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+Table of Contents

Orientation

Symmetry Operations

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

Operator Orientation
E {0,0,0} ,
i {0,0,0} ,

Different Settings

Character Table

E(1) i(1)
Ag 1 1
Au 1 −1

Product Table

Ag Au
Ag Ag Au
Au Au Ag

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 Ci Point group with orientation the form of the expansion coefficients is:

Expansion

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

Input format suitable for Mathematica (Quanty.nb)

Akm_Ci.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {-A[2, 1] + I*B[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1] + I*B[2, 1], k == 2 && m == 1}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {-A[4, 3] + I*B[4, 3], k == 4 && m == -3}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {-A[4, 1] + I*B[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1] + I*B[4, 1], k == 4 && m == 1}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 3] + I*B[4, 3], k == 4 && m == 3}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {-A[6, 5] + I*B[6, 5], k == 6 && m == -5}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {-A[6, 3] + I*B[6, 3], k == 6 && m == -3}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {-A[6, 1] + I*B[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {A[6, 1] + I*B[6, 1], k == 6 && m == 1}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 3] + I*B[6, 3], k == 6 && m == 3}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 5] + I*B[6, 5], k == 6 && m == 5}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_Ci.Quanty
Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1)) + (I)*(B(2,1))} , 
       {2, 1, A(2,1) + (I)*(B(2,1))} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1)*(A(4,1)) + (I)*(B(4,1))} , 
       {4, 1, A(4,1) + (I)*(B(4,1))} , 
       {4,-2, A(4,2) + (-I)*(B(4,2))} , 
       {4, 2, A(4,2) + (I)*(B(4,2))} , 
       {4,-3, (-1)*(A(4,3)) + (I)*(B(4,3))} , 
       {4, 3, A(4,3) + (I)*(B(4,3))} , 
       {4,-4, A(4,4) + (-I)*(B(4,4))} , 
       {4, 4, A(4,4) + (I)*(B(4,4))} , 
       {6, 0, A(6,0)} , 
       {6,-1, (-1)*(A(6,1)) + (I)*(B(6,1))} , 
       {6, 1, A(6,1) + (I)*(B(6,1))} , 
       {6,-2, A(6,2) + (-I)*(B(6,2))} , 
       {6, 2, A(6,2) + (I)*(B(6,2))} , 
       {6,-3, (-1)*(A(6,3)) + (I)*(B(6,3))} , 
       {6, 3, A(6,3) + (I)*(B(6,3))} , 
       {6,-4, A(6,4) + (-I)*(B(6,4))} , 
       {6, 4, A(6,4) + (I)*(B(6,4))} , 
       {6,-5, (-1)*(A(6,5)) + (I)*(B(6,5))} , 
       {6, 5, A(6,5) + (I)*(B(6,5))} , 
       {6,-6, A(6,6) + (-I)*(B(6,6))} , 
       {6, 6, A(6,6) + (I)*(B(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)000Asd(2,2)+iBsd(2,2)√5−Asd(2,1)+iBsd(2,1)√5Asd(2,0)√5−−Asd(2,1)+iBsd(2,1)√5Asd(2,2)−iBsd(2,2)√50000000
Y(1)−10App(0,0)−15App(2,0)15√3(−App(2,1)+iBpp(2,1))−15√6(App(2,2)−iBpp(2,2))000003(Apf(2,2)+iBpf(2,2))√35−Apf(4,2)+iBpf(4,2)3√21Apf(4,1)+iBpf(4,1)3√7−√635(Apf(2,1)+iBpf(2,1))35√27Apf(2,0)−13√27Apf(4,0)13√1021(−Apf(4,1)+iBpf(4,1))−3(−Apf(2,1)+iBpf(2,1))5√715√37(Apf(2,2)−iBpf(2,2))−13√57(Apf(4,2)−iBpf(4,2))13(−Apf(4,3)+iBpf(4,3))−2(Apf(4,4)−iBpf(4,4))3√3
Y(1)00−15√3(App(2,1)+iBpp(2,1))App(0,0)+25App(2,0)−15√3(−App(2,1)+iBpp(2,1))00000−Apf(4,3)+iBpf(4,3)3√3√335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3√7−25√67(Apf(2,1)+iBpf(2,1))−13√57(Apf(4,1)+iBpf(4,1))35√37Apf(2,0)+4Apf(4,0)3√21−25√67(−Apf(2,1)+iBpf(2,1))−13√57(−Apf(4,1)+iBpf(4,1))√335(Apf(2,2)−iBpf(2,2))+2(Apf(4,2)−iBpf(4,2))3√7−−Apf(4,3)+iBpf(4,3)3√3
Y(1)10−15√6(App(2,2)+iBpp(2,2))15√3(App(2,1)+iBpp(2,1))App(0,0)−15App(2,0)00000−2(Apf(4,4)+iBpf(4,4))3√313(Apf(4,3)+iBpf(4,3))15√37(Apf(2,2)+iBpf(2,2))−13√57(Apf(4,2)+iBpf(4,2))13√1021(Apf(4,1)+iBpf(4,1))−3(Apf(2,1)+iBpf(2,1))5√735√27Apf(2,0)−13√27Apf(4,0)−Apf(4,1)+iBpf(4,1)3√7−√635(−Apf(2,1)+iBpf(2,1))3(Apf(2,2)−iBpf(2,2))√35−Apf(4,2)−iBpf(4,2)3√21
Y(2)−2Asd(2,2)−iBsd(2,2)√5000Add(0,0)−27Add(2,0)+121Add(4,0)17√6(−Add(2,1)+iBdd(2,1))−121√5(−Add(4,1)+iBdd(4,1))17√53(Add(4,2)−iBdd(4,2))−27(Add(2,2)−iBdd(2,2))−13√57(−Add(4,3)+iBdd(4,3))13√107(Add(4,4)−iBdd(4,4))0000000
Y(2)−1−Asd(2,1)+iBsd(2,1)√5000121√5(Add(4,1)+iBdd(4,1))−17√6(Add(2,1)+iBdd(2,1))Add(0,0)+17Add(2,0)−421Add(4,0)17(−Add(2,1)+iBdd(2,1))+17√103(−Add(4,1)+iBdd(4,1))−17√6(Add(2,2)−iBdd(2,2))−221√10(Add(4,2)−iBdd(4,2))13√57(−Add(4,3)+iBdd(4,3))0000000
Y(2)0Asd(2,0)√500017√53(Add(4,2)+iBdd(4,2))−27(Add(2,2)+iBdd(2,2))17(−Add(2,1)−iBdd(2,1))−17√103(Add(4,1)+iBdd(4,1))Add(0,0)+27Add(2,0)+27Add(4,0)17(Add(2,1)−iBdd(2,1))−17√103(−Add(4,1)+iBdd(4,1))17√53(Add(4,2)−iBdd(4,2))−27(Add(2,2)−iBdd(2,2))0000000
Y(2)1Asd(2,1)+iBsd(2,1)√500013√57(Add(4,3)+iBdd(4,3))−17√6(Add(2,2)+iBdd(2,2))−221√10(Add(4,2)+iBdd(4,2))17(Add(2,1)+iBdd(2,1))+17√103(Add(4,1)+iBdd(4,1))Add(0,0)+17Add(2,0)−421Add(4,0)121√5(−Add(4,1)+iBdd(4,1))−17√6(−Add(2,1)+iBdd(2,1))0000000
Y(2)2Asd(2,2)+iBsd(2,2)√500013√107(Add(4,4)+iBdd(4,4))−13√57(Add(4,3)+iBdd(4,3))17√53(Add(4,2)+iBdd(4,2))−27(Add(2,2)+iBdd(2,2))17√6(Add(2,1)+iBdd(2,1))−121√5(Add(4,1)+iBdd(4,1))Add(0,0)−27Add(2,0)+121Add(4,0)0000000
Y(3)−303(Apf(2,2)−iBpf(2,2))√35−Apf(4,2)−iBpf(4,2)3√21−Apf(4,3)+iBpf(4,3)3√3−2(Apf(4,4)−iBpf(4,4))3√300000Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0)13(−Aff(2,1)+iBff(2,1))−111√103(−Aff(4,1)+iBff(4,1))+5429√7(−Aff(6,1)+iBff(6,1))−13√25(Aff(2,2)−iBff(2,2))+111√6(Aff(4,2)−iBff(4,2))−10429√7(Aff(6,2)−iBff(6,2))10143√73(−Aff(6,3)+iBff(6,3))−111√7(−Aff(4,3)+iBff(4,3))111√143(Aff(4,4)−iBff(4,4))−5143√703(Aff(6,4)−iBff(6,4))513√1433(−Aff(6,5)+iBff(6,5))−1013√733(Aff(6,6)−iBff(6,6))
Y(3)−20√635(−Apf(2,1)+iBpf(2,1))−−Apf(4,1)+iBpf(4,1)3√7√335(Apf(2,2)−iBpf(2,2))+2(Apf(4,2)−iBpf(4,2))3√713(Apf(4,3)−iBpf(4,3))0000013(−Aff(2,1)−iBff(2,1))+111√103(Aff(4,1)+iBff(4,1))−5429√7(Aff(6,1)+iBff(6,1))Aff(0,0)−733Aff(4,0)+10143Aff(6,0)−Aff(2,1)+iBff(2,1)√15+433√2(−Aff(4,1)+iBff(4,1))−5143√353(−Aff(6,1)+iBff(6,1))−2(Aff(2,2)−iBff(2,2))3√5−Aff(4,2)−iBff(4,2)11√3+20429√14(Aff(6,2)−iBff(6,2))−133√14(−Aff(4,3)+iBff(4,3))−5143√42(−Aff(6,3)+iBff(6,3))133√70(Aff(4,4)−iBff(4,4))+10143√14(Aff(6,4)−iBff(6,4))−513√1433(−Aff(6,5)+iBff(6,5))
Y(3)−1035√27Apf(2,0)−13√27Apf(4,0)25√67(−Apf(2,1)+iBpf(2,1))+13√57(−Apf(4,1)+iBpf(4,1))15√37(Apf(2,2)−iBpf(2,2))−13√57(Apf(4,2)−iBpf(4,2))00000−13√25(Aff(2,2)+iBff(2,2))+111√6(Aff(4,2)+iBff(4,2))−10429√7(Aff(6,2)+iBff(6,2))−Aff(2,1)+iBff(2,1)√15−433√2(Aff(4,1)+iBff(4,1))+5143√353(Aff(6,1)+iBff(6,1))Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0)115√2(−Aff(2,1)+iBff(2,1))+111√53(−Aff(4,1)+iBff(4,1))+25429√14(−Aff(6,1)+iBff(6,1))−25√23(Aff(2,2)−iBff(2,2))−233√10(Aff(4,2)−iBff(4,2))−10143√353(Aff(6,2)−iBff(6,2))133√14(−Aff(4,3)+iBff(4,3))+5143√42(−Aff(6,3)+iBff(6,3))111√143(Aff(4,4)−iBff(4,4))−5143√703(Aff(6,4)−iBff(6,4))
Y(3)003(Apf(2,1)+iBpf(2,1))5√7−13√1021(Apf(4,1)+iBpf(4,1))35√37Apf(2,0)+4Apf(4,0)3√213(−Apf(2,1)+iBpf(2,1))5√7−13√1021(−Apf(4,1)+iBpf(4,1))00000111√7(Aff(4,3)+iBff(4,3))−10143√73(Aff(6,3)+iBff(6,3))−2(Aff(2,2)+iBff(2,2))3√5−Aff(4,2)+iBff(4,2)11√3+20429√14(Aff(6,2)+iBff(6,2))−115√2(Aff(2,1)+iBff(2,1))−111√53(Aff(4,1)+iBff(4,1))−25429√14(Aff(6,1)+iBff(6,1))Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)−115√2(−Aff(2,1)+iBff(2,1))−111√53(−Aff(4,1)+iBff(4,1))−25429√14(−Aff(6,1)+iBff(6,1))−2(Aff(2,2)−iBff(2,2))3√5−Aff(4,2)−iBff(4,2)11√3+20429√14(Aff(6,2)−iBff(6,2))111√7(−Aff(4,3)+iBff(4,3))−10143√73(−Aff(6,3)+iBff(6,3))
Y(3)1015√37(Apf(2,2)+iBpf(2,2))−13√57(Apf(4,2)+iBpf(4,2))25√67(Apf(2,1)+iBpf(2,1))+13√57(Apf(4,1)+iBpf(4,1))35√27Apf(2,0)−13√27Apf(4,0)00000111√143(Aff(4,4)+iBff(4,4))−5143√703(Aff(6,4)+iBff(6,4))133√14(Aff(4,3)+iBff(4,3))+5143√42(Aff(6,3)+iBff(6,3))−25√23(Aff(2,2)+iBff(2,2))−233√10(Aff(4,2)+iBff(4,2))−10143√353(Aff(6,2)+iBff(6,2))115√2(Aff(2,1)+iBff(2,1))+111√53(Aff(4,1)+iBff(4,1))+25429√14(Aff(6,1)+iBff(6,1))Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0)−−Aff(2,1)+iBff(2,1)√15−433√2(−Aff(4,1)+iBff(4,1))+5143√353(−Aff(6,1)+iBff(6,1))−13√25(Aff(2,2)−iBff(2,2))+111√6(Aff(4,2)−iBff(4,2))−10429√7(Aff(6,2)−iBff(6,2))
Y(3)2013(−Apf(4,3)−iBpf(4,3))√335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3√7√635(Apf(2,1)+iBpf(2,1))−Apf(4,1)+iBpf(4,1)3√700000−513√1433(Aff(6,5)+iBff(6,5))133√70(Aff(4,4)+iBff(4,4))+10143√14(Aff(6,4)+iBff(6,4))−133√14(Aff(4,3)+iBff(4,3))−5143√42(Aff(6,3)+iBff(6,3))−2(Aff(2,2)+iBff(2,2))3√5−Aff(4,2)+iBff(4,2)11√3+20429√14(Aff(6,2)+iBff(6,2))Aff(2,1)+iBff(2,1)√15+433√2(Aff(4,1)+iBff(4,1))−5143√353(Aff(6,1)+iBff(6,1))Aff(0,0)−733Aff(4,0)+10143Aff(6,0)13(Aff(2,1)−iBff(2,1))+111√103(−Aff(4,1)+iBff(4,1))−5429√7(−Aff(6,1)+iBff(6,1))
Y(3)30−2(Apf(4,4)+iBpf(4,4))3√3Apf(4,3)+iBpf(4,3)3√33(Apf(2,2)+iBpf(2,2))√35−Apf(4,2)+iBpf(4,2)3√2100000−1013√733(Aff(6,6)+iBff(6,6))513√1433(Aff(6,5)+iBff(6,5))111√143(Aff(4,4)+iBff(4,4))−5143√703(Aff(6,4)+iBff(6,4))10143√73(Aff(6,3)+iBff(6,3))−111√7(Aff(4,3)+iBff(4,3))−13√25(Aff(2,2)+iBff(2,2))+111√6(Aff(4,2)+iBff(4,2))−10429√7(Aff(6,2)+iBff(6,2))13(Aff(2,1)+iBff(2,1))−111√103(Aff(4,1)+iBff(4,1))+5429√7(Aff(6,1)+iBff(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
px01√20−1√2000000000000
py0i√20i√2000000000000
pz0010000000000000
dx2−y200001√20001√20000000
d3z2−r20000001000000000
dyz00000i√20i√200000000
dxz000001√20−1√200000000
dxy0000i√2000−i√20000000
fxyz0000000000i√2000−i√20
fx(5x2−r2)000000000√540−√340√340−√54
fy(5y2−r2)000000000−i√540−i√340−i√340−i√54
fz(5z2−r2)0000000000001000
fx(y2−z2)000000000−√340−√540√540√34
fy(z2−x2)000000000−i√340i√540i√540−i√34
fz(x2−y2)00000000001√20001√20

