+Table of Contents
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Orientation Z
Symmetry Operations
In the Cs Point Group, with orientation Z there are the following symmetry operations
Operator | Orientation |
---|---|
E | {0,0,0} , |
σh | {0,0,1} , |
Different Settings
Character Table
E(1) | σh(1) | |
---|---|---|
A' | 1 | 1 |
A'' | 1 | −1 |
Product Table
A' | A'' | |
---|---|---|
A' | A' | A'' |
A'' | A'' | A' |
Sub Groups with compatible settings
Super Groups with compatible settings
Invariant Potential expanded on renormalized spherical Harmonics
Any potential (function) can be written in spherical coordinates as a sum over spherical harmonics V(→r)=∞∑k=0k∑m=−kAk,m(r)C(m)k(θ,ϕ). With C(m)k(θ,ϕ) a renormalised spherical harmonics C(m)k(θ,ϕ)=√4π2k+1Y(m)k(θ,ϕ). The allowed expansion coefficients Ak,m(r), or once evaluated for a given radial wave-function Ak,m=⟨ψ(r)|Ak,m(r)|ψ(r)⟩, such that $V(\vec{r}) is invariant under all symmetry operations of the Cs Point group with orientation Z are:
Input format suitable for Mathematica (Quanty.nb)
Ak,m={A(0,0)k=0∧m=0−A(1,1)+iAp(1,1)k=1∧m=−1A(1,1)+iAp(1,1)k=1∧m=1A(2,2)−iAp(2,2)k=2∧m=−2A(2,0)k=2∧m=0A(2,2)+iAp(2,2)k=2∧m=2−A(3,3)+iAp(3,3)k=3∧m=−3−A(3,1)+iAp(3,1)k=3∧m=−1A(3,1)+iAp(3,1)k=3∧m=1A(3,3)+iAp(3,3)k=3∧m=3A(4,4)−iAp(4,4)k=4∧m=−4A(4,2)−iAp(4,2)k=4∧m=−2A(4,0)k=4∧m=0A(4,2)+iAp(4,2)k=4∧m=2A(4,4)+iAp(4,4)k=4∧m=4−A(5,5)+iAp(5,5)k=5∧m=−5−A(5,3)+iAp(5,3)k=5∧m=−3−A(5,1)+iAp(5,1)k=5∧m=−1A(5,1)+iAp(5,1)k=5∧m=1A(5,3)+iAp(5,3)k=5∧m=3A(5,5)+iAp(5,5)k=5∧m=5A(6,6)−iAp(6,6)k=6∧m=−6A(6,4)−iAp(6,4)k=6∧m=−4A(6,2)−iAp(6,2)k=6∧m=−2A(6,0)k=6∧m=0A(6,2)+iAp(6,2)k=6∧m=2A(6,4)+iAp(6,4)k=6∧m=4A(6,6)+iAp(6,6)k=6∧m=6
Input format suitable for Quanty
- Akm_Cs_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1,-1, (-1)*(A(1,1)) + ((+1*I))*(Ap(1,1))} , {1, 1, A(1,1) + ((+1*I))*(Ap(1,1))} , {2, 0, A(2,0)} , {2,-2, A(2,2) + ((+-1*I))*(Ap(2,2))} , {2, 2, A(2,2) + ((+1*I))*(Ap(2,2))} , {3,-1, (-1)*(A(3,1)) + ((+1*I))*(Ap(3,1))} , {3, 1, A(3,1) + ((+1*I))*(Ap(3,1))} , {3,-3, (-1)*(A(3,3)) + ((+1*I))*(Ap(3,3))} , {3, 3, A(3,3) + ((+1*I))*(Ap(3,3))} , {4, 0, A(4,0)} , {4,-2, A(4,2) + ((+-1*I))*(Ap(4,2))} , {4, 2, A(4,2) + ((+1*I))*(Ap(4,2))} , {4,-4, A(4,4) + ((+-1*I))*(Ap(4,4))} , {4, 4, A(4,4) + ((+1*I))*(Ap(4,4))} , {5,-1, (-1)*(A(5,1)) + ((+1*I))*(Ap(5,1))} , {5, 1, A(5,1) + ((+1*I))*(Ap(5,1))} , {5,-3, (-1)*(A(5,3)) + ((+1*I))*(Ap(5,3))} , {5, 