{3}} \text{Add}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Add}(4,3) 0 0 \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) \color{darkred}{ 0 }\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }\color{darkred}{ 0 }
f_{\text{xyz}} \color{darkred}{ 0 } 0 0 -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }\color{darkred}{ 0 } \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) 0 0 \frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) 0 0 -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5)
f_{z\left(5z^2-r^2\right)} \color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} } \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) 0 \color{darkred}{ \frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)+\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)+\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}+\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) } 0 \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) 0 -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) 0
f_{x\left(5x^2-r^2\right)} \color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) } \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) 0 \color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,3)+\frac{5 \text{Adf}(5,1)}{6 \sqrt{14}}-\frac{5}{44} \sqrt{3} \text{Adf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{7}{15}} \text{Adf}(3,1)-\frac{\text{Adf}(3,3)}{6 \sqrt{7}}-\frac{5 \text{Adf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Adf}(5,3)+\frac{5}{44} \sqrt{5} \text{Adf}(5,5) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} } 0 \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) 0 -\frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) 0
f_{y\left(5y^2-r^2\right)} \color{darkred}{ 0 } 0 0 -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }\color{darkred}{ 0 } \frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) 0 0 \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) 0 0 -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6)
f_{z\left(x^2-y^2\right)} \color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) } \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) 0 \color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)+\frac{\text{Adf}(3,0)}{\sqrt{21}}-\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) } 0 -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) -\frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) 0 \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) \frac{\text{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) 0
f_{x\left(y^2-z^2\right)} \color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) } \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} 0 \color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{17}{22} \sqrt{\frac{5}{42}} \text{Adf}(5,1)+\frac{1}{132} \sqrt{5} \text{Adf}(5,3)-\frac{5}{44} \text{Adf}(5,5) }\color{darkred}{ -\sqrt{\frac{3}{14}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)-\frac{3}{22} \sqrt{\frac{5}{14}} \text{Adf}(5,1)-\frac{7}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,3)-\frac{5}{44} \sqrt{3} \text{Adf}(5,5) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) } 0 \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) 0 \frac{\text{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) 0
f_{y\left(z^2-x^2\right)} \color{darkred}{ 0 } 0 0 \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }\color{darkred}{ 0 } -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) 0 0 -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) 0 0 \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6)

Coupling for a single shell

Although the parameters A_{l'',l'}(k,m) uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters A_{l'',l'}(k,m) by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum l'' and l'.

Click on one of the subsections to expand it or

Potential for s orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \text{Eap} & k=0\land m=0 \\ 0 & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Eap, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{0, 0, Eap} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{0}^{(0)}}
{Y_{0}^{(0)}} \text{Eap}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

\text{s}
\text{s} \text{Eap}

Rotation matrix used

Rotation matrix used

{Y_{0}^{(0)}}
\text{s} 1

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Eap}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2 \sqrt{\pi }}
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2 \sqrt{\pi }}

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{3} (\text{Eapp}+\text{Eapx}+\text{Eapz}) & k=0\land m=0 \\ 0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\ -\frac{5 (\text{Eapp}-\text{Eapx})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\ \frac{5 \text{Mapzx}}{\sqrt{6}} & k=2\land m=-1 \\ -\frac{5}{6} (\text{Eapp}+\text{Eapx}-2 \text{Eapz}) & k=2\land m=0 \\ -\frac{5 \text{Mapzx}}{\sqrt{6}} & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eapp + Eapx + Eapz)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {(-5*(Eapp - Eapx))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}, {(5*Mapzx)/Sqrt[6], k == 2 && m == -1}, {(-5*(Eapp + Eapx - 2*Eapz))/6, k == 2 && m == 0}}, (-5*Mapzx)/Sqrt[6]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{0, 0, (1/3)*(Eapp + Eapx + Eapz)} , 
       {2, 0, (-5/6)*(Eapp + Eapx + (-2)*(Eapz))} , 
       {2, 1, (-5)*((1/(sqrt(6)))*(Mapzx))} , 
       {2,-1, (5)*((1/(sqrt(6)))*(Mapzx))} , 
       {2,-2, (-5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapx)))} , 
       {2, 2, (-5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapx)))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-1}^{(1)}} {Y_{0}^{(1)}} {Y_{1}^{(1)}}
{Y_{-1}^{(1)}} \frac{\text{Eapp}+\text{Eapx}}{2} \frac{\text{Mapzx}}{\sqrt{2}} \frac{\text{Eapp}-\text{Eapx}}{2}
{Y_{0}^{(1)}} \frac{\text{Mapzx}}{\sqrt{2}} \text{Eapz} -\frac{\text{Mapzx}}{\sqrt{2}}
{Y_{1}^{(1)}} \frac{\text{Eapp}-\text{Eapx}}{2} -\frac{\text{Mapzx}}{\sqrt{2}} \frac{\text{Eapp}+\text{Eapx}}{2}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

p_z p_x p_y
p_z \text{Eapz} \text{Mapzx} 0
p_x \text{Mapzx} \text{Eapx} 0
p_y 0 0 \text{Eapp}

Rotation matrix used

Rotation matrix used

{Y_{-1}^{(1)}} {Y_{0}^{(1)}} {Y_{1}^{(1)}}
p_z 0 1 0
p_x \frac{1}{\sqrt{2}} 0 -\frac{1}{\sqrt{2}}
p_y \frac{i}{\sqrt{2}} 0 \frac{i}{\sqrt{2}}

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Eapz}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} z
\text{Eapx}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} x
\text{Eapp}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} y

