| Both sides previous revisionPrevious revisionNext revision | Previous revision |
| physics_chemistry:point_groups:cs:orientation_z [2018/03/29 20:47] – Maurits W. Haverkort | physics_chemistry:point_groups:cs:orientation_z [2018/04/06 09:16] (current) – Maurits W. Haverkort |
|---|
| | ~~CLOSETOC~~ |
| | |
| ====== Orientation Z ====== | ====== Orientation Z ====== |
| |
| * [[physics_chemistry:point_groups:d3h:orientation_zy|Point Group D3h with orientation Zy]] | * [[physics_chemistry:point_groups:d3h:orientation_zy|Point Group D3h with orientation Zy]] |
| * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]] | * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:d5h:orientation_zx|Point Group D5h with orientation Zx]] |
| | * [[physics_chemistry:point_groups:d5h:orientation_zy|Point Group D5h with orientation Zy]] |
| * [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]] | * [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]] |
| * [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]] | * [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]] |
| ### | ### |
| |
| ==== Input format suitable for Mathematica (Quanty.nb) ==== | ==== Expansion ==== |
| |
| ### | ### |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | A(0,0) & k=0\land m=0 \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| A(2,0) & k=2\land m=0 \\ | A(2,0) & k=2\land m=0 \\ |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| A(4,0) & k=4\land m=0 \\ | A(4,0) & k=4\land m=0 \\ |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| A(6,0) & k=6\land m=0 \\ | A(6,0) & k=6\land m=0 \\ |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | A(6,6)+i B(6,6) & k=6\land m=6 |
| \end{cases}$$ | \end{cases}$$ |
| | |
| | ### |
| | |
| | ==== Input format suitable for Mathematica (Quanty.nb) ==== |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {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, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {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, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {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] |
| | |
| | </code> |
| |
| ### | ### |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{0, 0, A(0,0)} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| {2, 0, A(2,0)} , | {2, 0, A(2,0)} , |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | {3, 3, A(3,3) + (I)*(B(3,3))} , |
| {4, 0, A(4,0)} , | {4, 0, A(4,0)} , |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | {4,-2, A(4,2) + (-I)*(B(4,2))} , |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} , |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | {5, 5, A(5,5) + (I)*(B(5,5))} , |
| {6, 0, A(6,0)} , | {6, 0, A(6,0)} , |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | {6,-2, A(6,2) + (-I)*(B(6,2))} , |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| |
| </code> | </code> |
| ^fx(5x2−r2)|0|0|0|0|0|0|0|0|0|√54|0|−√34|0|√34|0|−√54| | ^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| | ^fy(5y2−r2)|0|0|0|0|0|0|0|0|0|−i√54|0|−i√34|0|−i√34|0|−i√54| |
| ^$ f_{x\left(5z^2-r^2\right)} |\color{darkred}{ 0 }| 0 | 0 | 0 |\color{darkred}{ 0 }|\color{darkred}{ 0 }|\color{darkred}{ 0 }|\color{darkred}{ 0 }|\color{darkred}{ 0 }| 0 | 0 | 0 | 1 | 0 | 0 | 0 $| | ^$ f_{z\left(5z^2-r^2\right)} |\color{darkred}{ 0 }| 0 | 0 | 0 |\color{darkred}{ 0 }|\color{darkred}{ 0 }|\color{darkred}{ 0 }|\color{darkred}{ 0 }|\color{darkred}{ 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| | ^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| | ^fy(z2−x2)|0|0|0|0|0|0|0|0|0|−i√34|0|i√54|0|i√54|0|−i√34| |
| ### | ### |
| |
| | ^ s ^ px ^ py ^ pz ^ dx2−y2 ^ d3z2−r2 ^ dyz ^ dxz ^ dxy ^ fxyz ^ fx(5x2−r2) ^ fy(5y2−r2) ^ $ f_{x\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)} $ ^ | | ^ s ^ px ^ py ^ pz ^ dx2−y2 ^ d3z2−r2 ^ dyz ^ dxz ^ dxy ^ fxyz ^ fx(5x2−r2) ^ fy(5y2−r2) ^ $ 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)} $ ^ |
| ^s|Ass(0,0)|−√23Asp(1,1)|√23Bsp(1,1)|0|√25Asd(2,2)|Asd(2,0)√5|0|0|−√25Bsd(2,2)|0|12√37Asf(3,1)−12√57Asf(3,3)|−12√37Bsf(3,1)−12√57Bsf(3,3)|0|12√57Asf(3,1)+12√37Asf(3,3)|12√57Bsf(3,1)−12√37Bsf(3,3)|0| | ^s|Ass(0,0)|−√23Asp(1,1)|√23Bsp(1,1)|0|√25Asd(2,2)|Asd(2,0)√5|0|0|−√25Bsd(2,2)|0|12√37Asf(3,1)−12√57Asf(3,3)|−12√37Bsf(3,1)−12√57Bsf(3,3)|0|12√57Asf(3,1)+12√37Asf(3,3)|12√57Bsf(3,1)−12√37Bsf(3,3)|0| |
| ^px|−√23Asp(1,1)|App(0,0)−15App(2,0)+15√6App(2,2)|−15√6Bpp(2,2)|0|−√25Apd(1,1)+17√35Apd(3,1)−37Apd(3,3)|√215Apd(1,1)−6Apd(3,1)7√5|0|0|√25Bpd(1,1)−17√35Bpd(3,1)+37Bpd(3,3)|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| | ^px|−√23Asp(1,1)|App(0,0)−15App(2,0)+15√6App(2,2)|−15√6Bpp(2,2)|0|−√25Apd(1,1)+17√35Apd(3,1)−37Apd(3,3)|√215Apd(1,1)−6Apd(3,1)7√5|0|0|√25Bpd(1,1)−17√35Bpd(3,1)+37Bpd(3,3)|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| |
| ^fx(5x2−r2)|12√37Asf(3,1)−12√57Asf(3,3)|−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)|0|−3√370Adf(1,1)+11Adf(3,1)6√35−Adf(3,3)2√21−533√27Adf(5,1)+5Adf(5,3)22√3−522√53Adf(5,5)|3Adf(1,1)√70+12√335Adf(3,1)+5Adf(3,3)6√7+511√314Adf(5,1)−533Adf(5,3)|0|0|−√635Bdf(1,1)−Bdf(3,1)6√35+Bdf(3,3)2√21+5Bdf(5,1)33√14−5Bdf(5,3)22√3+522√53Bdf(5,5)|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|Aff(2,0)√15+13√25Aff(2,2)−144√53Aff(4,0)+Aff(4,2)11√6+122√76Aff(4,4)−35572√53Aff(6,0)+85√7Aff(6,2)1716−5286√356Aff(6,4)−552√3511Aff(6,6)|Bff(2,2)3√10+211√23Bff(4,2)+111√143Bff(4,4)+5132√7Bff(6,2)−5143√703Bff(6,4)+552√3511Bff(6,6)|0| | ^fx(5x2−r2)|12√37Asf(3,1)−12√57Asf(3,3)|−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)|0|−3√370Adf(1,1)+11Adf(3,1)6√35−Adf(3,3)2√21−533√27Adf(5,1)+5Adf(5,3)22√3−522√53Adf(5,5)|3Adf(1,1)√70+12√335Adf(3,1)+5Adf(3,3)6√7+511√314Adf(5,1)−533Adf(5,3)|0|0|−√635Bdf(1,1)−Bdf(3,1)6√35+Bdf(3,3)2√21+5Bdf(5,1)33√14−5Bdf(5,3)22√3+522√53Bdf(5,5)|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|Aff(2,0)√15+13√25Aff(2,2)−144√53Aff(4,0)+Aff(4,2)11√6+122√76Aff(4,4)−35572√53Aff(6,0)+85√7Aff(6,2)1716−5286√356Aff(6,4)−552√3511Aff(6,6)|Bff(2,2)3√10+211√23Bff(4,2)+111√143Bff(4,4)+5132√7Bff(6,2)−5143√703Bff(6,4)+552√3511Bff(6,6)|0| |
| ^fy(5y2−r2)|−12√37Bsf(3,1)−12√57Bsf(3,3)|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)|0|−3√370Bdf(1,1)+11Bdf(3,1)6√35+Bdf(3,3)2√21−533√27Bdf(5,1)−5Bdf(5,3)22√3−522√53Bdf(5,5)|−3Bdf(1,1)√70−12√335Bdf(3,1)+5Bdf(3,3)6√7−511√314Bdf(5,1)−533Bdf(5,3)|0|0|√635Adf(1,1)+Adf(3,1)6√35+Adf(3,3)2√21−5Adf(5,1)33√14−5Adf(5,3)22√3−522√53Adf(5,5)|0|Bff(2,2)5√6−111√10Bff(4,2)−5572√353Bff(6,2)+2552√733Bff(6,6)|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)|0|−Bff(2,2)3√10−211√23Bff(4,2)+111√143Bff(4,4)−5132√7Bff(6,2)−5143√703Bff(6,4)−552√3511Bff(6,6)|−Aff(2,0)√15+13√25Aff(2,2)+144√53Aff(4,0)+Aff(4,2)11√6−122√76Aff(4,4)+35572√53Aff(6,0)+85√7Aff(6,2)1716+5286√356Aff(6,4)−552√3511Aff(6,6)|0| | ^fy(5y2−r2)|−12√37Bsf(3,1)−12√57Bsf(3,3)|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)|0|−3√370Bdf(1,1)+11Bdf(3,1)6√35+Bdf(3,3)2√21−533√27Bdf(5,1)−5Bdf(5,3)22√3−522√53Bdf(5,5)|−3Bdf(1,1)√70−12√335Bdf(3,1)+5Bdf(3,3)6√7−511√314Bdf(5,1)−533Bdf(5,3)|0|0|√635Adf(1,1)+Adf(3,1)6√35+Adf(3,3)2√21−5Adf(5,1)33√14−5Adf(5,3)22√3−522√53Adf(5,5)|0|Bff(2,2)5√6−111√10Bff(4,2)−5572√353Bff(6,2)+2552√733Bff(6,6)|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)|0|−Bff(2,2)3√10−211√23Bff(4,2)+111√143Bff(4,4)−5132√7Bff(6,2)−5143√703Bff(6,4)−552√3511Bff(6,6)|−Aff(2,0)√15+13√25Aff(2,2)+144√53Aff(4,0)+Aff(4,2)11√6−122√76Aff(4,4)+35572√53Aff(6,0)+85√7Aff(6,2)1716+5286√356Aff(6,4)−552√3511Aff(6,6)|0| |
| ^$ f_{x\left(5z^2-r^2\right)} |\color{darkred}{ 0 }| 0 | 0 | \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}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }|\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }|\color{darkred}{ 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) | 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_{z\left(5z^2-r^2\right)} |\color{darkred}{ 0 }| 0 | 0 | \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}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }|\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }|\color{darkred}{ 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) | 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) $| |
| ^fx(y2−z2)|12√57Asf(3,1)+12√37Asf(3,3)|−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|Adf(1,1)√14+Adf(3,1)2√21−16√57Adf(3,3)−111√1021Adf(5,1)+566√5Adf(5,3)+522Adf(5,5)|√314Adf(1,1)+Adf(3,1)2√7−12√521Adf(3,3)+511√514Adf(5,1)+111√53Adf(5,3)|0|0|√27Bdf(1,1)−12√37Bdf(3,1)+16√57Bdf(3,3)+111√1514Bdf(5,1)−566√5Bdf(5,3)−522Bdf(5,5)|0|Aff(2,0)√15+13√25Aff(2,2)−144√53Aff(4,0)+Aff(4,2)11√6+122√76Aff(4,4)−35572√53Aff(6,0)+85√7Aff(6,2)1716−5286√356Aff(6,4)−552√3511Aff(6,6)|−Bff(2,2)3√10−211√23Bff(4,2)+111√143Bff(4,4)−5132√7Bff(6,2)−5143√703Bff(6,4)−552√3511Bff(6,6)|0|Aff(0,0)+7132Aff(4,0)+733√52Aff(4,2)−122√352Aff(4,4)−544Aff(6,0)+5572√105Aff(6,2)+25286√72Aff(6,4)+552√2111Aff(6,6)|Bff(2,2)√6−133√10Bff(4,2)+35572√353Bff(6,2)−552√2111Bff(6,6)|0| | ^fx(y2−z2)|12√57Asf(3,1)+12√37Asf(3,3)|−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|Adf(1,1)√14+Adf(3,1)2√21−16√57Adf(3,3)−111√1021Adf(5,1)+566√5Adf(5,3)+522Adf(5,5)|√314Adf(1,1)+Adf(3,1)2√7−12√521Adf(3,3)+511√514Adf(5,1)+111√53Adf(5,3)|0|0|√27Bdf(1,1)−12√37Bdf(3,1)+16√57Bdf(3,3)+111√1514Bdf(5,1)−566√5Bdf(5,3)−522Bdf(5,5)|0|Aff(2,0)√15+13√25Aff(2,2)−144√53Aff(4,0)+Aff(4,2)11√6+122√76Aff(4,4)−35572√53Aff(6,0)+85√7Aff(6,2)1716−5286√356Aff(6,4)−552√3511Aff(6,6)|−Bff(2,2)3√10−211√23Bff(4,2)+111√143Bff(4,4)−5132√7Bff(6,2)−5143√703Bff(6,4)−552√3511Bff(6,6)|0|Aff(0,0)+7132Aff(4,0)+733√52Aff(4,2)−122√352Aff(4,4)−544Aff(6,0)+5572√105Aff(6,2)+25286√72Aff(6,4)+552√2111Aff(6,6)|Bff(2,2)√6−133√10Bff(4,2)+35572√353Bff(6,2)−552√2111Bff(6,6)|0| |
| ^fy(z2−x2)|12√57Bsf(3,1)−12√37Bsf(3,3)|√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|−Bdf(1,1)√14−Bdf(3,1)2√21−16√57Bdf(3,3)+111√1021Bdf(5,1)+566√5Bdf(5,3)−522Bdf(5,5)|√314Bdf(1,1)+Bdf(3,1)2√7+12√521Bdf(3,3)+511√514Bdf(5,1)−111√53Bdf(5,3)|0|0|√27Adf(1,1)−12√37Adf(3,1)−16√57Adf(3,3)+111√1514Adf(5,1)+566√5Adf(5,3)−522Adf(5,5)|0|Bff(2,2)3√10+211√23Bff(4,2)+111√143Bff(4,4)+5132√7Bff(6,2)−5143√703Bff(6,4)+552√3511Bff(6,6)|−Aff(2,0)√15+13√25Aff(2,2)+144√53Aff(4,0)+Aff(4,2)11√6−122√76Aff(4,4)+35572√53Aff(6,0)+85√7Aff(6,2)1716+5286√356Aff(6,4)−552√3511Aff(6,6)|0|Bff(2,2)√6−133√10Bff(4,2)+35572√353Bff(6,2)−552√2111Bff(6,6)|Aff(0,0)+7132Aff(4,0)−733√52Aff(4,2)−122√352Aff(4,4)−544Aff(6,0)−5572√105Aff(6,2)+25286√72Aff(6,4)−552√2111Aff(6,6)|0| | ^fy(z2−x2)|12√57Bsf(3,1)−12√37Bsf(3,3)|√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|−Bdf(1,1)√14−Bdf(3,1)2√21−16√57Bdf(3,3)+111√1021Bdf(5,1)+566√5Bdf(5,3)−522Bdf(5,5)|√314Bdf(1,1)+Bdf(3,1)2√7+12√521Bdf(3,3)+511√514Bdf(5,1)−111√53Bdf(5,3)|0|0|√27Adf(1,1)−12√37Adf(3,1)−16√57Adf(3,3)+111√1514Adf(5,1)+566√5Adf(5,3)−522Adf(5,5)|0|Bff(2,2)3√10+211√23Bff(4,2)+111√143Bff(4,4)+5132√7Bff(6,2)−5143√703Bff(6,4)+552√3511Bff(6,6)|−Aff(2,0)√15+13√25Aff(2,2)+144√53Aff(4,0)+Aff(4,2)11√6−122√76Aff(4,4)+35572√53Aff(6,0)+85√7Aff(6,2)1716+5286√356Aff(6,4)−552√3511Aff(6,6)|0|Bff(2,2)√6−133√10Bff(4,2)+35572√353Bff(6,2)−552√2111Bff(6,6)|Aff(0,0)+7132Aff(4,0)−733√52Aff(4,2)−122√352Aff(4,4)−544Aff(6,0)−5572√105Aff(6,2)+25286√72Aff(6,4)−552√2111Aff(6,6)|0| |
