In the C2 Point Group, with orientation Z there are the following symmetry operations
Operator | Orientation |
---|---|
E | {0,0,0} , |
C2 | {0,0,1} , |
E(1) | C2(1) | |
---|---|---|
A | 1 | 1 |
B | 1 | −1 |
A | B | |
---|---|---|
A | A | B |
B | B | A |
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 C2 Point group with orientation Z the form of the expansion coefficients is:
Ak,m={A(0,0)k=0∧m=0A(1,0)k=1∧m=0A(2,2)−iB(2,2)k=2∧m=−2A(2,0)k=2∧m=0A(2,2)+iB(2,2)k=2∧m=2A(3,2)−iB(3,2)k=3∧m=−2A(3,0)k=3∧m=0A(3,2)+iB(3,2)k=3∧m=2A(4,4)−iB(4,4)k=4∧m=−4A(4,2)−iB(4,2)k=4∧m=−2A(4,0)k=4∧m=0A(4,2)+iB(4,2)k=4∧m=2A(4,4)+iB(4,4)k=4∧m=4A(5,4)−iB(5,4)k=5∧m=−4A(5,2)−iB(5,2)k=5∧m=−2A(5,0)k=5∧m=0A(5,2)+iB(5,2)k=5∧m=2A(5,4)+iB(5,4)k=5∧m=4A(6,6)−iB(6,6)k=6∧m=−6A(6,4)−iB(6,4)k=6∧m=−4A(6,2)−iB(6,2)k=6∧m=−2A(6,0)k=6∧m=0A(6,2)+iB(6,2)k=6∧m=2A(6,4)+iB(6,4)k=6∧m=4A(6,6)+iB(6,6)k=6∧m=6
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[1, 0], k == 1 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}, {A[3, 2] + I*B[3, 2], k == 3 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {A[5, 2] - I*B[5, 2], k == 5 && m == -2}, {A[5, 0], k == 5 && m == 0}, {A[5, 2] + I*B[5, 2], k == 5 && m == 2}, {A[5, 4] + I*B[5, 4], k == 5 && m == 4}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0]
Akm = {{0, 0, A(0,0)} , {1, 0, A(1,0)} , {2, 0, A(2,0)} , {2,-2, A(2,2) + (-I)*(B(2,2))} , {2, 2, A(2,2) + (I)*(B(2,2))} , {3, 0, A(3,0)} , {3,-2, A(3,2) + (-I)*(B(3,2))} , {3, 2, A(3,2) + (I)*(B(3,2))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + (-I)*(B(4,2))} , {4, 2, A(4,2) + (I)*(B(4,2))} , {4,-4, A(4,4) + (-I)*(B(4,4))} , {4, 4, A(4,4) + (I)*(B(4,4))} , {5, 0, A(5,0)} , {5,-2, A(5,2) + (-I)*(B(5,2))} , {5, 2, A(5,2) + (I)*(B(5,2))} , {5,-4, A(5,4) + (-I)*(B(5,4))} , {5, 4, A(5,4) + (I)*(B(5,4))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + (-I)*(B(6,2))} , {6, 2, A(6,2) + (I)*(B(6,2))} , {6,-4, A(6,4) + (-I)*(B(6,4))} , {6, 4, A(6,4) + (I)*(B(6,4))} , {6,-6, A(6,6) + (-I)*(B(6,6))} , {6, 6, A(6,6) + (I)*(B(6,6))} }
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)0 | Ass(0,0) | 0 | Asp(1,0)√3 | 0 | Asd(2,2)+iBsd(2,2)√5 | 0 | Asd(2,0)√5 | 0 | Asd(2,2)−iBsd(2,2)√5 | 0 | Asf(3,2)+iBsf(3,2)√7 | 0 | Asf(3,0)√7 | 0 | Asf(3,2)−iBsf(3,2)√7 | 0 |
Y(1)−1 | 0 | App(0,0)−15App(2,0) | 0 | −15√6(App(2,2)−iBpp(2,2)) | 0 | Apd(1,0)√5−3Apd(3,0)7√5 | 0 | −17√6(Apd(3,2)−iBpd(3,2)) | 0 | 3(Apf(2,2)+iBpf(2,2))√35−Apf(4,2)+iBpf(4,2)3√21 | 0 | 35√27Apf(2,0)−13√27Apf(4,0) | 0 | 15√37(Apf(2,2)−iBpf(2,2))−13√57(Apf(4,2)−iBpf(4,2)) | 0 | −2(Apf(4,4)−iBpf(4,4))3√3 |