One particle coupling on a basis of symmetry adapted functions

After rotation we find

s px py pz dx2−y2 d3z2−r2 dyz dxz dxy fxyz fx(5x2−r2) fy(5y2−r2) fz(5z2−r2) fx(y2−z2) fy(z2−x2) fz(x2−y2)
sAss(0,0)000√25Asd(2,2)Asd(2,0)√5√25Bsd(2,1)−√25Asd(2,1)−√25Bsd(2,2)0000000
px0App(0,0)−15App(2,0)+15√6App(2,2)−15√6Bpp(2,2)−15√6App(2,1)00000√635Bpf(2,1)−Bpf(4,1)3√7+13Bpf(4,3)−310√37Apf(2,0)+9Apf(2,2)5√14+Apf(4,0)2√21−13√1021Apf(4,2)+13√56Apf(4,4)35√27Bpf(2,2)+13√542Bpf(4,2)+13√56Bpf(4,4)35√27Apf(2,1)−23√521Apf(4,1)−3Apf(2,0)2√35−√370Apf(2,2)+16√57Apf(4,0)−13√27Apf(4,2)−Apf(4,4)3√2√635Bpf(2,2)−Bpf(4,2)√14+Bpf(4,4)3√2−√635Apf(2,1)+Apf(4,1)3√7−13Apf(4,3)
py0−15√6Bpp(2,2)App(0,0)−15App(2,0)−15√6App(2,2)15√6Bpp(2,1)00000−√635Apf(2,1)+Apf(4,1)3√7+13Apf(4,3)35√27Bpf(2,2)+13√542Bpf(4,2)−13√56Bpf(4,4)−310√37Apf(2,0)−9Apf(2,2)5√14+Apf(4,0)2√21+13√1021Apf(4,2)+13√56Apf(4,4)23√521Bpf(4,1)−35√27Bpf(2,1)−√635Bpf(2,2)+Bpf(4,2)√14+Bpf(4,4)3√23Apf(2,0)2√35−√370Apf(2,2)−16√57Apf(4,0)−13√27Apf(4,2)+Apf(4,4)3√2−√635Bpf(2,1)+Bpf(4,1)3√7+13Bpf(4,3)
pz0−15√6App(2,1)15√6Bpp(2,1)App(0,0)+25App(2,0)00000−√635Bpf(2,2)−23√27Bpf(4,2)35√27Apf(2,1)+12√521Apf(4,1)−16√53Apf(4,3)−35√27Bpf(2,1)−12√521Bpf(4,1)−16√53Bpf(4,3)35√37Apf(2,0)+4Apf(4,0)3√21√635Apf(2,1)+5Apf(4,1)6√7+16Apf(4,3)√635Bpf(2,1)+5Bpf(4,1)6√7−16Bpf(4,3)√635Apf(2,2)+23√27Apf(4,2)
dx2−y2√25Asd(2,2)000Add(0,0)−27Add(2,0)+121Add(4,0)+13√107Add(4,4)17√103Add(4,2)−27√2Add(2,2)−17√6Bdd(2,1)+121√5Bdd(4,1)+13√57Bdd(4,3)−17√6Add(2,1)+121√5Add(4,1)−13√57Add(4,3)−13√107Bdd(4,4)0000000
d3z2−r2Asd(2,0)√500017√103Add(4,2)−27√2Add(2,2)Add(0,0)+27Add(2,0)+27Add(4,0) \frac{1}{7} \sqrt{2} \text{Bdd}(2,1)+\frac{2}{7} \sqrt{\frac{5}{3}} \text{Bdd}(4,1) -\frac{1}{7} \sqrt{2} \text{Add}(2,1)-\frac{2}{7} \sqrt{\frac{5}{3}} \text{Add}(4,1) \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }
d_{\text{yz}} \sqrt{\frac{2}{5}} \text{Bsd}(2,1) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } -\frac{1}{7} \sqrt{6} \text{Bdd}(2,1)+\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) \frac{1}{7} \sqrt{2} \text{Bdd}(2,1)+\frac{2}{7} \sqrt{\frac{5}{3}} \text{Bdd}(4,1) \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) -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) -\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) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }
d_{\text{xz}} -\sqrt{\frac{2}{5}} \text{Asd}(2,1) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 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) -\frac{1}{7} \sqrt{2} \text{Add}(2,1)-\frac{2}{7} \sqrt{\frac{5}{3}} \text{Add}(4,1) -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) \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) \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }
d_{\text{xy}} -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) -\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) \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) \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) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }
f_{\text{xyz}} \color{darkred}{ 0 } \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\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) -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) \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) \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) -\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) -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4)
f_{x\left(5x^2-r^2\right)} \color{darkred}{ 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) \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) \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) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) \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) \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) \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) \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) \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) -\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)
f_{y\left(5y^2-r^2\right)} \color{darkred}{ 0 } \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) -\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) -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1)-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Bpf}(4,3) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\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) \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) \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) -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) -\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) -\frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5)
f_{z\left(5z^2-r^2\right)} \color{darkred}{ 0 } \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) \frac{2}{3} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1) \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) \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) -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) \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)}{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{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) -\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)
f_{x\left(y^2-z^2\right)} \color{darkred}{ 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}} -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{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) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(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) -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) \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) \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) \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) \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)
f_{y\left(z^2-x^2\right)} \color{darkred}{ 0 } \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} \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}} \sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{5 \text{Bpf}(4,1)}{6 \sqrt{7}}-\frac{1}{6} \text{Bpf}(4,3) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\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) \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) -\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) \frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) \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) -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5)
f_{z\left(x^2-y^2\right)} \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) -\sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) -\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{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) -\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)}{\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) -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) \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)

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

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Ci.Quanty
Akm = {{0, 0, Eag} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

\text{s}
\text{s} \text{Eag}

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{Eag}
\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{Epxpx}+\text{Epypy}+\text{Epzpz}) & k=0\land m=0 \\ \frac{5 (\text{Epxpx}+2 i \text{Epxpy}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=-2 \\ \frac{5 (\text{Epxpz}+i \text{Epypz})}{\sqrt{6}} & k=2\land m=-1 \\ -\frac{5}{6} (\text{Epxpx}+\text{Epypy}-2 \text{Epzpz}) & k=2\land m=0 \\ -\frac{5 (\text{Epxpz}-i \text{Epypz})}{\sqrt{6}} & k=2\land m=1 \\ \frac{5 (\text{Epxpx}-2 i \text{Epxpy}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=2 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Ci.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Epxpx + Epypy + Epzpz)/3, k == 0 && m == 0}, {(5*(Epxpx + (2*I)*Epxpy - Epypy))/(2*Sqrt[6]), k == 2 && m == -2}, {(5*(Epxpz + I*Epypz))/Sqrt[6], k == 2 && m == -1}, {(-5*(Epxpx + Epypy - 2*Epzpz))/6, k == 2 && m == 0}, {(-5*(Epxpz - I*Epypz))/Sqrt[6], k == 2 && m == 1}, {(5*(Epxpx - (2*I)*Epxpy - Epypy))/(2*Sqrt[6]), k == 2 && m == 2}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Ci.Quanty
Akm = {{0, 0, (1/3)*(Epxpx + Epypy + Epzpz)} , 
       {2, 0, (-5/6)*(Epxpx + Epypy + (-2)*(Epzpz))} , 
       {2, 1, (-5)*((1/(sqrt(6)))*(Epxpz + (-I)*(Epypz)))} , 
       {2,-1, (5)*((1/(sqrt(6)))*(Epxpz + (I)*(Epypz)))} , 
       {2, 2, (5/2)*((1/(sqrt(6)))*(Epxpx + (-2*I)*(Epxpy) + (-1)*(Epypy)))} , 
       {2,-2, (5/2)*((1/(sqrt(6)))*(Epxpx + (2*I)*(Epxpy) + (-1)*(Epypy)))} }

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{Epxpx}+\text{Epypy}}{2} \frac{\text{Epxpz}+i \text{Epypz}}{\sqrt{2}} \frac{1}{2} (-\text{Epxpx}-2 i \text{Epxpy}+\text{Epypy})
{Y_{0}^{(1)}} \frac{\text{Epxpz}-i \text{Epypz}}{\sqrt{2}} \text{Epzpz} -\frac{\text{Epxpz}+i \text{Epypz}}{\sqrt{2}}
{Y_{1}^{(1)}} \frac{1}{2} (-\text{Epxpx}+2 i \text{Epxpy}+\text{Epypy}) -\frac{\text{Epxpz}-i \text{Epypz}}{\sqrt{2}} \frac{\text{Epxpx}+\text{Epypy}}{2}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

p_x p_y p_z
p_x \text{Epxpx} \text{Epxpy} \text{Epxpz}
p_y \text{Epxpy} \text{Epypy} \text{Epypz}
p_z \text{Epxpz} \text{Epypz} \text{Epzpz}

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Epxpx}
\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{Epypy}
\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
\text{Epzpz}
\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