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{5} (\text{Eappxy}+\text{Eappyz}+\text{Eapxz}+\text{Eapy2}+\text{Eapz2x2}) & k=0\land m=0 \\ 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ \frac{1}{4} \left(-\sqrt{6} \text{Eappyz}+\sqrt{6} \text{Eapxz}-\sqrt{6} \text{Eapy2}+\sqrt{6} \text{Eapz2x2}+2 \sqrt{2} \text{Mapz2x2y2}\right) & k=2\land (m=-2\lor m=2) \\ \frac{\sqrt{3} \text{Mappxyyz}-2 \text{Mapy2xz}}{\sqrt{2}} & k=2\land m=-1 \\ \frac{1}{2} \left(-2 \text{Eappxy}+\text{Eappyz}+\text{Eapxz}-\text{Eapy2}+\text{Eapz2x2}-2 \sqrt{3} \text{Mapz2x2y2}\right) & k=2\land m=0 \\ \sqrt{2} \text{Mapy2xz}-\sqrt{\frac{3}{2}} \text{Mappxyyz} & k=2\land m=1 \\ -\frac{3}{8} \sqrt{\frac{7}{10}} \left(4 \text{Eappxy}-3 \text{Eapy2}-\text{Eapz2x2}-2 \sqrt{3} \text{Mapz2x2y2}\right) & k=4\land (m=-4\lor m=4) \\ -\frac{3}{4} \sqrt{\frac{7}{5}} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) & k=4\land m=-3 \\ -\frac{3 \left(4 \text{Eappyz}-4 \text{Eapxz}-3 \text{Eapy2}+3 \text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right)}{4 \sqrt{10}} & k=4\land (m=-2\lor m=2) \\ -\frac{3 \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}-7 \text{Mapz2x2xz}\right)}{4 \sqrt{5}} & k=4\land m=-1 \\ \frac{3}{40} \left(4 \text{Eappxy}-16 \text{Eappyz}-16 \text{Eapxz}+9 \text{Eapy2}+19 \text{Eapz2x2}-10 \sqrt{3} \text{Mapz2x2y2}\right) & k=4\land m=0 \\ \frac{3 \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}-7 \text{Mapz2x2xz}\right)}{4 \sqrt{5}} & k=4\land m=1 \\ \frac{3}{4} \sqrt{\frac{7}{5}} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eappxy + Eappyz + Eapxz + Eapy2 + Eapz2x2)/5, k == 0 && m == 0}, {0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(-(Sqrt[6]*Eappyz) + Sqrt[6]*Eapxz - Sqrt[6]*Eapy2 + Sqrt[6]*Eapz2x2 + 2*Sqrt[2]*Mapz2x2y2)/4, k == 2 && (m == -2 || m == 2)}, {(Sqrt[3]*Mappxyyz - 2*Mapy2xz)/Sqrt[2], k == 2 && m == -1}, {(-2*Eappxy + Eappyz + Eapxz - Eapy2 + Eapz2x2 - 2*Sqrt[3]*Mapz2x2y2)/2, k == 2 && m == 0}, {-(Sqrt[3/2]*Mappxyyz) + Sqrt[2]*Mapy2xz, k == 2 && m == 1}, {(-3*Sqrt[7/10]*(4*Eappxy - 3*Eapy2 - Eapz2x2 - 2*Sqrt[3]*Mapz2x2y2))/8, k == 4 && (m == -4 || m == 4)}, {(-3*Sqrt[7/5]*(2*Mappxyyz + Sqrt[3]*Mapy2xz + Mapz2x2xz))/4, k == 4 && m == -3}, {(-3*(4*Eappyz - 4*Eapxz - 3*Eapy2 + 3*Eapz2x2 + 2*Sqrt[3]*Mapz2x2y2))/(4*Sqrt[10]), k == 4 && (m == -2 || m == 2)}, {(-3*(2*Mappxyyz + Sqrt[3]*Mapy2xz - 7*Mapz2x2xz))/(4*Sqrt[5]), k == 4 && m == -1}, {(3*(4*Eappxy - 16*Eappyz - 16*Eapxz + 9*Eapy2 + 19*Eapz2x2 - 10*Sqrt[3]*Mapz2x2y2))/40, k == 4 && m == 0}, {(3*(2*Mappxyyz + Sqrt[3]*Mapy2xz - 7*Mapz2x2xz))/(4*Sqrt[5]), k == 4 && m == 1}}, (3*Sqrt[7/5]*(2*Mappxyyz + Sqrt[3]*Mapy2xz + Mapz2x2xz))/4]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{0, 0, (1/5)*(Eappxy + Eappyz + Eapxz + Eapy2 + Eapz2x2)} , 
       {2, 0, (1/2)*((-2)*(Eappxy) + Eappyz + Eapxz + (-1)*(Eapy2) + Eapz2x2 + (-2)*((sqrt(3))*(Mapz2x2y2)))} , 
       {2,-1, (1/(sqrt(2)))*((sqrt(3))*(Mappxyyz) + (-2)*(Mapy2xz))} , 
       {2, 1, (-1)*((sqrt(3/2))*(Mappxyyz)) + (sqrt(2))*(Mapy2xz)} , 
       {2,-2, (1/4)*((-1)*((sqrt(6))*(Eappyz)) + (sqrt(6))*(Eapxz) + (-1)*((sqrt(6))*(Eapy2)) + (sqrt(6))*(Eapz2x2) + (2)*((sqrt(2))*(Mapz2x2y2)))} , 
       {2, 2, (1/4)*((-1)*((sqrt(6))*(Eappyz)) + (sqrt(6))*(Eapxz) + (-1)*((sqrt(6))*(Eapy2)) + (sqrt(6))*(Eapz2x2) + (2)*((sqrt(2))*(Mapz2x2y2)))} , 
       {4, 0, (3/40)*((4)*(Eappxy) + (-16)*(Eappyz) + (-16)*(Eapxz) + (9)*(Eapy2) + (19)*(Eapz2x2) + (-10)*((sqrt(3))*(Mapz2x2y2)))} , 
       {4,-1, (-3/4)*((1/(sqrt(5)))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + (-7)*(Mapz2x2xz)))} , 
       {4, 1, (3/4)*((1/(sqrt(5)))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + (-7)*(Mapz2x2xz)))} , 
       {4,-2, (-3/4)*((1/(sqrt(10)))*((4)*(Eappyz) + (-4)*(Eapxz) + (-3)*(Eapy2) + (3)*(Eapz2x2) + (2)*((sqrt(3))*(Mapz2x2y2))))} , 
       {4, 2, (-3/4)*((1/(sqrt(10)))*((4)*(Eappyz) + (-4)*(Eapxz) + (-3)*(Eapy2) + (3)*(Eapz2x2) + (2)*((sqrt(3))*(Mapz2x2y2))))} , 
       {4,-3, (-3/4)*((sqrt(7/5))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + Mapz2x2xz))} , 
       {4, 3, (3/4)*((sqrt(7/5))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + Mapz2x2xz))} , 
       {4,-4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapy2) + (-1)*(Eapz2x2) + (-2)*((sqrt(3))*(Mapz2x2y2))))} , 
       {4, 4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapy2) + (-1)*(Eapz2x2) + (-2)*((sqrt(3))*(Mapz2x2y2))))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{-2}^{(2)}} \frac{1}{8} \left(4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right) \frac{1}{4} \left(2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right) \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right) \frac{1}{4} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right)
{Y_{-1}^{(2)}} \frac{1}{4} \left(2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right) \frac{\text{Eappyz}+\text{Eapxz}}{2} \frac{1}{4} \left(\sqrt{6} \text{Mapz2x2xz}-\sqrt{2} \text{Mapy2xz}\right) \frac{\text{Eappyz}-\text{Eapxz}}{2} \frac{1}{4} \left(-2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right)
{Y_{0}^{(2)}} \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right) \frac{1}{4} \left(\sqrt{6} \text{Mapz2x2xz}-\sqrt{2} \text{Mapy2xz}\right) \frac{1}{4} \left(\text{Eapy2}+3 \text{Eapz2x2}-2 \sqrt{3} \text{Mapz2x2y2}\right) \frac{\text{Mapy2xz}-\sqrt{3} \text{Mapz2x2xz}}{2 \sqrt{2}} \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right)
{Y_{1}^{(2)}} \frac{1}{4} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) \frac{\text{Eappyz}-\text{Eapxz}}{2} \frac{\text{Mapy2xz}-\sqrt{3} \text{Mapz2x2xz}}{2 \sqrt{2}} \frac{\text{Eappyz}+\text{Eapxz}}{2} \frac{1}{4} \left(-2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right)
{Y_{2}^{(2)}} \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right) \frac{1}{4} \left(-2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right) \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right) \frac{1}{4} \left(-2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) \frac{1}{8} \left(4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right)

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{z^2-x^2} d_{3y^2-r^2} d_{\text{xy}} d_{\text{yz}} d_{\text{xz}}
d_{z^2-x^2} \text{Eapz2x2} \text{Mapz2x2y2} 0 0 \text{Mapz2x2xz}
d_{3y^2-r^2} \text{Mapz2x2y2} \text{Eapy2} 0 0 \text{Mapy2xz}
d_{\text{xy}} 0 0 \text{Eappxy} \text{Mappxyyz} 0
d_{\text{yz}} 0 0 \text{Mappxyyz} \text{Eappyz} 0
d_{\text{xz}} \text{Mapz2x2xz} \text{Mapy2xz} 0 0 \text{Eapxz}

Rotation matrix used

Rotation matrix used

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
d_{z^2-x^2} -\frac{1}{2 \sqrt{2}} 0 \frac{\sqrt{3}}{2} 0 -\frac{1}{2 \sqrt{2}}
d_{3y^2-r^2} -\frac{\sqrt{\frac{3}{2}}}{2} 0 -\frac{1}{2} 0 -\frac{\sqrt{\frac{3}{2}}}{2}
d_{\text{xy}} \frac{i}{\sqrt{2}} 0 0 0 -\frac{i}{\sqrt{2}}
d_{\text{yz}} 0 \frac{i}{\sqrt{2}} 0 \frac{i}{\sqrt{2}} 0
d_{\text{xz}} 0 \frac{1}{\sqrt{2}} 0 -\frac{1}{\sqrt{2}} 0

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Eapz2x2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{15}{\pi }} \left(-2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{8} \sqrt{\frac{15}{\pi }} \left(-x^2+y^2+3 z^2-1\right)
\text{Eapy2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} -\frac{1}{16} \sqrt{\frac{5}{\pi }} \left(6 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{8} \sqrt{\frac{5}{\pi }} \left(-3 x^2+3 y^2-3 z^2+1\right)
\text{Eappxy}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{15}{\pi }} x y
\text{Eappyz}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{15}{\pi }} y z
\text{Eapxz}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{15}{\pi }} x z