Y(1)0 | Asp(1,0)√3 | 0 | App(0,0)+25App(2,0) | 0 | 17√3(Apd(3,2)+iBpd(3,2)) | 0 | 2Apd(1,0)√15+37√35Apd(3,0) | 0 | 17√3(Apd(3,2)−iBpd(3,2)) | 0 | √335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3√7 | 0 | 35√37Apf(2,0)+4Apf(4,0)3√21 | 0 | √335(Apf(2,2)−iBpf(2,2))+2(Apf(4,2)−iBpf(4,2))3√7 | 0 |
Y(1)1 | 0 | −15√6(App(2,2)+iBpp(2,2)) | 0 | App(0,0)−15App(2,0) | 0 | −17√6(Apd(3,2)+iBpd(3,2)) | 0 | Apd(1,0)√5−3Apd(3,0)7√5 | 0 | −2(Apf(4,4)+iBpf(4,4))3√3 | 0 | 15√37(Apf(2,2)+iBpf(2,2))−13√57(Apf(4,2)+iBpf(4,2)) | 0 | 35√27Apf(2,0)−13√27Apf(4,0) | 0 | 3(Apf(2,2)−iBpf(2,2))√35−Apf(4,2)−iBpf(4,2)3√21 |
Y(2)−2 | Asd(2,2)−iBsd(2,2)√5 | 0 | 17√3(Apd(3,2)−iBpd(3,2)) | 0 | Add(0,0)−27Add(2,0)+121Add(4,0) | 0 | 17√53(Add(4,2)−iBdd(4,2))−27(Add(2,2)−iBdd(2,2)) | 0 | 13√107(Add(4,4)−iBdd(4,4)) | 0 | Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7 | 0 | 533(Adf(5,2)−iBdf(5,2))−2(Adf(3,2)−iBdf(3,2))3√7 | 0 | 111√10(Adf(5,4)−iBdf(5,4)) | 0 |
Y(2)−1 | 0 | Apd(1,0)√5−3Apd(3,0)7√5 | 0 | −17√6(Apd(3,2)−iBpd(3,2)) | 0 | Add(0,0)+17Add(2,0)−421Add(4,0) | 0 | −17√6(Add(2,2)−iBdd(2,2))−221√10(Add(4,2)−iBdd(4,2)) | 0 | 13√57(Adf(3,2)+iBdf(3,2))−133√5(Adf(5,2)+iBdf(5,2)) | 0 | 2√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0) | 0 | −Adf(3,2)−iBdf(3,2)√21−5(Adf(5,2)−iBdf(5,2))11√3 | 0 | −211√53(Adf(5,4)−iBdf(5,4)) |
Y(2)0 | Asd(2,0)√5 | 0 | 2Apd(1,0)√15+37√35Apd(3,0) | 0 | 17√53(Add(4,2)+iBdd(4,2))−27(Add(2,2)+iBdd(2,2)) | 0 | Add(0,0)+27Add(2,0)+27Add(4,0) | 0 | 17√53(Add(4,2)−iBdd(4,2))−27(Add(2,2)−iBdd(2,2)) | 0 | 111√5(Adf(5,2)+iBdf(5,2)) | 0 | 3Adf(1,0)√35+4Adf(3,0)3√35+1033√57Adf(5,0) | 0 | 111√5(Adf(5,2)−iBdf(5,2)) | 0 |
Y(2)1 | 0 | −17√6(Apd(3,2)+iBpd(3,2)) | 0 | Apd(1,0)√5−3Apd(3,0)7√5 | 0 | −17√6(Add(2,2)+iBdd(2,2))−221√10(Add(4,2)+iBdd(4,2)) | 0 | Add(0,0)+17Add(2,0)−421Add(4,0) | 0 | −211√53(Adf(5,4)+iBdf(5,4)) | 0 | −Adf(3,2)+iBdf(3,2)√21−5(Adf(5,2)+iBdf(5,2))11√3 | 0 | 2√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0) | 0 | 13√57(Adf(3,2)−iBdf(3,2))−133√5(Adf(5,2)−iBdf(5,2)) |
Y(2)2 | Asd(2,2)+iBsd(2,2)√5 | 0 | 17√3(Apd(3,2)+iBpd(3,2)) | 0 | 13√107(Add(4,4)+iBdd(4,4)) | 0 | 17√53(Add(4,2)+iBdd(4,2))−27(Add(2,2)+iBdd(2,2)) | 0 | Add(0,0)−27Add(2,0)+121Add(4,0) | 0 | 111√10(Adf(5,4)+iBdf(5,4)) | 0 | 533(Adf(5,2)+iBdf(5,2))−2(Adf(3,2)+iBdf(3,2))3√7 | 0 | Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7 | 0 |
Y(3)−3 | 0 | 3(Apf(2,2)−iBpf(2,2))√35−Apf(4,2)−iBpf(4,2)3√21 | 0 | −2(Apf(4,4)−iBpf(4,4))3√3 | 0 | 13√57(Adf(3,2)−iBdf(3,2))−133√5(Adf(5,2)−iBdf(5,2)) | 0 | −211√53(Adf(5,4)−iBdf(5,4)) | 0 | Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0) | 0 | −13√25(Aff(2,2)−iBff(2,2))+111√6(Aff(4,2)−iBff(4,2))−10429√7(Aff(6,2)−iBff(6,2)) | 0 | 111√143(Aff(4,4)−iBff(4,4))−5143√703(Aff(6,4)−iBff(6,4)) | 0 | −1013√733(Aff(6,6)−iBff(6,6)) |