3, A(5,3) + ((+1*I))*(Ap(5,3))} , {5,-5, (-1)*(A(5,5)) + ((+1*I))*(Ap(5,5))} , {5, 5, A(5,5) + ((+1*I))*(Ap(5,5))} , {6, 0, A(6,0)} , {6,-2, A(6,2) + ((+-1*I))*(Ap(6,2))} , {6, 2, A(6,2) + ((+1*I))*(Ap(6,2))} , {6,-4, A(6,4) + ((+-1*I))*(Ap(6,4))} , {6, 4, A(6,4) + ((+1*I))*(Ap(6,4))} , {6,-6, A(6,6) + ((+-1*I))*(Ap(6,6))} , {6, 6, A(6,6) + ((+1*I))*(Ap(6,6))} }
One particle coupling on a basis of spherical harmonics
Subsuperscript[Y,0,(0)] | Subsuperscript[Y,−1,(1)] | Subsuperscript[Y,0,(1)] | Subsuperscript[Y,1,(1)] | Subsuperscript[Y,−2,(2)] | Subsuperscript[Y,−1,(2)] | Subsuperscript[Y,0,(2)] | Subsuperscript[Y,1,(2)] | Subsuperscript[Y,2,(2)] | Subsuperscript[Y,−3,(3)] | Subsuperscript[Y,−2,(3)] | Subsuperscript[Y,−1,(3)] | Subsuperscript[Y,0,(3)] | Subsuperscript[Y,1,(3)] | Subsuperscript[Y,2,(3)] | Subsuperscript[Y,3,(3)] | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Subsuperscript[Y,0,(0)] | Ass(0,0) | −Asp(1,1)+iBsp(1,1)√3 | 0 | −−Asp(1,1)+iBsp(1,1)√3 | Asd(2,2)+iBsd(2,2)√5 | 0 | Asd(2,0)√5 | 0 | Asd(2,2)−iBsd(2,2)√5 | −Asf(3,3)+iBsf(3,3)√7 | 0 | −Asf(3,1)+iBsf(3,1)√7 | 0 | −−Asf(3,1)+iBsf(3,1)√7 | 0 | −−Asf(3,3)+iBsf(3,3)√7 |
Subsuperscript[Y,−1,(1)] | −Asp(1,1)+iBsp(1,1)√3 | App(0,0)−15App(2,0) | 0 | −15√6(App(2,2)−iBpp(2,2)) | 17√35(Apd(3,1)+iBpd(3,1))−√25(Apd(1,1)+iBpd(1,1)) | 0 | 37√25(−Apd(3,1)+iBpd(3,1))−−Apd(1,1)+iBpd(1,1)√15 | 0 | 37(−Apd(3,3)+iBpd(3,3)) | 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 |
Subsuperscript[Y,0,(1)] | 0 | 0 | App(0,0)+25App(2,0) | 0 | 0 | −Apd(1,1)+iBpd(1,1)√5−27√65(Apd(3,1)+iBpd(3,1)) | 0 | −−Apd(1,1)+iBpd(1,1)√5−27√65(−Apd(3,1)+iBpd(3,1)) | 0 | 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 |
Subsuperscript[Y,1,(1)] | Asp(1,1)+iBsp(1,1)√3 | −15√6(App(2,2)+iBpp(2,2)) | 0 | App(0,0)−15App(2,0) | 37(Apd(3,3)+iBpd(3,3)) | 0 | 37√25(Apd(3,1)+iBpd(3,1))−Apd(1,1)+iBpd(1,1)√15 | 0 | 17√35(−Apd(3,1)+iBpd(3,1))−√25(−Apd(1,1)+iBpd(1,1)) | −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 |
Subsuperscript[Y,−2,(2)] | Asd(2,2)−iBsd(2,2)√5 | √25(−Apd(1,1)+iBpd(1,1))−17√35(−Apd(3,1)+iBpd(3,1)) | 0 | −37(−Apd(3,3)+iBpd(3,3)) | 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)) | −√37(Adf(1,1)+iBdf(1,1))+13√27(Adf(3,1)+iBdf(3,1))−133√57(Adf(5,1)+iBdf(5,1)) | 0 | −−Adf(1,1)+iBdf(1,1)√35+2√2105(−Adf(3,1)+iBdf(3,1))−5(−Adf(5,1)+iBdf(5,1))11√21 | 0 | 13√27(−Adf(3,3)+iBdf(3,3))−533√2(−Adf(5,3)+iBdf(5,3)) | 0 | −511√23(−Adf(5,5)+iBdf(5,5)) |