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{Edx2y2dx2y2}+\text{Edxydxy}+\text{Edxzdxz}+\text{Edyzdyz}+\text{Edz2dz2}) & k=0\land m=0 \\ -\frac{4 \text{Edx2y2dz2}-\sqrt{3} \text{Edxzdxz}-2 i \sqrt{3} \text{Edyzdxz}+\sqrt{3} \text{Edyzdyz}+4 i \text{Edz2dxy}}{2 \sqrt{2}} & k=2\land m=-2 \\ \frac{\sqrt{3} \text{Edx2y2dxz}-i \sqrt{3} \text{Edx2y2dyz}+i \sqrt{3} \text{Edxzdxy}+\sqrt{3} \text{Edyzdxy}+\text{Edz2dxz}+i \text{Edz2dyz}}{\sqrt{2}} & k=2\land m=-1 \\ \frac{1}{2} (-2 \text{Edx2y2dx2y2}-2 \text{Edxydxy}+\text{Edxzdxz}+\text{Edyzdyz}+2 \text{Edz2dz2}) & k=2\land m=0 \\ -\frac{\sqrt{3} \text{Edx2y2dxz}+i \sqrt{3} \text{Edx2y2dyz}-i \sqrt{3} \text{Edxzdxy}+\sqrt{3} \text{Edyzdxy}+\text{Edz2dxz}-i \text{Edz2dyz}}{\sqrt{2}} & k=2\land m=1 \\ -\frac{4 \text{Edx2y2dz2}-\sqrt{3} \text{Edxzdxz}+2 i \sqrt{3} \text{Edyzdxz}+\sqrt{3} \text{Edyzdyz}-4 i \text{Edz2dxy}}{2 \sqrt{2}} & k=2\land m=2 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}+2 i \text{Edx2y2dxy}-\text{Edxydxy}) & k=4\land m=-4 \\ \frac{3}{2} \sqrt{\frac{7}{5}} (\text{Edx2y2dxz}+i (\text{Edx2y2dyz}+\text{Edxzdxy}+i \text{Edyzdxy})) & k=4\land m=-3 \\ \frac{3 \left(\sqrt{3} \text{Edx2y2dz2}+\text{Edxzdxz}+2 i \text{Edyzdxz}-\text{Edyzdyz}+i \sqrt{3} \text{Edz2dxy}\right)}{\sqrt{10}} & k=4\land m=-2 \\ -\frac{3 \left(\text{Edx2y2dxz}-i \text{Edx2y2dyz}+i \text{Edxzdxy}+\text{Edyzdxy}-2 \sqrt{3} \text{Edz2dxz}-2 i \sqrt{3} \text{Edz2dyz}\right)}{2 \sqrt{5}} & k=4\land m=-1 \\ \frac{3}{10} (\text{Edx2y2dx2y2}+\text{Edxydxy}-4 \text{Edxzdxz}-4 \text{Edyzdyz}+6 \text{Edz2dz2}) & k=4\land m=0 \\ \frac{3 \left(\text{Edx2y2dxz}+i \text{Edx2y2dyz}-i \text{Edxzdxy}+\text{Edyzdxy}-2 \sqrt{3} \text{Edz2dxz}+2 i \sqrt{3} \text{Edz2dyz}\right)}{2 \sqrt{5}} & k=4\land m=1 \\ \frac{3 \left(\sqrt{3} \text{Edx2y2dz2}+\text{Edxzdxz}-2 i \text{Edyzdxz}-\text{Edyzdyz}-i \sqrt{3} \text{Edz2dxy}\right)}{\sqrt{10}} & k=4\land m=2 \\ \frac{3}{2} \sqrt{\frac{7}{5}} (-\text{Edx2y2dxz}+i \text{Edx2y2dyz}+i \text{Edxzdxy}+\text{Edyzdxy}) & k=4\land m=3 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}-2 i \text{Edx2y2dxy}-\text{Edxydxy}) & k=4\land m=4 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Ci.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)/5, k == 0 && m == 0}, {-(4*Edx2y2dz2 - Sqrt[3]*Edxzdxz - (2*I)*Sqrt[3]*Edyzdxz + Sqrt[3]*Edyzdyz + (4*I)*Edz2dxy)/(2*Sqrt[2]), k == 2 && m == -2}, {(Sqrt[3]*Edx2y2dxz - I*Sqrt[3]*Edx2y2dyz + I*Sqrt[3]*Edxzdxy + Sqrt[3]*Edyzdxy + Edz2dxz + I*Edz2dyz)/Sqrt[2], k == 2 && m == -1}, {(-2*Edx2y2dx2y2 - 2*Edxydxy + Edxzdxz + Edyzdyz + 2*Edz2dz2)/2, k == 2 && m == 0}, {-((Sqrt[3]*Edx2y2dxz + I*Sqrt[3]*Edx2y2dyz - I*Sqrt[3]*Edxzdxy + Sqrt[3]*Edyzdxy + Edz2dxz - I*Edz2dyz)/Sqrt[2]), k == 2 && m == 1}, {-(4*Edx2y2dz2 - Sqrt[3]*Edxzdxz + (2*I)*Sqrt[3]*Edyzdxz + Sqrt[3]*Edyzdyz - (4*I)*Edz2dxy)/(2*Sqrt[2]), k == 2 && m == 2}, {(3*Sqrt[7/10]*(Edx2y2dx2y2 + (2*I)*Edx2y2dxy - Edxydxy))/2, k == 4 && m == -4}, {(3*Sqrt[7/5]*(Edx2y2dxz + I*(Edx2y2dyz + Edxzdxy + I*Edyzdxy)))/2, k == 4 && m == -3}, {(3*(Sqrt[3]*Edx2y2dz2 + Edxzdxz + (2*I)*Edyzdxz - Edyzdyz + I*Sqrt[3]*Edz2dxy))/Sqrt[10], k == 4 && m == -2}, {(-3*(Edx2y2dxz - I*Edx2y2dyz + I*Edxzdxy + Edyzdxy - 2*Sqrt[3]*Edz2dxz - (2*I)*Sqrt[3]*Edz2dyz))/(2*Sqrt[5]), k == 4 && m == -1}, {(3*(Edx2y2dx2y2 + Edxydxy - 4*Edxzdxz - 4*Edyzdyz + 6*Edz2dz2))/10, k == 4 && m == 0}, {(3*(Edx2y2dxz + I*Edx2y2dyz - I*Edxzdxy + Edyzdxy - 2*Sqrt[3]*Edz2dxz + (2*I)*Sqrt[3]*Edz2dyz))/(2*Sqrt[5]), k == 4 && m == 1}, {(3*(Sqrt[3]*Edx2y2dz2 + Edxzdxz - (2*I)*Edyzdxz - Edyzdyz - I*Sqrt[3]*Edz2dxy))/Sqrt[10], k == 4 && m == 2}, {(3*Sqrt[7/5]*(-Edx2y2dxz + I*Edx2y2dyz + I*Edxzdxy + Edyzdxy))/2, k == 4 && m == 3}, {(3*Sqrt[7/10]*(Edx2y2dx2y2 - (2*I)*Edx2y2dxy - Edxydxy))/2, k == 4 && m == 4}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Ci.Quanty
Akm = {{0, 0, (1/5)*(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)} , 
       {2, 0, (1/2)*((-2)*(Edx2y2dx2y2) + (-2)*(Edxydxy) + Edxzdxz + Edyzdyz + (2)*(Edz2dz2))} , 
       {2, 1, (-1)*((1/(sqrt(2)))*((sqrt(3))*(Edx2y2dxz) + (I)*((sqrt(3))*(Edx2y2dyz)) + (-I)*((sqrt(3))*(Edxzdxy)) + (sqrt(3))*(Edyzdxy) + Edz2dxz + (-I)*(Edz2dyz)))} , 
       {2,-1, (1/(sqrt(2)))*((sqrt(3))*(Edx2y2dxz) + (-I)*((sqrt(3))*(Edx2y2dyz)) + (I)*((sqrt(3))*(Edxzdxy)) + (sqrt(3))*(Edyzdxy) + Edz2dxz + (I)*(Edz2dyz))} , 
       {2,-2, (-1/2)*((1/(sqrt(2)))*((4)*(Edx2y2dz2) + (-1)*((sqrt(3))*(Edxzdxz)) + (-2*I)*((sqrt(3))*(Edyzdxz)) + (sqrt(3))*(Edyzdyz) + (4*I)*(Edz2dxy)))} , 
       {2, 2, (-1/2)*((1/(sqrt(2)))*((4)*(Edx2y2dz2) + (-1)*((sqrt(3))*(Edxzdxz)) + (2*I)*((sqrt(3))*(Edyzdxz)) + (sqrt(3))*(Edyzdyz) + (-4*I)*(Edz2dxy)))} , 
       {4, 0, (3/10)*(Edx2y2dx2y2 + Edxydxy + (-4)*(Edxzdxz) + (-4)*(Edyzdyz) + (6)*(Edz2dz2))} , 
       {4,-1, (-3/2)*((1/(sqrt(5)))*(Edx2y2dxz + (-I)*(Edx2y2dyz) + (I)*(Edxzdxy) + Edyzdxy + (-2)*((sqrt(3))*(Edz2dxz)) + (-2*I)*((sqrt(3))*(Edz2dyz))))} , 
       {4, 1, (3/2)*((1/(sqrt(5)))*(Edx2y2dxz + (I)*(Edx2y2dyz) + (-I)*(Edxzdxy) + Edyzdxy + (-2)*((sqrt(3))*(Edz2dxz)) + (2*I)*((sqrt(3))*(Edz2dyz))))} , 
       {4, 2, (3)*((1/(sqrt(10)))*((sqrt(3))*(Edx2y2dz2) + Edxzdxz + (-2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (-I)*((sqrt(3))*(Edz2dxy))))} , 
       {4,-2, (3)*((1/(sqrt(10)))*((sqrt(3))*(Edx2y2dz2) + Edxzdxz + (2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (I)*((sqrt(3))*(Edz2dxy))))} , 
       {4, 3, (3/2)*((sqrt(7/5))*((-1)*(Edx2y2dxz) + (I)*(Edx2y2dyz) + (I)*(Edxzdxy) + Edyzdxy))} , 
       {4,-3, (3/2)*((sqrt(7/5))*(Edx2y2dxz + (I)*(Edx2y2dyz + Edxzdxy + (I)*(Edyzdxy))))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (-2*I)*(Edx2y2dxy) + (-1)*(Edxydxy)))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (2*I)*(Edx2y2dxy) + (-1)*(Edxydxy)))} }

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{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} \frac{1}{2} (\text{Edx2y2dxz}-i \text{Edx2y2dyz}+i \text{Edxzdxy}+\text{Edyzdxy}) \frac{\text{Edx2y2dz2}+i \text{Edz2dxy}}{\sqrt{2}} \frac{1}{2} (-\text{Edx2y2dxz}-i (\text{Edx2y2dyz}+\text{Edxzdxy})+\text{Edyzdxy}) \frac{1}{2} (\text{Edx2y2dx2y2}+2 i \text{Edx2y2dxy}-\text{Edxydxy})
{Y_{-1}^{(2)}} \frac{1}{2} (\text{Edx2y2dxz}+i (\text{Edx2y2dyz}-\text{Edxzdxy})+\text{Edyzdxy}) \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} \frac{\text{Edz2dxz}+i \text{Edz2dyz}}{\sqrt{2}} \frac{1}{2} (-\text{Edxzdxz}-2 i \text{Edyzdxz}+\text{Edyzdyz}) \frac{1}{2} (\text{Edx2y2dxz}+i (\text{Edx2y2dyz}+\text{Edxzdxy}+i \text{Edyzdxy}))
{Y_{0}^{(2)}} \frac{\text{Edx2y2dz2}-i \text{Edz2dxy}}{\sqrt{2}} \frac{\text{Edz2dxz}-i \text{Edz2dyz}}{\sqrt{2}} \text{Edz2dz2} -\frac{\text{Edz2dxz}+i \text{Edz2dyz}}{\sqrt{2}} \frac{\text{Edx2y2dz2}+i \text{Edz2dxy}}{\sqrt{2}}
{Y_{1}^{(2)}} \frac{1}{2} (-\text{Edx2y2dxz}+i (\text{Edx2y2dyz}+\text{Edxzdxy})+\text{Edyzdxy}) \frac{1}{2} (-\text{Edxzdxz}+2 i \text{Edyzdxz}+\text{Edyzdyz}) -\frac{\text{Edz2dxz}-i \text{Edz2dyz}}{\sqrt{2}} \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} \frac{1}{2} (-\text{Edx2y2dxz}+i (\text{Edx2y2dyz}-\text{Edxzdxy}+i \text{Edyzdxy}))
{Y_{2}^{(2)}} \frac{1}{2} (\text{Edx2y2dx2y2}-2 i \text{Edx2y2dxy}-\text{Edxydxy}) \frac{1}{2} (\text{Edx2y2dxz}-i (\text{Edx2y2dyz}+\text{Edxzdxy})-\text{Edyzdxy}) \frac{\text{Edx2y2dz2}-i \text{Edz2dxy}}{\sqrt{2}} \frac{1}{2} (-\text{Edx2y2dxz}-i \text{Edx2y2dyz}+i \text{Edxzdxy}-\text{Edyzdxy}) \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{x^2-y^2} d_{3z^2-r^2} d_{\text{yz}} d_{\text{xz}} d_{\text{xy}}
d_{x^2-y^2} \text{Edx2y2dx2y2} \text{Edx2y2dz2} \text{Edx2y2dyz} \text{Edx2y2dxz} \text{Edx2y2dxy}
d_{3z^2-r^2} \text{Edx2y2dz2} \text{Edz2dz2} \text{Edz2dyz} \text{Edz2dxz} \text{Edz2dxy}
d_{\text{yz}} \text{Edx2y2dyz} \text{Edz2dyz} \text{Edyzdyz} \text{Edyzdxz} \text{Edyzdxy}
d_{\text{xz}} \text{Edx2y2dxz} \text{Edz2dxz} \text{Edyzdxz} \text{Edxzdxz} \text{Edxzdxy}
d_{\text{xy}} \text{Edx2y2dxy} \text{Edz2dxy} \text{Edyzdxy} \text{Edxzdxy} \text{Edxydxy}

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Edx2y2dx2y2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)
\text{Edz2dz2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)
\text{Edyzdyz}
\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{Edxzdxz}
\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
\text{Edxydxy}
\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

