Processing math: 56%

+Table of Contents

Orientation 111

This orientation is non-standard, but related to the orientation of the Oh pointgroup, which normally would be orrientated with the C3 axes in the 111 direction. We only show one of the options of the D3d subgroups of the Oh group with orientation XYZ.

Symmetry Operations

In the D3d Point Group, with orientation 111 there are the following symmetry operations

Operator Orientation
E {0,0,0} ,
C3 {1,1,1} , {−1,−1,−1} ,
C2 {1,−1,0} , {0,1,−1} , {1,0,−1} ,
i {0,0,0} ,
S6 {1,1,1} , {−1,−1,−1} ,
σd {1,−1,0} , {0,1,−1} , {1,0,−1} ,

Different Settings

Character Table

E(1) C3(2) C2(3) i(1) S6(2) σd(3)
A1g 1 1 1 1 1 1
A2g 1 1 −1 1 1 −1
Eg 2 −1 0 2 −1 0
A1u 1 1 1 −1 −1 −1
A2u 1 1 −1 −1 −1 1
Eu 2 −1 0 −2 1 0

Product Table

A1g A2g Eg A1u A2u Eu
A1g A1g A2g Eg A1u A2u Eu
A2g A2g A1g Eg A2u A1u Eu
Eg Eg Eg A1g+A2g+Eg Eu Eu A1u+A2u+Eu
A1u A1u A2u Eu A1g A2g Eg
A2u A2u A1u Eu A2g A1g Eg
Eu Eu Eu A1u+A2u+Eu Eg Eg A1g+A2g+Eg

Sub Groups with compatible settings

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Any potential (function) can be written as a sum over spherical harmonics. V(r,θ,ϕ)=∞∑k=0k∑m=−kAk,m(r)C(m)k(θ,ϕ) Here Ak,m(r) is a radial function and C(m)k(θ,ϕ) a renormalised spherical harmonics. C(m)k(θ,ϕ)=√4π2k+1Y(m)k(θ,ϕ) The presence of symmetry induces relations between the expansion coefficients such that V(r,θ,ϕ) is invariant under all symmetry operations. For the D3d Point group with orientation 111 the form of the expansion coefficients is:

Expansion

Ak,m={A(0,0)k=0∧m=0−iA(2,1)k=2∧m=−2(−1−i)A(2,1)k=2∧m=−1(1−i)A(2,1)k=2∧m=1iA(2,1)k=2∧m=2√514A(4,0)k=4∧(m=−4∨m=4)(−1+i)√7A(4,1)k=4∧m=−32i√2A(4,1)k=4∧m=−2(−1−i)A(4,1)k=4∧m=−1A(4,0)k=4∧m=0(1−i)A(4,1)k=4∧m=1−2i√2A(4,1)k=4∧m=2(1+i)√7A(4,1)k=4∧m=3−133i(8√22A(6,1)−√55B(6,2))k=6∧m=−6−(1+i)(A(6,1)+2√10B(6,2))√66k=6∧m=−5−√72A(6,0)k=6∧(m=−4∨m=4)(16−i6)(√10A(6,1)−4B(6,2))k=6∧m=−3−iB(6,2)k=6∧m=−2(−1−i)A(6,1)k=6∧m=−1A(6,0)k=6∧m=0(1−i)A(6,1)k=6∧m=1iB(6,2)k=6∧m=2(−16−i6)(√10A(6,1)−4B(6,2))k=6∧m=3(1−i)(A(6,1)+2√10B(6,2))√66k=6∧m=5133i(8√22A(6,1)−√55B(6,2))k=6∧m=6