Subsuperscript[Y,−1,(2)] | 0 | 0 | −Apd(1,1)+iBpd(1,1)√5+27√65(−Apd(3,1)+iBpd(3,1)) | 0 | 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 | 0 | −√27(Adf(1,1)+iBdf(1,1))−Adf(3,1)+iBdf(3,1)√21+211√1021(Adf(5,1)+iBdf(5,1)) | 0 | −√335(−Adf(1,1)+iBdf(1,1))+13√235(−Adf(3,1)+iBdf(3,1))+20(−Adf(5,1)+iBdf(5,1))33√7 | 0 | 13√57(−Adf(3,3)+iBdf(3,3))+433√5(−Adf(5,3)+iBdf(5,3)) | 0 |
Subsuperscript[Y,0,(2)] | Asd(2,0)√5 | Apd(1,1)+iBpd(1,1)√15−37√25(Apd(3,1)+iBpd(3,1)) | 0 | −Apd(1,1)+iBpd(1,1)√15−37√25(−Apd(3,1)+iBpd(3,1)) | 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)) | 13√57(Adf(3,3)+iBdf(3,3))−233√5(Adf(5,3)+iBdf(5,3)) | 0 | −√635(Adf(1,1)+iBdf(1,1))−Adf(3,1)+iBdf(3,1)√35−511√27(Adf(5,1)+iBdf(5,1)) | 0 | −√635(−Adf(1,1)+iBdf(1,1))−−Adf(3,1)+iBdf(3,1)√35−511√27(−Adf(5,1)+iBdf(5,1)) | 0 | 13√57(−Adf(3,3)+iBdf(3,3))−233√5(−Adf(5,3)+iBdf(5,3)) |
Subsuperscript[Y,1,(2)] | 0 | 0 | Apd(1,1)+iBpd(1,1)√5+27√65(Apd(3,1)+iBpd(3,1)) | 0 | 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 | 0 | 13√57(Adf(3,3)+iBdf(3,3))+433√5(Adf(5,3)+iBdf(5,3)) | 0 | −√335(Adf(1,1)+iBdf(1,1))+13√235(Adf(3,1)+iBdf(3,1))+20(Adf(5,1)+iBdf(5,1))33√7 | 0 | −√27(−Adf(1,1)+iBdf(1,1))−−Adf(3,1)+iBdf(3,1)√21+211√1021(−Adf(5,1)+iBdf(5,1)) | 0 |
Subsuperscript[Y,2,(2)] | Asd(2,2)+iBsd(2,2)√5 | −37(Apd(3,3)+iBpd(3,3)) | 0 | √25(Apd(1,1)+iBpd(1,1))−17√35(Apd(3,1)+iBpd(3,1)) | 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) | −511√23(Adf(5,5)+iBdf(5,5)) | 0 | 13√27(Adf(3,3)+iBdf(3,3))−533√2(Adf(5,3)+iBdf(5,3)) | 0 | −Adf(1,1)+iBdf(1,1)√35+2√2105(Adf(3,1)+iBdf(3,1))−5(Adf(5,1)+iBdf(5,1))11√21 | 0 | −√37(−Adf(1,1)+iBdf(1,1))+13√27(−Adf(3,1)+iBdf(3,1))−133√57(−Adf(5,1)+iBdf(5,1)) |
Subsuperscript[Y,−3,(3)] | −Asf(3,3)+iBsf(3,3)√7 | 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 | √37(−Adf(1,1)+iBdf(1,1))−13√27(−Adf(3,1)+iBdf(3,1))+133√57(−Adf(5,1)+iBdf(5,1)) | 0 | 233√5(−Adf(5,3)+iBdf(5,3))−13√57(−Adf(3,3)+iBdf(3,3)) | 0 | 511√23(−Adf(5,5)+iBdf(5,5)) | 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)) |
Subsuperscript[Y,−2,(3)] | 0 | 0 | √335(Apf(2,2)−iBpf(2,2))+2(Apf(4,2)−iBpf(4,2))3√7 | 0 | 0 | √27(−Adf(1,1)+iBdf(1,1))+−Adf(3,1)+iBdf(3,1)√21−211√1021(−Adf(5,1)+iBdf(5,1)) | 0 | −13√57(−Adf(3,3)+iBdf(3,3))−433√5(−Adf(5,3)+iBdf(5,3)) | 0 | 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 |