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{Efx3fx3}+\text{Efxy2z2fxy2z2}+\text{Efxyzfxyz}+\text{Efy3fy3}+\text{Efyz2x2fyz2x2}+\text{Efz3fz3}+\text{Efzx2y2fzx2y2}) & k=0\land m=0 \\ \frac{5}{56} \left(2 \left(\sqrt{6} \text{Efx3fx3}+\sqrt{10} \text{Efx3fxy2z2}-\sqrt{6} \text{Efy3fy3}+\sqrt{10} \text{Efy3fyz2x2}-2 \sqrt{10} \text{Efz3fzx2y2}\right)-i \left(\sqrt{6} \text{Efx3fy3}+\sqrt{10} \text{Efx3fyz2x2}+5 \sqrt{6} \text{Efxy2z2fyz2x2}+4 \sqrt{10} \text{Efxyzfz3}-\sqrt{10} \text{Efy3fxy2z2}\right)\right) & k=2\land m=-2 \\ \frac{5}{56} \left(-\sqrt{6} \text{Efx3fz3}+\sqrt{10} \text{Efx3fzx2y2}-5 \sqrt{6} \text{Efxy2z2fzx2y2}-i \left(4 \sqrt{10} \text{Efxyzfx3}+\sqrt{6} \text{Efy3fz3}+\sqrt{10} \text{Efy3fzx2y2}+5 \sqrt{6} \text{Efyz2x2fzx2y2}-\sqrt{10} \text{Efz3fyz2x2}\right)-4 \sqrt{10} \text{Efxyzfy3}-\sqrt{10} \text{Efz3fxy2z2}\right) & k=2\land m=-1 \\ -\frac{5}{14} \left(\text{Efx3fx3}-\sqrt{15} \text{Efx3fxy2z2}+\text{Efy3fy3}+\sqrt{15} \text{Efy3fyz2x2}-2 \text{Efz3fz3}\right) & k=2\land m=0 \\ \frac{5}{56} \left(\sqrt{6} \text{Efx3fz3}-\sqrt{10} \text{Efx3fzx2y2}+5 \sqrt{6} \text{Efxy2z2fzx2y2}-i \left(4 \sqrt{10} \text{Efxyzfx3}+\sqrt{6} \text{Efy3fz3}+\sqrt{10} \text{Efy3fzx2y2}+5 \sqrt{6} \text{Efyz2x2fzx2y2}-\sqrt{10} \text{Efz3fyz2x2}\right)+4 \sqrt{10} \text{Efxyzfy3}+\sqrt{10} \text{Efz3fxy2z2}\right) & k=2\land m=1 \\ \frac{5}{56} \left(2 \left(\sqrt{6} \text{Efx3fx3}+\sqrt{10} \text{Efx3fxy2z2}-\sqrt{6} \text{Efy3fy3}+\sqrt{10} \text{Efy3fyz2x2}-2 \sqrt{10} \text{Efz3fzx2y2}\right)+i \left(\sqrt{6} \text{Efx3fy3}+\sqrt{10} \text{Efx3fyz2x2}+5 \sqrt{6} \text{Efxy2z2fyz2x2}+4 \sqrt{10} \text{Efxyzfz3}-\sqrt{10} \text{Efy3fxy2z2}\right)\right) & k=2\land m=2 \\ \frac{3 \left(3 \sqrt{5} \text{Efx3fx3}+2 \sqrt{3} \text{Efx3fxy2z2}-8 i \sqrt{3} \text{Efx3fyz2x2}-3 \sqrt{5} \text{Efxy2z2fxy2z2}-4 \sqrt{5} \text{Efxyzfxyz}+8 i \sqrt{5} \text{Efxyzfzx2y2}-8 i \sqrt{3} \text{Efy3fxy2z2}+3 \sqrt{5} \text{Efy3fy3}-2 \sqrt{3} \text{Efy3fyz2x2}-3 \sqrt{5} \text{Efyz2x2fyz2x2}+4 \sqrt{5} \text{Efzx2y2fzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=-4 \\ -\frac{3 \left(3 \sqrt{5} \text{Efx3fz3}+\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \sqrt{3} \text{Efxyzfx3}+i \sqrt{5} \text{Efxyzfxy2z2}-\sqrt{3} \text{Efxyzfy3}+\sqrt{5} \text{Efxyzfyz2x2}-3 i \sqrt{5} \text{Efy3fz3}+i \sqrt{3} \text{Efy3fzx2y2}-i \sqrt{5} \text{Efyz2x2fzx2y2}-3 \sqrt{3} \text{Efz3fxy2z2}-3 i \sqrt{3} \text{Efz3fyz2x2}\right)}{4 \sqrt{7}} & k=4\land m=-3 \\ \frac{3}{56} \left(-3 \sqrt{10} \text{Efx3fx3}+2 \sqrt{6} \text{Efx3fxy2z2}+4 i \left(3 \sqrt{10} \text{Efx3fy3}-2 \sqrt{6} \text{Efx3fyz2x2}+\sqrt{10} \text{Efxy2z2fyz2x2}-\sqrt{6} \text{Efxyzfz3}+2 \sqrt{6} \text{Efy3fxy2z2}\right)+7 \sqrt{10} \text{Efxy2z2fxy2z2}+3 \sqrt{10} \text{Efy3fy3}+2 \sqrt{6} \text{Efy3fyz2x2}-7 \sqrt{10} \text{Efyz2x2fyz2x2}-4 \sqrt{6} \text{Efz3fzx2y2}\right) & k=4\land m=-2 \\ \frac{3}{28} \left(-3 \sqrt{5} \text{Efx3fz3}-9 \sqrt{3} \text{Efx3fzx2y2}-\sqrt{5} \text{Efxy2z2fzx2y2}+i \left(\sqrt{3} \text{Efxyzfx3}-7 \sqrt{5} \text{Efxyzfxy2z2}-3 \sqrt{5} \text{Efy3fz3}+9 \sqrt{3} \text{Efy3fzx2y2}-\sqrt{5} \text{Efyz2x2fzx2y2}+5 \sqrt{3} \text{Efz3fyz2x2}\right)+\sqrt{3} \text{Efxyzfy3}+7 \sqrt{5} \text{Efxyzfyz2x2}-5 \sqrt{3} \text{Efz3fxy2z2}\right) & k=4\land m=-1 \\ \frac{3}{56} \left(9 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+7 \text{Efxy2z2fxy2z2}-28 \text{Efxyzfxyz}+9 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+7 \text{Efyz2x2fyz2x2}+24 \text{Efz3fz3}-28 \text{Efzx2y2fzx2y2}\right) & k=4\land m=0 \\ \frac{3}{28} \left(3 \sqrt{5} \text{Efx3fz3}+9 \sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \left(\sqrt{3} \text{Efxyzfx3}-7 \sqrt{5} \text{Efxyzfxy2z2}-3 \sqrt{5} \text{Efy3fz3}+9 \sqrt{3} \text{Efy3fzx2y2}-\sqrt{5} \text{Efyz2x2fzx2y2}+5 \sqrt{3} \text{Efz3fyz2x2}\right)-\sqrt{3} \text{Efxyzfy3}-7 \sqrt{5} \text{Efxyzfyz2x2}+5 \sqrt{3} \text{Efz3fxy2z2}\right) & k=4\land m=1 \\ \frac{3}{56} \left(-3 \sqrt{10} \text{Efx3fx3}+2 \sqrt{6} \text{Efx3fxy2z2}-4 i \left(3 \sqrt{10} \text{Efx3fy3}-2 \sqrt{6} \text{Efx3fyz2x2}+\sqrt{10} \text{Efxy2z2fyz2x2}-\sqrt{6} \text{Efxyzfz3}+2 \sqrt{6} \text{Efy3fxy2z2}\right)+7 \sqrt{10} \text{Efxy2z2fxy2z2}+3 \sqrt{10} \text{Efy3fy3}+2 \sqrt{6} \text{Efy3fyz2x2}-7 \sqrt{10} \text{Efyz2x2fyz2x2}-4 \sqrt{6} \text{Efz3fzx2y2}\right) & k=4\land m=2 \\ \frac{3 \left(3 \sqrt{5} \text{Efx3fz3}+\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}-i \left(\sqrt{3} \text{Efxyzfx3}+\sqrt{5} \text{Efxyzfxy2z2}-3 \sqrt{5} \text{Efy3fz3}+\sqrt{3} \text{Efy3fzx2y2}-\sqrt{5} \text{Efyz2x2fzx2y2}-3 \sqrt{3} \text{Efz3fyz2x2}\right)-\sqrt{3} \text{Efxyzfy3}+\sqrt{5} \text{Efxyzfyz2x2}-3 \sqrt{3} \text{Efz3fxy2z2}\right)}{4 \sqrt{7}} & k=4\land m=3 \\ \frac{3 \left(3 \sqrt{5} \text{Efx3fx3}+2 \sqrt{3} \text{Efx3fxy2z2}+8 i \sqrt{3} \text{Efx3fyz2x2}-3 \sqrt{5} \text{Efxy2z2fxy2z2}-4 \sqrt{5} \text{Efxyzfxyz}-8 i \sqrt{5} \text{Efxyzfzx2y2}+8 i \sqrt{3} \text{Efy3fxy2z2}+3 \sqrt{5} \text{Efy3fy3}-2 \sqrt{3} \text{Efy3fyz2x2}-3 \sqrt{5} \text{Efyz2x2fyz2x2}+4 \sqrt{5} \text{Efzx2y2fzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=4 \\ \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Efx3fx3}-6 \sqrt{5} \text{Efx3fxy2z2}-10 i \sqrt{3} \text{Efx3fy3}-6 i \sqrt{5} \text{Efx3fyz2x2}+3 \sqrt{3} \text{Efxy2z2fxy2z2}+6 i \sqrt{3} \text{Efxy2z2fyz2x2}+6 i \sqrt{5} \text{Efy3fxy2z2}-5 \sqrt{3} \text{Efy3fy3}-6 \sqrt{5} \text{Efy3fyz2x2}-3 \sqrt{3} \text{Efyz2x2fyz2x2}\right) & k=6\land m=-6 \\ \frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}+i \sqrt{15} \text{Efxyzfx3}-3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}-i \sqrt{15} \text{Efy3fzx2y2}-3 i \text{Efyz2x2fzx2y2}\right) & k=6\land m=-5 \\ -\frac{13 \left(15 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}-8 i \sqrt{15} \text{Efx3fyz2x2}-15 \text{Efxy2z2fxy2z2}+24 \text{Efxyzfxyz}-48 i \text{Efxyzfzx2y2}-8 i \sqrt{15} \text{Efy3fxy2z2}+15 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}-15 \text{Efyz2x2fyz2x2}-24 \text{Efzx2y2fzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=-4 \\ \frac{13 \left(2 \sqrt{15} \text{Efx3fz3}-9 \text{Efx3fzx2y2}-3 \sqrt{15} \text{Efxy2z2fzx2y2}-9 i \text{Efxyzfx3}-3 i \sqrt{15} \text{Efxyzfxy2z2}+9 \text{Efxyzfy3}-3 \sqrt{15} \text{Efxyzfyz2x2}-2 i \sqrt{15} \text{Efy3fz3}-9 i \text{Efy3fzx2y2}+3 i \sqrt{15} \text{Efyz2x2fzx2y2}-6 \text{Efz3fxy2z2}-6 i \text{Efz3fyz2x2}\right)}{40 \sqrt{7}} & k=6\land m=-3 \\ \frac{13 \left(5 \sqrt{15} \text{Efx3fx3}+34 \text{Efx3fxy2z2}+2 i \sqrt{15} \text{Efx3fy3}-26 i \text{Efx3fyz2x2}+3 \sqrt{15} \text{Efxy2z2fxy2z2}-14 i \sqrt{15} \text{Efxy2z2fyz2x2}+64 i \text{Efxyzfz3}+26 i \text{Efy3fxy2z2}-5 \sqrt{15} \text{Efy3fy3}+34 \text{Efy3fyz2x2}-3 \sqrt{15} \text{Efyz2x2fyz2x2}+64 \text{Efz3fzx2y2}\right)}{160 \sqrt{7}} & k=6\land m=-2 \\ \frac{13}{280} \left(-5 \sqrt{42} \text{Efx3fz3}+2 \sqrt{70} \text{Efx3fzx2y2}+2 \sqrt{42} \text{Efxy2z2fzx2y2}+i \left(\sqrt{70} \text{Efxyzfx3}+3 \sqrt{42} \text{Efxyzfxy2z2}-5 \sqrt{42} \text{Efy3fz3}-2 \sqrt{70} \text{Efy3fzx2y2}+2 \sqrt{42} \text{Efyz2x2fzx2y2}+5 \sqrt{70} \text{Efz3fyz2x2}\right)+\sqrt{70} \text{Efxyzfy3}-3 \sqrt{42} \text{Efxyzfyz2x2}-5 \sqrt{70} \text{Efz3fxy2z2}\right) & k=6\land m=-1 \\ -\frac{13}{560} \left(25 \text{Efx3fx3}+14 \sqrt{15} \text{Efx3fxy2z2}+39 \text{Efxy2z2fxy2z2}-24 \text{Efxyzfxyz}+25 \text{Efy3fy3}-14 \sqrt{15} \text{Efy3fyz2x2}+39 \text{Efyz2x2fyz2x2}-80 \text{Efz3fz3}-24 \text{Efzx2y2fzx2y2}\right) & k=6\land m=0 \\ \frac{13}{280} \left(5 \sqrt{42} \text{Efx3fz3}-2 \sqrt{70} \text{Efx3fzx2y2}-2 \sqrt{42} \text{Efxy2z2fzx2y2}+i \left(\sqrt{70} \text{Efxyzfx3}+3 \sqrt{42} \text{Efxyzfxy2z2}-5 \sqrt{42} \text{Efy3fz3}-2 \sqrt{70} \text{Efy3fzx2y2}+2 \sqrt{42} \text{Efyz2x2fzx2y2}+5 \sqrt{70} \text{Efz3fyz2x2}\right)-\sqrt{70} \text{Efxyzfy3}+3 \sqrt{42} \text{Efxyzfyz2x2}+5 \sqrt{70} \text{Efz3fxy2z2}\right) & k=6\land m=1 \\ \frac{13 \left(5 \sqrt{15} \text{Efx3fx3}+34 \text{Efx3fxy2z2}-2 i \sqrt{15} \text{Efx3fy3}+26 i \text{Efx3fyz2x2}+3 \sqrt{15} \text{Efxy2z2fxy2z2}+14 i \sqrt{15} \text{Efxy2z2fyz2x2}-64 i \text{Efxyzfz3}-26 i \text{Efy3fxy2z2}-5 \sqrt{15} \text{Efy3fy3}+34 \text{Efy3fyz2x2}-3 \sqrt{15} \text{Efyz2x2fyz2x2}+64 \text{Efz3fzx2y2}\right)}{160 \sqrt{7}} & k=6\land m=2 \\ -\frac{13 \left(2 \sqrt{15} \text{Efx3fz3}-9 \text{Efx3fzx2y2}-3 \sqrt{15} \text{Efxy2z2fzx2y2}+9 i \text{Efxyzfx3}+3 i \sqrt{15} \text{Efxyzfxy2z2}+9 \text{Efxyzfy3}-3 \sqrt{15} \text{Efxyzfyz2x2}+2 i \sqrt{15} \text{Efy3fz3}+9 i \text{Efy3fzx2y2}-3 i \sqrt{15} \text{Efyz2x2fzx2y2}-6 \text{Efz3fxy2z2}+6 i \text{Efz3fyz2x2}\right)}{40 \sqrt{7}} & k=6\land m=3 \\ -\frac{13 \left(15 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+8 i \sqrt{15} \text{Efx3fyz2x2}-15 \text{Efxy2z2fxy2z2}+24 \text{Efxyzfxyz}+48 i \text{Efxyzfzx2y2}+8 i \sqrt{15} \text{Efy3fxy2z2}+15 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}-15 \text{Efyz2x2fyz2x2}-24 \text{Efzx2y2fzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=4 \\ -\frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}-i \sqrt{15} \text{Efxyzfx3}+3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}+i \sqrt{15} \text{Efy3fzx2y2}+3 i \text{Efyz2x2fzx2y2}\right) & k=6\land m=5 \\ \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Efx3fx3}-6 \sqrt{5} \text{Efx3fxy2z2}+10 i \sqrt{3} \text{Efx3fy3}+6 i \sqrt{5} \text{Efx3fyz2x2}+3 \sqrt{3} \text{Efxy2z2fxy2z2}-6 i \sqrt{3} \text{Efxy2z2fyz2x2}-6 i \sqrt{5} \text{Efy3fxy2z2}-5 \sqrt{3} \text{Efy3fy3}-6 \sqrt{5} \text{Efy3fyz2x2}-3 \sqrt{3} \text{Efyz2x2fyz2x2}\right) & k=6\land m=6 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Ci.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Efx3fx3 + Efxy2z2fxy2z2 + Efxyzfxyz + Efy3fy3 + Efyz2x2fyz2x2 + Efz3fz3 + Efzx2y2fzx2y2)/7, k == 0 && m == 0}, {(5*((-I)*(Sqrt[6]*Efx3fy3 + Sqrt[10]*Efx3fyz2x2 + 5*Sqrt[6]*Efxy2z2fyz2x2 + 4*Sqrt[10]*Efxyzfz3 - Sqrt[10]*Efy3fxy2z2) + 2*(Sqrt[6]*Efx3fx3 + Sqrt[10]*Efx3fxy2z2 - Sqrt[6]*Efy3fy3 + Sqrt[10]*Efy3fyz2x2 - 2*Sqrt[10]*Efz3fzx2y2)))/56, k == 2 && m == -2}, {(5*(-(Sqrt[6]*Efx3fz3) + Sqrt[10]*Efx3fzx2y2 - 5*Sqrt[6]*Efxy2z2fzx2y2 - 4*Sqrt[10]*Efxyzfy3 - Sqrt[10]*Efz3fxy2z2 - I*(4*Sqrt[10]*Efxyzfx3 + Sqrt[6]*Efy3fz3 + Sqrt[10]*Efy3fzx2y2 + 5*Sqrt[6]*Efyz2x2fzx2y2 - Sqrt[10]*Efz3fyz2x2)))/56, k == 2 && m == -1}, {(-5*(Efx3fx3 - Sqrt[15]*Efx3fxy2z2 + Efy3fy3 + Sqrt[15]*Efy3fyz2x2 - 2*Efz3fz3))/14, k == 2 && m == 0}, {(5*(Sqrt[6]*Efx3fz3 - Sqrt[10]*Efx3fzx2y2 + 5*Sqrt[6]*Efxy2z2fzx2y2 + 4*Sqrt[10]*Efxyzfy3 + Sqrt[10]*Efz3fxy2z2 - I*(4*Sqrt[10]*Efxyzfx3 + Sqrt[6]*Efy3fz3 + Sqrt[10]*Efy3fzx2y2 + 5*Sqrt[6]*Efyz2x2fzx2y2 - Sqrt[10]*Efz3fyz2x2)))/56, k == 2 && m == 1}, {(5*(I*(Sqrt[6]*Efx3fy3 + Sqrt[10]*Efx3fyz2x2 + 5*Sqrt[6]*Efxy2z2fyz2x2 + 4*Sqrt[10]*Efxyzfz3 - Sqrt[10]*Efy3fxy2z2) + 2*(Sqrt[6]*Efx3fx3 + Sqrt[10]*Efx3fxy2z2 - Sqrt[6]*Efy3fy3 + Sqrt[10]*Efy3fyz2x2 - 2*Sqrt[10]*Efz3fzx2y2)))/56, k == 2 && m == 2}, {(3*(3*Sqrt[5]*Efx3fx3 + 2*Sqrt[3]*Efx3fxy2z2 - (8*I)*Sqrt[3]*Efx3fyz2x2 - 3*Sqrt[5]*Efxy2z2fxy2z2 - 4*Sqrt[5]*Efxyzfxyz + (8*I)*Sqrt[5]*Efxyzfzx2y2 - (8*I)*Sqrt[3]*Efy3fxy2z2 + 3*Sqrt[5]*Efy3fy3 - 2*Sqrt[3]*Efy3fyz2x2 - 3*Sqrt[5]*Efyz2x2fyz2x2 + 4*Sqrt[5]*Efzx2y2fzx2y2))/(8*Sqrt[14]), k == 4 && m == -4}, {(-3*(3*Sqrt[5]*Efx3fz3 + Sqrt[3]*Efx3fzx2y2 + Sqrt[5]*Efxy2z2fzx2y2 + I*Sqrt[3]*Efxyzfx3 + I*Sqrt[5]*Efxyzfxy2z2 - Sqrt[3]*Efxyzfy3 + Sqrt[5]*Efxyzfyz2x2 - (3*I)*Sqrt[5]*Efy3fz3 + I*Sqrt[3]*Efy3fzx2y2 - I*Sqrt[5]*Efyz2x2fzx2y2 - 3*Sqrt[3]*Efz3fxy2z2 - (3*I)*Sqrt[3]*Efz3fyz2x2))/(4*Sqrt[7]), k == 4 && m == -3}, {(3*(-3*Sqrt[10]*Efx3fx3 + 2*Sqrt[6]*Efx3fxy2z2 + 7*Sqrt[10]*Efxy2z2fxy2z2 + (4*I)*(3*Sqrt[10]*Efx3fy3 - 2*Sqrt[6]*Efx3fyz2x2 + Sqrt[10]*Efxy2z2fyz2x2 - Sqrt[6]*Efxyzfz3 + 2*Sqrt[6]*Efy3fxy2z2) + 3*Sqrt[10]*Efy3fy3 + 2*Sqrt[6]*Efy3fyz2x2 - 7*Sqrt[10]*Efyz2x2fyz2x2 - 4*Sqrt[6]*Efz3fzx2y2))/56, k == 4 && m == -2}, {(3*(-3*Sqrt[5]*Efx3fz3 - 9*Sqrt[3]*Efx3fzx2y2 - Sqrt[5]*Efxy2z2fzx2y2 + Sqrt[3]*Efxyzfy3 + 7*Sqrt[5]*Efxyzfyz2x2 - 5*Sqrt[3]*Efz3fxy2z2 + I*(Sqrt[3]*Efxyzfx3 - 7*Sqrt[5]*Efxyzfxy2z2 - 3*Sqrt[5]*Efy3fz3 + 9*Sqrt[3]*Efy3fzx2y2 - Sqrt[5]*Efyz2x2fzx2y2 + 5*Sqrt[3]*Efz3fyz2x2)))/28, k == 4 && m == -1}, {(3*(9*Efx3fx3 - 2*Sqrt[15]*Efx3fxy2z2 + 7*Efxy2z2fxy2z2 - 28*Efxyzfxyz + 9*Efy3fy3 + 2*Sqrt[15]*Efy3fyz2x2 + 7*Efyz2x2fyz2x2 + 24*Efz3fz3 - 28*Efzx2y2fzx2y2))/56, k == 4 && m == 0}, {(3*(3*Sqrt[5]*Efx3fz3 + 9*Sqrt[3]*Efx3fzx2y2 + Sqrt[5]*Efxy2z2fzx2y2 - Sqrt[3]*Efxyzfy3 - 7*Sqrt[5]*Efxyzfyz2x2 + 5*Sqrt[3]*Efz3fxy2z2 + I*(Sqrt[3]*Efxyzfx3 - 7*Sqrt[5]*Efxyzfxy2z2 - 3*Sqrt[5]*Efy3fz3 + 9*Sqrt[3]*Efy3fzx2y2 - Sqrt[5]*Efyz2x2fzx2y2 + 5*Sqrt[3]*Efz3fyz2x2)))/28, k == 4 && m == 1}, {(3*(-3*Sqrt[10]*Efx3fx3 + 2*Sqrt[6]*Efx3fxy2z2 + 7*Sqrt[10]*Efxy2z2fxy2z2 - (4*I)*(3*Sqrt[10]*Efx3fy3 - 2*Sqrt[6]*Efx3fyz2x2 + Sqrt[10]*Efxy2z2fyz2x2 - Sqrt[6]*Efxyzfz3 + 2*Sqrt[6]*Efy3fxy2z2) + 3*Sqrt[10]*Efy3fy3 + 2*Sqrt[6]*Efy3fyz2x2 - 7*Sqrt[10]*Efyz2x2fyz2x2 - 4*Sqrt[6]*Efz3fzx2y2))/56, k == 4 && m == 2}, {(3*(3*Sqrt[5]*Efx3fz3 + Sqrt[3]*Efx3fzx2y2 + Sqrt[5]*Efxy2z2fzx2y2 - Sqrt[3]*Efxyzfy3 + Sqrt[5]*Efxyzfyz2x2 - 3*Sqrt[3]*Efz3fxy2z2 - I*(Sqrt[3]*Efxyzfx3 + Sqrt[5]*Efxyzfxy2z2 - 3*Sqrt[5]*Efy3fz3 + Sqrt[3]*Efy3fzx2y2 - Sqrt[5]*Efyz2x2fzx2y2 - 3*Sqrt[3]*Efz3fyz2x2)))/(4*Sqrt[7]), k == 4 && m == 3}, {(3*(3*Sqrt[5]*Efx3fx3 + 