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{7} (\text{Eappxyz}+\text{Eappy3}+\text{Eappyz2x2}+\text{Eapx3}+\text{Eapxy2z2}+\text{Eapz3}+\text{Eapzx2y2}) & k=0\land m=0 \\ 0 & (k\neq 2\land k\neq 4\land k\neq 6)\lor (k\neq 4\land k\neq 6\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (k\neq 6\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4)\lor (m\neq -6\land m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5\land m\neq 6) \\ \frac{5}{28} \left(-\sqrt{6} \text{Eappy3}+\sqrt{6} \text{Eapx3}+\sqrt{10} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2}-2 \text{Mapz3zx2y2})\right) & k=2\land (m=-2\lor m=2) \\ -\frac{5}{56} \left(4 \sqrt{10} \text{Mappxyzy3}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapz3x3}+\sqrt{10} \text{Mapz3xy2z2}+5 \sqrt{6} \text{Mapzx2y2xy2z2}\right) & k=2\land m=-1 \\ -\frac{5}{14} \left(\text{Eappy3}+\text{Eapx3}-2 \text{Eapz3}+\sqrt{15} \text{Mappy3yz2x2}-\sqrt{15} \text{Mapx3xy2z2}\right) & k=2\land m=0 \\ \frac{5}{56} \left(4 \sqrt{10} \text{Mappxyzy3}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapz3x3}+\sqrt{10} \text{Mapz3xy2z2}+5 \sqrt{6} \text{Mapzx2y2xy2z2}\right) & k=2\land m=1 \\ -\frac{3 \left(4 \sqrt{5} \text{Eappxyz}-3 \sqrt{5} \text{Eappy3}+3 \sqrt{5} \text{Eappyz2x2}-3 \sqrt{5} \text{Eapx3}+3 \sqrt{5} \text{Eapxy2z2}-4 \sqrt{5} \text{Eapzx2y2}+2 \sqrt{3} \text{Mappy3yz2x2}-2 \sqrt{3} \text{Mapx3xy2z2}\right)}{8 \sqrt{14}} & k=4\land (m=-4\lor m=4) \\ \frac{3 \sqrt{3} \text{Mappxyzy3}-3 \left(\sqrt{5} \text{Mappxyzyz2x2}+\sqrt{3} \text{Mapx3zx2y2}+3 \sqrt{5} \text{Mapz3x3}-3 \sqrt{3} \text{Mapz3xy2z2}+\sqrt{5} \text{Mapzx2y2xy2z2}\right)}{4 \sqrt{7}} & k=4\land m=-3 \\ \frac{3}{56} \left(3 \sqrt{10} \text{Eappy3}-7 \sqrt{10} \text{Eappyz2x2}-3 \sqrt{10} \text{Eapx3}+7 \sqrt{10} \text{Eapxy2z2}+2 \sqrt{6} \text{Mappy3yz2x2}+2 \sqrt{6} \text{Mapx3xy2z2}-4 \sqrt{6} \text{Mapz3zx2y2}\right) & k=4\land (m=-2\lor m=2) \\ \frac{3}{28} \left(\sqrt{3} \text{Mappxyzy3}+7 \sqrt{5} \text{Mappxyzyz2x2}-9 \sqrt{3} \text{Mapx3zx2y2}-3 \sqrt{5} \text{Mapz3x3}-5 \sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapzx2y2xy2z2}\right) & k=4\land m=-1 \\ -\frac{3}{56} \left(28 \text{Eappxyz}-9 \text{Eappy3}-7 \text{Eappyz2x2}-9 \text{Eapx3}-7 \text{Eapxy2z2}-24 \text{Eapz3}+28 \text{Eapzx2y2}-2 \sqrt{15} \text{Mappy3yz2x2}+2 \sqrt{15} \text{Mapx3xy2z2}\right) & k=4\land m=0 \\ -\frac{3}{28} \left(\sqrt{3} \text{Mappxyzy3}+7 \sqrt{5} \text{Mappxyzyz2x2}-9 \sqrt{3} \text{Mapx3zx2y2}-3 \sqrt{5} \text{Mapz3x3}-5 \sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapzx2y2xy2z2}\right) & k=4\land m=1 \\ \frac{3 \left(-\sqrt{3} \text{Mappxyzy3}+\sqrt{5} \text{Mappxyzyz2x2}+\sqrt{3} \text{Mapx3zx2y2}+3 \sqrt{5} \text{Mapz3x3}-3 \sqrt{3} \text{Mapz3xy2z2}+\sqrt{5} \text{Mapzx2y2xy2z2}\right)}{4 \sqrt{7}} & k=4\land m=3 \\ -\frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappy3}+3 \sqrt{3} \text{Eappyz2x2}-5 \sqrt{3} \text{Eapx3}-3 \sqrt{3} \text{Eapxy2z2}+6 \sqrt{5} \text{Mappy3yz2x2}+6 \sqrt{5} \text{Mapx3xy2z2}\right) & k=6\land (m=-6\lor m=6) \\ \frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}\right) & k=6\land m=-5 \\ -\frac{13 \left(24 \text{Eappxyz}+15 \text{Eappy3}-15 \text{Eappyz2x2}+15 \text{Eapx3}-15 \text{Eapxy2z2}-24 \text{Eapzx2y2}-2 \sqrt{15} \text{Mappy3yz2x2}+2 \sqrt{15} \text{Mapx3xy2z2}\right)}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ \frac{13 \left(9 \text{Mappxyzy3}-3 \sqrt{15} \text{Mappxyzyz2x2}-9 \text{Mapx3zx2y2}+2 \sqrt{15} \text{Mapz3x3}-6 \text{Mapz3xy2z2}-3 \sqrt{15} \text{Mapzx2y2xy2z2}\right)}{40 \sqrt{7}} & k=6\land m=-3 \\ -\frac{13 \left(5 \sqrt{15} \text{Eappy3}+3 \sqrt{15} \text{Eappyz2x2}-5 \sqrt{15} \text{Eapx3}-3 \sqrt{15} \text{Eapxy2z2}-34 \text{Mappy3yz2x2}-34 \text{Mapx3xy2z2}-64 \text{Mapz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\ \frac{13}{280} \left(\sqrt{70} \text{Mappxyzy3}-3 \sqrt{42} \text{Mappxyzyz2x2}+2 \sqrt{70} \text{Mapx3zx2y2}-5 \sqrt{42} \text{Mapz3x3}-5 \sqrt{70} \text{Mapz3xy2z2}+2 \sqrt{42} \text{Mapzx2y2xy2z2}\right) & k=6\land m=-1 \\ \frac{13}{560} \left(24 \text{Eappxyz}-25 \text{Eappy3}-39 \text{Eappyz2x2}-25 \text{Eapx3}-39 \text{Eapxy2z2}+80 \text{Eapz3}+24 \text{Eapzx2y2}+14 \sqrt{15} \text{Mappy3yz2x2}-14 \sqrt{15} \text{Mapx3xy2z2}\right) & k=6\land m=0 \\ -\frac{13}{280} \left(\sqrt{70} \text{Mappxyzy3}-3 \sqrt{42} \text{Mappxyzyz2x2}+2 \sqrt{70} \text{Mapx3zx2y2}-5 \sqrt{42} \text{Mapz3x3}-5 \sqrt{70} \text{Mapz3xy2z2}+2 \sqrt{42} \text{Mapzx2y2xy2z2}\right) & k=6\land m=1 \\ -\frac{13 \left(9 \text{Mappxyzy3}-3 \sqrt{15} \text{Mappxyzyz2x2}-9 \text{Mapx3zx2y2}+2 \sqrt{15} \text{Mapz3x3}-6 \text{Mapz3xy2z2}-3 \sqrt{15} \text{Mapzx2y2xy2z2}\right)}{40 \sqrt{7}} & k=6\land m=3 \\ -\frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}\right) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eappxyz + Eappy3 + Eappyz2x2 + Eapx3 + Eapxy2z2 + Eapz3 + Eapzx2y2)/7, k == 0 && m == 0}, {0, (k != 2 && k != 4 && k != 6) || (k != 4 && k != 6 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (k != 6 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4) || (m != -6 && m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5 && m != 6)}, {(5*(-(Sqrt[6]*Eappy3) + Sqrt[6]*Eapx3 + Sqrt[10]*(Mappy3yz2x2 + Mapx3xy2z2 - 2*Mapz3zx2y2)))/28, k == 2 && (m == -2 || m == 2)}, {(-5*(4*Sqrt[10]*Mappxyzy3 - Sqrt[10]*Mapx3zx2y2 + Sqrt[6]*Mapz3x3 + Sqrt[10]*Mapz3xy2z2 + 5*Sqrt[6]*Mapzx2y2xy2z2))/56, k == 2 && m == -1}, {(-5*(Eappy3 + Eapx3 - 2*Eapz3 + Sqrt[15]*Mappy3yz2x2 - Sqrt[15]*Mapx3xy2z2))/14, k == 2 && m == 0}, {(5*(4*Sqrt[10]*Mappxyzy3 - Sqrt[10]*Mapx3zx2y2 + Sqrt[6]*Mapz3x3 + Sqrt[10]*Mapz3xy2z2 + 5*Sqrt[6]*Mapzx2y2xy2z2))/56, k == 2 && m == 1}, {(-3*(4*Sqrt[5]*Eappxyz - 3*Sqrt[5]*Eappy3 + 3*Sqrt[5]*Eappyz2x2 - 3*Sqrt[5]*Eapx3 + 3*Sqrt[5]*Eapxy2z2 - 4*Sqrt[5]*Eapzx2y2 + 2*Sqrt[3]*Mappy3yz2x2 - 2*Sqrt[3]*Mapx3xy2z2))/(8*Sqrt[14]), k == 4 && (m == -4 || m == 4)}, {(3*Sqrt[3]*Mappxyzy3 - 3*(Sqrt[5]*Mappxyzyz2x2 + Sqrt[3]*Mapx3zx2y2 + 3*Sqrt[5]*Mapz3x3 - 3*Sqrt[3]*Mapz3xy2z2 + Sqrt[5]*Mapzx2y2xy2z2))/(4*Sqrt[7]), k == 4 && m == -3}, {(3*(3*Sqrt[10]*Eappy3 - 7*Sqrt[10]*Eappyz2x2 - 3*Sqrt[10]*Eapx3 + 7*Sqrt[10]*Eapxy2z2 + 2*Sqrt[6]*Mappy3yz2x2 + 2*Sqrt[6]*Mapx3xy2z2 - 4*Sqrt[6]*Mapz3zx2y2))/56, k == 4 && (m == -2 || m == 2)}, {(3*(Sqrt[3]*Mappxyzy3 + 7*Sqrt[5]*Mappxyzyz2x2 - 9*Sqrt[3]*Mapx3zx2y2 - 3*Sqrt[5]*Mapz3x3 - 5*Sqrt[3]*Mapz3xy2z2 - Sqrt[5]*Mapzx2y2xy2z2))/28, k == 4 && m == -1}, {(-3*(28*Eappxyz - 9*Eappy3 - 7*Eappyz2x2 - 9*Eapx3 - 7*Eapxy2z2 - 24*Eapz3 + 28*Eapzx2y2 - 2*Sqrt[15]*Mappy3yz2x2 + 2*Sqrt[15]*Mapx3xy2z2))/56, k == 4 && m == 0}, {(-3*(Sqrt[3]*Mappxyzy3 + 7*Sqrt[5]*Mappxyzyz2x2 - 9*Sqrt[3]*Mapx3zx2y2 - 3*Sqrt[5]*Mapz3x3 - 5*Sqrt[3]*Mapz3xy2z2 - Sqrt[5]*Mapzx2y2xy2z2))/28, k == 4 && m == 1}, {(3*(-(Sqrt[3]*Mappxyzy3) + Sqrt[5]*Mappxyzyz2x2 + Sqrt[3]*Mapx3zx2y2 + 3*Sqrt[5]*Mapz3x3 - 3*Sqrt[3]*Mapz3xy2z2 + Sqrt[5]*Mapzx2y2xy2z2))/(4*Sqrt[7]), k == 4 && m == 3}, {(-13*Sqrt[11/7]*(5*Sqrt[3]*Eappy3 + 3*Sqrt[3]*Eappyz2x2 - 5*Sqrt[3]*Eapx3 - 3*Sqrt[3]*Eapxy2z2 + 6*Sqrt[5]*Mappy3yz2x2 + 6*Sqrt[5]*Mapx3xy2z2))/160, k == 6 && (m == -6 || m == 6)}, {(13*Sqrt[11/7]*(Sqrt[15]*Mappxyzy3 + 3*Mappxyzyz2x2 + Sqrt[15]*Mapx3zx2y2 - 3*Mapzx2y2xy2z2))/40, k == 6 && m == -5}, {(-13*(24*Eappxyz + 15*Eappy3 - 15*Eappyz2x2 + 15*Eapx3 - 15*Eapxy2z2 - 24*Eapzx2y2 - 2*Sqrt[15]*Mappy3yz2x2 + 2*Sqrt[15]*Mapx3xy2z2))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(13*(9*Mappxyzy3 - 3*Sqrt[15]*Mappxyzyz2x2 - 9*Mapx3zx2y2 + 2*Sqrt[15]*Mapz3x3 - 6*Mapz3xy2z2 - 3*Sqrt[15]*Mapzx2y2xy2z2))/(40*Sqrt[7]), k == 6 && m == -3}, {(-13*(5*Sqrt[15]*Eappy3 + 3*Sqrt[15]*Eappyz2x2 - 5*Sqrt[15]*Eapx3 - 3*Sqrt[15]*Eapxy2z2 - 34*Mappy3yz2x2 - 34*Mapx3xy2z2 - 64*Mapz3zx2y2))/(160*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {(13*(Sqrt[70]*Mappxyzy3 - 3*Sqrt[42]*Mappxyzyz2x2 + 2*Sqrt[70]*Mapx3zx2y2 - 5*Sqrt[42]*Mapz3x3 - 5*Sqrt[70]*Mapz3xy2z2 + 2*Sqrt[42]*Mapzx2y2xy2z2))/280, k == 6 && m == -1}, {(13*(24*Eappxyz - 25*Eappy3 - 39*Eappyz2x2 - 25*Eapx3 - 39*Eapxy2z2 + 80*Eapz3 + 24*Eapzx2y2 + 14*Sqrt[15]*Mappy3yz2x2 - 14*Sqrt[15]*Mapx3xy2z2))/560, k == 6 && m == 0}, {(-13*(Sqrt[70]*Mappxyzy3 - 3*Sqrt[42]*Mappxyzyz2x2 + 2*Sqrt[70]*Mapx3zx2y2 - 5*Sqrt[42]*Mapz3x3 - 5*Sqrt[70]*Mapz3xy2z2 + 2*Sqrt[42]*Mapzx2y2xy2z2))/280, k == 6 && m == 1}, {(-13*(9*Mappxyzy3 - 3*Sqrt[15]*Mappxyzyz2x2 - 9*Mapx3zx2y2 + 2*Sqrt[15]*Mapz3x3 - 6*Mapz3xy2z2 - 3*Sqrt[15]*Mapzx2y2xy2z2))/(40*Sqrt[7]), k == 6 && m == 3}}, (-13*Sqrt[11/7]*(Sqrt[15]*Mappxyzy3 + 3*Mappxyzyz2x2 + Sqrt[15]*Mapx3zx2y2 - 3*Mapzx2y2xy2z2))/40]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{0, 0, (1/7)*(Eappxyz + Eappy3 + Eappyz2x2 + Eapx3 + Eapxy2z2 + Eapz3 + Eapzx2y2)} , 
       {2, 0, (-5/14)*(Eappy3 + Eapx3 + (-2)*(Eapz3) + (sqrt(15))*(Mappy3yz2x2) + (-1)*((sqrt(15))*(Mapx3xy2z2)))} , 
       {2,-1, (-5/56)*((4)*((sqrt(10))*(Mappxyzy3)) + (-1)*((sqrt(10))*(Mapx3zx2y2)) + (sqrt(6))*(Mapz3x3) + (sqrt(10))*(Mapz3xy2z2) + (5)*((sqrt(6))*(Mapzx2y2xy2z2)))} , 