| |
| Although the parameters Al″ 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'. | 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 <hiddenSwitch expand all> |
| |
| ### | ### |
| ==== Potential for s orbitals ==== | ==== Potential for s orbitals ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| \text{Eag} & k=0\land m=0 \\ | \text{Ap} & k=0\land m=0 \\ |
| 0 & \text{True} | 0 & \text{True} |
| \end{cases}$$ | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{Ap, k == 0 && m == 0}}, 0] |
| | |
| | </code> |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, Eag} } | Akm = {{0, 0, Ap} } |
| |
| </code> | </code> |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| |
| ### | ### |
| |
| | ^ {Y_{0}^{(0)}} ^ | | ^ {Y_{0}^{(0)}} ^ |
| ^ {Y_{0}^{(0)}} |$ \text{Eag} $| | ^ {Y_{0}^{(0)}} |$ \text{Ap} $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| |
| ### | ### |
| |
| | ^ \text{s} ^ | | ^ \text{s} ^ |
| ^ \text{s} |$ \text{Eag} $| | ^ \text{s} |$ \text{Ap} $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **Rotation matrix used** > | </hidden> |
| | <hidden **Rotation matrix used** > |
| |
| ### | ### |
| |
| TODO | | $ ^ {Y_{0}^{(0)}} $ ^ |
| | ^ \text{s} | 1 | |
| |
| ### | ### |
| |
| </hidden><hidden **Irriducible representations and their onsite energy** > | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| |
| ### | ### |
| |
| TODO | ^ ^\text{Ap} | {{:physics_chemistry:pointgroup:cs_z_orb_0_1.png?150}} | |
| | |\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 }} | ::: | |
| |
| ### | ### |
| |
| </hidden>==== Potential for p orbitals ==== | </hidden> |
| | ==== Potential for p orbitals ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| \frac{1}{3} (\text{Epxpx}+\text{Epypy}+\text{Epzpz}) & k=0\land m=0 \\ | \frac{1}{3} (\text{Eapp}+\text{Eapx}+\text{Eapy}) & k=0\land m=0 \\ |
| \frac{5 (\text{Epxpx}+2 i \text{Epypx}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=-2 \\ | 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ |
| \frac{5 (\text{Epzpx}+i \text{Epypz})}{\sqrt{6}} & k=2\land m=-1 \\ | \frac{5 (\text{Eapx}-\text{Eapy}+2 i \text{Mapxy})}{2 \sqrt{6}} & k=2\land m=-2 \\ |
| -\frac{5}{6} (\text{Epxpx}+\text{Epypy}-2 \text{Epzpz}) & k=2\land m=0 \\ | \frac{5}{6} (2 \text{Eapp}-\text{Eapx}-\text{Eapy}) & k=2\land m=0 \\ |
| \frac{5 i (\text{Epypz}+i \text{Epzpx})}{\sqrt{6}} & k=2\land m=1 \\ | \frac{5 (\text{Eapx}-\text{Eapy}-2 i \text{Mapxy})}{2 \sqrt{6}} & \text{True} |
| \frac{5 (\text{Epxpx}-2 i \text{Epypx}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=2 | |
| \end{cases}$$ | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eapp + Eapx + Eapy)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != 0 && m != 2)}, {(5*(Eapx - Eapy + (2*I)*Mapxy))/(2*Sqrt[6]), k == 2 && m == -2}, {(5*(2*Eapp - Eapx - Eapy))/6, k == 2 && m == 0}}, (5*(Eapx - Eapy - (2*I)*Mapxy))/(2*Sqrt[6])] |
| | |
| | </code> |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, (1/3)*(Epxpx + Epypy + Epzpz)} , | Akm = {{0, 0, (1/3)*(Eapp + Eapx + Eapy)} , |
| {2, 0, (-5/6)*(Epxpx + Epypy + (-2)*(Epzpz))} , | {2, 0, (5/6)*((2)*(Eapp) + (-1)*(Eapx) + (-1)*(Eapy))} , |
| {2,-1, (5)*((1/(sqrt(6)))*((I)*(Epypz) + Epzpx))} , | {2, 2, (5/2)*((1/(sqrt(6)))*(Eapx + (-1)*(Eapy) + (-2*I)*(Mapxy)))} , |
| {2, 1, (5*I)*((1/(sqrt(6)))*(Epypz + (I)*(Epzpx)))} , | {2,-2, (5/2)*((1/(sqrt(6)))*(Eapx + (-1)*(Eapy) + (2*I)*(Mapxy)))} } |
| {2, 2, (5/2)*((1/(sqrt(6)))*(Epxpx + (-2*I)*(Epypx) + (-1)*(Epypy)))} , | |
| {2,-2, (5/2)*((1/(sqrt(6)))*(Epxpx + (2*I)*(Epypx) + (-1)*(Epypy)))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| |
| ### | ### |
| |
| | ^ {Y_{-1}^{(1)}} ^ {Y_{0}^{(1)}} ^ {Y_{1}^{(1)}} ^ | | ^ {Y_{-1}^{(1)}} ^ {Y_{0}^{(1)}} ^ {Y_{1}^{(1)}} ^ |
| ^ {Y_{-1}^{(1)}} |$ \frac{\text{Epxpx}+\text{Epypy}}{2} | \frac{\text{Epzpx}+i \text{Epypz}}{\sqrt{2}} | \frac{1}{2} (-\text{Epxpx}-2 i \text{Epypx}+\text{Epypy}) $| | ^ {Y_{-1}^{(1)}} |$ \frac{\text{Eapx}+\text{Eapy}}{2} | 0 | \frac{1}{2} (-\text{Eapx}+\text{Eapy}-2 i \text{Mapxy}) $| |
| ^ {Y_{0}^{(1)}} |$ \frac{\text{Epzpx}-i \text{Epypz}}{\sqrt{2}} | \text{Epzpz} | -\frac{\text{Epzpx}+i \text{Epypz}}{\sqrt{2}} $| | ^ {Y_{0}^{(1)}} |$ 0 | \text{Eapp} | 0 $| |
| ^ {Y_{1}^{(1)}} |$ \frac{1}{2} (-\text{Epxpx}+2 i \text{Epypx}+\text{Epypy}) | \frac{i (\text{Epypz}+i \text{Epzpx})}{\sqrt{2}} | \frac{\text{Epxpx}+\text{Epypy}}{2} $| | ^ {Y_{1}^{(1)}} |$ \frac{1}{2} (-\text{Eapx}+\text{Eapy}+2 i \text{Mapxy}) | 0 | \frac{\text{Eapx}+\text{Eapy}}{2} $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| |
| ### | ### |
| |
| | ^ p_x ^ p_y ^ p_z ^ | | ^ p_x ^ p_y ^ p_z ^ |
| ^ p_x |$ \text{Epxpx} | \text{Epypx} | \text{Epzpx} $| | ^ p_x |$ \text{Eapx} | \text{Mapxy} | 0 $| |
| ^ p_y |$ \text{Epypx} | \text{Epypy} | \text{Epypz} $| | ^ p_y |$ \text{Mapxy} | \text{Eapy} | 0 $| |
| ^ p_z |$ \text{Epzpx} | \text{Epypz} | \text{Epzpz} $| | ^ p_z |$ 0 | 0 | \text{Eapp} $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **Rotation matrix used** > | </hidden> |
| | <hidden **Rotation matrix used** > |
| |
| ### | ### |
| |
| TODO | | $ ^ {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 | |
| |
| ### | ### |
| |
| </hidden><hidden **Irriducible representations and their onsite energy** > | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| |
| ### | ### |
| |
| TODO | ^ ^\text{Eapx} | {{:physics_chemistry:pointgroup:cs_z_orb_1_1.png?150}} | |
| | |\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{Eapy} | {{:physics_chemistry:pointgroup:cs_z_orb_1_2.png?150}} | |
| | |\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{Eapp} | {{:physics_chemistry:pointgroup:cs_z_orb_1_3.png?150}} | |
| | |\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 | ::: | |
| |
| ### | ### |
| |
| </hidden>==== Potential for d orbitals ==== | </hidden> |
| | ==== Potential for d orbitals ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| \frac{1}{5} (\text{Edx2y2dx2y2}+\text{Edxydxy}+\text{Edxzdxz}+\text{Edyzdyz}+\text{Edz2dz2}) & k=0\land m=0 \\ | \frac{1}{5} (\text{Eappxz}+\text{Eappyz}+\text{Eapx2y2}+\text{Eapxy}+\text{Eapz2}) & k=0\land m=0 \\ |
| \frac{-4 i \text{Edxydz2}+\sqrt{3} \text{Edxzdxz}+2 i \sqrt{3} \text{Edyzdxz}-\sqrt{3} \text{Edyzdyz}-4 \text{Edz2dx2y2}}{2 \sqrt{2}} & k=2\land m=-2 \\ | 0 & (k\neq 4\land (k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\ |
| \frac{i \sqrt{3} \text{Edxydxz}+\sqrt{3} \text{Edxydyz}+\sqrt{3} \text{Edxzdx2y2}-i \sqrt{3} \text{Edyzdx2y2}+i \text{Edyzdz2}+\text{Edz2dxz}}{\sqrt{2}} & k=2\land m=-1 \\ | \frac{\sqrt{3} \text{Eappxz}-\sqrt{3} \text{Eappyz}+2 i \sqrt{3} \text{Mappyzxz}-4 \text{Mapx2y2z2}-4 i \text{Mapz2xy}}{2 \sqrt{2}} & k=2\land m=-2 \\ |
| -\text{Edx2y2dx2y2}-\text{Edxydxy}+\frac{\text{Edxzdxz}+\text{Edyzdyz}}{2}+\text{Edz2dz2} & k=2\land m=0 \\ | \frac{1}{2} (\text{Eappxz}+\text{Eappyz}-2 (\text{Eapx2y2}+\text{Eapxy}-\text{Eapz2})) & k=2\land m=0 \\ |
| \frac{i \left(\sqrt{3} \text{Edxydxz}+i \sqrt{3} \text{Edxydyz}+i \sqrt{3} \text{Edxzdx2y2}-\sqrt{3} \text{Edyzdx2y2}+\text{Edyzdz2}+i \text{Edz2dxz}\right)}{\sqrt{2}} & k=2\land m=1 \\ | \frac{\sqrt{3} \text{Eappxz}-\sqrt{3} \text{Eappyz}-2 i \sqrt{3} \text{Mappyzxz}-4 \text{Mapx2y2z2}+4 i \text{Mapz2xy}}{2 \sqrt{2}} & k=2\land m=2 \\ |
| \frac{4 i \text{Edxydz2}+\sqrt{3} \text{Edxzdxz}-2 i \sqrt{3} \text{Edyzdxz}-\sqrt{3} \text{Edyzdyz}-4 \text{Edz2dx2y2}}{2 \sqrt{2}} & k=2\land m=2 \\ | \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eapx2y2}-\text{Eapxy}+2 i \text{Mapx2y2xy}) & k=4\land m=-4 \\ |
| \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}+2 i \text{Edxydx2y2}-\text{Edxydxy}) & k=4\land m=-4 \\ | \frac{3 \left(\text{Eappxz}-\text{Eappyz}+2 i \text{Mappyzxz}+\sqrt{3} \text{Mapx2y2z2}+i \sqrt{3} \text{Mapz2xy}\right)}{\sqrt{10}} & k=4\land m=-2 \\ |
| \frac{3}{2} \sqrt{\frac{7}{5}} (i \text{Edxydxz}-\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}) & k=4\land m=-3 \\ | -\frac{3}{10} (4 \text{Eappxz}+4 \text{Eappyz}-\text{Eapx2y2}-\text{Eapxy}-6 \text{Eapz2}) & k=4\land m=0 \\ |
| \frac{3 \left(i \sqrt{3} \text{Edxydz2}+\text{Edxzdxz}+2 i \text{Edyzdxz}-\text{Edyzdyz}+\sqrt{3} \text{Edz2dx2y2}\right)}{\sqrt{10}} & k=4\land m=-2 \\ | \frac{3 \left(\text{Eappxz}-\text{Eappyz}-2 i \text{Mappyzxz}+\sqrt{3} \text{Mapx2y2z2}-i \sqrt{3} \text{Mapz2xy}\right)}{\sqrt{10}} & k=4\land m=2 \\ |
| \frac{3 \left(-i \text{Edxydxz}-\text{Edxydyz}-\text{Edxzdx2y2}+i \text{Edyzdx2y2}+2 i \sqrt{3} \text{Edyzdz2}+2 \sqrt{3} \text{Edz2dxz}\right)}{2 \sqrt{5}} & k=4\land m=-1 \\ | \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eapx2y2}-\text{Eapxy}-2 i \text{Mapx2y2xy}) & \text{True} |
| \frac{3}{10} (\text{Edx2y2dx2y2}+\text{Edxydxy}-4 (\text{Edxzdxz}+\text{Edyzdyz})+6 \text{Edz2dz2}) & k=4\land m=0 \\ | |
| \frac{3 \left(-i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}+2 i \sqrt{3} \text{Edyzdz2}-2 \sqrt{3} \text{Edz2dxz}\right)}{2 \sqrt{5}} & k=4\land m=1 \\ | |
| \frac{3 \left(-i \sqrt{3} \text{Edxydz2}+\text{Edxzdxz}-2 i \text{Edyzdxz}-\text{Edyzdyz}+\sqrt{3} \text{Edz2dx2y2}\right)}{\sqrt{10}} & k=4\land m=2 \\ | |
| \frac{3}{2} \sqrt{\frac{7}{5}} (i \text{Edxydxz}+\text{Edxydyz}-\text{Edxzdx2y2}+i \text{Edyzdx2y2}) & k=4\land m=3 \\ | |
| \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}-2 i \text{Edxydx2y2}-\text{Edxydxy}) & k=4\land m=4 | |
| \end{cases}$$ | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eappxz + Eappyz + Eapx2y2 + Eapxy + Eapz2)/5, k == 0 && m == 0}, {0, (k != 4 && (k != 2 || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {(Sqrt[3]*Eappxz - Sqrt[3]*Eappyz + (2*I)*Sqrt[3]*Mappyzxz - 4*Mapx2y2z2 - (4*I)*Mapz2xy)/(2*Sqrt[2]), k == 2 && m == -2}, {(Eappxz + Eappyz - 2*(Eapx2y2 + Eapxy - Eapz2))/2, k == 2 && m == 0}, {(Sqrt[3]*Eappxz - Sqrt[3]*Eappyz - (2*I)*Sqrt[3]*Mappyzxz - 4*Mapx2y2z2 + (4*I)*Mapz2xy)/(2*Sqrt[2]), k == 2 && m == 2}, {(3*Sqrt[7/10]*(Eapx2y2 - Eapxy + (2*I)*Mapx2y2xy))/2, k == 4 && m == -4}, {(3*(Eappxz - Eappyz + (2*I)*Mappyzxz + Sqrt[3]*Mapx2y2z2 + I*Sqrt[3]*Mapz2xy))/Sqrt[10], k == 4 && m == -2}, {(-3*(4*Eappxz + 4*Eappyz - Eapx2y2 - Eapxy - 6*Eapz2))/10, k == 4 && m == 0}, {(3*(Eappxz - Eappyz - (2*I)*Mappyzxz + Sqrt[3]*Mapx2y2z2 - I*Sqrt[3]*Mapz2xy))/Sqrt[10], k == 4 && m == 2}}, (3*Sqrt[7/10]*(Eapx2y2 - Eapxy - (2*I)*Mapx2y2xy))/2] |
| | |
| | </code> |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, (1/5)*(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)} , | Akm = {{0, 0, (1/5)*(Eappxz + Eappyz + Eapx2y2 + Eapxy + Eapz2)} , |
| {2, 0, (-1)*(Edx2y2dx2y2) + (-1)*(Edxydxy) + (1/2)*(Edxzdxz + Edyzdyz) + Edz2dz2} , | {2, 0, (1/2)*(Eappxz + Eappyz + (-2)*(Eapx2y2 + Eapxy + (-1)*(Eapz2)))} , |
| {2,-1, (1/(sqrt(2)))*((I)*((sqrt(3))*(Edxydxz)) + (sqrt(3))*(Edxydyz) + (sqrt(3))*(Edxzdx2y2) + (-I)*((sqrt(3))*(Edyzdx2y2)) + (I)*(Edyzdz2) + Edz2dxz)} , | {2, 2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Eappxz) + (-1)*((sqrt(3))*(Eappyz)) + (-2*I)*((sqrt(3))*(Mappyzxz)) + (-4)*(Mapx2y2z2) + (4*I)*(Mapz2xy)))} , |