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(-I)*A[2, 1], k == 2 && m == -2}, {(-1 - I)*A[2, 1], k == 2 && m == -1}, {(1 - I)*A[2, 1], k == 2 && m == 1}, {I*A[2, 1], k == 2 && m == 2}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}, {(-1 + I)*Sqrt[7]*A[4, 1], k == 4 && m == -3}, {(2*I)*Sqrt[2]*A[4, 1], k == 4 && m == -2}, {(-1 - I)*A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {(1 - I)*A[4, 1], k == 4 && m == 1}, {(-2*I)*Sqrt[2]*A[4, 1], k == 4 && m == 2}, {(1 + I)*Sqrt[7]*A[4, 1], k == 4 && m == 3}, {(-I/33)*(8*Sqrt[22]*A[6, 1] - Sqrt[55]*B[6, 2]), k == 6 && m == -6}, {((-1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6, 2]))/Sqrt[66], k == 6 && m == -5}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {(1/6 - I/6)*(Sqrt[10]*A[6, 1] - 4*B[6, 2]), k == 6 && m == -3}, {(-I)*B[6, 2], k == 6 && m == -2}, {(-1 - I)*A[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {(1 - I)*A[6, 1], k == 6 && m == 1}, {I*B[6, 2], k == 6 && m == 2}, {(-1/6 - I/6)*(Sqrt[10]*A[6, 1] - 4*B[6, 2]), k == 6 && m == 3}, {((1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6, 2]))/Sqrt[66], k == 6 && m == 5}, {(I/33)*(8*Sqrt[22]*A[6, 1] - Sqrt[55]*B[6, 2]), k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_D3d_111.Quanty
Akm = {{0, 0, A(0,0)} , 
       {2,-1, (-1+-1*I)*(A(2,1))} , 
       {2, 1, (1+-1*I)*(A(2,1))} , 
       {2,-2, (-I)*(A(2,1))} , 
       {2, 2, (I)*(A(2,1))} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1+-1*I)*(A(4,1))} , 
       {4, 1, (1+-1*I)*(A(4,1))} , 
       {4, 2, (-2*I)*((sqrt(2))*(A(4,1)))} , 
       {4,-2, (2*I)*((sqrt(2))*(A(4,1)))} , 
       {4,-3, (-1+1*I)*((sqrt(7))*(A(4,1)))} , 
       {4, 3, (1+1*I)*((sqrt(7))*(A(4,1)))} , 
       {4,-4, (sqrt(5/14))*(A(4,0))} , 
       {4, 4, (sqrt(5/14))*(A(4,0))} , 
       {6, 0, A(6,0)} , 
       {6,-1, (-1+-1*I)*(A(6,1))} , 
       {6, 1, (1+-1*I)*(A(6,1))} , 
       {6,-2, (-I)*(B(6,2))} , 
       {6, 2, (I)*(B(6,2))} , 
       {6, 3, (-1/6+-1/6*I)*((sqrt(10))*(A(6,1)) + (-4)*(B(6,2)))} , 
       {6,-3, (1/6+-1/6*I)*((sqrt(10))*(A(6,1)) + (-4)*(B(6,2)))} , 
       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6,-5, (-1+-1*I)*((1/(sqrt(66)))*(A(6,1) + (2)*((sqrt(10))*(B(6,2)))))} , 
       {6, 5, (1+-1*I)*((1/(sqrt(66)))*(A(6,1) + (2)*((sqrt(10))*(B(6,2)))))} , 
       {6,-6, (-1/33*I)*((8)*((sqrt(22))*(A(6,1))) + (-1)*((sqrt(55))*(B(6,2))))} , 
       {6, 6, (1/33*I)*((8)*((sqrt(22))*(A(6,1))) + (-1)*((sqrt(55))*(B(6,2))))} }