       {2, 1, (5/56)*((4)*((sqrt(10))*(Mappxyzy3)) + (-1)*((sqrt(10))*(Mapx3zx2y2)) + (sqrt(6))*(Mapz3x3) + (sqrt(10))*(Mapz3xy2z2) + (5)*((sqrt(6))*(Mapzx2y2xy2z2)))} , 
       {2,-2, (5/28)*((-1)*((sqrt(6))*(Eappy3)) + (sqrt(6))*(Eapx3) + (sqrt(10))*(Mappy3yz2x2 + Mapx3xy2z2 + (-2)*(Mapz3zx2y2)))} , 
       {2, 2, (5/28)*((-1)*((sqrt(6))*(Eappy3)) + (sqrt(6))*(Eapx3) + (sqrt(10))*(Mappy3yz2x2 + Mapx3xy2z2 + (-2)*(Mapz3zx2y2)))} , 
       {4, 0, (-3/56)*((28)*(Eappxyz) + (-9)*(Eappy3) + (-7)*(Eappyz2x2) + (-9)*(Eapx3) + (-7)*(Eapxy2z2) + (-24)*(Eapz3) + (28)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappy3yz2x2)) + (2)*((sqrt(15))*(Mapx3xy2z2)))} , 
       {4, 1, (-3/28)*((sqrt(3))*(Mappxyzy3) + (7)*((sqrt(5))*(Mappxyzyz2x2)) + (-9)*((sqrt(3))*(Mapx3zx2y2)) + (-3)*((sqrt(5))*(Mapz3x3)) + (-5)*((sqrt(3))*(Mapz3xy2z2)) + (-1)*((sqrt(5))*(Mapzx2y2xy2z2)))} , 
       {4,-1, (3/28)*((sqrt(3))*(Mappxyzy3) + (7)*((sqrt(5))*(Mappxyzyz2x2)) + (-9)*((sqrt(3))*(Mapx3zx2y2)) + (-3)*((sqrt(5))*(Mapz3x3)) + (-5)*((sqrt(3))*(Mapz3xy2z2)) + (-1)*((sqrt(5))*(Mapzx2y2xy2z2)))} , 
       {4,-2, (3/56)*((3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (-3)*((sqrt(10))*(Eapx3)) + (7)*((sqrt(10))*(Eapxy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (2)*((sqrt(6))*(Mapx3xy2z2)) + (-4)*((sqrt(6))*(Mapz3zx2y2)))} , 
       {4, 2, (3/56)*((3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (-3)*((sqrt(10))*(Eapx3)) + (7)*((sqrt(10))*(Eapxy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (2)*((sqrt(6))*(Mapx3xy2z2)) + (-4)*((sqrt(6))*(Mapz3zx2y2)))} , 
       {4,-3, (1/4)*((1/(sqrt(7)))*((3)*((sqrt(3))*(Mappxyzy3)) + (-3)*((sqrt(5))*(Mappxyzyz2x2) + (sqrt(3))*(Mapx3zx2y2) + (3)*((sqrt(5))*(Mapz3x3)) + (-3)*((sqrt(3))*(Mapz3xy2z2)) + (sqrt(5))*(Mapzx2y2xy2z2))))} , 
       {4, 3, (3/4)*((1/(sqrt(7)))*((-1)*((sqrt(3))*(Mappxyzy3)) + (sqrt(5))*(Mappxyzyz2x2) + (sqrt(3))*(Mapx3zx2y2) + (3)*((sqrt(5))*(Mapz3x3)) + (-3)*((sqrt(3))*(Mapz3xy2z2)) + (sqrt(5))*(Mapzx2y2xy2z2)))} , 
       {4,-4, (-3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Eappxyz)) + (-3)*((sqrt(5))*(Eappy3)) + (3)*((sqrt(5))*(Eappyz2x2)) + (-3)*((sqrt(5))*(Eapx3)) + (3)*((sqrt(5))*(Eapxy2z2)) + (-4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappy3yz2x2)) + (-2)*((sqrt(3))*(Mapx3xy2z2))))} , 
       {4, 4, (-3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Eappxyz)) + (-3)*((sqrt(5))*(Eappy3)) + (3)*((sqrt(5))*(Eappyz2x2)) + (-3)*((sqrt(5))*(Eapx3)) + (3)*((sqrt(5))*(Eapxy2z2)) + (-4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappy3yz2x2)) + (-2)*((sqrt(3))*(Mapx3xy2z2))))} , 
       {6, 0, (13/560)*((24)*(Eappxyz) + (-25)*(Eappy3) + (-39)*(Eappyz2x2) + (-25)*(Eapx3) + (-39)*(Eapxy2z2) + (80)*(Eapz3) + (24)*(Eapzx2y2) + (14)*((sqrt(15))*(Mappy3yz2x2)) + (-14)*((sqrt(15))*(Mapx3xy2z2)))} , 
       {6, 1, (-13/280)*((sqrt(70))*(Mappxyzy3) + (-3)*((sqrt(42))*(Mappxyzyz2x2)) + (2)*((sqrt(70))*(Mapx3zx2y2)) + (-5)*((sqrt(42))*(Mapz3x3)) + (-5)*((sqrt(70))*(Mapz3xy2z2)) + (2)*((sqrt(42))*(Mapzx2y2xy2z2)))} , 
       {6,-1, (13/280)*((sqrt(70))*(Mappxyzy3) + (-3)*((sqrt(42))*(Mappxyzyz2x2)) + (2)*((sqrt(70))*(Mapx3zx2y2)) + (-5)*((sqrt(42))*(Mapz3x3)) + (-5)*((sqrt(70))*(Mapz3xy2z2)) + (2)*((sqrt(42))*(Mapzx2y2xy2z2)))} , 
       {6,-2, (-13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappy3)) + (3)*((sqrt(15))*(Eappyz2x2)) + (-5)*((sqrt(15))*(Eapx3)) + (-3)*((sqrt(15))*(Eapxy2z2)) + (-34)*(Mappy3yz2x2) + (-34)*(Mapx3xy2z2) + (-64)*(Mapz3zx2y2)))} , 
       {6, 2, (-13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappy3)) + (3)*((sqrt(15))*(Eappyz2x2)) + (-5)*((sqrt(15))*(Eapx3)) + (-3)*((sqrt(15))*(Eapxy2z2)) + (-34)*(Mappy3yz2x2) + (-34)*(Mapx3xy2z2) + (-64)*(Mapz3zx2y2)))} , 
       {6, 3, (-13/40)*((1/(sqrt(7)))*((9)*(Mappxyzy3) + (-3)*((sqrt(15))*(Mappxyzyz2x2)) + (-9)*(Mapx3zx2y2) + (2)*((sqrt(15))*(Mapz3x3)) + (-6)*(Mapz3xy2z2) + (-3)*((sqrt(15))*(Mapzx2y2xy2z2))))} , 
       {6,-3, (13/40)*((1/(sqrt(7)))*((9)*(Mappxyzy3) + (-3)*((sqrt(15))*(Mappxyzyz2x2)) + (-9)*(Mapx3zx2y2) + (2)*((sqrt(15))*(Mapz3x3)) + (-6)*(Mapz3xy2z2) + (-3)*((sqrt(15))*(Mapzx2y2xy2z2))))} , 
       {6,-4, (-13/80)*((1/(sqrt(14)))*((24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (15)*(Eapx3) + (-15)*(Eapxy2z2) + (-24)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappy3yz2x2)) + (2)*((sqrt(15))*(Mapx3xy2z2))))} , 
       {6, 4, (-13/80)*((1/(sqrt(14)))*((24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (15)*(Eapx3) + (-15)*(Eapxy2z2) + (-24)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappy3yz2x2)) + (2)*((sqrt(15))*(Mapx3xy2z2))))} , 
       {6, 5, (-13/40)*((sqrt(11/7))*((sqrt(15))*(Mappxyzy3) + (3)*(Mappxyzyz2x2) + (sqrt(15))*(Mapx3zx2y2) + (-3)*(Mapzx2y2xy2z2)))} , 
       {6,-5, (13/40)*((sqrt(11/7))*((sqrt(15))*(Mappxyzy3) + (3)*(Mappxyzyz2x2) + (sqrt(15))*(Mapx3zx2y2) + (-3)*(Mapzx2y2xy2z2)))} , 
       {6,-6, (-13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappy3)) + (3)*((sqrt(3))*(Eappyz2x2)) + (-5)*((sqrt(3))*(Eapx3)) + (-3)*((sqrt(3))*(Eapxy2z2)) + (6)*((sqrt(5))*(Mappy3yz2x2)) + (6)*((sqrt(5))*(Mapx3xy2z2))))} , 
       {6, 6, (-13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappy3)) + (3)*((sqrt(3))*(Eappyz2x2)) + (-5)*((sqrt(3))*(Eapx3)) + (-3)*((sqrt(3))*(Eapxy2z2)) + (6)*((sqrt(5))*(Mappy3yz2x2)) + (6)*((sqrt(5))*(Mapx3xy2z2))))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-3}^{(3)}} \frac{1}{16} \left(5 \text{Eappy3}+3 \text{Eappyz2x2}+5 \text{Eapx3}+3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mapx3xy2z2})\right) \frac{1}{8} \left(-\sqrt{10} \text{Mappxyzy3}-\sqrt{6} \text{Mappxyzyz2x2}+\sqrt{10} \text{Mapx3zx2y2}-\sqrt{6} \text{Mapzx2y2xy2z2}\right) \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) \frac{1}{4} \left(\sqrt{5} \text{Mapz3x3}-\sqrt{3} \text{Mapz3xy2z2}\right) \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right) \frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}} \frac{1}{16} \left(5 \text{Eappy3}+3 \text{Eappyz2x2}-5 \text{Eapx3}-3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right)
{Y_{-2}^{(3)}} \frac{1}{8} \left(-\sqrt{10} \text{Mappxyzy3}-\sqrt{6} \text{Mappxyzyz2x2}+\sqrt{10} \text{Mapx3zx2y2}-\sqrt{6} \text{Mapzx2y2xy2z2}\right) \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{\text{Mapz3zx2y2}}{\sqrt{2}} \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} -\frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}}
{Y_{-1}^{(3)}} \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}+3 \text{Eapx3}+5 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mapx3xy2z2}-\text{Mappy3yz2x2})\right) \frac{1}{4} \left(-\sqrt{3} \text{Mapz3x3}-\sqrt{5} \text{Mapz3xy2z2}\right) \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}-3 \text{Eapx3}-5 \text{Eapxy2z2}-2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right)
{Y_{0}^{(3)}} \frac{1}{4} \left(\sqrt{5} \text{Mapz3x3}-\sqrt{3} \text{Mapz3xy2z2}\right) \frac{\text{Mapz3zx2y2}}{\sqrt{2}} \frac{1}{4} \left(-\sqrt{3} \text{Mapz3x3}-\sqrt{5} \text{Mapz3xy2z2}\right) \text{Eapz3} \frac{1}{4} \left(\sqrt{3} \text{Mapz3x3}+\sqrt{5} \text{Mapz3xy2z2}\right) \frac{\text{Mapz3zx2y2}}{\sqrt{2}} \frac{1}{4} \left(\sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapz3x3}\right)
{Y_{1}^{(3)}} \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right) \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}-3 \text{Eapx3}-5 \text{Eapxy2z2}-2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) \frac{1}{4} \left(\sqrt{3} \text{Mapz3x3}+\sqrt{5} \text{Mapz3xy2z2}\right) \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}+3 \text{Eapx3}+5 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mapx3xy2z2}-\text{Mappy3yz2x2})\right) \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right)
{Y_{2}^{(3)}} \frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}} \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{\text{Mapz3zx2y2}}{\sqrt{2}} \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} \frac{1}{8} \left(\sqrt{10} \text{Mappxyzy3}+\sqrt{6} \text{Mappxyzyz2x2}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapzx2y2xy2z2}\right)
{Y_{3}^{(3)}} \frac{1}{16} \left(5 \text{Eappy3}+3 \text{Eappyz2x2}-5 \text{Eapx3}-3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) -\frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}} \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right) \frac{1}{4} \left(\sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapz3x3}\right) \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) \frac{1}{8} \left(\sqrt{10} \text{Mappxyzy3}+\sqrt{6} \text{Mappxyzyz2x2}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapzx2y2xy2z2}\right) \frac{1}{16} \left(5 \text{Eappy3}+3 \text{Eappyz2x2}+5 \text{Eapx3}+3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mapx3xy2z2})\right)