2*Sqrt[3]*Efx3fxy2z2 + (8*I)*Sqrt[3]*Efx3fyz2x2 - 3*Sqrt[5]*Efxy2z2fxy2z2 - 4*Sqrt[5]*Efxyzfxyz - (8*I)*Sqrt[5]*Efxyzfzx2y2 + (8*I)*Sqrt[3]*Efy3fxy2z2 + 3*Sqrt[5]*Efy3fy3 - 2*Sqrt[3]*Efy3fyz2x2 - 3*Sqrt[5]*Efyz2x2fyz2x2 + 4*Sqrt[5]*Efzx2y2fzx2y2))/(8*Sqrt[14]), k == 4 && m == 4}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Efx3fx3 - 6*Sqrt[5]*Efx3fxy2z2 - (10*I)*Sqrt[3]*Efx3fy3 - (6*I)*Sqrt[5]*Efx3fyz2x2 + 3*Sqrt[3]*Efxy2z2fxy2z2 + (6*I)*Sqrt[3]*Efxy2z2fyz2x2 + (6*I)*Sqrt[5]*Efy3fxy2z2 - 5*Sqrt[3]*Efy3fy3 - 6*Sqrt[5]*Efy3fyz2x2 - 3*Sqrt[3]*Efyz2x2fyz2x2))/160, k == 6 && m == -6}, {(13*Sqrt[11/7]*(Sqrt[15]*Efx3fzx2y2 - 3*Efxy2z2fzx2y2 + I*Sqrt[15]*Efxyzfx3 - (3*I)*Efxyzfxy2z2 + Sqrt[15]*Efxyzfy3 + 3*Efxyzfyz2x2 - I*Sqrt[15]*Efy3fzx2y2 - (3*I)*Efyz2x2fzx2y2))/40, k == 6 && m == -5}, {(-13*(15*Efx3fx3 + 2*Sqrt[15]*Efx3fxy2z2 - (8*I)*Sqrt[15]*Efx3fyz2x2 - 15*Efxy2z2fxy2z2 + 24*Efxyzfxyz - (48*I)*Efxyzfzx2y2 - (8*I)*Sqrt[15]*Efy3fxy2z2 + 15*Efy3fy3 - 2*Sqrt[15]*Efy3fyz2x2 - 15*Efyz2x2fyz2x2 - 24*Efzx2y2fzx2y2))/(80*Sqrt[14]), k == 6 && m == -4}, {(13*(2*Sqrt[15]*Efx3fz3 - 9*Efx3fzx2y2 - 3*Sqrt[15]*Efxy2z2fzx2y2 - (9*I)*Efxyzfx3 - (3*I)*Sqrt[15]*Efxyzfxy2z2 + 9*Efxyzfy3 - 3*Sqrt[15]*Efxyzfyz2x2 - (2*I)*Sqrt[15]*Efy3fz3 - (9*I)*Efy3fzx2y2 + (3*I)*Sqrt[15]*Efyz2x2fzx2y2 - 6*Efz3fxy2z2 - (6*I)*Efz3fyz2x2))/(40*Sqrt[7]), k == 6 && m == -3}, {(13*(5*Sqrt[15]*Efx3fx3 + 34*Efx3fxy2z2 + (2*I)*Sqrt[15]*Efx3fy3 - (26*I)*Efx3fyz2x2 + 3*Sqrt[15]*Efxy2z2fxy2z2 - (14*I)*Sqrt[15]*Efxy2z2fyz2x2 + (64*I)*Efxyzfz3 + (26*I)*Efy3fxy2z2 - 5*Sqrt[15]*Efy3fy3 + 34*Efy3fyz2x2 - 3*Sqrt[15]*Efyz2x2fyz2x2 + 64*Efz3fzx2y2))/(160*Sqrt[7]), k == 6 && m == -2}, {(13*(-5*Sqrt[42]*Efx3fz3 + 2*Sqrt[70]*Efx3fzx2y2 + 2*Sqrt[42]*Efxy2z2fzx2y2 + Sqrt[70]*Efxyzfy3 - 3*Sqrt[42]*Efxyzfyz2x2 - 5*Sqrt[70]*Efz3fxy2z2 + I*(Sqrt[70]*Efxyzfx3 + 3*Sqrt[42]*Efxyzfxy2z2 - 5*Sqrt[42]*Efy3fz3 - 2*Sqrt[70]*Efy3fzx2y2 + 2*Sqrt[42]*Efyz2x2fzx2y2 + 5*Sqrt[70]*Efz3fyz2x2)))/280, k == 6 && m == -1}, {(-13*(25*Efx3fx3 + 14*Sqrt[15]*Efx3fxy2z2 + 39*Efxy2z2fxy2z2 - 24*Efxyzfxyz + 25*Efy3fy3 - 14*Sqrt[15]*Efy3fyz2x2 + 39*Efyz2x2fyz2x2 - 80*Efz3fz3 - 24*Efzx2y2fzx2y2))/560, k == 6 && m == 0}, {(13*(5*Sqrt[42]*Efx3fz3 - 2*Sqrt[70]*Efx3fzx2y2 - 2*Sqrt[42]*Efxy2z2fzx2y2 - Sqrt[70]*Efxyzfy3 + 3*Sqrt[42]*Efxyzfyz2x2 + 5*Sqrt[70]*Efz3fxy2z2 + I*(Sqrt[70]*Efxyzfx3 + 3*Sqrt[42]*Efxyzfxy2z2 - 5*Sqrt[42]*Efy3fz3 - 2*Sqrt[70]*Efy3fzx2y2 + 2*Sqrt[42]*Efyz2x2fzx2y2 + 5*Sqrt[70]*Efz3fyz2x2)))/280, k == 6 && m == 1}, {(13*(5*Sqrt[15]*Efx3fx3 + 34*Efx3fxy2z2 - (2*I)*Sqrt[15]*Efx3fy3 + (26*I)*Efx3fyz2x2 + 3*Sqrt[15]*Efxy2z2fxy2z2 + (14*I)*Sqrt[15]*Efxy2z2fyz2x2 - (64*I)*Efxyzfz3 - (26*I)*Efy3fxy2z2 - 5*Sqrt[15]*Efy3fy3 + 34*Efy3fyz2x2 - 3*Sqrt[15]*Efyz2x2fyz2x2 + 64*Efz3fzx2y2))/(160*Sqrt[7]), k == 6 && m == 2}, {(-13*(2*Sqrt[15]*Efx3fz3 - 9*Efx3fzx2y2 - 3*Sqrt[15]*Efxy2z2fzx2y2 + (9*I)*Efxyzfx3 + (3*I)*Sqrt[15]*Efxyzfxy2z2 + 9*Efxyzfy3 - 3*Sqrt[15]*Efxyzfyz2x2 + (2*I)*Sqrt[15]*Efy3fz3 + (9*I)*Efy3fzx2y2 - (3*I)*Sqrt[15]*Efyz2x2fzx2y2 - 6*Efz3fxy2z2 + (6*I)*Efz3fyz2x2))/(40*Sqrt[7]), k == 6 && m == 3}, {(-13*(15*Efx3fx3 + 2*Sqrt[15]*Efx3fxy2z2 + (8*I)*Sqrt[15]*Efx3fyz2x2 - 15*Efxy2z2fxy2z2 + 24*Efxyzfxyz + (48*I)*Efxyzfzx2y2 + (8*I)*Sqrt[15]*Efy3fxy2z2 + 15*Efy3fy3 - 2*Sqrt[15]*Efy3fyz2x2 - 15*Efyz2x2fyz2x2 - 24*Efzx2y2fzx2y2))/(80*Sqrt[14]), k == 6 && m == 4}, {(-13*Sqrt[11/7]*(Sqrt[15]*Efx3fzx2y2 - 3*Efxy2z2fzx2y2 - I*Sqrt[15]*Efxyzfx3 + (3*I)*Efxyzfxy2z2 + Sqrt[15]*Efxyzfy3 + 3*Efxyzfyz2x2 + I*Sqrt[15]*Efy3fzx2y2 + (3*I)*Efyz2x2fzx2y2))/40, k == 6 && m == 5}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Efx3fx3 - 6*Sqrt[5]*Efx3fxy2z2 + (10*I)*Sqrt[3]*Efx3fy3 + (6*I)*Sqrt[5]*Efx3fyz2x2 + 3*Sqrt[3]*Efxy2z2fxy2z2 - (6*I)*Sqrt[3]*Efxy2z2fyz2x2 - (6*I)*Sqrt[5]*Efy3fxy2z2 - 5*Sqrt[3]*Efy3fy3 - 6*Sqrt[5]*Efy3fyz2x2 - 3*Sqrt[3]*Efyz2x2fyz2x2))/160, k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Ci.Quanty
Akm = {{0, 0, (1/7)*(Efx3fx3 + Efxy2z2fxy2z2 + Efxyzfxyz + Efy3fy3 + Efyz2x2fyz2x2 + Efz3fz3 + Efzx2y2fzx2y2)} , 
       {2, 0, (-5/14)*(Efx3fx3 + (-1)*((sqrt(15))*(Efx3fxy2z2)) + Efy3fy3 + (sqrt(15))*(Efy3fyz2x2) + (-2)*(Efz3fz3))} , 
       {2,-1, (5/56)*((-1)*((sqrt(6))*(Efx3fz3)) + (sqrt(10))*(Efx3fzx2y2) + (-5)*((sqrt(6))*(Efxy2z2fzx2y2)) + (-4)*((sqrt(10))*(Efxyzfy3)) + (-1)*((sqrt(10))*(Efz3fxy2z2)) + (-I)*((4)*((sqrt(10))*(Efxyzfx3)) + (sqrt(6))*(Efy3fz3) + (sqrt(10))*(Efy3fzx2y2) + (5)*((sqrt(6))*(Efyz2x2fzx2y2)) + (-1)*((sqrt(10))*(Efz3fyz2x2))))} , 
       {2, 1, (5/56)*((sqrt(6))*(Efx3fz3) + (-1)*((sqrt(10))*(Efx3fzx2y2)) + (5)*((sqrt(6))*(Efxy2z2fzx2y2)) + (4)*((sqrt(10))*(Efxyzfy3)) + (sqrt(10))*(Efz3fxy2z2) + (-I)*((4)*((sqrt(10))*(Efxyzfx3)) + (sqrt(6))*(Efy3fz3) + (sqrt(10))*(Efy3fzx2y2) + (5)*((sqrt(6))*(Efyz2x2fzx2y2)) + (-1)*((sqrt(10))*(Efz3fyz2x2))))} , 
       {2,-2, (5/56)*((-I)*((sqrt(6))*(Efx3fy3) + (sqrt(10))*(Efx3fyz2x2) + (5)*((sqrt(6))*(Efxy2z2fyz2x2)) + (4)*((sqrt(10))*(Efxyzfz3)) + (-1)*((sqrt(10))*(Efy3fxy2z2))) + (2)*((sqrt(6))*(Efx3fx3) + (sqrt(10))*(Efx3fxy2z2) + (-1)*((sqrt(6))*(Efy3fy3)) + (sqrt(10))*(Efy3fyz2x2) + (-2)*((sqrt(10))*(Efz3fzx2y2))))} , 
       {2, 2, (5/56)*((I)*((sqrt(6))*(Efx3fy3) + (sqrt(10))*(Efx3fyz2x2) + (5)*((sqrt(6))*(Efxy2z2fyz2x2)) + (4)*((sqrt(10))*(Efxyzfz3)) + (-1)*((sqrt(10))*(Efy3fxy2z2))) + (2)*((sqrt(6))*(Efx3fx3) + (sqrt(10))*(Efx3fxy2z2) + (-1)*((sqrt(6))*(Efy3fy3)) + (sqrt(10))*(Efy3fyz2x2) + (-2)*((sqrt(10))*(Efz3fzx2y2))))} , 
       {4, 0, (3/56)*((9)*(Efx3fx3) + (-2)*((sqrt(15))*(Efx3fxy2z2)) + (7)*(Efxy2z2fxy2z2) + (-28)*(Efxyzfxyz) + (9)*(Efy3fy3) + (2)*((sqrt(15))*(Efy3fyz2x2)) + (7)*(Efyz2x2fyz2x2) + (24)*(Efz3fz3) + (-28)*(Efzx2y2fzx2y2))} , 
       {4,-1, (3/28)*((-3)*((sqrt(5))*(Efx3fz3)) + (-9)*((sqrt(3))*(Efx3fzx2y2)) + (-1)*((sqrt(5))*(Efxy2z2fzx2y2)) + (sqrt(3))*(Efxyzfy3) + (7)*((sqrt(5))*(Efxyzfyz2x2)) + (-5)*((sqrt(3))*(Efz3fxy2z2)) + (I)*((sqrt(3))*(Efxyzfx3) + (-7)*((sqrt(5))*(Efxyzfxy2z2)) + (-3)*((sqrt(5))*(Efy3fz3)) + (9)*((sqrt(3))*(Efy3fzx2y2)) + (-1)*((sqrt(5))*(Efyz2x2fzx2y2)) + (5)*((sqrt(3))*(Efz3fyz2x2))))} , 
       {4, 1, (3/28)*((3)*((sqrt(5))*(Efx3fz3)) + (9)*((sqrt(3))*(Efx3fzx2y2)) + (sqrt(5))*(Efxy2z2fzx2y2) + (-1)*((sqrt(3))*(Efxyzfy3)) + (-7)*((sqrt(5))*(Efxyzfyz2x2)) + (5)*((sqrt(3))*(Efz3fxy2z2)) + (I)*((sqrt(3))*(Efxyzfx3) + (-7)*((sqrt(5))*(Efxyzfxy2z2)) + (-3)*((sqrt(5))*(Efy3fz3)) + (9)*((sqrt(3))*(Efy3fzx2y2)) + (-1)*((sqrt(5))*(Efyz2x2fzx2y2)) + (5)*((sqrt(3))*(Efz3fyz2x2))))} , 
       {4, 2, (3/56)*((-3)*((sqrt(10))*(Efx3fx3)) + (2)*((sqrt(6))*(Efx3fxy2z2)) + (7)*((sqrt(10))*(Efxy2z2fxy2z2)) + (-4*I)*((3)*((sqrt(10))*(Efx3fy3)) + (-2)*((sqrt(6))*(Efx3fyz2x2)) + (sqrt(10))*(Efxy2z2fyz2x2) + (-1)*((sqrt(6))*(Efxyzfz3)) + (2)*((sqrt(6))*(Efy3fxy2z2))) + (3)*((sqrt(10))*(Efy3fy3)) + (2)*((sqrt(6))*(Efy3fyz2x2)) + (-7)*((sqrt(10))*(Efyz2x2fyz2x2)) + (-4)*((sqrt(6))*(Efz3fzx2y2)))} , 
       {4,-2, (3/56)*((-3)*((sqrt(10))*(Efx3fx3)) + (2)*((sqrt(6))*(Efx3fxy2z2)) + (7)*((sqrt(10))*(Efxy2z2fxy2z2)) + (4*I)*((3)*((sqrt(10))*(Efx3fy3)) + (-2)*((sqrt(6))*(Efx3fyz2x2)) + (sqrt(10))*(Efxy2z2fyz2x2) + (-1)*((sqrt(6))*(Efxyzfz3)) + (2)*((sqrt(6))*(Efy3fxy2z2))) + (3)*((sqrt(10))*(Efy3fy3)) + (2)*((sqrt(6))*(Efy3fyz2x2)) + (-7)*((sqrt(10))*(Efyz2x2fyz2x2)) + (-4)*((sqrt(6))*(Efz3fzx2y2)))} , 
       {4, 3, (3/4)*((1/(sqrt(7)))*((3)*((sqrt(5))*(Efx3fz3)) + (sqrt(3))*(Efx3fzx2y2) + (sqrt(5))*(Efxy2z2fzx2y2) + (-1)*((sqrt(3))*(Efxyzfy3)) + (sqrt(5))*(Efxyzfyz2x2) + (-3)*((sqrt(3))*(Efz3fxy2z2)) + (-I)*((sqrt(3))*(Efxyzfx3) + (sqrt(5))*(Efxyzfxy2z2) + (-3)*((sqrt(5))*(Efy3fz3)) + (sqrt(3))*(Efy3fzx2y2) + (-1)*((sqrt(5))*(Efyz2x2fzx2y2)) + (-3)*((sqrt(3))*(Efz3fyz2x2)))))} , 
       {4,-3, (-3/4)*((1/(sqrt(7)))*((3)*((sqrt(5))*(Efx3fz3)) + (sqrt(3))*(Efx3fzx2y2) + (sqrt(5))*(Efxy2z2fzx2y2) + (I)*((sqrt(3))*(Efxyzfx3)) + (I)*((sqrt(5))*(Efxyzfxy2z2)) + (-1)*((sqrt(3))*(Efxyzfy3)) + (sqrt(5))*(Efxyzfyz2x2) + (-3*I)*((sqrt(5))*(Efy3fz3)) + (I)*((sqrt(3))*(Efy3fzx2y2)) + (-I)*((sqrt(5))*(Efyz2x2fzx2y2)) + (-3)*((sqrt(3))*(Efz3fxy2z2)) + (-3*I)*((sqrt(3))*(Efz3fyz2x2))))} , 
       {4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Efx3fx3)) + (2)*((sqrt(3))*(Efx3fxy2z2)) + (-8*I)*((sqrt(3))*(Efx3fyz2x2)) + (-3)*((sqrt(5))*(Efxy2z2fxy2z2)) + (-4)*((sqrt(5))*(Efxyzfxyz)) + (8*I)*((sqrt(5))*(Efxyzfzx2y2)) + (-8*I)*((sqrt(3))*(Efy3fxy2z2)) + (3)*((sqrt(5))*(Efy3fy3)) + (-2)*((sqrt(3))*(Efy3fyz2x2)) + (-3)*((sqrt(5))*(Efyz2x2fyz2x2)) + (4)*((sqrt(5))*(Efzx2y2fzx2y2))))} , 
       {4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Efx3fx3)) + (2)*((sqrt(3))*(Efx3fxy2z2)) + (8*I)*((sqrt(3))*(Efx3fyz2x2)) + (-3)*((sqrt(5))*(Efxy2z2fxy2z2)) + (-4)*((sqrt(5))*(Efxyzfxyz)) + (-8*I)*((sqrt(5))*(Efxyzfzx2y2)) + (8*I)*((sqrt(3))*(Efy3fxy2z2)) + (3)*((sqrt(5))*(Efy3fy3)) + (-2)*((sqrt(3))*(Efy3fyz2x2)) + (-3)*((sqrt(5))*(Efyz2x2fyz2x2)) + (4)*((sqrt(5))*(Efzx2y2fzx2y2))))} , 
       {6, 0, (-13/560)*((25)*(Efx3fx3) + (14)*((sqrt(15))*(Efx3fxy2z2)) + (39)*(Efxy2z2fxy2z2) + (-24)*(Efxyzfxyz) + (25)*(Efy3fy3) + (-14)*((sqrt(15))*(Efy3fyz2x2)) + (39)*(Efyz2x2fyz2x2) + (-80)*(Efz3fz3) + (-24)*(Efzx2y2fzx2y2))} , 
       {6, 1, (13/280)*((5)*((sqrt(42))*(Efx3fz3)) + (-2)*((sqrt(70))*(Efx3fzx2y2)) + (-2)*((sqrt(42))*(Efxy2z2fzx2y2)) + (-1)*((sqrt(70))*(Efxyzfy3)) + (3)*((sqrt(42))*(Efxyzfyz2x2)) + (5)*((sqrt(70))*(Efz3fxy2z2)) + (I)*((sqrt(70))*(Efxyzfx3) + (3)*((sqrt(42))*(Efxyzfxy2z2)) + (-5)*((sqrt(42))*(Efy3fz3)) + (-2)*((sqrt(70))*(Efy3fzx2y2)) + (2)*((sqrt(42))*(Efyz2x2fzx2y2)) + (5)*((sqrt(70))*(Efz3fyz2x2))))} , 
       {6,-1, (13/280)*((-5)*((sqrt(42))*(Efx3fz3)) + (2)*((sqrt(70))*(Efx3fzx2y2)) + (2)*((sqrt(42))*(Efxy2z2fzx2y2)) + (sqrt(70))*(Efxyzfy3) + (-3)*((sqrt(42))*(Efxyzfyz2x2)) + (-5)*((sqrt(70))*(Efz3fxy2z2)) + (I)*((sqrt(70))*(Efxyzfx3) + (3)*((sqrt(42))*(Efxyzfxy2z2)) + (-5)*((sqrt(42))*(Efy3fz3)) + (-2)*((sqrt(70))*(Efy3fzx2y2)) + (2)*((sqrt(42))*(Efyz2x2fzx2y2)) + (5)*((sqrt(70))*(Efz3fyz2x2))))} , 
       {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Efx3fx3)) + (34)*(Efx3fxy2z2) + (-2*I)*((sqrt(15))*(Efx3fy3)) + (26*I)*(Efx3fyz2x2) + (3)*((sqrt(15))*(Efxy2z2fxy2z2)) + (14*I)*((sqrt(15))*(Efxy2z2fyz2x2)) + (-64*I)*(Efxyzfz3) + (-26*I)*(Efy3fxy2z2) + (-5)*((sqrt(15))*(Efy3fy3)) + (34)*(Efy3fyz2x2) + (-3)*((sqrt(15))*(Efyz2x2fyz2x2)) + (64)*(Efz3fzx2y2)))} , 
       {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Efx3fx3)) + (34)*(Efx3fxy2z2) + (2*I)*((sqrt(15))*(Efx3fy3)) + (-26*I)*(Efx3fyz2x2) + (3)*((sqrt(15))*(Efxy2z2fxy2z2)) + (-14*I)*((sqrt(15))*(Efxy2z2fyz2x2)) + (64*I)*(Efxyzfz3) + (26*I)*(Efy3fxy2z2) + (-5)*((sqrt(15))*(Efy3fy3)) + (34)*(Efy3fyz2x2) + (-3)*((sqrt(15))*(Efyz2x2fyz2x2)) + (64)*(Efz3fzx2y2)))} , 
       {6, 3, (-13/40)*((1/(sqrt(7)))*((2)*((sqrt(15))*(Efx3fz3)) + (-9)*(Efx3fzx2y2) + (-3)*((sqrt(15))*(Efxy2z2fzx2y2)) + (9*I)*(Efxyzfx3) + (3*I)*((sqrt(15))*(Efxyzfxy2z2)) + (9)*(Efxyzfy3) + (-3)*((sqrt(15))*(Efxyzfyz2x2)) + (2*I)*((sqrt(15))*(Efy3fz3)) + (9*I)*(Efy3fzx2y2) + (-3*I)*((sqrt(15))*(Efyz2x2fzx2y2)) + (-6)*(Efz3fxy2z2) + (6*I)*(Efz3fyz2x2)))} , 
       {6,-3, (13/40)*((1/(sqrt(7)))*((2)*((sqrt(15))*(Efx3fz3)) + (-9)*(Efx3fzx2y2) + (-3)*((sqrt(15))*(Efxy2z2fzx2y2)) + (-9*I)*(Efxyzfx3) + (-3*I)*((sqrt(15))*(Efxyzfxy2z2)) + (9)*(Efxyzfy3) + (-3)*((sqrt(15))*(Efxyzfyz2x2)) + (-2*I)*((sqrt(15))*(Efy3fz3)) + (-9*I)*(Efy3fzx2y2) + (3*I)*((sqrt(15))*(Efyz2x2fzx2y2)) + (-6)*(Efz3fxy2z2) + (-6*I)*(Efz3fyz2x2)))} , 
       {6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Efx3fx3) + (2)*((sqrt(15))*(Efx3fxy2z2)) + (-8*I)*((sqrt(15))*(Efx3fyz2x2)) + (-15)*(Efxy2z2fxy2z2) + (24)*(Efxyzfxyz) + (-48*I)*(Efxyzfzx2y2) + (-8*I)*((sqrt(15))*(Efy3fxy2z2)) + (15)*(Efy3fy3) + (-2)*((sqrt(15))*(Efy3fyz2x2)) + (-15)*(Efyz2x2fyz2x2) + (-24)*(Efzx2y2fzx2y2)))} , 
       {6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Efx3fx3) + (2)*((sqrt(15))*(Efx3fxy2z2)) + (8*I)*((sqrt(15))*(Efx3fyz2x2)) + (-15)*(Efxy2z2fxy2z2) + (24)*(Efxyzfxyz) + (48*I)*(Efxyzfzx2y2) + (8*I)*((sqrt(15))*(Efy3fxy2z2)) + (15)*(Efy3fy3) + (-2)*((sqrt(15))*(Efy3fyz2x2)) + (-15)*(Efyz2x2fyz2x2) + (-24)*(Efzx2y2fzx2y2)))} , 
       {6, 5, (-13/40)*((sqrt(11/7))*((sqrt(15))*(Efx3fzx2y2) + (-3)*(Efxy2z2fzx2y2) + (-I)*((sqrt(15))*(Efxyzfx3)) + (3*I)*(Efxyzfxy2z2) + (sqrt(15))*(Efxyzfy3) + (3)*(Efxyzfyz2x2) + (I)*((sqrt(15))*(Efy3fzx2y2)) + (3*I)*(Efyz2x2fzx2y2)))} , 
       {6,-5, (13/40)*((sqrt(11/7))*((sqrt(15))*(Efx3fzx2y2) + (-3)*(Efxy2z2fzx2y2) + (I)*((sqrt(15))*(Efxyzfx3)) + (-3*I)*(Efxyzfxy2z2) + (sqrt(15))*(Efxyzfy3) + (3)*(Efxyzfyz2x2) + (-I)*((sqrt(15))*(Efy3fzx2y2)) + (-3*I)*(Efyz2x2fzx2y2)))} , 
       {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Efx3fx3)) + (-6)*((sqrt(5))*(Efx3fxy2z2)) + (-10*I)*((sqrt(3))*(Efx3fy3)) + (-6*I)*((sqrt(5))*(Efx3fyz2x2)) + (3)*((sqrt(3))*(Efxy2z2fxy2z2)) + (6*I)*((sqrt(3))*(Efxy2z2fyz2x2)) + (6*I)*((sqrt(5))*(Efy3fxy2z2)) + (-5)*((sqrt(3))*(Efy3fy3)) + (-6)*((sqrt(5))*(Efy3fyz2x2)) + (-3)*((sqrt(3))*(Efyz2x2fyz2x2))))} , 
       {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Efx3fx3)) + (-6)*((sqrt(5))*(Efx3fxy2z2)) + (10*I)*((sqrt(3))*(Efx3fy3)) + (6*I)*((sqrt(5))*(Efx3fyz2x2)) + (3)*((sqrt(3))*(Efxy2z2fxy2z2)) + (-6*I)*((sqrt(3))*(Efxy2z2fyz2x2)) + (-6*I)*((sqrt(5))*(Efy3fxy2z2)) + (-5)*((sqrt(3))*(Efy3fy3)) + (-6)*((sqrt(5))*(Efy3fyz2x2)) + (-3)*((sqrt(3))*(Efyz2x2fyz2x2))))} }