Subsuperscript[Y,−1,(3)] | −Asf(3,1)+iBsf(3,1)√7 | 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)) | Adf(1,1)+iBdf(1,1)√35−2√2105(Adf(3,1)+iBdf(3,1))+5(Adf(5,1)+iBdf(5,1))11√21 | 0 | √635(−Adf(1,1)+iBdf(1,1))+−Adf(3,1)+iBdf(3,1)√35+511√27(−Adf(5,1)+iBdf(5,1)) | 0 | 533√2(−Adf(5,3)+iBdf(5,3))−13√27(−Adf(3,3)+iBdf(3,3)) | −13√25(Aff(2,2)+iBff(2,2))+111√6(Aff(4,2)+iBff(4,2))−10429√7(Aff(6,2)+iBff(6,2)) | 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)) |
Subsuperscript[Y,0,(3)] | 0 | 0 | 35√37Apf(2,0)+4Apf(4,0)3√21 | 0 | 0 | √335(Adf(1,1)+iBdf(1,1))−13√235(Adf(3,1)+iBdf(3,1))−20(Adf(5,1)+iBdf(5,1))33√7 | 0 | √335(−Adf(1,1)+iBdf(1,1))−13√235(−Adf(3,1)+iBdf(3,1))−20(−Adf(5,1)+iBdf(5,1))33√7 | 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 | 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 |
Subsuperscript[Y,1,(3)] | Asf(3,1)+iBsf(3,1)√7 | 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) | 533√2(Adf(5,3)+iBdf(5,3))−13√27(Adf(3,3)+iBdf(3,3)) | 0 | √635(Adf(1,1)+iBdf(1,1))+Adf(3,1)+iBdf(3,1)√35+511√27(Adf(5,1)+iBdf(5,1)) | 0 | −Adf(1,1)+iBdf(1,1)√35−2√2105(−Adf(3,1)+iBdf(3,1))+5(−Adf(5,1)+iBdf(5,1))11√21 | 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)) |
Subsuperscript[Y,2,(3)] | 0 | 0 | √335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3√7 | 0 | 0 | −13√57(Adf(3,3)+iBdf(3,3))−433√5(Adf(5,3)+iBdf(5,3)) | 0 | √27(Adf(1,1)+iBdf(1,1))+Adf(3,1)+iBdf(3,1)√21−211√1021(Adf(5,1)+iBdf(5,1)) | 0 | 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 |
Subsuperscript[Y,3,(3)] | Asf(3,3)+iBsf(3,3)√7 | −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 | 511√23(Adf(5,5)+iBdf(5,5)) | 0 | 233√5(Adf(5,3)+iBdf(5,3))−13√57(Adf(3,3)+iBdf(3,3)) | 0 | √37(Adf(1,1)+iBdf(1,1))−13√27(Adf(3,1)+iBdf(3,1))+133√57(Adf(5,1)+iBdf(5,1)) | −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) |
Rotation matrix to symmetry adapted functions (choice is not unique)
One particle coupling on a basis of symmetry adapted functions
Potential for s orbitals
Potential for p orbitals
Potential for d orbitals
Potential for f orbitals
Potential for s-p orbital mixing
Potential for s-d orbital mixing
Potential for s-f orbital mixing
Potential for p-d orbital mixing
Potential for p-f orbital mixing
Potential for d-f orbital mixing
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∞v | D∞h |