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{z\left(5z^2-r^2\right)} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(x^2-y^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)}
f_{\text{xyz}} \text{Eappxyz} 0 0 \text{Mappxyzy3} 0 0 \text{Mappxyzyz2x2}
f_{z\left(5z^2-r^2\right)} 0 \text{Eapz3} \text{Mapz3x3} 0 \text{Mapz3zx2y2} \text{Mapz3xy2z2} 0
f_{x\left(5x^2-r^2\right)} 0 \text{Mapz3x3} \text{Eapx3} 0 \text{Mapx3zx2y2} \text{Mapx3xy2z2} 0
f_{y\left(5y^2-r^2\right)} \text{Mappxyzy3} 0 0 \text{Eappy3} 0 0 \text{Mappy3yz2x2}
f_{z\left(x^2-y^2\right)} 0 \text{Mapz3zx2y2} \text{Mapx3zx2y2} 0 \text{Eapzx2y2} \text{Mapzx2y2xy2z2} 0
f_{x\left(y^2-z^2\right)} 0 \text{Mapz3xy2z2} \text{Mapx3xy2z2} 0 \text{Mapzx2y2xy2z2} \text{Eapxy2z2} 0
f_{y\left(z^2-x^2\right)} \text{Mappxyzyz2x2} 0 0 \text{Mappy3yz2x2} 0 0 \text{Eappyz2x2}