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{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+3 \text{Efxy2z2fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right) \frac{1}{8} \left(\sqrt{10} \text{Efx3fzx2y2}-\sqrt{2} \left(\sqrt{3} \text{Efxy2z2fzx2y2}+i \sqrt{5} \text{Efxyzfx3}-i \sqrt{3} \text{Efxyzfxy2z2}+\sqrt{5} \text{Efxyzfy3}+\sqrt{3} \text{Efxyzfyz2x2}+i \left(\sqrt{5} \text{Efy3fzx2y2}+\sqrt{3} \text{Efyz2x2fzx2y2}\right)\right)\right) \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}+2 i \sqrt{15} \text{Efx3fy3}-2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}+2 i \sqrt{15} \text{Efxy2z2fyz2x2}+2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) \frac{1}{4} \left(\sqrt{5} \text{Efx3fz3}-i \sqrt{5} \text{Efy3fz3}-\sqrt{3} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right) \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}-8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}-8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) \frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}+i \sqrt{15} \text{Efxyzfx3}-3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}-i \left(\sqrt{15} \text{Efy3fzx2y2}+3 \text{Efyz2x2fzx2y2}\right)}{4 \sqrt{6}} \frac{1}{16} \left(-5 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+10 i \text{Efx3fy3}+2 i \sqrt{15} \text{Efx3fyz2x2}-3 \text{Efxy2z2fxy2z2}-6 i \text{Efxy2z2fyz2x2}-2 i \sqrt{15} \text{Efy3fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right)
{Y_{-2}^{(3)}} \frac{1}{8} \left(\sqrt{10} \text{Efx3fzx2y2}-\sqrt{6} \text{Efxy2z2fzx2y2}+i \left(\sqrt{10} \text{Efxyzfx3}-\sqrt{6} \text{Efxyzfxy2z2}+i \sqrt{10} \text{Efxyzfy3}+i \sqrt{6} \text{Efxyzfyz2x2}+\sqrt{10} \text{Efy3fzx2y2}+\sqrt{6} \text{Efyz2x2fzx2y2}\right)\right) \frac{\text{Efxyzfxyz}+\text{Efzx2y2fzx2y2}}{2} -\frac{\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \sqrt{3} \text{Efxyzfx3}+i \sqrt{5} \text{Efxyzfxy2z2}+\sqrt{3} \text{Efxyzfy3}-\sqrt{5} \text{Efxyzfyz2x2}-i \sqrt{3} \text{Efy3fzx2y2}+i \sqrt{5} \text{Efyz2x2fzx2y2}}{4 \sqrt{2}} \frac{\text{Efz3fzx2y2}+i \text{Efxyzfz3}}{\sqrt{2}} \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}-\sqrt{6} \text{Efxyzfy3}+\sqrt{10} \text{Efxyzfyz2x2}+i \sqrt{6} \text{Efy3fzx2y2}-i \sqrt{10} \text{Efyz2x2fzx2y2}\right) \frac{1}{2} (-\text{Efxyzfxyz}+2 i \text{Efxyzfzx2y2}+\text{Efzx2y2fzx2y2}) -\frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}+i \sqrt{15} \text{Efxyzfx3}-3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}-i \left(\sqrt{15} \text{Efy3fzx2y2}+3 \text{Efyz2x2fzx2y2}\right)}{4 \sqrt{6}}
{Y_{-1}^{(3)}} \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}-2 i \sqrt{15} \text{Efx3fy3}+2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}-2 i \sqrt{15} \text{Efxy2z2fyz2x2}-2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) \frac{1}{8} \left(-\sqrt{6} \text{Efx3fzx2y2}-\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}-\sqrt{6} \text{Efxyzfy3}+\sqrt{10} \text{Efxyzfyz2x2}-i \sqrt{6} \text{Efy3fzx2y2}+i \sqrt{10} \text{Efyz2x2fzx2y2}\right) \frac{1}{16} \left(3 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+5 \text{Efxy2z2fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) \frac{1}{4} \left(-\sqrt{3} \text{Efx3fz3}-i \sqrt{3} \text{Efy3fz3}-\sqrt{5} (\text{Efz3fxy2z2}-i \text{Efz3fyz2x2})\right) \frac{1}{16} \left(-3 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}-6 i \text{Efx3fy3}+2 i \sqrt{15} \text{Efx3fyz2x2}-5 \text{Efxy2z2fxy2z2}+10 i \text{Efxy2z2fyz2x2}-2 i \sqrt{15} \text{Efy3fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) -\frac{\sqrt{3} \text{Efx3fzx2y2}+\sqrt{5} \text{Efxy2z2fzx2y2}+i \sqrt{3} \text{Efxyzfx3}+i \sqrt{5} \text{Efxyzfxy2z2}-\sqrt{3} \text{Efxyzfy3}+\sqrt{5} \text{Efxyzfyz2x2}+i \sqrt{3} \text{Efy3fzx2y2}-i \sqrt{5} \text{Efyz2x2fzx2y2}}{4 \sqrt{2}} \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}-8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}-8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right)
{Y_{0}^{(3)}} \frac{1}{4} \left(\sqrt{5} \text{Efx3fz3}+i \left(\sqrt{5} \text{Efy3fz3}+\sqrt{3} (\text{Efz3fyz2x2}+i \text{Efz3fxy2z2})\right)\right) \frac{\text{Efz3fzx2y2}-i \text{Efxyzfz3}}{\sqrt{2}} \frac{1}{4} \left(-\sqrt{3} \text{Efx3fz3}+i \sqrt{3} \text{Efy3fz3}-\sqrt{5} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right) \text{Efz3fz3} \frac{1}{4} \left(\sqrt{3} \text{Efx3fz3}+i \sqrt{3} \text{Efy3fz3}+\sqrt{5} (\text{Efz3fxy2z2}-i \text{Efz3fyz2x2})\right) \frac{\text{Efz3fzx2y2}+i \text{Efxyzfz3}}{\sqrt{2}} \frac{1}{4} \left(-\sqrt{5} \text{Efx3fz3}+i \sqrt{5} \text{Efy3fz3}+\sqrt{3} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right)
{Y_{1}^{(3)}} \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}+8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}+8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}-i \sqrt{6} \text{Efxyzfx3}-i \sqrt{10} \text{Efxyzfxy2z2}-\sqrt{6} \text{Efxyzfy3}+\sqrt{10} \text{Efxyzfyz2x2}-i \sqrt{6} \text{Efy3fzx2y2}+i \sqrt{10} \text{Efyz2x2fzx2y2}\right) \frac{1}{16} \left(-3 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+6 i \text{Efx3fy3}-2 i \sqrt{15} \text{Efx3fyz2x2}-5 \text{Efxy2z2fxy2z2}-10 i \text{Efxy2z2fyz2x2}+2 i \sqrt{15} \text{Efy3fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) \frac{1}{4} \left(\sqrt{3} \text{Efx3fz3}-i \sqrt{3} \text{Efy3fz3}+\sqrt{5} (\text{Efz3fxy2z2}+i \text{Efz3fyz2x2})\right) \frac{1}{16} \left(3 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}+5 \text{Efxy2z2fxy2z2}+3 \text{Efy3fy3}-2 \sqrt{15} \text{Efy3fyz2x2}+5 \text{Efyz2x2fyz2x2}\right) \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}+\sqrt{6} \text{Efxyzfy3}-\sqrt{10} \text{Efxyzfyz2x2}-i \sqrt{6} \text{Efy3fzx2y2}+i \sqrt{10} \text{Efyz2x2fzx2y2}\right) \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}+2 i \sqrt{15} \text{Efx3fy3}-2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}+2 i \sqrt{15} \text{Efxy2z2fyz2x2}+2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right)
{Y_{2}^{(3)}} \frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}-i \sqrt{15} \text{Efxyzfx3}+3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}+i \sqrt{15} \text{Efy3fzx2y2}+3 i \text{Efyz2x2fzx2y2}}{4 \sqrt{6}} \frac{1}{2} (-\text{Efxyzfxyz}-2 i \text{Efxyzfzx2y2}+\text{Efzx2y2fzx2y2}) \frac{1}{8} \left(-\sqrt{6} \text{Efx3fzx2y2}-\sqrt{10} \text{Efxy2z2fzx2y2}+i \sqrt{6} \text{Efxyzfx3}+i \sqrt{10} \text{Efxyzfxy2z2}+\sqrt{6} \text{Efxyzfy3}-\sqrt{10} \text{Efxyzfyz2x2}+i \sqrt{6} \text{Efy3fzx2y2}-i \sqrt{10} \text{Efyz2x2fzx2y2}\right) \frac{\text{Efz3fzx2y2}-i \text{Efxyzfz3}}{\sqrt{2}} \frac{1}{8} \left(\sqrt{6} \text{Efx3fzx2y2}+\sqrt{10} \text{Efxy2z2fzx2y2}-i \sqrt{6} \text{Efxyzfx3}-i \sqrt{10} \text{Efxyzfxy2z2}+\sqrt{6} \text{Efxyzfy3}-\sqrt{10} \text{Efxyzfyz2x2}+i \sqrt{6} \text{Efy3fzx2y2}-i \sqrt{10} \text{Efyz2x2fzx2y2}\right) \frac{\text{Efxyzfxyz}+\text{Efzx2y2fzx2y2}}{2} \frac{1}{8} \left(-\sqrt{10} \text{Efx3fzx2y2}+\sqrt{6} \text{Efxy2z2fzx2y2}+i \sqrt{10} \text{Efxyzfx3}-i \sqrt{6} \text{Efxyzfxy2z2}+\sqrt{10} \text{Efxyzfy3}+\sqrt{6} \text{Efxyzfyz2x2}+i \left(\sqrt{10} \text{Efy3fzx2y2}+\sqrt{6} \text{Efyz2x2fzx2y2}\right)\right)
{Y_{3}^{(3)}} \frac{1}{16} \left(-5 \text{Efx3fx3}+2 \sqrt{15} \text{Efx3fxy2z2}-10 i \text{Efx3fy3}-2 i \sqrt{15} \text{Efx3fyz2x2}-3 \text{Efxy2z2fxy2z2}+6 i \text{Efxy2z2fyz2x2}+2 i \sqrt{15} \text{Efy3fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right) -\frac{\sqrt{15} \text{Efx3fzx2y2}-3 \text{Efxy2z2fzx2y2}-i \sqrt{15} \text{Efxyzfx3}+3 i \text{Efxyzfxy2z2}+\sqrt{15} \text{Efxyzfy3}+3 \text{Efxyzfyz2x2}+i \sqrt{15} \text{Efy3fzx2y2}+3 i \text{Efyz2x2fzx2y2}}{4 \sqrt{6}} \frac{1}{16} \left(\sqrt{15} \text{Efx3fx3}+2 \text{Efx3fxy2z2}+8 i \text{Efx3fyz2x2}-\sqrt{15} \text{Efxy2z2fxy2z2}+8 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) \frac{1}{4} \left(-\sqrt{5} \text{Efx3fz3}-i \sqrt{5} \text{Efy3fz3}+\sqrt{3} (\text{Efz3fxy2z2}-i \text{Efz3fyz2x2})\right) \frac{1}{16} \left(-\sqrt{15} \text{Efx3fx3}-2 \text{Efx3fxy2z2}-2 i \sqrt{15} \text{Efx3fy3}+2 i \text{Efx3fyz2x2}+\sqrt{15} \text{Efxy2z2fxy2z2}-2 i \sqrt{15} \text{Efxy2z2fyz2x2}-2 i \text{Efy3fxy2z2}+\sqrt{15} \text{Efy3fy3}-2 \text{Efy3fyz2x2}-\sqrt{15} \text{Efyz2x2fyz2x2}\right) \frac{1}{8} \left(-\sqrt{10} \text{Efx3fzx2y2}+\sqrt{6} \text{Efxy2z2fzx2y2}-i \sqrt{10} \text{Efxyzfx3}+i \sqrt{6} \text{Efxyzfxy2z2}+\sqrt{10} \text{Efxyzfy3}+\sqrt{6} \text{Efxyzfyz2x2}-i \left(\sqrt{10} \text{Efy3fzx2y2}+\sqrt{6} \text{Efyz2x2fzx2y2}\right)\right) \frac{1}{16} \left(5 \text{Efx3fx3}-2 \sqrt{15} \text{Efx3fxy2z2}+3 \text{Efxy2z2fxy2z2}+5 \text{Efy3fy3}+2 \sqrt{15} \text{Efy3fyz2x2}+3 \text{Efyz2x2fyz2x2}\right)