| {2, 1, (I)*((1/(sqrt(2)))*((sqrt(3))*(Edxydxz) + (I)*((sqrt(3))*(Edxydyz)) + (I)*((sqrt(3))*(Edxzdx2y2)) + (-1)*((sqrt(3))*(Edyzdx2y2)) + Edyzdz2 + (I)*(Edz2dxz)))} , | {2,-2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Eappxz) + (-1)*((sqrt(3))*(Eappyz)) + (2*I)*((sqrt(3))*(Mappyzxz)) + (-4)*(Mapx2y2z2) + (-4*I)*(Mapz2xy)))} , |
| {2,-2, (1/2)*((1/(sqrt(2)))*((-4*I)*(Edxydz2) + (sqrt(3))*(Edxzdxz) + (2*I)*((sqrt(3))*(Edyzdxz)) + (-1)*((sqrt(3))*(Edyzdyz)) + (-4)*(Edz2dx2y2)))} , | {4, 0, (-3/10)*((4)*(Eappxz) + (4)*(Eappyz) + (-1)*(Eapx2y2) + (-1)*(Eapxy) + (-6)*(Eapz2))} , |
| {2, 2, (1/2)*((1/(sqrt(2)))*((4*I)*(Edxydz2) + (sqrt(3))*(Edxzdxz) + (-2*I)*((sqrt(3))*(Edyzdxz)) + (-1)*((sqrt(3))*(Edyzdyz)) + (-4)*(Edz2dx2y2)))} , | {4, 2, (3)*((1/(sqrt(10)))*(Eappxz + (-1)*(Eappyz) + (-2*I)*(Mappyzxz) + (sqrt(3))*(Mapx2y2z2) + (-I)*((sqrt(3))*(Mapz2xy))))} , |
| {4, 0, (3/10)*(Edx2y2dx2y2 + Edxydxy + (-4)*(Edxzdxz + Edyzdyz) + (6)*(Edz2dz2))} , | {4,-2, (3)*((1/(sqrt(10)))*(Eappxz + (-1)*(Eappyz) + (2*I)*(Mappyzxz) + (sqrt(3))*(Mapx2y2z2) + (I)*((sqrt(3))*(Mapz2xy))))} , |
| {4,-1, (3/2)*((1/(sqrt(5)))*((-I)*(Edxydxz) + (-1)*(Edxydyz) + (-1)*(Edxzdx2y2) + (I)*(Edyzdx2y2) + (2*I)*((sqrt(3))*(Edyzdz2)) + (2)*((sqrt(3))*(Edz2dxz))))} , | {4, 4, (3/2)*((sqrt(7/10))*(Eapx2y2 + (-1)*(Eapxy) + (-2*I)*(Mapx2y2xy)))} , |
| {4, 1, (3/2)*((1/(sqrt(5)))*((-I)*(Edxydxz) + Edxydyz + Edxzdx2y2 + (I)*(Edyzdx2y2) + (2*I)*((sqrt(3))*(Edyzdz2)) + (-2)*((sqrt(3))*(Edz2dxz))))} , | {4,-4, (3/2)*((sqrt(7/10))*(Eapx2y2 + (-1)*(Eapxy) + (2*I)*(Mapx2y2xy)))} } |
| {4, 2, (3)*((1/(sqrt(10)))*((-I)*((sqrt(3))*(Edxydz2)) + Edxzdxz + (-2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (sqrt(3))*(Edz2dx2y2)))} , | |
| {4,-2, (3)*((1/(sqrt(10)))*((I)*((sqrt(3))*(Edxydz2)) + Edxzdxz + (2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (sqrt(3))*(Edz2dx2y2)))} , | |
| {4,-3, (3/2)*((sqrt(7/5))*((I)*(Edxydxz) + (-1)*(Edxydyz) + Edxzdx2y2 + (I)*(Edyzdx2y2)))} , | |
| {4, 3, (3/2)*((sqrt(7/5))*((I)*(Edxydxz) + Edxydyz + (-1)*(Edxzdx2y2) + (I)*(Edyzdx2y2)))} , | |
| {4, 4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (-2*I)*(Edxydx2y2) + (-1)*(Edxydxy)))} , | |
| {4,-4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (2*I)*(Edxydx2y2) + (-1)*(Edxydxy)))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **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)}} ^ {Y_{-1}^{(2)}} ^ {Y_{0}^{(2)}} ^ {Y_{1}^{(2)}} ^ {Y_{2}^{(2)}} ^ |
| ^ {Y_{-2}^{(2)}} |$ \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} | \frac{1}{2} (i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}-i \text{Edyzdx2y2}) | \frac{\text{Edz2dx2y2}+i \text{Edxydz2}}{\sqrt{2}} | -\frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}-\text{Edxzdx2y2})+\text{Edyzdx2y2}) | \frac{1}{2} (\text{Edx2y2dx2y2}+2 i \text{Edxydx2y2}-\text{Edxydxy}) $| | ^ {Y_{-2}^{(2)}} |$ \frac{\text{Eapx2y2}+\text{Eapxy}}{2} | 0 | \frac{\text{Mapx2y2z2}+i \text{Mapz2xy}}{\sqrt{2}} | 0 | \frac{1}{2} (\text{Eapx2y2}-\text{Eapxy}+2 i \text{Mapx2y2xy}) $| |
| ^ {Y_{-1}^{(2)}} |$ \frac{1}{2} (-i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}) | \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} | \frac{\text{Edz2dxz}+i \text{Edyzdz2}}{\sqrt{2}} | \frac{1}{2} (-\text{Edxzdxz}-2 i \text{Edyzdxz}+\text{Edyzdyz}) | \frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}-\text{Edxzdx2y2})+\text{Edyzdx2y2}) $| | ^ {Y_{-1}^{(2)}} |$ 0 | \frac{\text{Eappxz}+\text{Eappyz}}{2} | 0 | \frac{1}{2} (-\text{Eappxz}+\text{Eappyz}-2 i \text{Mappyzxz}) | 0 $| |
| ^ {Y_{0}^{(2)}} |$ \frac{\text{Edz2dx2y2}-i \text{Edxydz2}}{\sqrt{2}} | \frac{\text{Edz2dxz}-i \text{Edyzdz2}}{\sqrt{2}} | \text{Edz2dz2} | \frac{-\text{Edz2dxz}-i \text{Edyzdz2}}{\sqrt{2}} | \frac{\text{Edz2dx2y2}+i \text{Edxydz2}}{\sqrt{2}} $| | ^ {Y_{0}^{(2)}} |$ \frac{\text{Mapx2y2z2}-i \text{Mapz2xy}}{\sqrt{2}} | 0 | \text{Eapz2} | 0 | \frac{\text{Mapx2y2z2}+i \text{Mapz2xy}}{\sqrt{2}} $| |
| ^ {Y_{1}^{(2)}} |$ \frac{1}{2} i (\text{Edxydxz}-i \text{Edxydyz}+i \text{Edxzdx2y2}+\text{Edyzdx2y2}) | \frac{1}{2} (-\text{Edxzdxz}+2 i \text{Edyzdxz}+\text{Edyzdyz}) | \frac{i (\text{Edyzdz2}+i \text{Edz2dxz})}{\sqrt{2}} | \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} | -\frac{1}{2} i (\text{Edxydxz}-i (\text{Edxydyz}+\text{Edxzdx2y2})-\text{Edyzdx2y2}) $| | ^ {Y_{1}^{(2)}} |$ 0 | \frac{1}{2} (-\text{Eappxz}+\text{Eappyz}+2 i \text{Mappyzxz}) | 0 | \frac{\text{Eappxz}+\text{Eappyz}}{2} | 0 $| |
| ^ {Y_{2}^{(2)}} |$ \frac{1}{2} (\text{Edx2y2dx2y2}-2 i \text{Edxydx2y2}-\text{Edxydxy}) | -\frac{1}{2} i (\text{Edxydxz}-i \text{Edxydyz}+i \text{Edxzdx2y2}+\text{Edyzdx2y2}) | \frac{\text{Edz2dx2y2}-i \text{Edxydz2}}{\sqrt{2}} | \frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2})) | \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} $| | ^ {Y_{2}^{(2)}} |$ \frac{1}{2} (\text{Eapx2y2}-\text{Eapxy}-2 i \text{Mapx2y2xy}) | 0 | \frac{\text{Mapx2y2z2}-i \text{Mapz2xy}}{\sqrt{2}} | 0 | \frac{\text{Eapx2y2}+\text{Eapxy}}{2} $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| |
| ### | ### |
| |
| | ^ d_{x^2-y^2} ^ d_{3z^2-r^2} ^ d_{\text{yz}} ^ d_{\text{xz}} ^ d_{\text{xy}} ^ | | ^ d_{x^2-y^2} ^ d_{3z^2-r^2} ^ d_{\text{yz}} ^ d_{\text{xz}} ^ d_{\text{xy}} ^ |
| ^ d_{x^2-y^2} |$ \text{Edx2y2dx2y2} | \text{Edz2dx2y2} | \text{Edyzdx2y2} | \text{Edxzdx2y2} | \text{Edxydx2y2} $| | ^ d_{x^2-y^2} |$ \text{Eapx2y2} | \text{Mapx2y2z2} | 0 | 0 | \text{Mapx2y2xy} $| |
| ^ d_{3z^2-r^2} |$ \text{Edz2dx2y2} | \text{Edz2dz2} | \text{Edyzdz2} | \text{Edz2dxz} | \text{Edxydz2} $| | ^ d_{3z^2-r^2} |$ \text{Mapx2y2z2} | \text{Eapz2} | 0 | 0 | \text{Mapz2xy} $| |
| ^ d_{\text{yz}} |$ \text{Edyzdx2y2} | \text{Edyzdz2} | \text{Edyzdyz} | \text{Edyzdxz} | \text{Edxydyz} $| | ^ d_{\text{yz}} |$ 0 | 0 | \text{Eappyz} | \text{Mappyzxz} | 0 $| |
| ^ d_{\text{xz}} |$ \text{Edxzdx2y2} | \text{Edz2dxz} | \text{Edyzdxz} | \text{Edxzdxz} | \text{Edxydxz} $| | ^ d_{\text{xz}} |$ 0 | 0 | \text{Mappyzxz} | \text{Eappxz} | 0 $| |
| ^ d_{\text{xy}} |$ \text{Edxydx2y2} | \text{Edxydz2} | \text{Edxydyz} | \text{Edxydxz} | \text{Edxydxy} $| | ^ d_{\text{xy}} |$ \text{Mapx2y2xy} | \text{Mapz2xy} | 0 | 0 | \text{Eapxy} $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **Rotation matrix used** > | </hidden> |
| | <hidden **Rotation matrix used** > |
| |
| ### | ### |
| |
| TODO | | $ ^ {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}} | |
| |
| ### | ### |
| |
| </hidden><hidden **Irriducible representations and their onsite energy** > | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| |
| ### | ### |
| |
| TODO | ^ ^\text{Eapx2y2} | {{:physics_chemistry:pointgroup:cs_z_orb_2_1.png?150}} | |
| | |\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{Eapz2} | {{:physics_chemistry:pointgroup:cs_z_orb_2_2.png?150}} | |
| | |\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{Eappyz} | {{:physics_chemistry:pointgroup:cs_z_orb_2_3.png?150}} | |
| | |\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{Eappxz} | {{:physics_chemistry:pointgroup:cs_z_orb_2_4.png?150}} | |
| | |\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{Eapxy} | {{:physics_chemistry:pointgroup:cs_z_orb_2_5.png?150}} | |
| | |\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 | ::: | |
| |
| ### | ### |
| |
| </hidden>==== Potential for f orbitals ==== | </hidden> |
| | ==== Potential for f orbitals ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | \frac{1}{7} (\text{Eappx3}+\text{Eappxy2z2}+\text{Eappxyz}+\text{Eappy3}+\text{Eappyz2x2}+\text{Eappz3}+\text{Eappzx2y2}) & k=0\land m=0 \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | 0 & (k\neq 6\land (((k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2))\land k\neq 4)\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4)))\lor (m\neq -6\land m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4\land m\neq 6) \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | \frac{5 \left(2 \sqrt{3} \text{Eappx3}-2 \sqrt{3} \text{Eappy3}+2 \sqrt{5} \text{Mappx3xy2z2}-i \sqrt{3} \text{Mappx3y3}-i \sqrt{5} \text{Mappx3yz2x2}-5 i \sqrt{3} \text{Mappxy2z2yz2x2}-4 i \sqrt{5} \text{Mappxyzz3}+i \sqrt{5} \text{Mappy3xy2z2}+2 \sqrt{5} \text{Mappy3yz2x2}-4 \sqrt{5} \text{Mappz3zx2y2}\right)}{28 \sqrt{2}} & k=2\land m=-2 \\ |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | -\frac{5}{14} \left(\text{Eappx3}+\text{Eappy3}-2 \text{Eappz3}-\sqrt{15} \text{Mappx3xy2z2}+\sqrt{15} \text{Mappy3yz2x2}\right) & k=2\land m=0 \\ |
| A(2,0) & k=2\land m=0 \\ | \frac{5 \left(2 \sqrt{3} \text{Eappx3}-2 \sqrt{3} \text{Eappy3}+2 \sqrt{5} \text{Mappx3xy2z2}+i \sqrt{3} \text{Mappx3y3}+i \sqrt{5} \text{Mappx3yz2x2}+5 i \sqrt{3} \text{Mappxy2z2yz2x2}+4 i \sqrt{5} \text{Mappxyzz3}-i \sqrt{5} \text{Mappy3xy2z2}+2 \sqrt{5} \text{Mappy3yz2x2}-4 \sqrt{5} \text{Mappz3zx2y2}\right)}{28 \sqrt{2}} & k=2\land m=2 \\ |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | \frac{3 \left(3 \sqrt{5} \text{Eappx3}-3 \sqrt{5} \text{Eappxy2z2}-4 \sqrt{5} \text{Eappxyz}+3 \sqrt{5} \text{Eappy3}-3 \sqrt{5} \text{Eappyz2x2}+4 \sqrt{5} \text{Eappzx2y2}+2 \sqrt{3} \text{Mappx3xy2z2}-8 i \sqrt{3} \text{Mappx3yz2x2}+8 i \sqrt{5} \text{Mappxyzzx2y2}-8 i \sqrt{3} \text{Mappy3xy2z2}-2 \sqrt{3} \text{Mappy3yz2x2}\right)}{8 \sqrt{14}} & k=4\land m=-4 \\ |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | \frac{3}{56} \left(-3 \sqrt{10} \text{Eappx3}+7 \sqrt{10} \text{Eappxy2z2}+3 \sqrt{10} \text{Eappy3}-7 \sqrt{10} \text{Eappyz2x2}+2 \sqrt{6} \text{Mappx3xy2z2}+12 i \sqrt{10} \text{Mappx3y3}-8 i \sqrt{6} \text{Mappx3yz2x2}+4 i \sqrt{10} \text{Mappxy2z2yz2x2}-4 i \sqrt{6} \text{Mappxyzz3}+8 i \sqrt{6} \text{Mappy3xy2z2}+2 \sqrt{6} \text{Mappy3yz2x2}-4 \sqrt{6} \text{Mappz3zx2y2}\right) & k=4\land m=-2 \\ |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | \frac{3}{56} \left(9 \text{Eappx3}+7 \text{Eappxy2z2}-28 \text{Eappxyz}+9 \text{Eappy3}+7 \text{Eappyz2x2}+24 \text{Eappz3}-28 \text{Eappzx2y2}-2 \sqrt{15} \text{Mappx3xy2z2}+2 \sqrt{15} \text{Mappy3yz2x2}\right) & k=4\land m=0 \\ |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | \frac{3}{56} \left(-3 \sqrt{10} \text{Eappx3}+7 \sqrt{10} \text{Eappxy2z2}+3 \sqrt{10} \text{Eappy3}-7 \sqrt{10} \text{Eappyz2x2}+2 \sqrt{6} \text{Mappx3xy2z2}-12 i \sqrt{10} \text{Mappx3y3}+8 i \sqrt{6} \text{Mappx3yz2x2}-4 i \sqrt{10} \text{Mappxy2z2yz2x2}+4 i \sqrt{6} \text{Mappxyzz3}-8 i \sqrt{6} \text{Mappy3xy2z2}+2 \sqrt{6} \text{Mappy3yz2x2}-4 \sqrt{6} \text{Mappz3zx2y2}\right) & k=4\land m=2 \\ |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | \frac{3 \left(3 \sqrt{5} \text{Eappx3}-3 \sqrt{5} \text{Eappxy2z2}-4 \sqrt{5} \text{Eappxyz}+3 \sqrt{5} \text{Eappy3}-3 \sqrt{5} \text{Eappyz2x2}+4 \sqrt{5} \text{Eappzx2y2}+2 \sqrt{3} \text{Mappx3xy2z2}+8 i \sqrt{3} \text{Mappx3yz2x2}-8 i \sqrt{5} \text{Mappxyzzx2y2}+8 i \sqrt{3} \text{Mappy3xy2z2}-2 \sqrt{3} \text{Mappy3yz2x2}\right)}{8 \sqrt{14}} & k=4\land m=4 \\ |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappx3}+3 \sqrt{3} \text{Eappxy2z2}-5 \sqrt{3} \text{Eappy3}-3 \sqrt{3} \text{Eappyz2x2}-6 \sqrt{5} \text{Mappx3xy2z2}-10 i \sqrt{3} \text{Mappx3y3}-6 i \sqrt{5} \text{Mappx3yz2x2}+6 i \sqrt{3} \text{Mappxy2z2yz2x2}+6 i \sqrt{5} \text{Mappy3xy2z2}-6 \sqrt{5} \text{Mappy3yz2x2}\right) & k=6\land m=-6 \\ |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | -\frac{13 \left(15 \text{Eappx3}-15 \text{Eappxy2z2}+24 \text{Eappxyz}+15 \text{Eappy3}-15 \text{Eappyz2x2}-24 \text{Eappzx2y2}+2 \sqrt{15} \text{Mappx3xy2z2}-8 i \sqrt{15} \text{Mappx3yz2x2}-48 i \text{Mappxyzzx2y2}-8 i \sqrt{15} \text{Mappy3xy2z2}-2 \sqrt{15} \text{Mappy3yz2x2}\right)}{80 \sqrt{14}} & k=6\land m=-4 \\ |