Rotation matrix used

Rotation matrix used

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
f_{\text{xyz}} 0 \frac{i}{\sqrt{2}} 0 0 0 -\frac{i}{\sqrt{2}} 0
f_{z\left(5z^2-r^2\right)} 0 0 0 1 0 0 0
f_{x\left(5x^2-r^2\right)} \frac{\sqrt{5}}{4} 0 -\frac{\sqrt{3}}{4} 0 \frac{\sqrt{3}}{4} 0 -\frac{\sqrt{5}}{4}
f_{y\left(5y^2-r^2\right)} -\frac{i \sqrt{5}}{4} 0 -\frac{i \sqrt{3}}{4} 0 -\frac{i \sqrt{3}}{4} 0 -\frac{i \sqrt{5}}{4}
f_{z\left(x^2-y^2\right)} 0 \frac{1}{\sqrt{2}} 0 0 0 \frac{1}{\sqrt{2}} 0
f_{x\left(y^2-z^2\right)} -\frac{\sqrt{3}}{4} 0 -\frac{\sqrt{5}}{4} 0 \frac{\sqrt{5}}{4} 0 \frac{\sqrt{3}}{4}
f_{y\left(z^2-x^2\right)} -\frac{i \sqrt{3}}{4} 0 \frac{i \sqrt{5}}{4} 0 \frac{i \sqrt{5}}{4} 0 -\frac{i \sqrt{3}}{4}

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Eappxyz}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{105}{\pi }} x y z
\text{Eapz3}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)
\text{Eapx3}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)
\text{Eappy3}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} -\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)
\text{Eapzx2y2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)
\text{Eapxy2z2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} -\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} -\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)
\text{Eappyz2x2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-p orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & k\neq 1\lor (m\neq -1\land m\neq 0\land m\neq 1) \\ -A(1,1) & k=1\land m=-1 \\ A(1,0) & k=1\land m=0 \\ A(1,1) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 1 || (m != -1 && m != 0 && m != 1)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}}, A[1, 1]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{1, 0, A(1,0)} , 
       {1,-1, (-1)*(A(1,1))} , 
       {1, 1, A(1,1)} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-1}^{(1)}} {Y_{0}^{(1)}} {Y_{1}^{(1)}}
{Y_{0}^{(0)}} -\frac{A(1,1)}{\sqrt{3}} \frac{A(1,0)}{\sqrt{3}} \frac{A(1,1)}{\sqrt{3}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

p_z p_x p_y
\text{s} \frac{A(1,0)}{\sqrt{3}} -\sqrt{\frac{2}{3}} A(1,1) 0

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\ A(2,2) & k=2\land (m=-2\lor m=2) \\ -A(2,1) & k=2\land m=-1 \\ A(2,0) & k=2\land m=0 \\ A(2,1) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {-A[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}}, A[2, 1]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1))} , 
       {2, 1, A(2,1)} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{0}^{(0)}} \frac{A(2,2)}{\sqrt{5}} -\frac{A(2,1)}{\sqrt{5}} \frac{A(2,0)}{\sqrt{5}} \frac{A(2,1)}{\sqrt{5}} \frac{A(2,2)}{\sqrt{5}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{z^2-x^2} d_{3y^2-r^2} d_{\text{xy}} d_{\text{yz}} d_{\text{xz}}
\text{s} \frac{1}{10} \left(\sqrt{15} A(2,0)-\sqrt{10} A(2,2)\right) -\frac{A(2,0)+\sqrt{6} A(2,2)}{2 \sqrt{5}} 0 0 -\sqrt{\frac{2}{5}} A(2,1)