One particle coupling on a basis of spherical harmonics

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

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

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

Y(0)0 Y(1)−1 Y(1)0 Y(1)1 Y(2)−2 Y(2)−1 Y(2)0 Y(2)1 Y(2)2 Y(3)−3 Y(3)−2 Y(3)−1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
Y(0)0Ass(0,0)000iAsd(2,1)√5−(1−i)Asd(2,1)√50(1+i)Asd(2,1)√5−iAsd(2,1)√50000000
Y(1)−10App(0,0)(−15−i5)√3App(2,1)15i√6App(2,1)000003iApf(2,1)√35+23i√221Apf(4,1)(13−i3)Apf(4,1)√7−(1−i)√635Apf(2,1)−13√27Apf(4,0)(35+3i5)Apf(2,1)√7−(13+i3)√1021Apf(4,1)−15i√37Apf(2,1)−23i√107Apf(4,1)(−13+i3)√7Apf(4,1)−13√1021Apf(4,0)
Y(1)00(−15+i5)√3App(2,1)App(0,0)(15+i5)√3App(2,1)00000(−13−i3)√73Apf(4,1)i√335Apf(2,1)−43i√27Apf(4,1)(−25+2i5)√67Apf(2,1)−(13−i3)√57Apf(4,1)4Apf(4,0)3√21(25+2i5)√67Apf(2,1)+(13+i3)√57Apf(4,1)43i√27Apf(4,1)−i√335Apf(2,1)(13−i3)√73Apf(4,1)
Y(1)10−15i√6App(2,1)(15−i5)√3App(2,1)App(0,0)00000−13√1021Apf(4,0)(13+i3)√7Apf(4,1)15i√37Apf(2,1)+23i√107Apf(4,1)(13−i3)√1021Apf(4,1)−(35−3i5)Apf(2,1)√7−13√27Apf(4,0)(1+i)√635Apf(2,1)−(13+i3)Apf(4,1)√7−3iApf(2,1)√35−23i√221Apf(4,1)
Y(2)−2−iAsd(2,1)√5000Add(0,0)+121Add(4,0)(121+i21)√5Add(4,1)−(17+i7)√6Add(2,1)27iAdd(2,1)+27i√103Add(4,1)(13−i3)√5Add(4,1)521Add(4,0)0000000
Y(2)−1−(1+i)Asd(2,1)√5000(121−i21)√5Add(4,1)−(17−i7)√6Add(2,1)Add(0,0)−421Add(4,0)(−17−i7)Add(2,1)−(17+i7)√103Add(4,1)17i√6Add(2,1)−821i√5Add(4,1)(−13+i3)√5Add(4,1)0000000
Y(2)00000−27iAdd(2,1)−27i√103Add(4,1)(−17+i7)Add(2,1)−(17−i7)√103Add(4,1)Add(0,0)+27Add(4,0)(17+i7)Add(2,1)+(17+i7)√103Add(4,1)27iAdd(2,1)+27i√103Add(4,1)0000000
Y(2)1(1−i)Asd(2,1)√5000(13+i3)√5Add(4,1)821i√5Add(4,1)−17i√6Add(2,1)(17−i7)Add(2,1)+(17−i7)√103Add(4,1)Add(0,0)−421Add(4,0)(17+i7)√6Add(2,1)−(121+i21)√5Add(4,1)0000000
Y(2)2iAsd(2,1)√5000521Add(4,0)(−13−i3)√5Add(4,1)−27iAdd(2,1)−27i√103Add(4,1)(17−i7)√6Add(2,1)−(121−i21)√5Add(4,1)Add(0,0)+121Add(4,0)0000000
Y(3)−30−3iApf(2,1)√35−23i√221Apf(4,1)(−13+i3)√73Apf(4,1)−13√1021Apf(4,0)00000Aff(0,0)+111Aff(4,0)−5429Aff(6,0)(−13−i3)Aff(2,1)+(111+i11)√103Aff(4,1)−(5429+5i429)√7Aff(6,1)13i√25Aff(2,1)+411i√3Aff(4,1)+10429i√7Bff(6,2)(5429−5i429)√73(√10Aff(6,1)−4Bff(6,2))+(711−7i11)Aff(4,1)111√53Aff(4,0)+35143√53Aff(6,0)(−5429−5i429)√7(Aff(6,1)+2√10Bff(6,2))10429i√733(8√22Aff(6,1)−√55Bff(6,2))