Y(3)−2 | Asf(3,2)−iBsf(3,2)√7 | 0 | √335(Apf(2,2)−iBpf(2,2))+2(Apf(4,2)−iBpf(4,2))3√7 | 0 | Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7 | 0 | 111√5(Adf(5,2)−iBdf(5,2)) | 0 | 111√10(Adf(5,4)−iBdf(5,4)) | 0 | Aff(0,0)−733Aff(4,0)+10143Aff(6,0) | 0 | −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)) | 0 | 133√70(Aff(4,4)−iBff(4,4))+10143√14(Aff(6,4)−iBff(6,4)) | 0 |
Y(3)−1 | 0 | 35√27Apf(2,0)−13√27Apf(4,0) | 0 | 15√37(Apf(2,2)−iBpf(2,2))−13√57(Apf(4,2)−iBpf(4,2)) | 0 | 2√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0) | 0 | −Adf(3,2)−iBdf(3,2)√21−5(Adf(5,2)−iBdf(5,2))11√3 | 0 | −13√25(Aff(2,2)+iBff(2,2))+111√6(Aff(4,2)+iBff(4,2))−10429√7(Aff(6,2)+iBff(6,2)) | 0 | Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0) | 0 | −25√23(Aff(2,2)−iBff(2,2))−233√10(Aff(4,2)−iBff(4,2))−10143√353(Aff(6,2)−iBff(6,2)) | 0 | 111√143(Aff(4,4)−iBff(4,4))−5143√703(Aff(6,4)−iBff(6,4)) |
Y(3)0 | Asf(3,0)√7 | 0 | 35√37Apf(2,0)+4Apf(4,0)3√21 | 0 | 533(Adf(5,2)+iBdf(5,2))−2(Adf(3,2)+iBdf(3,2))3√7 | 0 | 3Adf(1,0)√35+4Adf(3,0)3√35+1033√57Adf(5,0) | 0 | 533(Adf(5,2)−iBdf(5,2))−2(Adf(3,2)−iBdf(3,2))3√7 | 0 | −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)) | 0 | Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0) | 0 | −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)) | 0 |
Y(3)1 | 0 | 15√37(Apf(2,2)+iBpf(2,2))−13√57(Apf(4,2)+iBpf(4,2)) | 0 | 35√27Apf(2,0)−13√27Apf(4,0) | 0 | −Adf(3,2)+iBdf(3,2)√21−5(Adf(5,2)+iBdf(5,2))11√3 | 0 | 2√235Adf(1,0)+13√235Adf(3,0)−533√107Adf(5,0) | 0 | 111√143(Aff(4,4)+iBff(4,4))−5143√703(Aff(6,4)+iBff(6,4)) | 0 | −25√23(Aff(2,2)+iBff(2,2))−233√10(Aff(4,2)+iBff(4,2))−10143√353(Aff(6,2)+iBff(6,2)) | 0 | Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0) | 0 | −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)2 | Asf(3,2)+iBsf(3,2)√7 | 0 | √335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3√7 | 0 | 111√10(Adf(5,4)+iBdf(5,4)) | 0 | 111√5(Adf(5,2)+iBdf(5,2)) | 0 | Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7 | 0 | 133√70(Aff(4,4)+iBff(4,4))+10143√14(Aff(6,4)+iBff(6,4)) | 0 | −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)) | 0 | Aff(0,0)−733Aff(4,0)+10143Aff(6,0) | 0 |
Y(3)3 | 0 | −2(Apf(4,4)+iBpf(4,4))3√3 | 0 | 3(Apf(2,2)+iBpf(2,2))√35−Apf(4,2)+iBpf(4,2)3√21 | 0 | −211√53(Adf(5,4)+iBdf(5,4)) | 0 | 13√57(Adf(3,2)+iBdf(3,2))−133√5(Adf(5,2)+iBdf(5,2)) | 0 | −1013√733(Aff(6,6)+iBff(6,6)) | 0 | 111√143(Aff(4,4)+iBff(4,4))−5143√703(Aff(6,4)+iBff(6,4)) | 0 | −13√25(Aff(2,2)+iBff(2,2))+111√6(Aff(4,2)+iBff(4,2))−10429√7(Aff(6,2)+iBff(6,2)) | 0 | Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0) |