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3) \\ -A(3,3) & k=3\land m=-3 \\ A(3,2) & k=3\land (m=-2\lor m=2) \\ -A(3,1) & k=3\land m=-1 \\ A(3,0) & k=3\land m=0 \\ A(3,1) & k=3\land m=1 \\ A(3,3) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3)}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}}, A[3, 3]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{3, 0, A(3,0)} , 
       {3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, A(3,2)} , 
       {3, 2, A(3,2)} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{0}^{(0)}} -\frac{A(3,3)}{\sqrt{7}} \frac{A(3,2)}{\sqrt{7}} -\frac{A(3,1)}{\sqrt{7}} \frac{A(3,0)}{\sqrt{7}} \frac{A(3,1)}{\sqrt{7}} \frac{A(3,2)}{\sqrt{7}} \frac{A(3,3)}{\sqrt{7}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{z\left(5z^2-r^2\right)} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(x^2-y^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)}
\text{s} 0 \frac{A(3,0)}{\sqrt{7}} \frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right) 0 \sqrt{\frac{2}{7}} A(3,2) \frac{1}{14} \left(\sqrt{35} A(3,1)+\sqrt{21} A(3,3)\right) 0

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & (k\neq 3\land (k\neq 1\lor (m\neq -1\land m\neq 0\land m\neq 1)))\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3) \\ -A(1,1) & k=1\land m=-1 \\ A(1,0) & k=1\land m=0 \\ A(1,1) & k=1\land m=1 \\ -A(3,3) & k=3\land m=-3 \\ A(3,2) & k=3\land (m=-2\lor m=2) \\ -A(3,1) & k=3\land m=-1 \\ A(3,0) & k=3\land m=0 \\ A(3,1) & k=3\land m=1 \\ A(3,3) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || (m != -1 && m != 0 && m != 1))) || (m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}, {A[1, 1], k == 1 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}}, A[3, 3]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{1, 0, A(1,0)} , 
       {1,-1, (-1)*(A(1,1))} , 
       {1, 1, A(1,1)} , 
       {3, 0, A(3,0)} , 
       {3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, A(3,2)} , 
       {3, 2, A(3,2)} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{-1}^{(1)}} \frac{1}{35} \left(\sqrt{15} A(3,1)-7 \sqrt{10} A(1,1)\right) \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} \frac{A(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} A(3,1) -\frac{1}{7} \sqrt{6} A(3,2) -\frac{3}{7} A(3,3)
{Y_{0}^{(1)}} \frac{1}{7} \sqrt{3} A(3,2) -\frac{7 A(1,1)+2 \sqrt{6} A(3,1)}{7 \sqrt{5}} \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} \frac{A(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} A(3,1) \frac{1}{7} \sqrt{3} A(3,2)
{Y_{1}^{(1)}} \frac{3}{7} A(3,3) -\frac{1}{7} \sqrt{6} A(3,2) \frac{3}{7} \sqrt{\frac{2}{5}} A(3,1)-\frac{A(1,1)}{\sqrt{15}} \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} \sqrt{\frac{2}{5}} A(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} A(3,1)

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{z^2-x^2} d_{3y^2-r^2} d_{\text{xy}} d_{\text{yz}} d_{\text{xz}}
p_z \frac{1}{70} \left(14 \sqrt{5} A(1,0)+9 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) -\frac{14 A(1,0)+9 A(3,0)}{14 \sqrt{15}}-\frac{3 A(3,2)}{7 \sqrt{2}} 0 0 -\sqrt{\frac{2}{5}} A(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} A(3,1)
p_x \sqrt{\frac{2}{5}} A(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} A(3,1)+\frac{3}{14} A(3,3) \frac{1}{210} \left(14 \sqrt{30} A(1,1)+9 \sqrt{5} A(3,1)+45 \sqrt{3} A(3,3)\right) 0 0 \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)+5 \sqrt{6} A(3,2)\right)
p_y 0 0 \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right) \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) 0

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ A(2,2) & k=2\land (m=-2\lor m=2) \\ -A(2,1) & k=2\land m=-1 \\ A(2,0) & k=2\land m=0 \\ A(2,1) & k=2\land m=1 \\ A(4,4) & k=4\land (m=-4\lor m=4) \\ -A(4,3) & k=4\land m=-3 \\ A(4,2) & k=4\land (m=-2\lor m=2) \\ -A(4,1) & k=4\land m=-1 \\ A(4,0) & k=4\land m=0 \\ A(4,1) & k=4\land m=1 \\ A(4,3) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {-A[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1], k == 2 && m == 1}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {-A[4, 3], k == 4 && m == -3}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {-A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1], k == 4 && m == 1}}, A[4, 3]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1))} , 
       {2, 1, A(2,1)} , 
       {2,-2, A(2,2)} , 
       {2, 2, A(2,2)} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1)*(A(4,1))} , 
       {4, 1, A(4,1)} , 
       {4,-2, A(4,2)} , 
       {4, 2, A(4,2)} , 
       {4,-3, (-1)*(A(4,3))} , 
       {4, 3, A(4,3)} , 
       {4,-4, A(4,4)} , 
       {4, 4, A(4,4)} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-1}^{(1)}} \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} \frac{A(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} A(2,1) \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) \frac{27 A(2,1)-5 \sqrt{30} A(4,1)}{45 \sqrt{7}} \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) -\frac{1}{3} A(4,3) -\frac{2 A(4,4)}{3 \sqrt{3}}
{Y_{0}^{(1)}} -\frac{A(4,3)}{3 \sqrt{3}} \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} \frac{1}{105} \left(-6 \sqrt{42} A(2,1)-5 \sqrt{35} A(4,1)\right) \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} \frac{1}{105} \left(6 \sqrt{42} A(2,1)+5 \sqrt{35} A(4,1)\right) \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} \frac{A(4,3)}{3 \sqrt{3}}
{Y_{1}^{(1)}} -\frac{2 A(4,4)}{3 \sqrt{3}} \frac{1}{3} A(4,3) \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) \frac{5 \sqrt{30} A(4,1)-27 A(2,1)}{45 \sqrt{7}} \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) \sqrt{\frac{6}{35}} A(2,1)-\frac{A(4,1)}{3 \sqrt{7}} \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{z\left(5z^2-r^2\right)} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(x^2-y^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)}
p_z 0 \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} \frac{1}{630} \left(54 \sqrt{14} A(2,1)+5 \sqrt{15} \left(3 \sqrt{7} A(4,1)-7 A(4,3)\right)\right) 0 \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) \sqrt{\frac{6}{35}} A(2,1)+\frac{5 A(4,1)}{6 \sqrt{7}}+\frac{1}{6} A(4,3) 0
p_x 0 \frac{3}{5} \sqrt{\frac{2}{7}} A(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} A(4,1) \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) 0 \frac{1}{21} \left(\sqrt{7} A(4,1)-7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1) \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) 0
p_y \frac{1}{21} \left(\sqrt{7} A(4,1)+7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1) 0 0 \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) 0 0 \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right)