Y(3)−20(13+i3)Apf(4,1)√7−(1+i)√635Apf(2,1)43i√27Apf(4,1)−i√335Apf(2,1)(13−i3)√7Apf(4,1)00000(−13+i3)Aff(2,1)+(111−i11)√103Aff(4,1)−(5429−5i429)√7Aff(6,1)Aff(0,0)−733Aff(4,0)+10143Aff(6,0)−(1+i)Aff(2,1)√15−(433+4i33)√2Aff(4,1)+(5143+5i143)√353Aff(6,1)2iAff(2,1)3√5−211i√23Aff(4,1)−20429i√14Bff(6,2)(733−7i33)√2Aff(4,1)−(5143−5i143)√76(√10Aff(6,1)−4Bff(6,2))533Aff(4,0)−70143Aff(6,0)(5429+5i429)√7(Aff(6,1)+2√10Bff(6,2))
Y(3)−10−13√27Apf(4,0)(−25−2i5)√67Apf(2,1)−(13+i3)√57Apf(4,1)−15i√37Apf(2,1)−23i√107Apf(4,1)00000−13i√25Aff(2,1)−411i√3Aff(4,1)−10429i√7Bff(6,2)−(1−i)Aff(2,1)√15−(433−4i33)√2Aff(4,1)+(5143−5i143)√353Aff(6,1)Aff(0,0)+133Aff(4,0)−25143Aff(6,0)(−115−i15)√2Aff(2,1)−(111+i11)√53Aff(4,1)−(25429+25i429)√14Aff(6,1)25i√23Aff(2,1)−833i√5Aff(4,1)+10143i√353Bff(6,2)(5143−5i143)√76(√10Aff(6,1)−4Bff(6,2))−(733−7i33)√2Aff(4,1)111√53Aff(4,0)+35143√53Aff(6,0)
Y(3)00(35−3i5)Apf(2,1)√7−(13−i3)√1021Apf(4,1)4Apf(4,0)3√21(13+i3)√1021Apf(4,1)−(35+3i5)Apf(2,1)√700000(5429+5i429)√73(√10Aff(6,1)−4Bff(6,2))+(711+7i11)Aff(4,1)−2iAff(2,1)3√5+211i√23Aff(4,1)+20429i√14Bff(6,2)(−115+i15)√2Aff(2,1)−(111−i11)√53Aff(4,1)−(25429−25i429)√14Aff(6,1)Aff(0,0)+211Aff(4,0)+100429Aff(6,0)(115+i15)√2Aff(2,1)+(111+i11)√53Aff(4,1)+(25429+25i429)√14Aff(6,1)2iAff(2,1)3√5−211i√23Aff(4,1)−20429i√14Bff(6,2)(−711+7i11)Aff(4,1)−(5429−5i429)√73(√10Aff(6,1)−4Bff(6,2))
Y(3)1015i√37Apf(2,1)+23i√107Apf(4,1)(25−2i5)√67Apf(2,1)+(13−i3)√57Apf(4,1)−13√27Apf(4,0)00000111√53Aff(4,0)+35143√53Aff(6,0)(733+7i33)√2Aff(4,1)−(5143+5i143)√76(√10Aff(6,1)−4Bff(6,2))−25i√23Aff(2,1)+833i√5Aff(4,1)−10143i√353Bff(6,2)(115−i15)√2Aff(2,1)+(111−i11)√53Aff(4,1)+(25429−25i429)√14Aff(6,1)Aff(0,0)+133Aff(4,0)−25143Aff(6,0)(1+i)Aff(2,1)√15+(433+4i33)√2Aff(4,1)−(5143+5i143)√353Aff(6,1)13i√25Aff(2,1)+411i√3Aff(4,1)+10429i√7Bff(6,2)
Y(3)20(−13−i3)√7Apf(4,1)i√335Apf(2,1)−43i√27Apf(4,1)(1−i)√635Apf(2,1)−(13−i3)Apf(4,1)√700000(−5429+5i429)√7(Aff(6,1)+2√10Bff(6,2))533Aff(4,0)−70143Aff(6,0)(5143+5i143)√76(√10Aff(6,1)−4Bff(6,2))−(733+7i33)√2Aff(4,1)−2iAff(2,1)3√5+211i√23Aff(4,1)+20429i√14Bff(6,2)(1−i)Aff(2,1)√15+(433−4i33)√2Aff(4,1)−(5143−5i143)√353Aff(6,1)Aff(0,0)−733Aff(4,0)+10143Aff(6,0)(13+i3)Aff(2,1)−(111+i11)√103Aff(4,1)+(5429+5i429)√7Aff(6,1)
Y(3)30−13√1021Apf(4,0)(13+i3)√73Apf(4,1)3iApf(2,1)√35+23i√221Apf(4,1)00000−10429i√733(8√22Aff(6,1)−√55Bff(6,2))(5429−5i429)√7(Aff(6,1)+2√10Bff(6,2))111√53Aff(4,0)+35143√53Aff(6,0)(−711−7i11)Aff(4,1)−(5429+5i429)√73(√10Aff(6,1)−4Bff(6,2))−13i√25Aff(2,1)−411i√3Aff(4,1)−10429i√7Bff(6,2)(13−i3)Aff(2,1)−(111−i11)√103Aff(4,1)+(5429−5i429)√7Aff(6,1)Aff(0,0)+111Aff(4,0)−5429Aff(6,0)