| A(4,0) & k=4\land m=0 \\ | \frac{13 \left(5 \sqrt{15} \text{Eappx3}+3 \sqrt{15} \text{Eappxy2z2}-5 \sqrt{15} \text{Eappy3}-3 \sqrt{15} \text{Eappyz2x2}+34 \text{Mappx3xy2z2}+2 i \sqrt{15} \text{Mappx3y3}-26 i \text{Mappx3yz2x2}-14 i \sqrt{15} \text{Mappxy2z2yz2x2}+64 i \text{Mappxyzz3}+26 i \text{Mappy3xy2z2}+34 \text{Mappy3yz2x2}+64 \text{Mappz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=-2 \\ |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | -\frac{13}{560} \left(25 \text{Eappx3}+39 \text{Eappxy2z2}-24 \text{Eappxyz}+25 \text{Eappy3}+39 \text{Eappyz2x2}-80 \text{Eappz3}-24 \text{Eappzx2y2}+14 \sqrt{15} \text{Mappx3xy2z2}-14 \sqrt{15} \text{Mappy3yz2x2}\right) & k=6\land m=0 \\ |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | \frac{13 \left(5 \sqrt{15} \text{Eappx3}+3 \sqrt{15} \text{Eappxy2z2}-5 \sqrt{15} \text{Eappy3}-3 \sqrt{15} \text{Eappyz2x2}+34 \text{Mappx3xy2z2}-2 i \sqrt{15} \text{Mappx3y3}+26 i \text{Mappx3yz2x2}+14 i \sqrt{15} \text{Mappxy2z2yz2x2}-64 i \text{Mappxyzz3}-26 i \text{Mappy3xy2z2}+34 \text{Mappy3yz2x2}+64 \text{Mappz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=2 \\ |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | -\frac{13 \left(15 \text{Eappx3}-15 \text{Eappxy2z2}+24 \text{Eappxyz}+15 \text{Eappy3}-15 \text{Eappyz2x2}-24 \text{Eappzx2y2}+2 \sqrt{15} \text{Mappx3xy2z2}+8 i \sqrt{15} \text{Mappx3yz2x2}+48 i \text{Mappxyzzx2y2}+8 i \sqrt{15} \text{Mappy3xy2z2}-2 \sqrt{15} \text{Mappy3yz2x2}\right)}{80 \sqrt{14}} & k=6\land m=4 \\ |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappx3}+3 \sqrt{3} \text{Eappxy2z2}-5 \sqrt{3} \text{Eappy3}-3 \sqrt{3} \text{Eappyz2x2}-6 \sqrt{5} \text{Mappx3xy2z2}+10 i \sqrt{3} \text{Mappx3y3}+6 i \sqrt{5} \text{Mappx3yz2x2}-6 i \sqrt{3} \text{Mappxy2z2yz2x2}-6 i \sqrt{5} \text{Mappy3xy2z2}-6 \sqrt{5} \text{Mappy3yz2x2}\right) & \text{True} |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | |
| A(6,0) & k=6\land m=0 \\ | |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | |
| \end{cases}$$ | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eappx3 + Eappxy2z2 + Eappxyz + Eappy3 + Eappyz2x2 + Eappz3 + Eappzx2y2)/7, k == 0 && m == 0}, {0, (k != 6 && (((k != 2 || (m != -2 && m != 0 && m != 2)) && k != 4) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4))) || (m != -6 && m != -4 && m != -2 && m != 0 && m != 2 && m != 4 && m != 6)}, {(5*(2*Sqrt[3]*Eappx3 - 2*Sqrt[3]*Eappy3 + 2*Sqrt[5]*Mappx3xy2z2 - I*Sqrt[3]*Mappx3y3 - I*Sqrt[5]*Mappx3yz2x2 - (5*I)*Sqrt[3]*Mappxy2z2yz2x2 - (4*I)*Sqrt[5]*Mappxyzz3 + I*Sqrt[5]*Mappy3xy2z2 + 2*Sqrt[5]*Mappy3yz2x2 - 4*Sqrt[5]*Mappz3zx2y2))/(28*Sqrt[2]), k == 2 && m == -2}, {(-5*(Eappx3 + Eappy3 - 2*Eappz3 - Sqrt[15]*Mappx3xy2z2 + Sqrt[15]*Mappy3yz2x2))/14, k == 2 && m == 0}, {(5*(2*Sqrt[3]*Eappx3 - 2*Sqrt[3]*Eappy3 + 2*Sqrt[5]*Mappx3xy2z2 + I*Sqrt[3]*Mappx3y3 + I*Sqrt[5]*Mappx3yz2x2 + (5*I)*Sqrt[3]*Mappxy2z2yz2x2 + (4*I)*Sqrt[5]*Mappxyzz3 - I*Sqrt[5]*Mappy3xy2z2 + 2*Sqrt[5]*Mappy3yz2x2 - 4*Sqrt[5]*Mappz3zx2y2))/(28*Sqrt[2]), k == 2 && m == 2}, {(3*(3*Sqrt[5]*Eappx3 - 3*Sqrt[5]*Eappxy2z2 - 4*Sqrt[5]*Eappxyz + 3*Sqrt[5]*Eappy3 - 3*Sqrt[5]*Eappyz2x2 + 4*Sqrt[5]*Eappzx2y2 + 2*Sqrt[3]*Mappx3xy2z2 - (8*I)*Sqrt[3]*Mappx3yz2x2 + (8*I)*Sqrt[5]*Mappxyzzx2y2 - (8*I)*Sqrt[3]*Mappy3xy2z2 - 2*Sqrt[3]*Mappy3yz2x2))/(8*Sqrt[14]), k == 4 && m == -4}, {(3*(-3*Sqrt[10]*Eappx3 + 7*Sqrt[10]*Eappxy2z2 + 3*Sqrt[10]*Eappy3 - 7*Sqrt[10]*Eappyz2x2 + 2*Sqrt[6]*Mappx3xy2z2 + (12*I)*Sqrt[10]*Mappx3y3 - (8*I)*Sqrt[6]*Mappx3yz2x2 + (4*I)*Sqrt[10]*Mappxy2z2yz2x2 - (4*I)*Sqrt[6]*Mappxyzz3 + (8*I)*Sqrt[6]*Mappy3xy2z2 + 2*Sqrt[6]*Mappy3yz2x2 - 4*Sqrt[6]*Mappz3zx2y2))/56, k == 4 && m == -2}, {(3*(9*Eappx3 + 7*Eappxy2z2 - 28*Eappxyz + 9*Eappy3 + 7*Eappyz2x2 + 24*Eappz3 - 28*Eappzx2y2 - 2*Sqrt[15]*Mappx3xy2z2 + 2*Sqrt[15]*Mappy3yz2x2))/56, k == 4 && m == 0}, {(3*(-3*Sqrt[10]*Eappx3 + 7*Sqrt[10]*Eappxy2z2 + 3*Sqrt[10]*Eappy3 - 7*Sqrt[10]*Eappyz2x2 + 2*Sqrt[6]*Mappx3xy2z2 - (12*I)*Sqrt[10]*Mappx3y3 + (8*I)*Sqrt[6]*Mappx3yz2x2 - (4*I)*Sqrt[10]*Mappxy2z2yz2x2 + (4*I)*Sqrt[6]*Mappxyzz3 - (8*I)*Sqrt[6]*Mappy3xy2z2 + 2*Sqrt[6]*Mappy3yz2x2 - 4*Sqrt[6]*Mappz3zx2y2))/56, k == 4 && m == 2}, {(3*(3*Sqrt[5]*Eappx3 - 3*Sqrt[5]*Eappxy2z2 - 4*Sqrt[5]*Eappxyz + 3*Sqrt[5]*Eappy3 - 3*Sqrt[5]*Eappyz2x2 + 4*Sqrt[5]*Eappzx2y2 + 2*Sqrt[3]*Mappx3xy2z2 + (8*I)*Sqrt[3]*Mappx3yz2x2 - (8*I)*Sqrt[5]*Mappxyzzx2y2 + (8*I)*Sqrt[3]*Mappy3xy2z2 - 2*Sqrt[3]*Mappy3yz2x2))/(8*Sqrt[14]), k == 4 && m == 4}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eappx3 + 3*Sqrt[3]*Eappxy2z2 - 5*Sqrt[3]*Eappy3 - 3*Sqrt[3]*Eappyz2x2 - 6*Sqrt[5]*Mappx3xy2z2 - (10*I)*Sqrt[3]*Mappx3y3 - (6*I)*Sqrt[5]*Mappx3yz2x2 + (6*I)*Sqrt[3]*Mappxy2z2yz2x2 + (6*I)*Sqrt[5]*Mappy3xy2z2 - 6*Sqrt[5]*Mappy3yz2x2))/160, k == 6 && m == -6}, {(-13*(15*Eappx3 - 15*Eappxy2z2 + 24*Eappxyz + 15*Eappy3 - 15*Eappyz2x2 - 24*Eappzx2y2 + 2*Sqrt[15]*Mappx3xy2z2 - (8*I)*Sqrt[15]*Mappx3yz2x2 - (48*I)*Mappxyzzx2y2 - (8*I)*Sqrt[15]*Mappy3xy2z2 - 2*Sqrt[15]*Mappy3yz2x2))/(80*Sqrt[14]), k == 6 && m == -4}, {(13*(5*Sqrt[15]*Eappx3 + 3*Sqrt[15]*Eappxy2z2 - 5*Sqrt[15]*Eappy3 - 3*Sqrt[15]*Eappyz2x2 + 34*Mappx3xy2z2 + (2*I)*Sqrt[15]*Mappx3y3 - (26*I)*Mappx3yz2x2 - (14*I)*Sqrt[15]*Mappxy2z2yz2x2 + (64*I)*Mappxyzz3 + (26*I)*Mappy3xy2z2 + 34*Mappy3yz2x2 + 64*Mappz3zx2y2))/(160*Sqrt[7]), k == 6 && m == -2}, {(-13*(25*Eappx3 + 39*Eappxy2z2 - 24*Eappxyz + 25*Eappy3 + 39*Eappyz2x2 - 80*Eappz3 - 24*Eappzx2y2 + 14*Sqrt[15]*Mappx3xy2z2 - 14*Sqrt[15]*Mappy3yz2x2))/560, k == 6 && m == 0}, {(13*(5*Sqrt[15]*Eappx3 + 3*Sqrt[15]*Eappxy2z2 - 5*Sqrt[15]*Eappy3 - 3*Sqrt[15]*Eappyz2x2 + 34*Mappx3xy2z2 - (2*I)*Sqrt[15]*Mappx3y3 + (26*I)*Mappx3yz2x2 + (14*I)*Sqrt[15]*Mappxy2z2yz2x2 - (64*I)*Mappxyzz3 - (26*I)*Mappy3xy2z2 + 34*Mappy3yz2x2 + 64*Mappz3zx2y2))/(160*Sqrt[7]), k == 6 && m == 2}, {(-13*(15*Eappx3 - 15*Eappxy2z2 + 24*Eappxyz + 15*Eappy3 - 15*Eappyz2x2 - 24*Eappzx2y2 + 2*Sqrt[15]*Mappx3xy2z2 + (8*I)*Sqrt[15]*Mappx3yz2x2 + (48*I)*Mappxyzzx2y2 + (8*I)*Sqrt[15]*Mappy3xy2z2 - 2*Sqrt[15]*Mappy3yz2x2))/(80*Sqrt[14]), k == 6 && m == 4}}, (13*Sqrt[11/7]*(5*Sqrt[3]*Eappx3 + 3*Sqrt[3]*Eappxy2z2 - 5*Sqrt[3]*Eappy3 - 3*Sqrt[3]*Eappyz2x2 - 6*Sqrt[5]*Mappx3xy2z2 + (10*I)*Sqrt[3]*Mappx3y3 + (6*I)*Sqrt[5]*Mappx3yz2x2 - (6*I)*Sqrt[3]*Mappxy2z2yz2x2 - (6*I)*Sqrt[5]*Mappy3xy2z2 - 6*Sqrt[5]*Mappy3yz2x2))/160] |
| | |
| | </code> |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{0, 0, (1/7)*(Eappx3 + Eappxy2z2 + Eappxyz + Eappy3 + Eappyz2x2 + Eappz3 + Eappzx2y2)} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {2, 0, (-5/14)*(Eappx3 + Eappy3 + (-2)*(Eappz3) + (-1)*((sqrt(15))*(Mappx3xy2z2)) + (sqrt(15))*(Mappy3yz2x2))} , |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | {2,-2, (5/28)*((1/(sqrt(2)))*((2)*((sqrt(3))*(Eappx3)) + (-2)*((sqrt(3))*(Eappy3)) + (2)*((sqrt(5))*(Mappx3xy2z2)) + (-I)*((sqrt(3))*(Mappx3y3)) + (-I)*((sqrt(5))*(Mappx3yz2x2)) + (-5*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (-4*I)*((sqrt(5))*(Mappxyzz3)) + (I)*((sqrt(5))*(Mappy3xy2z2)) + (2)*((sqrt(5))*(Mappy3yz2x2)) + (-4)*((sqrt(5))*(Mappz3zx2y2))))} , |
| {2, 0, A(2,0)} , | {2, 2, (5/28)*((1/(sqrt(2)))*((2)*((sqrt(3))*(Eappx3)) + (-2)*((sqrt(3))*(Eappy3)) + (2)*((sqrt(5))*(Mappx3xy2z2)) + (I)*((sqrt(3))*(Mappx3y3)) + (I)*((sqrt(5))*(Mappx3yz2x2)) + (5*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (4*I)*((sqrt(5))*(Mappxyzz3)) + (-I)*((sqrt(5))*(Mappy3xy2z2)) + (2)*((sqrt(5))*(Mappy3yz2x2)) + (-4)*((sqrt(5))*(Mappz3zx2y2))))} , |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | {4, 0, (3/56)*((9)*(Eappx3) + (7)*(Eappxy2z2) + (-28)*(Eappxyz) + (9)*(Eappy3) + (7)*(Eappyz2x2) + (24)*(Eappz3) + (-28)*(Eappzx2y2) + (-2)*((sqrt(15))*(Mappx3xy2z2)) + (2)*((sqrt(15))*(Mappy3yz2x2)))} , |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | {4, 2, (3/56)*((-3)*((sqrt(10))*(Eappx3)) + (7)*((sqrt(10))*(Eappxy2z2)) + (3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (2)*((sqrt(6))*(Mappx3xy2z2)) + (-12*I)*((sqrt(10))*(Mappx3y3)) + (8*I)*((sqrt(6))*(Mappx3yz2x2)) + (-4*I)*((sqrt(10))*(Mappxy2z2yz2x2)) + (4*I)*((sqrt(6))*(Mappxyzz3)) + (-8*I)*((sqrt(6))*(Mappy3xy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (-4)*((sqrt(6))*(Mappz3zx2y2)))} , |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | {4,-2, (3/56)*((-3)*((sqrt(10))*(Eappx3)) + (7)*((sqrt(10))*(Eappxy2z2)) + (3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (2)*((sqrt(6))*(Mappx3xy2z2)) + (12*I)*((sqrt(10))*(Mappx3y3)) + (-8*I)*((sqrt(6))*(Mappx3yz2x2)) + (4*I)*((sqrt(10))*(Mappxy2z2yz2x2)) + (-4*I)*((sqrt(6))*(Mappxyzz3)) + (8*I)*((sqrt(6))*(Mappy3xy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (-4)*((sqrt(6))*(Mappz3zx2y2)))} , |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | {4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eappy3)) + (-3)*((sqrt(5))*(Eappyz2x2)) + (4)*((sqrt(5))*(Eappzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (-8*I)*((sqrt(3))*(Mappx3yz2x2)) + (8*I)*((sqrt(5))*(Mappxyzzx2y2)) + (-8*I)*((sqrt(3))*(Mappy3xy2z2)) + (-2)*((sqrt(3))*(Mappy3yz2x2))))} , |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | {4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eappy3)) + (-3)*((sqrt(5))*(Eappyz2x2)) + (4)*((sqrt(5))*(Eappzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (8*I)*((sqrt(3))*(Mappx3yz2x2)) + (-8*I)*((sqrt(5))*(Mappxyzzx2y2)) + (8*I)*((sqrt(3))*(Mappy3xy2z2)) + (-2)*((sqrt(3))*(Mappy3yz2x2))))} , |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | {6, 0, (-13/560)*((25)*(Eappx3) + (39)*(Eappxy2z2) + (-24)*(Eappxyz) + (25)*(Eappy3) + (39)*(Eappyz2x2) + (-80)*(Eappz3) + (-24)*(Eappzx2y2) + (14)*((sqrt(15))*(Mappx3xy2z2)) + (-14)*((sqrt(15))*(Mappy3yz2x2)))} , |