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(5z^2-r^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)} f_{z\left(x^2-y^2\right)}
f_{\text{xyz}} \text{Efxyzfxyz} \text{Efxyzfx3} \text{Efxyzfy3} \text{Efxyzfz3} \text{Efxyzfxy2z2} \text{Efxyzfyz2x2} \text{Efxyzfzx2y2}
f_{x\left(5x^2-r^2\right)} \text{Efxyzfx3} \text{Efx3fx3} \text{Efx3fy3} \text{Efx3fz3} \text{Efx3fxy2z2} \text{Efx3fyz2x2} \text{Efx3fzx2y2}
f_{y\left(5y^2-r^2\right)} \text{Efxyzfy3} \text{Efx3fy3} \text{Efy3fy3} \text{Efy3fz3} \text{Efy3fxy2z2} \text{Efy3fyz2x2} \text{Efy3fzx2y2}
f_{z\left(5z^2-r^2\right)} \text{Efxyzfz3} \text{Efx3fz3} \text{Efy3fz3} \text{Efz3fz3} \text{Efz3fxy2z2} \text{Efz3fyz2x2} \text{Efz3fzx2y2}
f_{x\left(y^2-z^2\right)} \text{Efxyzfxy2z2} \text{Efx3fxy2z2} \text{Efy3fxy2z2} \text{Efz3fxy2z2} \text{Efxy2z2fxy2z2} \text{Efxy2z2fyz2x2} \text{Efxy2z2fzx2y2}
f_{y\left(z^2-x^2\right)} \text{Efxyzfyz2x2} \text{Efx3fyz2x2} \text{Efy3fyz2x2} \text{Efz3fyz2x2} \text{Efxy2z2fyz2x2} \text{Efyz2x2fyz2x2} \text{Efyz2x2fzx2y2}
f_{z\left(x^2-y^2\right)} \text{Efxyzfzx2y2} \text{Efx3fzx2y2} \text{Efy3fzx2y2} \text{Efz3fzx2y2} \text{Efxy2z2fzx2y2} \text{Efyz2x2fzx2y2} \text{Efzx2y2fzx2y2}