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 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
s | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
px | 0 | 1√2 | 0 | −1√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
py | 0 | i√2 | 0 | i√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
pz | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
dx2−y2 | 0 | 0 | 0 | 0 | 1√2 | 0 | 0 | 0 | 1√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
d3z2−r2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
dyz | 0 | 0 | 0 | 0 | 0 | i√2 | 0 | i√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
dxz | 0 | 0 | 0 | 0 | 0 | 1√2 | 0 | −1√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
dxy | 0 | 0 | 0 | 0 | i√2 | 0 | 0 | 0 | −i√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
fxyz | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i√2 | 0 | 0 | 0 | −i√2 | 0 |
fx(5x2−r2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √54 | 0 | −√34 | 0 | √34 | 0 | −√54 |
fy(5y2−r2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −i√54 | 0 | −i√34 | 0 | −i√34 | 0 | −i√54 |
fz(5z2−r2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
fx(y2−z2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −√34 | 0 | −√54 | 0 | √54 | 0 | √34 |
fy(z2−x2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −i√34 | 0 | i√54 | 0 | i√54 | 0 | −i√34 |
fz(x2−y2) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1√2 | 0 | 0 | 0 | 1√2 | 0 |
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) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
s | Ass(0,0) | 0 | 0 | Asp(1,0)√3 | √25Asd(2,2) | Asd(2,0)√5 | 0 | 0 | −√25Bsd(2,2) | −√27Bsf(3,2) | 0 | 0 | Asf(3,0)√7 | 0 | 0 | √27Asf(3,2) |
px | 0 | App(0,0)−15App(2,0)+15√6App(2,2) | −15√6Bpp(2,2) | 0 | 0 | 0 | −17√6Bpd(3,2) | Apd(1,0)√5−3Apd(3,0)7√5+17√6Apd(3,2) | 0 | 0 | −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) | 0 | −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 | 0 |
py | 0 | −15√6Bpp(2,2) | App(0,0)−15App(2,0)−15√6App(2,2) | 0 | 0 | 0 | Apd(1,0)√5−3Apd(3,0)7√5−17√6Apd(3,2) | −17√6Bpd(3,2) | 0 | 0 | 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) | 0 | −√635Bpf(2,2)+Bpf(4,2)√14+Bpf(4,4)3√2 | 3Apf(2,0)2√35−√370Apf(2,2)−16√57Apf(4,0)−13√27Apf(4,2)+Apf(4,4)3√2 | 0 |
pz | Asp(1,0)√3 | 0 | 0 | App(0,0)+25App(2,0) | 17√6Apd(3,2) | 2Apd(1,0)√15+37√35Apd(3,0) | 0 | 0 | −17√6Bpd(3,2) | −√635Bpf(2,2)−23√27Bpf(4,2) | 0 | 0 | 35√37Apf(2,0)+4Apf(4,0)3√21 | 0 | 0 | √635Apf(2,2)+23√27Apf(4,2) |