| {4, 0, A(4,0)} , | {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eappy3)) + (-3)*((sqrt(15))*(Eappyz2x2)) + (34)*(Mappx3xy2z2) + (-2*I)*((sqrt(15))*(Mappx3y3)) + (26*I)*(Mappx3yz2x2) + (14*I)*((sqrt(15))*(Mappxy2z2yz2x2)) + (-64*I)*(Mappxyzz3) + (-26*I)*(Mappy3xy2z2) + (34)*(Mappy3yz2x2) + (64)*(Mappz3zx2y2)))} , |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eappy3)) + (-3)*((sqrt(15))*(Eappyz2x2)) + (34)*(Mappx3xy2z2) + (2*I)*((sqrt(15))*(Mappx3y3)) + (-26*I)*(Mappx3yz2x2) + (-14*I)*((sqrt(15))*(Mappxy2z2yz2x2)) + (64*I)*(Mappxyzz3) + (26*I)*(Mappy3xy2z2) + (34)*(Mappy3yz2x2) + (64)*(Mappz3zx2y2)))} , |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | {6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (-24)*(Eappzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (-8*I)*((sqrt(15))*(Mappx3yz2x2)) + (-48*I)*(Mappxyzzx2y2) + (-8*I)*((sqrt(15))*(Mappy3xy2z2)) + (-2)*((sqrt(15))*(Mappy3yz2x2))))} , |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | {6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (-24)*(Eappzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (8*I)*((sqrt(15))*(Mappx3yz2x2)) + (48*I)*(Mappxyzzx2y2) + (8*I)*((sqrt(15))*(Mappy3xy2z2)) + (-2)*((sqrt(15))*(Mappy3yz2x2))))} , |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eappy3)) + (-3)*((sqrt(3))*(Eappyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (-10*I)*((sqrt(3))*(Mappx3y3)) + (-6*I)*((sqrt(5))*(Mappx3yz2x2)) + (6*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (6*I)*((sqrt(5))*(Mappy3xy2z2)) + (-6)*((sqrt(5))*(Mappy3yz2x2))))} , |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eappy3)) + (-3)*((sqrt(3))*(Eappyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (10*I)*((sqrt(3))*(Mappx3y3)) + (6*I)*((sqrt(5))*(Mappx3yz2x2)) + (-6*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (-6*I)*((sqrt(5))*(Mappy3xy2z2)) + (-6)*((sqrt(5))*(Mappy3yz2x2))))} } |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | |
| {6, 0, A(6,0)} , | |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **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)}} ^ {Y_{-2}^{(3)}} ^ {Y_{-1}^{(3)}} ^ {Y_{0}^{(3)}} ^ {Y_{1}^{(3)}} ^ {Y_{2}^{(3)}} ^ {Y_{3}^{(3)}} ^ |
| ^ {Y_{-3}^{(3)}} |$ A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0) | 0 | -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)-i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i \text{Ap}(6,2)) | 0 | \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)-i \text{Ap}(6,4))\right) | 0 | -\frac{10}{13} \sqrt{\frac{7}{33}} (A(6,6)-i \text{Ap}(6,6)) $| | ^ {Y_{-3}^{(3)}} |$ \frac{1}{16} \left(5 \text{Eappx3}+3 \text{Eappxy2z2}+5 \text{Eappy3}+3 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mappx3xy2z2})\right) | 0 | \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}+2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}+i \text{Mappy3yz2x2}\right)\right) | 0 | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}-4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eappy3}+3 \text{Eappyz2x2}+2 \left(\sqrt{15} \text{Mappx3xy2z2}+5 i \text{Mappx3y3}+i \sqrt{15} \text{Mappx3yz2x2}-3 i \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3yz2x2}-i \text{Mappy3xy2z2})\right)\right) $| |
| ^ {Y_{-2}^{(3)}} | 0 |$ A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0) | 0 | -\frac{2 (A(2,2)-i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i \text{Ap}(6,2)) | 0 | \frac{1}{429} \sqrt{14} \left(13 \sqrt{5} (A(4,4)-i \text{Ap}(4,4))+30 (A(6,4)-i \text{Ap}(6,4))\right) | 0 $| | ^ {Y_{-2}^{(3)}} | 0 |$ \frac{\text{Eappxyz}+\text{Eappzx2y2}}{2} | 0 | \frac{\text{Mappz3zx2y2}+i \text{Mappxyzz3}}{\sqrt{2}} | 0 | \frac{1}{2} (-\text{Eappxyz}+\text{Eappzx2y2}+2 i \text{Mappxyzzx2y2}) | 0 $| |
| ^ {Y_{-1}^{(3)}} |$ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)+i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i \text{Ap}(6,2)) | 0 | A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0) | 0 | \frac{2 \left(-143 \sqrt{6} (A(2,2)-i \text{Ap}(2,2))-65 \sqrt{10} (A(4,2)-i \text{Ap}(4,2))-25 \sqrt{105} (A(6,2)-i \text{Ap}(6,2))\right)}{2145} | 0 | \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)-i \text{Ap}(6,4))\right) $| | ^ {Y_{-1}^{(3)}} |$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}-2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}-i \text{Mappy3yz2x2}\right)\right) | 0 | \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}-2 \left(\sqrt{15} \text{Mappx3xy2z2}+3 i \text{Mappx3y3}-i \sqrt{15} \text{Mappx3yz2x2}-5 i \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3yz2x2}+i \text{Mappy3xy2z2})\right)\right) | 0 | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}-4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) $| |
| ^ {Y_{0}^{(3)}} | 0 |$ -\frac{2 (A(2,2)+i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i \text{Ap}(6,2)) | 0 | A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0)) | 0 | -\frac{2 (A(2,2)-i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i \text{Ap}(6,2)) | 0 $| | ^ {Y_{0}^{(3)}} | 0 |$ \frac{\text{Mappz3zx2y2}-i \text{Mappxyzz3}}{\sqrt{2}} | 0 | \text{Eappz3} | 0 | \frac{\text{Mappz3zx2y2}+i \text{Mappxyzz3}}{\sqrt{2}} | 0 $| |
| ^$ {Y_{1}^{(3)}} | \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)+i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)+i \text{Ap}(6,4))\right) | 0 | \frac{2 \left(-143 \sqrt{6} (A(2,2)+i \text{Ap}(2,2))-65 \sqrt{10} (A(4,2)+i \text{Ap}(4,2))-25 \sqrt{105} (A(6,2)+i \text{Ap}(6,2))\right)}{2145} | 0 | A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0) | 0 | -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)-i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i \text{Ap}(6,2)) $| | ^$ {Y_{1}^{(3)}} | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}+4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}-2 \sqrt{15} \text{Mappx3xy2z2}+2 i \left(3 \text{Mappx3y3}-\sqrt{15} \text{Mappx3yz2x2}-5 \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3xy2z2}+i \text{Mappy3yz2x2})\right)\right) | 0 | \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}+2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}+i \text{Mappy3yz2x2}\right)\right) $| |
| ^ {Y_{2}^{(3)}} | 0 |$ \frac{1}{429} \sqrt{14} \left(13 \sqrt{5} (A(4,4)+i \text{Ap}(4,4))+30 (A(6,4)+i \text{Ap}(6,4))\right) | 0 | -\frac{2 (A(2,2)+i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i \text{Ap}(6,2)) | 0 | A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0) | 0 $| | ^ {Y_{2}^{(3)}} | 0 |$ \frac{1}{2} (-\text{Eappxyz}+\text{Eappzx2y2}-2 i \text{Mappxyzzx2y2}) | 0 | \frac{\text{Mappz3zx2y2}-i \text{Mappxyzz3}}{\sqrt{2}} | 0 | \frac{\text{Eappxyz}+\text{Eappzx2y2}}{2} | 0 $| |
| ^ {Y_{3}^{(3)}} |$ -\frac{10}{13} \sqrt{\frac{7}{33}} (A(6,6)+i \text{Ap}(6,6)) | 0 | \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)+i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)+i \text{Ap}(6,4))\right) | 0 | -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)+i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i \text{Ap}(6,2)) | 0 | A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0) $| | ^ {Y_{3}^{(3)}} |$ \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eappy3}+3 \text{Eappyz2x2}+2 \left(\sqrt{15} \text{Mappx3xy2z2}-5 i \text{Mappx3y3}-i \sqrt{15} \text{Mappx3yz2x2}+3 i \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3yz2x2}+i \text{Mappy3xy2z2})\right)\right) | 0 | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}+4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}-2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}-i \text{Mappy3yz2x2}\right)\right) | 0 | \frac{1}{16} \left(5 \text{Eappx3}+3 \text{Eappxy2z2}+5 \text{Eappy3}+3 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mappx3xy2z2})\right) $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **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_{x\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}} ^ 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}} |$ \frac{1}{429} \left(429 A(0,0)-91 A(4,0)-13 \sqrt{70} A(4,4)+30 A(6,0)-30 \sqrt{14} A(6,4)\right) | 0 | 0 | \frac{286 \sqrt{10} \text{Ap}(2,2)+65 \sqrt{6} \text{Ap}(4,2)-200 \sqrt{7} \text{Ap}(6,2)}{2145} | 0 | 0 | -\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} \text{Ap}(4,4)+30 \text{Ap}(6,4)\right) $| | ^ f_{\text{xyz}} |$ \text{Eappxyz} | 0 | 0 | \text{Mappxyzz3} | 0 | 0 | \text{Mappxyzzx2y2} $| |
| ^ f_{x\left(5x^2-r^2\right)} | 0 |$ \frac{8580 A(0,0)-1144 A(2,0)+1144 \sqrt{6} A(2,2)+585 A(4,0)-390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)+125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)+125 \sqrt{231} A(6,6)}{8580} | \frac{286 \sqrt{6} \text{Ap}(2,2)-780 \sqrt{10} \text{Ap}(4,2)-25 \sqrt{105} \text{Ap}(6,2)+125 \sqrt{231} \text{Ap}(6,6)}{8580} | 0 | \frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)-26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)-85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)+15 \sqrt{385} A(6,6)\right)}{8580} | \frac{286 \sqrt{10} \text{Ap}(2,2)+5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)-20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} | 0 $| | ^ f_{x\left(5x^2-r^2\right)} | 0 |$ \text{Eappx3} | \text{Mappx3y3} | 0 | \text{Mappx3xy2z2} | \text{Mappx3yz2x2} | 0 $| |
| ^ f_{y\left(5y^2-r^2\right)} | 0 |$ \frac{286 \sqrt{6} \text{Ap}(2,2)-780 \sqrt{10} \text{Ap}(4,2)-25 \sqrt{105} \text{Ap}(6,2)+125 \sqrt{231} \text{Ap}(6,6)}{8580} | \frac{8580 A(0,0)-1144 A(2,0)-1144 \sqrt{6} A(2,2)+585 A(4,0)+390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)-125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)-125 \sqrt{231} A(6,6)}{8580} | 0 | \frac{-286 \sqrt{10} \text{Ap}(2,2)-5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(-52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)+20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} | \frac{-572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)+5 \left(13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{8580} | 0 $| | ^ f_{y\left(5y^2-r^2\right)} | 0 |$ \text{Mappx3y3} | \text{Eappy3} | 0 | \text{Mappy3xy2z2} | \text{Mappy3yz2x2} | 0 $| |
| ^$ f_{x\left(5z^2-r^2\right)} | \frac{286 \sqrt{10} \text{Ap}(2,2)+65 \sqrt{6} \text{Ap}(4,2)-200 \sqrt{7} \text{Ap}(6,2)}{2145} | 0 | 0 | A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0)) | 0 | 0 | \frac{-286 \sqrt{10} A(2,2)-65 \sqrt{6} A(4,2)+200 \sqrt{7} A(6,2)}{2145} $| | ^$ f_{z\left(5z^2-r^2\right)} | \text{Mappxyzz3} | 0 | 0 | \text{Eappz3} | 0 | 0 | \text{Mappz3zx2y2} $| |
| ^ f_{x\left(y^2-z^2\right)} | 0 |$ \frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)-26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)-85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)+15 \sqrt{385} A(6,6)\right)}{8580} | \frac{-286 \sqrt{10} \text{Ap}(2,2)-5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(-52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)+20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} | 0 | \frac{1716 A(0,0)+91 A(4,0)+182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)+15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)+15 \sqrt{231} A(6,6)}{1716} | \frac{286 \sqrt{6} \text{Ap}(2,2)-52 \sqrt{10} \text{Ap}(4,2)+35 \sqrt{105} \text{Ap}(6,2)-15 \sqrt{231} \text{Ap}(6,6)}{1716} | 0 $| | ^ f_{x\left(y^2-z^2\right)} | 0 |$ \text{Mappx3xy2z2} | \text{Mappy3xy2z2} | 0 | \text{Eappxy2z2} | \text{Mappxy2z2yz2x2} | 0 $| |
| ^ f_{y\left(z^2-x^2\right)} | 0 |$ \frac{286 \sqrt{10} \text{Ap}(2,2)+5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)-20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} | \frac{-572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)+5 \left(13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{8580} | 0 | \frac{286 \sqrt{6} \text{Ap}(2,2)-52 \sqrt{10} \text{Ap}(4,2)+35 \sqrt{105} \text{Ap}(6,2)-15 \sqrt{231} \text{Ap}(6,6)}{1716} | \frac{1716 A(0,0)+91 A(4,0)-182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)-15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)-15 \sqrt{231} A(6,6)}{1716} | 0 $| | ^ f_{y\left(z^2-x^2\right)} | 0 |$ \text{Mappx3yz2x2} | \text{Mappy3yz2x2} | 0 | \text{Mappxy2z2yz2x2} | \text{Eappyz2x2} | 0 $| |