Potential for d-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & (k\neq 1\land k\neq 3\land k\neq 5)\lor (k\neq 3\land k\neq 5\land m\neq -1\land m\neq 0\land m\neq 1)\lor (k\neq 5\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3)\lor (m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5) \\ -A(1,1) & k=1\land m=-1 \\ A(1,0) & k=1\land m=0 \\ A(1,1) & k=1\land m=1 \\ -A(3,3) & k=3\land m=-3 \\ A(3,2) & k=3\land (m=-2\lor m=2) \\ -A(3,1) & k=3\land m=-1 \\ A(3,0) & k=3\land m=0 \\ A(3,1) & k=3\land m=1 \\ A(3,3) & k=3\land m=3 \\ -A(5,5) & k=5\land m=-5 \\ A(5,4) & k=5\land (m=-4\lor m=4) \\ -A(5,3) & k=5\land m=-3 \\ A(5,2) & k=5\land (m=-2\lor m=2) \\ -A(5,1) & k=5\land m=-1 \\ A(5,0) & k=5\land m=0 \\ A(5,1) & k=5\land m=1 \\ A(5,3) & k=5\land m=3 \\ A(5,5) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_Y.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 1 && k != 3 && k != 5) || (k != 3 && k != 5 && m != -1 && m != 0 && m != 1) || (k != 5 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3) || (m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}, {A[1, 1], k == 1 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}, {A[3, 3], k == 3 && m == 3}, {-A[5, 5], k == 5 && m == -5}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {-A[5, 3], k == 5 && m == -3}, {A[5, 2], k == 5 && (m == -2 || m == 2)}, {-A[5, 1], k == 5 && m == -1}, {A[5, 0], k == 5 && m == 0}, {A[5, 1], k == 5 && m == 1}, {A[5, 3], k == 5 && m == 3}}, A[5, 5]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_Y.Quanty
Akm = {{1, 0, A(1,0)} , 
       {1,-1, (-1)*(A(1,1))} , 
       {1, 1, A(1,1)} , 
       {3, 0, A(3,0)} , 
       {3,-1, (-1)*(A(3,1))} , 
       {3, 1, A(3,1)} , 
       {3,-2, A(3,2)} , 
       {3, 2, A(3,2)} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} , 
       {5, 0, A(5,0)} , 
       {5,-1, (-1)*(A(5,1))} , 
       {5, 1, A(5,1)} , 
       {5,-2, A(5,2)} , 
       {5, 2, A(5,2)} , 
       {5,-3, (-1)*(A(5,3))} , 
       {5, 3, A(5,3)} , 
       {5,-4, A(5,4)} , 
       {5, 4, A(5,4)} , 
       {5,-5, (-1)*(A(5,5))} , 
       {5, 5, A(5,5)} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-2}^{(2)}} \frac{1}{231} \left(-33 \sqrt{21} A(1,1)+11 \sqrt{14} A(3,1)-\sqrt{35} A(5,1)\right) \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} \frac{A(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} A(3,1)+\frac{5 A(5,1)}{11 \sqrt{21}} \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} \frac{5}{33} \sqrt{2} A(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} A(3,3) \frac{1}{11} \sqrt{10} A(5,4) \frac{5}{11} \sqrt{\frac{2}{3}} A(5,5)
{Y_{-1}^{(2)}} \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) -\sqrt{\frac{2}{7}} A(1,1)-\frac{A(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} A(5,1) \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) \sqrt{\frac{3}{35}} A(1,1)-\frac{1}{3} \sqrt{\frac{2}{35}} A(3,1)-\frac{20 A(5,1)}{33 \sqrt{7}} -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} -\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,3)+28 A(5,3)\right) -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4)
{Y_{0}^{(2)}} \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)-\frac{2}{33} \sqrt{5} A(5,3) \frac{1}{11} \sqrt{5} A(5,2) -\sqrt{\frac{6}{35}} A(1,1)-\frac{A(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} A(5,1) \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} \sqrt{\frac{6}{35}} A(1,1)+\frac{A(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} A(5,1) \frac{1}{11} \sqrt{5} A(5,2) \frac{2}{33} \sqrt{5} A(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)
{Y_{1}^{(2)}} -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)+\frac{4}{33} \sqrt{5} A(5,3) -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} -\sqrt{\frac{3}{35}} A(1,1)+\frac{1}{3} \sqrt{\frac{2}{35}} A(3,1)+\frac{20 A(5,1)}{33 \sqrt{7}} \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) \sqrt{\frac{2}{7}} A(1,1)+\frac{A(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} A(5,1) \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right)
{Y_{2}^{(2)}} -\frac{5}{11} \sqrt{\frac{2}{3}} A(5,5) \frac{1}{11} \sqrt{10} A(5,4) \frac{1}{3} \sqrt{\frac{2}{7}} A(3,3)-\frac{5}{33} \sqrt{2} A(5,3) \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} -\frac{A(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} A(3,1)-\frac{5 A(5,1)}{11 \sqrt{21}} \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} \frac{1}{231} \left(33 \sqrt{21} A(1,1)-11 \sqrt{14} A(3,1)+\sqrt{35} A(5,1)\right)

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{z\left(5z^2-r^2\right)} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(x^2-y^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)}
d_{z^2-x^2} 0 \frac{\sqrt{105} (99 A(1,0)+44 A(3,0)+50 A(5,0))+5 \sqrt{2} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)}{2310} 3 \sqrt{\frac{3}{70}} A(1,1)-\frac{A(3,1)}{6 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{3}{7}} A(3,3)+\frac{5 A(5,1)}{6 \sqrt{14}}-\frac{5}{44} \sqrt{3} A(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} A(5,5) 0 \frac{\sqrt{14} (-33 A(1,0)+22 A(3,0)-5 A(5,0))+42 \sqrt{15} A(5,2)-42 \sqrt{5} A(5,4)}{462 \sqrt{2}} \frac{1}{924} \left(66 \sqrt{14} A(1,1)+22 \sqrt{21} A(3,1)-22 \sqrt{35} A(3,3)+17 \sqrt{210} A(5,1)+7 \sqrt{5} A(5,3)-105 A(5,5)\right) 0
d_{3y^2-r^2} 0 \frac{5 \sqrt{6} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)-\sqrt{35} (99 A(1,0)+44 A(3,0)+50 A(5,0))}{2310} \frac{198 \sqrt{70} A(1,1)-154 \sqrt{105} A(3,1)-5 \left(22 \sqrt{7} A(3,3)+5 \sqrt{42} A(5,1)+35 A(5,3)-105 \sqrt{5} A(5,5)\right)}{4620} 0 \frac{\sqrt{42} (-33 A(1,0)+22 A(3,0)-5 A(5,0))-42 \sqrt{5} A(5,2)-42 \sqrt{15} A(5,4)}{462 \sqrt{2}} \frac{1}{924} \left(-66 \sqrt{42} A(1,1)-66 \sqrt{7} A(3,1)+22 \sqrt{105} A(3,3)-9 \sqrt{70} A(5,1)-49 \sqrt{15} A(5,3)-105 \sqrt{3} A(5,5)\right) 0
d_{\text{xy}} \frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)-21 \sqrt{10} A(5,4)\right) 0 0 \frac{66 \sqrt{210} A(1,1)+11 \sqrt{35} A(3,1)+5 \left(11 \sqrt{21} A(3,3)-5 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)-35 \sqrt{15} A(5,5)\right)}{2310} 0 0 \frac{1}{462} \left(66 \sqrt{14} A(1,1)-33 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)+3 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)-105 A(5,5)\right)
d_{\text{yz}} \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)+28 A(5,3)\right)\right) 0 0 \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(-11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)+70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} 0 0 \frac{1}{462} \left(66 \sqrt{7} A(1,0)+11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)-25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)+21 \sqrt{10} A(5,4)\right)
d_{\text{xz}} 0 \sqrt{\frac{6}{35}} A(1,1)-\frac{2 A(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} A(5,1) \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)-70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} 0 \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(-11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)-28 A(5,3)\right)\right) \frac{1}{462} \left(-66 \sqrt{7} A(1,0)-11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)+25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)-21 \sqrt{10} A(5,4)\right) 0

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups C1 Cs Ci
Cn groups C2 C3 C4 C5 C6 C7 C8
Dn groups D2 D3 D4 D5 D6 D7 D8
Cnv groups C2v C3v C4v C5v C6v C7v C8v
Cnh groups C2h C3h C4h C5h C6h
Dnh groups D2h D3h D4h D5h D6h D7h D8h
Dnd groups D2d D3d D4d D5d D6d D7d D8d
Sn groups S2 S4 S6 S8 S10 S12
Cubic groups T Th Td O Oh I Ih
Linear groups C\inftyv D\inftyh

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