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

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

Y(0)0 Y(1)−1 Y(1)0 Y(1)1 Y(2)−2 Y(2)−1 Y(2)0 Y(2)1 Y(2)2 Y(3)−3 Y(3)−2 Y(3)−1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
s1000000000000000
px+y+z01+i√61√3−1−i√6000000000000
px−y012−i20−12−i2000000000000
p3z−r0−12+i2√3√2312−i2√3000000000000
dyz+xz+xy0000i√61+i√60−1−i√6−i√60000000
dyz−xz00000−12+i2012+i200000000
d2xy−xz−yz0000i√3−12+i2√3012−i2√3−i√30000000
dx2−y200001√20001√20000000
d3z2−r20000001000000000
fxyz0000000000i√2000−i√20
fx3+y3+z3000000000(14−i4)√530−14−i41√314−i40(−14−i4)√53
fx3−y3000000000(14+i4)√520(−14+i4)√320(14+i4)√320(−14+i4)√52
f2z3−x3−y3000000000(−14+i4)√56014+i4√2√23−14−i4√20(14+i4)√56
f(y2−z2)x+(z2−x2)y+(x2−y2)z000000000−14−i41√6(−14+i4)√530(14+i4)√531√614−i4
f−(y2−z2)x−(z2−x2)y+2(x2−y2)z00000000014+i4√21√3(14−i4)√560(−14−i4)√561√3−14−i4√2
f−(y2−z2)x+(z2−x2)y000000000(14−i4)√320(14+i4)√520(−14+i4)√520(−14−i4)√32

One particle coupling on a basis of symmetry adapted functions

After rotation we find

s px+y+z px−y p3z−r dyz+xz+xy dyz−xz d2xy−xz−yz dx2−y2 d3z2−r2 fxyz fx3+y3+z3 fx3−y3 f2z3−x3−y3 f(y2−z2)x+(z2−x2)y+(x2−y2)z f−(y2−z2)x−(z2−x2)y+2(x2−y2)z f−(y2−z2)x+(z2−x2)y
sAss(0,0)000−√65Asd(2,1)00000000000
px+y+z0App(0,0)−25√6App(2,1)00000008Apf(4,1)√21−3√235Apf(2,1)65√27Apf(2,1)+4Apf(4,0)3√21−43√521Apf(4,1)00000
px−y00App(0,0)+15√6App(2,1)00000000−35√27Apf(2,1)+4Apf(4,0)3√21+23√521Apf(4,1)00−3√235Apf(2,1)−2√37Apf(4,1)0
p3z−r000App(0,0)+15√6App(2,1)00000000−35√27Apf(2,1)+4Apf(4,0)3√21+23√521Apf(4,1)00−3√235Apf(2,1)−2√37Apf(4,1)
dyz+xz+xy−√65Asd(2,1)000Add(0,0)−27√6Add(2,1)−421Add(4,0)+1621√5Add(4,1)00000000000
dyz−xz00000Add(0,0)+17√6Add(2,1)−421Add(4,0)−821√5Add(4,1)027√3Add(2,1)+27√10Add(4,1)00000000
d2xy−xz−yz000000Add(0,0)+17√6Add(2,1)−421Add(4,0)−821√5Add(4,1)027√3Add(2,1)+27√10Add(4,1)0000000
dx2−y20000027√3Add(2,1)+27√10Add(4,1)0Add(0,0)+27Add(4,0)00000000
d3z2−r200000027√3Add(2,1)+27√10Add(4,1)0Add(0,0)+27Add(4,0)0000000
fxyz08Apf(4,1)√21−3√235Apf(2,1)0000000Aff(0,0)−411Aff(4,0)+80143Aff(6,0)2√215Aff(2,1)−411Aff(4,1)−40143√73Bff(6,2)00000
fx3+y3+z3065√27Apf(2,1)+4Apf(4,0)3√21−43√521Apf(4,1)00000002√215Aff(2,1)−411Aff(4,1)−40143√73Bff(6,2)Aff(0,0)+15√23Aff(2,1)+211Aff(4,0)+811√5Aff(4,1)+100429Aff(6,0)+100429√143Aff(6,1)−20429√353Bff(6,2)00000
fx3−y300−35√27Apf(2,1)+4Apf(4,0)3√21+23√521Apf(4,1)00000000Aff(0,0)−Aff(2,1)5√6+211Aff(4,0)−411√5Aff(4,1)+100429Aff(6,0)−50429√143Aff(6,1)+10429√353Bff(6,2)00−Aff(2,1)√30+811Aff(4,1)−10143√703Aff(6,1)−10143√73Bff(6,2) 0
f_{2z^3-x^3-y^3} \color{darkred}{ 0 } 0 0 -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } 0 0 0 \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{2}{11} \text{Aff}(4,0)-\frac{4}{11} \sqrt{5} \text{Aff}(4,1)+\frac{100}{429} \text{Aff}(6,0)-\frac{50}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{10}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,2) 0 0 -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2)
f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} \color{darkred}{ 0 } 0 0 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } 0 0 0 0 \text{Aff}(0,0)+\sqrt{\frac{2}{3}} \text{Aff}(2,1)-\frac{2}{33} \text{Aff}(4,0)+\frac{8}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)-\frac{20}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{20}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) 0 0
f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} \color{darkred}{ 0 } 0 -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } 0 0 -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) 0 0 \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) 0
f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} \color{darkred}{ 0 } 0 0 -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ 0 } 0 0 0 -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) 0 0 \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2)