| ^ f_{z\left(x^2-y^2\right)} |$ -\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} \text{Ap}(4,4)+30 \text{Ap}(6,4)\right) | 0 | 0 | \frac{-286 \sqrt{10} A(2,2)-65 \sqrt{6} A(4,2)+200 \sqrt{7} A(6,2)}{2145} | 0 | 0 | \frac{1}{429} \left(429 A(0,0)-91 A(4,0)+13 \sqrt{70} A(4,4)+30 A(6,0)+30 \sqrt{14} A(6,4)\right) $| | ^ f_{z\left(x^2-y^2\right)} |$ \text{Mappxyzzx2y2} | 0 | 0 | \text{Mappz3zx2y2} | 0 | 0 | \text{Eappzx2y2} $| |
| |
| |
| ### | ### |
| |
| </hidden><hidden **Rotation matrix used** > | </hidden> |
| | <hidden **Rotation matrix used** > |
| |
| ### | ### |
| |
| TODO | | $ ^ {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 | |
| |
| ### | ### |
| |
| </hidden><hidden **Irriducible representations and their onsite energy** > | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| |
| ### | ### |
| |
| TODO | ^ ^\text{Eappxyz} | {{:physics_chemistry:pointgroup:cs_z_orb_3_1.png?150}} | |
| | |\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{Eappx3} | {{:physics_chemistry:pointgroup:cs_z_orb_3_2.png?150}} | |
| | |\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right) | ::: | |
| | |\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right) | ::: | |
| | ^ ^\text{Eappy3} | {{:physics_chemistry:pointgroup:cs_z_orb_3_3.png?150}} | |
| | |\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{Eappz3} | {{:physics_chemistry:pointgroup:cs_z_orb_3_4.png?150}} | |
| | |\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{Eappxy2z2} | {{:physics_chemistry:pointgroup:cs_z_orb_3_5.png?150}} | |
| | |\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} |-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right) | ::: | |
| | |\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} |-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right) | ::: | |
| | ^ ^\text{Eappyz2x2} | {{:physics_chemistry:pointgroup:cs_z_orb_3_6.png?150}} | |
| | |\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{Eappzx2y2} | {{:physics_chemistry:pointgroup:cs_z_orb_3_7.png?150}} | |
| | |\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) | ::: | |
| |
| ### | ### |
| |
| </hidden>===== Coupling between two shells ===== | </hidden> |
| | ===== Coupling between two shells ===== |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
| | |
| | ### |
| |
| ==== Potential for s-p orbital mixing ==== | ==== Potential for s-p orbital mixing ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | 0 & k\neq 1\lor (m\neq -1\land m\neq 1) \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | A(1,1)+i B(1,1) & \text{True} |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | |
| A(2,0) & k=2\land m=0 \\ | |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | |
| A(4,0) & k=4\land m=0 \\ | |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | |
| A(6,0) & k=6\land m=0 \\ | |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | |
| \end{cases}$$ | \end{cases}$$ |
| |
| ### | ### |
| |
| <\hidden><hidden **Input format suitable for Quanty** > | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 1 || (m != -1 && m != 1)}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}}, A[1, 1] + I*B[1, 1]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {1, 1, A(1,1) + (I)*(B(1,1))} } |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | |
| {2, 0, A(2,0)} , | |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | |
| {4, 0, A(4,0)} , | |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | |
| {6, 0, A(6,0)} , | |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| |
| ### | ### |
| |
| | ^ {Y_{-1}^{(1)}} ^ {Y_{0}^{(1)}} ^ {Y_{1}^{(1)}} ^ | | ^ {Y_{-1}^{(1)}} ^ {Y_{0}^{(1)}} ^ {Y_{1}^{(1)}} ^ |
| ^ {Y_{0}^{(0)}} |$ -\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{3}} | 0 | \frac{A(1,1)-i \text{Ap}(1,1)}{\sqrt{3}} $| | ^ {Y_{0}^{(0)}} |$ -\frac{A(1,1)+i B(1,1)}{\sqrt{3}} | 0 | \frac{A(1,1)-i B(1,1)}{\sqrt{3}} $| |
| |
| |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| |
| ### | ### |
| |
| | ^ p_x ^ p_y ^ p_z ^ | | ^ p_x ^ p_y ^ p_z ^ |
| ^ \text{s} | -\sqrt{\frac{2}{3}} A(1,1) |$ \sqrt{\frac{2}{3}} \text{Ap}(1,1) | 0 $| | ^ \text{s} | -\sqrt{\frac{2}{3}} A(1,1) |$ \sqrt{\frac{2}{3}} B(1,1) | 0 $| |
| |
| |
| ### | ### |
| |
| </hidden>==== Potential for s-d orbital mixing ==== | </hidden> |
| | ==== Potential for s-d orbital mixing ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | |
| A(2,0) & k=2\land m=0 \\ | A(2,0) & k=2\land m=0 \\ |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | A(2,2)+i B(2,2) & \text{True} |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | |
| A(4,0) & k=4\land m=0 \\ | |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | |
| A(6,0) & k=6\land m=0 \\ | |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | |
| \end{cases}$$ | \end{cases}$$ |
| |
| ### | ### |
| |
| <\hidden><hidden **Input format suitable for Quanty** > | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != 0 && m != 2)}, {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]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{2, 0, A(2,0)} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | {2, 2, A(2,2) + (I)*(B(2,2))} } |
| {2, 0, A(2,0)} , | |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | |
| {4, 0, A(4,0)} , | |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | |
| {6, 0, A(6,0)} , | |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **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)}} ^ {Y_{-1}^{(2)}} ^ {Y_{0}^{(2)}} ^ {Y_{1}^{(2)}} ^ {Y_{2}^{(2)}} ^ |
| ^ {Y_{0}^{(0)}} |$ \frac{A(2,2)+i \text{Ap}(2,2)}{\sqrt{5}} | 0 | \frac{A(2,0)}{\sqrt{5}} | 0 | \frac{A(2,2)-i \text{Ap}(2,2)}{\sqrt{5}} $| | ^ {Y_{0}^{(0)}} |$ \frac{A(2,2)+i B(2,2)}{\sqrt{5}} | 0 | \frac{A(2,0)}{\sqrt{5}} | 0 | \frac{A(2,2)-i B(2,2)}{\sqrt{5}} $| |
| |
| |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| |
| ### | ### |
| |
| | ^ d_{x^2-y^2} ^ d_{3z^2-r^2} ^ d_{\text{yz}} ^ d_{\text{xz}} ^ d_{\text{xy}} ^ | | ^ d_{x^2-y^2} ^ d_{3z^2-r^2} ^ d_{\text{yz}} ^ d_{\text{xz}} ^ d_{\text{xy}} ^ |
| ^ \text{s} | \sqrt{\frac{2}{5}} A(2,2) | \frac{A(2,0)}{\sqrt{5}} | 0 | 0 |$ -\sqrt{\frac{2}{5}} \text{Ap}(2,2) $| | ^ \text{s} | \sqrt{\frac{2}{5}} A(2,2) | \frac{A(2,0)}{\sqrt{5}} | 0 | 0 |$ -\sqrt{\frac{2}{5}} B(2,2) $| |
| |
| |
| ### | ### |
| |
| </hidden>==== Potential for s-f orbital mixing ==== | </hidden> |
| | ==== Potential for s-f orbital mixing ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | 0 & k\neq 3\lor (m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3) \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| A(2,0) & k=2\land m=0 \\ | A(3,3)+i B(3,3) & \text{True} |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | |
| A(4,0) & k=4\land m=0 \\ | |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | |
| A(6,0) & k=6\land m=0 \\ | |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | |
| \end{cases}$$ | \end{cases}$$ |
| |
| ### | ### |
| |
| <\hidden><hidden **Input format suitable for Quanty** > | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != -1 && m != 1 && m != 3)}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}}, A[3, 3] + I*B[3, 3]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , |
| {2, 0, A(2,0)} , | {3, 3, A(3,3) + (I)*(B(3,3))} } |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | |
| {4, 0, A(4,0)} , | |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | |
| {6, 0, A(6,0)} , | |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **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)}} ^ {Y_{-2}^{(3)}} ^ {Y_{-1}^{(3)}} ^ {Y_{0}^{(3)}} ^ {Y_{1}^{(3)}} ^ {Y_{2}^{(3)}} ^ {Y_{3}^{(3)}} ^ |
| ^ {Y_{0}^{(0)}} |$ -\frac{A(3,3)+i \text{Ap}(3,3)}{\sqrt{7}} | 0 | -\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{7}} | 0 | \frac{A(3,1)-i \text{Ap}(3,1)}{\sqrt{7}} | 0 | \frac{A(3,3)-i \text{Ap}(3,3)}{\sqrt{7}} $| | ^ {Y_{0}^{(0)}} |$ -\frac{A(3,3)+i B(3,3)}{\sqrt{7}} | 0 | -\frac{A(3,1)+i B(3,1)}{\sqrt{7}} | 0 | \frac{A(3,1)-i B(3,1)}{\sqrt{7}} | 0 | \frac{A(3,3)-i B(3,3)}{\sqrt{7}} $| |
| |
| |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **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_{x\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}} ^ 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)} $ ^ |
| ^ \text{s} | 0 | \frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right) |$ -\frac{\sqrt{3} \text{Ap}(3,1)+\sqrt{5} \text{Ap}(3,3)}{2 \sqrt{7}} | 0 | \frac{\sqrt{5} A(3,1)+\sqrt{3} A(3,3)}{2 \sqrt{7}} | \frac{1}{14} \left(\sqrt{35} \text{Ap}(3,1)-\sqrt{21} \text{Ap}(3,3)\right) | 0 $| | ^ \text{s} | 0 | \frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right) |$ -\frac{\sqrt{3} B(3,1)+\sqrt{5} B(3,3)}{2 \sqrt{7}} | 0 | \frac{\sqrt{5} A(3,1)+\sqrt{3} A(3,3)}{2 \sqrt{7}} | \frac{1}{14} \left(\sqrt{35} B(3,1)-\sqrt{21} B(3,3)\right) | 0 $| |
| |
| |
| ### | ### |
| |
| </hidden>==== Potential for p-d orbital mixing ==== | </hidden> |
| | ==== Potential for p-d orbital mixing ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | 0 & (k\neq 3\land (k\neq 1\lor (m\neq -1\land m\neq 1)))\lor (m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3) \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| A(2,0) & k=2\land m=0 \\ | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | A(3,3)+i B(3,3) & \text{True} |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | |
| A(4,0) & k=4\land m=0 \\ | |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | |
| A(6,0) & k=6\land m=0 \\ | |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | |
| \end{cases}$$ | \end{cases}$$ |
| |
| ### | ### |
| |
| <\hidden><hidden **Input format suitable for Quanty** > | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || (m != -1 && m != 1))) || (m != -3 && m != -1 && m != 1 && m != 3)}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}}, A[3, 3] + I*B[3, 3]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| {2, 0, A(2,0)} , | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | {3, 3, A(3,3) + (I)*(B(3,3))} } |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | |
| {4, 0, A(4,0)} , | |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | |
| {6, 0, A(6,0)} , | |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **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)}} ^ {Y_{-1}^{(2)}} ^ {Y_{0}^{(2)}} ^ {Y_{1}^{(2)}} ^ {Y_{2}^{(2)}} ^ |
| ^ {Y_{-1}^{(1)}} |$ \frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)+i \text{Ap}(3,1))-\sqrt{\frac{2}{5}} (A(1,1)+i \text{Ap}(1,1)) | 0 | \frac{1}{105} \left(7 \sqrt{15} (A(1,1)-i \text{Ap}(1,1))-9 \sqrt{10} (A(3,1)-i \text{Ap}(3,1))\right) | 0 | -\frac{3}{7} (A(3,3)-i \text{Ap}(3,3)) $| | ^ {Y_{-1}^{(1)}} |$ \frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)+i B(3,1))-\sqrt{\frac{2}{5}} (A(1,1)+i B(1,1)) | 0 | \frac{1}{105} \left(7 \sqrt{15} (A(1,1)-i B(1,1))-9 \sqrt{10} (A(3,1)-i B(3,1))\right) | 0 | -\frac{3}{7} (A(3,3)-i B(3,3)) $| |