dx2−y2 | √25Asd(2,2) | 0 | 0 | 17√6Apd(3,2) | Add(0,0)−27Add(2,0)+121Add(4,0)+13√107Add(4,4) | 17√103Add(4,2)−27√2Add(2,2) | 0 | 0 | −13√107Bdd(4,4) | −111√10Bdf(5,4) | 0 | 0 | 533√2Adf(5,2)−23√27Adf(3,2) | 0 | 0 | Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7+111√10Adf(5,4) |
d3z2−r2 | Asd(2,0)√5 | 0 | 0 | 2Apd(1,0)√15+37√35Apd(3,0) | 17√103Add(4,2)−27√2Add(2,2) | Add(0,0)+27Add(2,0)+27Add(4,0) | 0 | 0 | 27√2Bdd(2,2)−17√103Bdd(4,2) | −111√10Bdf(5,2) | 0 | 0 | 3Adf(1,0)√35+4Adf(3,0)3√35+1033√57Adf(5,0) | 0 | 0 | 111√10Adf(5,2) |
dyz | 0 | −17√6Bpd(3,2) | Apd(1,0)√5−3Apd(3,0)7√5−17√6Apd(3,2) | 0 | 0 | 0 | Add(0,0)+17Add(2,0)−17√6Add(2,2)−421Add(4,0)−221√10Add(4,2) | −17√6Bdd(2,2)−221√10Bdd(4,2) | 0 | 0 | 23√27Bdf(3,2)+5Bdf(5,2)33√2−5Bdf(5,4)11√6 | −√335Adf(1,0)−Adf(3,0)2√105−Adf(3,2)3√14+522√521Adf(5,0)+533√2Adf(5,2)+5Adf(5,4)11√6 | 0 | 111√152Bdf(5,2)+111√52Bdf(5,4) | Adf(1,0)√7+Adf(3,0)6√7−√542Adf(3,2)−25Adf(5,0)66√7−111√103Adf(5,2)+111√52Adf(5,4) | 0 |
dxz | 0 | Apd(1,0)√5−3Apd(3,0)7√5+17√6Apd(3,2) | −17√6Bpd(3,2) | 0 | 0 | 0 | −17√6Bdd(2,2)−221√10Bdd(4,2) | Add(0,0)+17Add(2,0)+17√6Add(2,2)−421Add(4,0)+221√10Add(4,2) | 0 | 0 | −√335Adf(1,0)−Adf(3,0)2√105+Adf(3,2)3√14+522√521Adf(5,0)−533√2Adf(5,2)+5Adf(5,4)11√6 | 23√27Bdf(3,2)+5Bdf(5,2)33√2+5Bdf(5,4)11√6 | 0 | −Adf(1,0)√7−Adf(3,0)6√7−√542Adf(3,2)+25Adf(5,0)66√7−111√103Adf(5,2)−111√52Adf(5,4) | 111√52Bdf(5,4)−111√152Bdf(5,2) | 0 |
dxy | −√25Bsd(2,2) | 0 | 0 | −17√6Bpd(3,2) | −13√107Bdd(4,4) | 27√2Bdd(2,2)−17√103Bdd(4,2) | 0 | 0 | Add(0,0)−27Add(2,0)+121Add(4,0)−13√107Add(4,4) | Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7−111√10Adf(5,4) | 0 | 0 | 23√27Bdf(3,2)−533√2Bdf(5,2) | 0 | 0 | −111√10Bdf(5,4) |
fxyz | −√27Bsf(3,2) | 0 | 0 | −√635Bpf(2,2)−23√27Bpf(4,2) | −111√10Bdf(5,4) | −111√10Bdf(5,2) | 0 | 0 | Adf(1,0)√7−2Adf(3,0)3√7+5Adf(5,0)33√7−111√10Adf(5,4) | Aff(0,0)−733Aff(4,0)−133√70Aff(4,4)+10143Aff(6,0)−10143√14Aff(6,4) | 0 | 0 | 23√25Bff(2,2)+111√23Bff(4,2)−40429√7Bff(6,2) | 0 | 0 | −133√70Bff(4,4)−10143√14Bff(6,4) |
fx(5x2−r2) | 0 | −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) | 0 | 0 | 0 | 23√27Bdf(3,2)+5Bdf(5,2)33√2−5Bdf(5,4)11√6 | −√335Adf(1,0)−Adf(3,0)2√105+Adf(3,2)3√14+522√521Adf(5,0)−533√2Adf(5,2)+5Adf(5,4)11√6 | 0 | 0 | Aff(0,0)−215Aff(2,0)+25√23Aff(2,2)+344Aff(4,0)−111√52Aff(4,2)+122√352Aff(4,4)−125Aff(6,0)1716+25572√353Aff(6,2)−25286√72Aff(6,4)+2552√733Aff(6,6) | Bff(2,2)5√6−111√10Bff(4,2)−5572√353Bff(6,2)+2552√733Bff(6,6) | 0 | \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) | \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) | 0 |
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) | 0 | \color{darkred}{ 0 } | \color{darkred}{ 0 } | \color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} } | \color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}+\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} } | \color{darkred}{ 0 } | 0 | \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) | 0 | -\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) | 0 |
f_{z\left(5z^2-r^2\right)} | \color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} } | 0 | 0 | \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} | \color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) } | \color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) } | \color{darkred}{ 0 } | \color{darkred}{ 0 } | \color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) } | \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) | 0 | 0 | \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) | 0 | 0 | -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) |
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}} | 0 | \color{darkred}{ 0 } | \color{darkred}{ 0 } | \color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) } | \color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) } | \color{darkred}{ 0 } | 0 | \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) | -\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) | 0 | \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) | \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) | 0 |
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}} | 0 | \color{darkred}{ 0 } | \color{darkred}{ 0 } | \color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) } | \color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) } | \color{darkred}{ 0 } | 0 | \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) | 0 | \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) | 0 |
f_{z\left(x^2-y^2\right)} | \color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) } | 0 | 0 | \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) | \color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) } | \color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) } | \color{darkred}{ 0 } | \color{darkred}{ 0 } | \color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) } | -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) | 0 | 0 | -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) | 0 | 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) |
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'.
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Click on one of the subsections to expand it or
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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 |