Coupling for a single shell

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

Click on one of the subsections to expand it or

Potential for s orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \text{Ea1g} & k=0\land m=0 \\ 0 & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_111.Quanty
Akm = {{0, 0, Ea1g} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{0}^{(0)}}
{Y_{0}^{(0)}} \text{Ea1g}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

\text{s}
\text{s} \text{Ea1g}

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Ea1g}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2 \sqrt{\pi }}
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2 \sqrt{\pi }}

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\ \frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=-2 \\ \frac{\left(\frac{5}{3}+\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=-1 \\ -\frac{\left(\frac{5}{3}-\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=1 \\ -\frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=2 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(((5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == -2}, {((5/3 + (5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == -1}, {((-5/3 + (5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == 1}, {(((-5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == 2}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_111.Quanty
Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , 
       {2, 1, (-5/3+5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
       {2,-1, (5/3+5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
       {2, 2, (-5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
       {2,-2, (5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-1}^{(1)}} {Y_{0}^{(1)}} {Y_{1}^{(1)}}
{Y_{-1}^{(1)}} \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) \frac{1}{3} \sqrt[4]{-1} (\text{Ea2u}-\text{Eeu}) -\frac{1}{3} i (\text{Ea2u}-\text{Eeu})
{Y_{0}^{(1)}} \frac{1}{3} (-1)^{3/4} (\text{Eeu}-\text{Ea2u}) \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) \frac{1}{3} \sqrt[4]{-1} (\text{Eeu}-\text{Ea2u})
{Y_{1}^{(1)}} \frac{1}{3} i (\text{Ea2u}-\text{Eeu}) \frac{1}{3} (-1)^{3/4} (\text{Ea2u}-\text{Eeu}) \frac{1}{3} (\text{Ea2u}+2 \text{Eeu})

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

p_{x+y+z} p_{x-y} p_{3z-r}
p_{x+y+z} \text{Ea2u} 0 0
p_{x-y} 0 \text{Eeu} 0
p_{3z-r} 0 0 \text{Eeu}

Rotation matrix used

Rotation matrix used

{Y_{-1}^{(1)}} {Y_{0}^{(1)}} {Y_{1}^{(1)}}
p_{x+y+z} \frac{1+i}{\sqrt{6}} \frac{1}{\sqrt{3}} -\frac{1-i}{\sqrt{6}}
p_{x-y} \frac{1}{2}-\frac{i}{2} 0 -\frac{1}{2}-\frac{i}{2}
p_{3z-r} -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} \sqrt{\frac{2}{3}} \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Ea2u}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta )}{2 \sqrt{\pi }}
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{x+y+z}{2 \sqrt{\pi }}
\text{Eeu}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi ))
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{2 \pi }} (x-y)
\text{Eeu}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} -\frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta )}{2 \sqrt{2 \pi }}
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} -\frac{x+y-2 z}{2 \sqrt{2 \pi }}