| ^ {Y_{0}^{(1)}} | 0 |$ -\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (A(3,1)+i \text{Ap}(3,1)) | 0 | \frac{7 A(1,1)+2 \sqrt{6} A(3,1)-i \left(7 \text{Ap}(1,1)+2 \sqrt{6} \text{Ap}(3,1)\right)}{7 \sqrt{5}} | 0 $| | ^ {Y_{0}^{(1)}} | 0 |$ -\frac{A(1,1)+i B(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (A(3,1)+i B(3,1)) | 0 | \frac{7 A(1,1)+2 \sqrt{6} A(3,1)-i \left(7 B(1,1)+2 \sqrt{6} B(3,1)\right)}{7 \sqrt{5}} | 0 $| |
| ^ {Y_{1}^{(1)}} |$ \frac{3}{7} (A(3,3)+i \text{Ap}(3,3)) | 0 | \frac{3}{7} \sqrt{\frac{2}{5}} (A(3,1)+i \text{Ap}(3,1))-\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{15}} | 0 | \sqrt{\frac{2}{5}} (A(1,1)-i \text{Ap}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)-i \text{Ap}(3,1)) $| | ^ {Y_{1}^{(1)}} |$ \frac{3}{7} (A(3,3)+i B(3,3)) | 0 | \frac{3}{7} \sqrt{\frac{2}{5}} (A(3,1)+i B(3,1))-\frac{A(1,1)+i B(1,1)}{\sqrt{15}} | 0 | \sqrt{\frac{2}{5}} (A(1,1)-i B(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)-i B(3,1)) $| |
| |
| |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| |
| ### | ### |
| |
| | ^ d_{x^2-y^2} ^ d_{3z^2-r^2} ^ d_{\text{yz}} ^ d_{\text{xz}} ^ d_{\text{xy}} ^ | | ^ d_{x^2-y^2} ^ d_{3z^2-r^2} ^ d_{\text{yz}} ^ d_{\text{xz}} ^ d_{\text{xy}} ^ |
| ^ p_x | \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)-15 A(3,3)\right) | \sqrt{\frac{2}{15}} A(1,1)-\frac{6 A(3,1)}{7 \sqrt{5}} | 0 | 0 |$ \sqrt{\frac{2}{5}} \text{Ap}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Ap}(3,1)+\frac{3}{7} \text{Ap}(3,3) $| | ^ p_x | \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)-15 A(3,3)\right) | \sqrt{\frac{2}{15}} A(1,1)-\frac{6 A(3,1)}{7 \sqrt{5}} | 0 | 0 |$ \sqrt{\frac{2}{5}} B(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} B(3,1)+\frac{3}{7} B(3,3) $| |
| ^ p_y |$ \frac{1}{35} \left(-7 \sqrt{10} \text{Ap}(1,1)+\sqrt{15} \text{Ap}(3,1)+15 \text{Ap}(3,3)\right) | \frac{6 \text{Ap}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Ap}(1,1) | 0 | 0 | \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right) $| | ^ p_y |$ \frac{1}{35} \left(-7 \sqrt{10} B(1,1)+\sqrt{15} B(3,1)+15 B(3,3)\right) | \frac{6 B(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} B(1,1) | 0 | 0 | \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right) $| |
| ^ p_z | 0 | 0 |$ \sqrt{\frac{2}{5}} \text{Ap}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Ap}(3,1) | -\frac{1}{7} \sqrt{\frac{2}{5}} \left(7 A(1,1)+2 \sqrt{6} A(3,1)\right) | 0 $| | ^ p_z | 0 | 0 |$ \sqrt{\frac{2}{5}} B(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} B(3,1) | -\frac{1}{7} \sqrt{\frac{2}{5}} \left(7 A(1,1)+2 \sqrt{6} A(3,1)\right) | 0 $| |
| |
| |
| ### | ### |
| |
| </hidden>==== Potential for p-f orbital mixing ==== | </hidden> |
| | ==== Potential for p-f orbital mixing ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | 0 & (k\neq 4\land (k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | |
| A(2,0) & k=2\land m=0 \\ | A(2,0) & k=2\land m=0 \\ |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | |
| A(4,0) & k=4\land m=0 \\ | A(4,0) & k=4\land m=0 \\ |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | A(4,4)+i B(4,4) & \text{True} |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | |
| A(6,0) & k=6\land m=0 \\ | |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | |
| \end{cases}$$ | \end{cases}$$ |
| |
| ### | ### |
| |
| <\hidden><hidden **Input format suitable for Quanty** > | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {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[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]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{2, 0, A(2,0)} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| {2, 0, A(2,0)} , | |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | |
| {4, 0, A(4,0)} , | {4, 0, A(4,0)} , |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | {4,-2, A(4,2) + (-I)*(B(4,2))} , |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | {4, 4, A(4,4) + (I)*(B(4,4))} } |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | |
| {6, 0, A(6,0)} , | |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **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)}} ^ {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 \text{Ap}(2,2))}{\sqrt{35}}-\frac{A(4,2)+i \text{Ap}(4,2)}{3 \sqrt{21}} | 0 | \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) | 0 | \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)-i \text{Ap}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i \text{Ap}(4,2)) | 0 | -\frac{2 (A(4,4)-i \text{Ap}(4,4))}{3 \sqrt{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}} | 0 | \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) | 0 | \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)) | 0 | -\frac{2 (A(4,4)-i B(4,4))}{3 \sqrt{3}} $| |
| ^ {Y_{0}^{(1)}} | 0 |$ \sqrt{\frac{3}{35}} (A(2,2)+i \text{Ap}(2,2))+\frac{2 (A(4,2)+i \text{Ap}(4,2))}{3 \sqrt{7}} | 0 | \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} | 0 | \sqrt{\frac{3}{35}} (A(2,2)-i \text{Ap}(2,2))+\frac{2 (A(4,2)-i \text{Ap}(4,2))}{3 \sqrt{7}} | 0 $| | ^ {Y_{0}^{(1)}} | 0 |$ \sqrt{\frac{3}{35}} (A(2,2)+i B(2,2))+\frac{2 (A(4,2)+i B(4,2))}{3 \sqrt{7}} | 0 | \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} | 0 | \sqrt{\frac{3}{35}} (A(2,2)-i B(2,2))+\frac{2 (A(4,2)-i B(4,2))}{3 \sqrt{7}} | 0 $| |
| ^ {Y_{1}^{(1)}} |$ -\frac{2 (A(4,4)+i \text{Ap}(4,4))}{3 \sqrt{3}} | 0 | \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i \text{Ap}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i \text{Ap}(4,2)) | 0 | \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) | 0 | \frac{3 (A(2,2)-i \text{Ap}(2,2))}{\sqrt{35}}-\frac{A(4,2)-i \text{Ap}(4,2)}{3 \sqrt{21}} $| | ^ {Y_{1}^{(1)}} |$ -\frac{2 (A(4,4)+i B(4,4))}{3 \sqrt{3}} | 0 | \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)) | 0 | \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) | 0 | \frac{3 (A(2,2)-i B(2,2))}{\sqrt{35}}-\frac{A(4,2)-i B(4,2)}{3 \sqrt{21}} $| |
| |
| |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **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_{x\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}} ^ 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 | 0 | \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) |$ \frac{1}{630} \left(54 \sqrt{14} \text{Ap}(2,2)+5 \sqrt{30} \left(\sqrt{7} \text{Ap}(4,2)+7 \text{Ap}(4,4)\right)\right) | 0 | \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) | \sqrt{\frac{6}{35}} \text{Ap}(2,2)-\frac{\text{Ap}(4,2)}{\sqrt{14}}+\frac{\text{Ap}(4,4)}{3 \sqrt{2}} | 0 $| | ^ p_x | 0 | \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) |$ \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) | 0 | \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) | \sqrt{\frac{6}{35}} B(2,2)-\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} | 0 $| |
| ^ p_y | 0 |$ \frac{1}{630} \left(54 \sqrt{14} \text{Ap}(2,2)+5 \sqrt{30} \left(\sqrt{7} \text{Ap}(4,2)-7 \text{Ap}(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) | 0 | -\sqrt{\frac{6}{35}} \text{Ap}(2,2)+\frac{\text{Ap}(4,2)}{\sqrt{14}}+\frac{\text{Ap}(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) | 0 $| | ^ p_y | 0 |$ \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) | 0 | -\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) | 0 $| |
| ^ p_z |$ -\sqrt{\frac{6}{35}} \text{Ap}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Ap}(4,2) | 0 | 0 | \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} | 0 | 0 | \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $| | ^ p_z |$ -\sqrt{\frac{6}{35}} B(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} B(4,2) | 0 | 0 | \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} | 0 | 0 | \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $| |
| |
| |
| ### | ### |
| |
| </hidden>==== Potential for d-f orbital mixing ==== | </hidden> |
| | ==== Potential for d-f orbital mixing ==== |
| |
| <hidden **Input format suitable for Mathematica (Quanty.nb)** > | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| |
| ### | ### |
| |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | 0 & (k\neq 5\land (((k\neq 1\lor (m\neq -1\land m\neq 1))\land k\neq 3)\lor (m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3)))\lor (m\neq -5\land m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3\land m\neq 5) \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| A(2,0) & k=2\land m=0 \\ | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| A(4,0) & k=4\land m=0 \\ | A(5,5)+i B(5,5) & \text{True} |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | |
| A(6,0) & k=6\land m=0 \\ | |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | |
| \end{cases}$$ | \end{cases}$$ |
| |
| ### | ### |
| |
| <\hidden><hidden **Input format suitable for Quanty** > | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 5 && (((k != 1 || (m != -1 && m != 1)) && k != 3) || (m != -3 && m != -1 && m != 1 && m != 3))) || (m != -5 && m != -3 && m != -1 && m != 1 && m != 3 && m != 5)}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}}, A[5, 5] + I*B[5, 5]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| |
| ### | ### |
| <code Quanty Akm_Cs_Z.Quanty> | <code Quanty Akm_Cs_Z.Quanty> |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} , | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| {1, 1, A(1,1) + (I)*(Ap(1,1))} , | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| {2, 0, A(2,0)} , | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| {2,-2, A(2,2) + (-I)*(Ap(2,2))} , | {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , |
| {2, 2, A(2,2) + (I)*(Ap(2,2))} , | {3, 3, A(3,3) + (I)*(B(3,3))} , |
| {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} , | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| {3, 1, A(3,1) + (I)*(Ap(3,1))} , | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} , | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| {3, 3, A(3,3) + (I)*(Ap(3,3))} , | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| {4, 0, A(4,0)} , | {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} , |
| {4,-2, A(4,2) + (-I)*(Ap(4,2))} , | {5, 5, A(5,5) + (I)*(B(5,5))} } |
| {4, 2, A(4,2) + (I)*(Ap(4,2))} , | |
| {4,-4, A(4,4) + (-I)*(Ap(4,4))} , | |
| {4, 4, A(4,4) + (I)*(Ap(4,4))} , | |
| {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} , | |
| {5, 1, A(5,1) + (I)*(Ap(5,1))} , | |
| {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} , | |
| {5, 3, A(5,3) + (I)*(Ap(5,3))} , | |
| {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} , | |
| {5, 5, A(5,5) + (I)*(Ap(5,5))} , | |
| {6, 0, A(6,0)} , | |
| {6,-2, A(6,2) + (-I)*(Ap(6,2))} , | |
| {6, 2, A(6,2) + (I)*(Ap(6,2))} , | |
| {6,-4, A(6,4) + (-I)*(Ap(6,4))} , | |
| {6, 4, A(6,4) + (I)*(Ap(6,4))} , | |
| {6,-6, A(6,6) + (-I)*(Ap(6,6))} , | |
| {6, 6, A(6,6) + (I)*(Ap(6,6))} } | |
| |
| </code> | </code> |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** > | </hidden> |