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_{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(5z^2-r^2\right)} 0 0 0 1 0 0 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}
f_{z\left(x^2-y^2\right)} 0 \frac{1}{\sqrt{2}} 0 0 0 \frac{1}{\sqrt{2}} 0

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Efxyzfxyz}
\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{Efx3fx3}
\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{Efy3fy3}
\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{Efz3fz3}
\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{Efxy2z2fxy2z2}
\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{Efyz2x2fyz2x2}
\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)
\text{Efzx2y2fzx2y2}
\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)

Coupling between two shells

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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=0\land m=0 \\ A(2,2)-i B(2,2) & k=2\land m=-2 \\ -A(2,1)+i B(2,1) & k=2\land m=-1 \\ A(2,0) & k=2\land m=0 \\ A(2,1)+i B(2,1) & k=2\land m=1 \\ A(2,2)+i B(2,2) & k=2\land m=2 \end{cases}

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Input format suitable for Mathematica (Quanty.nb)

Akm_Ci.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {-A[2, 1] + I*B[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1] + I*B[2, 1], k == 2 && m == 1}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Ci.Quanty
Akm = {{2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1)) + (I)*(B(2,1))} , 
       {2, 1, A(2,1) + (I)*(B(2,1))} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(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)+i B(2,2)}{\sqrt{5}} -\frac{A(2,1)+i B(2,1)}{\sqrt{5}} \frac{A(2,0)}{\sqrt{5}} \frac{A(2,1)-i B(2,1)}{\sqrt{5}} \frac{A(2,2)-i B(2,2)}{\sqrt{5}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{x^2-y^2} d_{3z^2-r^2} d_{\text{yz}} d_{\text{xz}} d_{\text{xy}}
\text{s} \sqrt{\frac{2}{5}} A(2,2) \frac{A(2,0)}{\sqrt{5}} \sqrt{\frac{2}{5}} B(2,1) -\sqrt{\frac{2}{5}} A(2,1) -\sqrt{\frac{2}{5}} B(2,2)

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=0\land m=0 \\ A(2,2)-i B(2,2) & k=2\land m=-2 \\ -A(2,1)+i B(2,1) & k=2\land m=-1 \\ A(2,0) & k=2\land m=0 \\ A(2,1)+i B(2,1) & k=2\land m=1 \\ A(2,2)+i B(2,2) & k=2\land m=2 \\ A(4,4)-i B(4,4) & k=4\land m=-4 \\ -A(4,3)+i B(4,3) & k=4\land m=-3 \\ A(4,2)-i B(4,2) & k=4\land m=-2 \\ -A(4,1)+i B(4,1) & k=4\land m=-1 \\ A(4,0) & k=4\land m=0 \\ A(4,1)+i B(4,1) & k=4\land m=1 \\ A(4,2)+i B(4,2) & k=4\land m=2 \\ A(4,3)+i B(4,3) & k=4\land m=3 \\ A(4,4)+i B(4,4) & k=4\land m=4 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Ci.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {-A[2, 1] + I*B[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1] + I*B[2, 1], k == 2 && m == 1}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {-A[4, 3] + I*B[4, 3], k == 4 && m == -3}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {-A[4, 1] + I*B[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1] + I*B[4, 1], k == 4 && m == 1}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 3] + I*B[4, 3], k == 4 && m == 3}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Ci.Quanty
Akm = {{2, 0, A(2,0)} , 
       {2,-1, (-1)*(A(2,1)) + (I)*(B(2,1))} , 
       {2, 1, A(2,1) + (I)*(B(2,1))} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1)*(A(4,1)) + (I)*(B(4,1))} , 
       {4, 1, A(4,1) + (I)*(B(4,1))} , 
       {4,-2, A(4,2) + (-I)*(B(4,2))} , 
       {4, 2, A(4,2) + (I)*(B(4,2))} , 
       {4,-3, (-1)*(A(4,3)) + (I)*(B(4,3))} , 
       {4, 3, A(4,3) + (I)*(B(4,3))} , 
       {4,-4, A(4,4) + (-I)*(B(4,4))} , 
       {4, 4, A(4,4) + (I)*(B(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)+i B(2,2))}{\sqrt{35}}-\frac{A(4,2)+i B(4,2)}{3 \sqrt{21}} \frac{A(4,1)+i B(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} (A(2,1)+i B(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)-27 i B(2,1)+5 i \sqrt{30} B(4,1)}{45 \sqrt{7}} \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)-i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i B(4,2)) \frac{1}{3} (-A(4,3)+i B(4,3)) -\frac{2 (A(4,4)-i B(4,4))}{3 \sqrt{3}}
{Y_{0}^{(1)}} -\frac{A(4,3)+i B(4,3)}{3 \sqrt{3}} \sqrt{\frac{3}{35}} (A(2,2)+i B(2,2))+\frac{2 (A(4,2)+i B(4,2))}{3 \sqrt{7}} -\frac{2}{5} \sqrt{\frac{6}{7}} (A(2,1)+i B(2,1))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,1)+i B(4,1)) \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} \frac{2}{5} \sqrt{\frac{6}{7}} (A(2,1)-i B(2,1))+\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,1)-i B(4,1)) \sqrt{\frac{3}{35}} (A(2,2)-i B(2,2))+\frac{2 (A(4,2)-i B(4,2))}{3 \sqrt{7}} \frac{A(4,3)-i B(4,3)}{3 \sqrt{3}}
{Y_{1}^{(1)}} -\frac{2 (A(4,4)+i B(4,4))}{3 \sqrt{3}} \frac{1}{3} (A(4,3)+i B(4,3)) \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i B(4,2)) \frac{5 \sqrt{30} (A(4,1)+i B(4,1))-27 (A(2,1)+i B(2,1))}{45 \sqrt{7}} \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) \frac{1}{105} \left(3 \sqrt{210} (A(2,1)-i B(2,1))-5 \sqrt{7} (A(4,1)-i B(4,1))\right) \frac{3 (A(2,2)-i B(2,2))}{\sqrt{35}}-\frac{A(4,2)-i B(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_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(5z^2-r^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)} f_{z\left(x^2-y^2\right)}
p_x \sqrt{\frac{6}{35}} B(2,1)-\frac{B(4,1)}{3 \sqrt{7}}+\frac{1}{3} B(4,3) \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) \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)+7 B(4,4)\right)\right) \frac{3}{5} \sqrt{\frac{2}{7}} A(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} A(4,1) \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) \sqrt{\frac{6}{35}} B(2,2)-\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} \frac{1}{21} \left(\sqrt{7} A(4,1)-7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1)
p_y \frac{1}{21} \left(\sqrt{7} A(4,1)+7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1) \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)-7 B(4,4)\right)\right) \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) \frac{1}{45} \sqrt{\frac{2}{7}} \left(5 \sqrt{30} B(4,1)-27 B(2,1)\right) -\sqrt{\frac{6}{35}} B(2,2)+\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} \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) \frac{1}{21} \left(\sqrt{7} B(4,1)+7 B(4,3)\right)-\sqrt{\frac{6}{35}} B(2,1)
p_z -\sqrt{\frac{6}{35}} B(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} B(4,2) \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) \frac{1}{630} \left(-54 \sqrt{14} B(2,1)-5 \sqrt{15} \left(3 \sqrt{7} B(4,1)+7 B(4,3)\right)\right) \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} \sqrt{\frac{6}{35}} A(2,1)+\frac{5 A(4,1)}{6 \sqrt{7}}+\frac{1}{6} A(4,3) \sqrt{\frac{6}{35}} B(2,1)+\frac{5 B(4,1)}{6 \sqrt{7}}-\frac{1}{6} B(4,3) \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2)

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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