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea1g}+2 (\text{Eeg}\pi +\text{Eeg}\sigma )) & k=0\land m=0 \\ \frac{1}{6} i \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg}\pi -4 \sqrt{3} \text{Meg}\right) & k=2\land m=-2 \\ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg}\pi -4 \sqrt{3} \text{Meg}\right) & k=2\land m=-1 \\ \left(\frac{1}{6}-\frac{i}{6}\right) \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg}\pi +4 \sqrt{3} \text{Meg}\right) & k=2\land m=1 \\ \frac{1}{6} i \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg}\pi +4 \sqrt{3} \text{Meg}\right) & k=2\land m=2 \\ -\frac{1}{2} \sqrt{\frac{7}{10}} (\text{Ea1g}+2 \text{Eeg}\pi -3 \text{Eeg}\sigma ) & k=4\land (m=-4\lor m=4) \\ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right) & k=4\land m=-3 \\ \frac{i \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=-2 \\ -\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=-1 \\ -\frac{7}{10} (\text{Ea1g}+2 \text{Eeg}\pi -3 \text{Eeg}\sigma ) & k=4\land m=0 \\ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=1 \\ -\frac{i \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=2 \\ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right) & k=4\land m=3 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea1g + 2*(Eeg\[Pi] + Eeg\[Sigma]))/5, k == 0 && m == 0}, {(I/6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg), k == 2 && m == -2}, {(1/6 + I/6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg), k == 2 && m == -1}, {(1/6 - I/6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg), k == 2 && m == 1}, {(I/6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg), k == 2 && m == 2}, {-(Sqrt[7/10]*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/2, k == 4 && (m == -4 || m == 4)}, {(-1/4 + I/4)*Sqrt[7/5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg), k == 4 && m == -3}, {(I*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[10], k == 4 && m == -2}, {((-1/4 - I/4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[5], k == 4 && m == -1}, {(-7*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/10, k == 4 && m == 0}, {((1/4 - I/4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[5], k == 4 && m == 1}, {((-I)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[10], k == 4 && m == 2}, {(1/4 + I/4)*Sqrt[7/5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg), k == 4 && m == 3}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_111.Quanty
Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg\[Pi] + Eeg\[Sigma]))} , 
       {2, 1, (1/6+-1/6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , 
       {2,-1, (1/6+1/6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , 
       {2, 2, (1/6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , 
       {2,-2, (1/6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , 
       {4, 0, (-7/10)*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma]))} , 
       {4,-1, (-1/4+-1/4*I)*((1/(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4, 1, (1/4+-1/4*I)*((1/(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4, 2, (-I)*((1/(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4,-2, (I)*((1/(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4,-3, (-1/4+1/4*I)*((sqrt(7/5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4, 3, (1/4+1/4*I)*((sqrt(7/5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4,-4, (-1/2)*((sqrt(7/10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} , 
       {4, 4, (-1/2)*((sqrt(7/10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{-2}^{(2)}} \frac{1}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) \frac{i \text{Meg}}{\sqrt{3}} \left(\frac{1}{12}-\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $})
{Y_{-1}^{(2)}} \left(-\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) \text{Meg} \text{Root}\left[36 \text{\#$1}^4+1$,1\right] -\frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) \left(-\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)
{Y_{0}^{(2)}} -\frac{i \text{Meg}}{\sqrt{3}} \frac{(-1)^{3/4} \text{Meg}}{\sqrt{6}} \text{Eeg$\sigma $} \frac{\sqrt[4]{-1} \text{Meg}}{\sqrt{6}} \frac{i \text{Meg}}{\sqrt{3}}
{Y_{1}^{(2)}} \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) \frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) \text{Meg} \text{Root}\left[36 \text{\#$1}^4+1$,3\right] \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)
{Y_{2}^{(2)}} \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) -\frac{i \text{Meg}}{\sqrt{3}} \left(\frac{1}{12}-\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) \frac{1}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $})

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{\text{yz}+\text{xz}+\text{xy}} d_{\text{yz}-\text{xz}} d_{2\text{xy}-\text{xz}-\text{yz}} d_{x^2-y^2} d_{3z^2-r^2}
d_{\text{yz}+\text{xz}+\text{xy}} \text{Ea1g} 0 0 0 \frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Meg} \left(\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(1+i)\right)}{\sqrt{2}}
d_{\text{yz}-\text{xz}} 0 \text{Eeg$\pi $} 0 \text{Meg} \left(-\frac{1}{2}-\frac{i}{2}\right) \text{Meg} \left(\text{Root}\left[36 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]\right)
d_{2\text{xy}-\text{xz}-\text{yz}} 0 0 \text{Eeg$\pi $} 0 \left(\frac{1}{6}+\frac{i}{6}\right) \text{Meg} \left(i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(2-2 i)\right)
d_{x^2-y^2} 0 \text{Meg} 0 \text{Eeg$\sigma $} 0
d_{3z^2-r^2} 0 0 \text{Meg} 0 \text{Eeg$\sigma $}