| | <hidden **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)}} ^ {Y_{-2}^{(3)}} ^ {Y_{-1}^{(3)}} ^ {Y_{0}^{(3)}} ^ {Y_{1}^{(3)}} ^ {Y_{2}^{(3)}} ^ {Y_{3}^{(3)}} ^ |
| ^ {Y_{-2}^{(2)}} |$ -\sqrt{\frac{3}{7}} (A(1,1)+i \text{Ap}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)+i \text{Ap}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)+i \text{Ap}(5,1)) | 0 | \frac{33 \sqrt{35} (A(1,1)-i \text{Ap}(1,1))-22 \sqrt{210} (A(3,1)-i \text{Ap}(3,1))+25 \sqrt{21} (A(5,1)-i \text{Ap}(5,1))}{1155} | 0 | \frac{5}{33} \sqrt{2} (A(5,3)-i \text{Ap}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)-i \text{Ap}(3,3)) | 0 | \frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)-i \text{Ap}(5,5)) $| | ^ {Y_{-2}^{(2)}} |$ -\sqrt{\frac{3}{7}} (A(1,1)+i B(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)+i B(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)+i B(5,1)) | 0 | \frac{33 \sqrt{35} (A(1,1)-i B(1,1))-22 \sqrt{210} (A(3,1)-i B(3,1))+25 \sqrt{21} (A(5,1)-i B(5,1))}{1155} | 0 | \frac{5}{33} \sqrt{2} (A(5,3)-i B(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)-i B(3,3)) | 0 | \frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)-i B(5,5)) $| |
| ^ {Y_{-1}^{(2)}} | 0 |$ -\sqrt{\frac{2}{7}} (A(1,1)+i \text{Ap}(1,1))-\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (A(5,1)+i \text{Ap}(5,1)) | 0 | \frac{33 \sqrt{105} (A(1,1)-i \text{Ap}(1,1))-11 \sqrt{70} (A(3,1)-i \text{Ap}(3,1))-100 \sqrt{7} (A(5,1)-i \text{Ap}(5,1))}{1155} | 0 | -\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i \text{Ap}(3,3))-\frac{4}{33} \sqrt{5} (A(5,3)-i \text{Ap}(5,3)) | 0 $| | ^ {Y_{-1}^{(2)}} | 0 |$ -\sqrt{\frac{2}{7}} (A(1,1)+i B(1,1))-\frac{A(3,1)+i B(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (A(5,1)+i B(5,1)) | 0 | \frac{33 \sqrt{105} (A(1,1)-i B(1,1))-11 \sqrt{70} (A(3,1)-i B(3,1))-100 \sqrt{7} (A(5,1)-i B(5,1))}{1155} | 0 | -\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i B(3,3))-\frac{4}{33} \sqrt{5} (A(5,3)-i B(5,3)) | 0 $| |
| ^ {Y_{0}^{(2)}} |$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i \text{Ap}(3,3))-\frac{2}{33} \sqrt{5} (A(5,3)+i \text{Ap}(5,3)) | 0 | -\sqrt{\frac{6}{35}} (A(1,1)+i \text{Ap}(1,1))-\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (A(5,1)+i \text{Ap}(5,1)) | 0 | \frac{1}{385} \left(11 \sqrt{210} (A(1,1)-i \text{Ap}(1,1))+11 \sqrt{35} (A(3,1)-i \text{Ap}(3,1))+25 \sqrt{14} (A(5,1)-i \text{Ap}(5,1))\right) | 0 | \frac{2}{33} \sqrt{5} (A(5,3)-i \text{Ap}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i \text{Ap}(3,3)) $| | ^ {Y_{0}^{(2)}} |$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i B(3,3))-\frac{2}{33} \sqrt{5} (A(5,3)+i B(5,3)) | 0 | -\sqrt{\frac{6}{35}} (A(1,1)+i B(1,1))-\frac{A(3,1)+i B(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (A(5,1)+i B(5,1)) | 0 | \frac{1}{385} \left(11 \sqrt{210} (A(1,1)-i B(1,1))+11 \sqrt{35} (A(3,1)-i B(3,1))+25 \sqrt{14} (A(5,1)-i B(5,1))\right) | 0 | \frac{2}{33} \sqrt{5} (A(5,3)-i B(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i B(3,3)) $| |
| ^ {Y_{1}^{(2)}} | 0 |$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i \text{Ap}(3,3))+\frac{4}{33} \sqrt{5} (A(5,3)+i \text{Ap}(5,3)) | 0 | -\sqrt{\frac{3}{35}} (A(1,1)+i \text{Ap}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (A(3,1)+i \text{Ap}(3,1))+\frac{20 (A(5,1)+i \text{Ap}(5,1))}{33 \sqrt{7}} | 0 | \frac{1}{231} \left(33 \sqrt{14} (A(1,1)-i \text{Ap}(1,1))+11 \sqrt{21} (A(3,1)-i \text{Ap}(3,1))-2 \sqrt{210} (A(5,1)-i \text{Ap}(5,1))\right) | 0 $| | ^ {Y_{1}^{(2)}} | 0 |$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i B(3,3))+\frac{4}{33} \sqrt{5} (A(5,3)+i B(5,3)) | 0 | -\sqrt{\frac{3}{35}} (A(1,1)+i B(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (A(3,1)+i B(3,1))+\frac{20 (A(5,1)+i B(5,1))}{33 \sqrt{7}} | 0 | \frac{1}{231} \left(33 \sqrt{14} (A(1,1)-i B(1,1))+11 \sqrt{21} (A(3,1)-i B(3,1))-2 \sqrt{210} (A(5,1)-i B(5,1))\right) | 0 $| |
| ^ {Y_{2}^{(2)}} |$ -\frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)+i \text{Ap}(5,5)) | 0 | \frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)+i \text{Ap}(3,3))-\frac{5}{33} \sqrt{2} (A(5,3)+i \text{Ap}(5,3)) | 0 | -\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (A(3,1)+i \text{Ap}(3,1))-\frac{5 (A(5,1)+i \text{Ap}(5,1))}{11 \sqrt{21}} | 0 | \sqrt{\frac{3}{7}} (A(1,1)-i \text{Ap}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)-i \text{Ap}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)-i \text{Ap}(5,1)) $| | ^ {Y_{2}^{(2)}} |$ -\frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)+i B(5,5)) | 0 | \frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)+i B(3,3))-\frac{5}{33} \sqrt{2} (A(5,3)+i B(5,3)) | 0 | -\frac{A(1,1)+i B(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (A(3,1)+i B(3,1))-\frac{5 (A(5,1)+i B(5,1))}{11 \sqrt{21}} | 0 | \sqrt{\frac{3}{7}} (A(1,1)-i B(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)-i B(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)-i B(5,1)) $| |
| |
| |
| ### | ### |
| |
| <\hidden><hidden **The Hamiltonian on a basis of symmetric functions** > | </hidden> |
| | <hidden **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_{x\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}} ^ 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)} $ ^ |
| ^ d_{x^2-y^2} | 0 | \frac{-99 \sqrt{210} A(1,1)+121 \sqrt{35} A(3,1)-5 \left(11 \sqrt{21} A(3,3)+10 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)+35 \sqrt{15} A(5,5)\right)}{2310} |$ \frac{-99 \sqrt{210} \text{Ap}(1,1)+121 \sqrt{35} \text{Ap}(3,1)+55 \sqrt{21} \text{Ap}(3,3)-50 \sqrt{14} \text{Ap}(5,1)-175 \sqrt{3} \text{Ap}(5,3)-175 \sqrt{15} \text{Ap}(5,5)}{2310} | 0 | \frac{1}{462} \left(33 \sqrt{14} A(1,1)+11 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)-2 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)+105 A(5,5)\right) | \frac{1}{462} \left(-33 \sqrt{14} \text{Ap}(1,1)-11 \sqrt{21} \text{Ap}(3,1)-11 \sqrt{35} \text{Ap}(3,3)+2 \sqrt{210} \text{Ap}(5,1)+35 \sqrt{5} \text{Ap}(5,3)-105 \text{Ap}(5,5)\right) | 0 $| | ^ d_{x^2-y^2} | 0 | \frac{-99 \sqrt{210} A(1,1)+121 \sqrt{35} A(3,1)-5 \left(11 \sqrt{21} A(3,3)+10 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)+35 \sqrt{15} A(5,5)\right)}{2310} |$ \frac{-99 \sqrt{210} B(1,1)+121 \sqrt{35} B(3,1)+55 \sqrt{21} B(3,3)-50 \sqrt{14} B(5,1)-175 \sqrt{3} B(5,3)-175 \sqrt{15} B(5,5)}{2310} | 0 | \frac{1}{462} \left(33 \sqrt{14} A(1,1)+11 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)-2 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)+105 A(5,5)\right) | \frac{1}{462} \left(-33 \sqrt{14} B(1,1)-11 \sqrt{21} B(3,1)-11 \sqrt{35} B(3,3)+2 \sqrt{210} B(5,1)+35 \sqrt{5} B(5,3)-105 B(5,5)\right) | 0 $| |
| ^ d_{3z^2-r^2} | 0 | \frac{99 \sqrt{70} A(1,1)+33 \sqrt{105} A(3,1)+275 \sqrt{7} A(3,3)+75 \sqrt{42} A(5,1)-350 A(5,3)}{2310} |$ \frac{-99 \sqrt{70} \text{Ap}(1,1)-33 \sqrt{105} \text{Ap}(3,1)+275 \sqrt{7} \text{Ap}(3,3)-75 \sqrt{42} \text{Ap}(5,1)-350 \text{Ap}(5,3)}{2310} | 0 | \frac{1}{462} \left(33 \sqrt{42} A(1,1)+33 \sqrt{7} A(3,1)-11 \sqrt{105} A(3,3)+15 \sqrt{70} A(5,1)+14 \sqrt{15} A(5,3)\right) | \frac{1}{462} \left(33 \sqrt{42} \text{Ap}(1,1)+33 \sqrt{7} \text{Ap}(3,1)+11 \sqrt{105} \text{Ap}(3,3)+15 \sqrt{70} \text{Ap}(5,1)-14 \sqrt{15} \text{Ap}(5,3)\right) | 0 $| | ^ d_{3z^2-r^2} | 0 | \frac{99 \sqrt{70} A(1,1)+33 \sqrt{105} A(3,1)+275 \sqrt{7} A(3,3)+75 \sqrt{42} A(5,1)-350 A(5,3)}{2310} |$ \frac{-99 \sqrt{70} B(1,1)-33 \sqrt{105} B(3,1)+275 \sqrt{7} B(3,3)-75 \sqrt{42} B(5,1)-350 B(5,3)}{2310} | 0 | \frac{1}{462} \left(33 \sqrt{42} A(1,1)+33 \sqrt{7} A(3,1)-11 \sqrt{105} A(3,3)+15 \sqrt{70} A(5,1)+14 \sqrt{15} A(5,3)\right) | \frac{1}{462} \left(33 \sqrt{42} B(1,1)+33 \sqrt{7} B(3,1)+11 \sqrt{105} B(3,3)+15 \sqrt{70} B(5,1)-14 \sqrt{15} B(5,3)\right) | 0 $| |
| ^ d_{\text{yz}} | \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)+28 A(5,3)\right)\right) | 0 | 0 |$ -\sqrt{\frac{6}{35}} \text{Ap}(1,1)+\frac{2 \text{Ap}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Ap}(5,1) | 0 | 0 | \frac{1}{231} \left(-33 \sqrt{14} \text{Ap}(1,1)-11 \sqrt{21} \text{Ap}(3,1)+\sqrt{5} \left(11 \sqrt{7} \text{Ap}(3,3)+2 \sqrt{42} \text{Ap}(5,1)+28 \text{Ap}(5,3)\right)\right) $| | ^ d_{\text{yz}} | \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)+28 A(5,3)\right)\right) | 0 | 0 |$ -\sqrt{\frac{6}{35}} B(1,1)+\frac{2 B(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} B(5,1) | 0 | 0 | \frac{1}{231} \left(-33 \sqrt{14} B(1,1)-11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)+2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) $| |
| ^ d_{\text{xz}} |$ \frac{1}{231} \left(33 \sqrt{14} \text{Ap}(1,1)+11 \sqrt{21} \text{Ap}(3,1)+\sqrt{5} \left(11 \sqrt{7} \text{Ap}(3,3)-2 \sqrt{42} \text{Ap}(5,1)+28 \text{Ap}(5,3)\right)\right) | 0 | 0 | \sqrt{\frac{6}{35}} A(1,1)-\frac{2 A(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} A(5,1) | 0 | 0 | \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(-11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)-28 A(5,3)\right)\right) $| | ^ d_{\text{xz}} |$ \frac{1}{231} \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)-2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) | 0 | 0 | \sqrt{\frac{6}{35}} A(1,1)-\frac{2 A(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} A(5,1) | 0 | 0 | \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(-11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)-28 A(5,3)\right)\right) $| |
| ^ d_{\text{xy}} | 0 |$ \frac{-66 \sqrt{210} \text{Ap}(1,1)-11 \sqrt{35} \text{Ap}(3,1)+5 \left(11 \sqrt{21} \text{Ap}(3,3)+5 \sqrt{14} \text{Ap}(5,1)-35 \sqrt{3} \text{Ap}(5,3)+35 \sqrt{15} \text{Ap}(5,5)\right)}{2310} | \frac{66 \sqrt{210} A(1,1)+11 \sqrt{35} A(3,1)+5 \left(11 \sqrt{21} A(3,3)-5 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)-35 \sqrt{15} A(5,5)\right)}{2310} | 0 | \frac{1}{462} \left(66 \sqrt{14} \text{Ap}(1,1)-33 \sqrt{21} \text{Ap}(3,1)+11 \sqrt{35} \text{Ap}(3,3)+3 \sqrt{210} \text{Ap}(5,1)-35 \sqrt{5} \text{Ap}(5,3)-105 \text{Ap}(5,5)\right) | \frac{1}{462} \left(66 \sqrt{14} A(1,1)-33 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)+3 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)-105 A(5,5)\right) | 0 $| | ^ d_{\text{xy}} | 0 |$ \frac{-66 \sqrt{210} B(1,1)-11 \sqrt{35} B(3,1)+5 \left(11 \sqrt{21} B(3,3)+5 \sqrt{14} B(5,1)-35 \sqrt{3} B(5,3)+35 \sqrt{15} B(5,5)\right)}{2310} | \frac{66 \sqrt{210} A(1,1)+11 \sqrt{35} A(3,1)+5 \left(11 \sqrt{21} A(3,3)-5 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)-35 \sqrt{15} A(5,5)\right)}{2310} | 0 | \frac{1}{462} \left(66 \sqrt{14} B(1,1)-33 \sqrt{21} B(3,1)+11 \sqrt{35} B(3,3)+3 \sqrt{210} B(5,1)-35 \sqrt{5} B(5,3)-105 B(5,5)\right) | \frac{1}{462} \left(66 \sqrt{14} A(1,1)-33 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)+3 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)-105 A(5,5)\right) | 0 $| |
| |
| |
| |
| </hidden> | </hidden> |
| | |
| ===== Table of several point groups ===== | ===== Table of several point groups ===== |
| |
| |
| ### | ### |
| |