Rotation matrix used

Rotation matrix used

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
d_{\text{yz}+\text{xz}+\text{xy}} \frac{i}{\sqrt{6}} \frac{1+i}{\sqrt{6}} 0 -\frac{1-i}{\sqrt{6}} -\frac{i}{\sqrt{6}}
d_{\text{yz}-\text{xz}} 0 -\frac{1}{2}+\frac{i}{2} 0 \frac{1}{2}+\frac{i}{2} 0
d_{2\text{xy}-\text{xz}-\text{yz}} \frac{i}{\sqrt{3}} -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} 0 \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} -\frac{i}{\sqrt{3}}
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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Ea1g}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)
\text{Eeg$\pi $}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{2 \pi }} \sin (2 \theta ) (\sin (\phi )-\cos (\phi ))
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{15}{2 \pi }} z (y-x)
\text{Eeg$\pi $}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)
\text{Eeg$\sigma $}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)
\text{Eeg$\sigma $}
\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)

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 (\text{Eeu1}+\text{Eeu2})) & k=0\land m=0 \\ -\frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=-2 \\ -\frac{\left(\frac{5}{28}+\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=-1 \\ \frac{\left(\frac{5}{28}-\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=1 \\ \frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=2 \\ -\frac{1}{4} \sqrt{\frac{5}{14}} (\text{Ea1u}+6 \text{Ea2u1}-3 \text{Ea2u2}-6 \text{Eeu1}+2 \text{Eeu2}) & k=4\land (m=-4\lor m=4) \\ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=-3 \\ \frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=-2 \\ \left(-\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=-1 \\ \frac{1}{4} (-\text{Ea1u}-6 \text{Ea2u1}+3 \text{Ea2u2}+6 \text{Eeu1}-2 \text{Eeu2}) & k=4\land m=0 \\ \left(\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=1 \\ -\frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=2 \\ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=3 \\ \frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=-6 \\ \left(-\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=-5 \\ \frac{13 (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ -\frac{\left(\frac{13}{40}-\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=-3 \\ -\frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=-2 \\ \frac{\left(\frac{13}{20}+\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=-1 \\ -\frac{13}{280} (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2}) & k=6\land m=0 \\ -\frac{\left(\frac{13}{20}-\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=1 \\ \frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=2 \\ \frac{\left(\frac{13}{40}+\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=3 \\ \left(\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=5 \\ -\frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=6 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea1u + Ea2u1 + Ea2u2 + 2*(Eeu1 + Eeu2))/7, k == 0 && m == 0}, {(((-5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == -2}, {((-5/28 - (5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == -1}, {((5/28 - (5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == 1}, {(((5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == 2}, {-(Sqrt[5/14]*(Ea1u + 6*Ea2u1 - 3*Ea2u2 - 6*Eeu1 + 2*Eeu2))/4, k == 4 && (m == -4 || m == 4)}, {((-1/4 + I/4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[7], k == 4 && m == -3}, {((I/7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[2], k == 4 && m == -2}, {(-1/28 - I/28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == -1}, {(-Ea1u - 6*Ea2u1 + 3*Ea2u2 + 6*Eeu1 - 2*Eeu2)/4, k == 4 && m == 0}, {(1/28 - I/28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == 1}, {((-I/7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[2], k == 4 && m == 2}, {((1/4 + I/4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[7], k == 4 && m == 3}, {((13*I)/80)*Sqrt[11/21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu), k == 6 && m == -6}, {(-13/40 - (13*I)/40)*Sqrt[11/7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == -5}, {(13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {((-13/40 + (13*I)/40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/Sqrt[21], k == 6 && m == -3}, {(((-13*I)/80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/Sqrt[21], k == 6 && m == -2}, {((13/20 + (13*I)/20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/Sqrt[42], k == 6 && m == -1}, {(-13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/280, k == 6 && m == 0}, {((-13/20 + (13*I)/20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/Sqrt[42], k == 6 && m == 1}, {(((13*I)/80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/Sqrt[21], k == 6 && m == 2}, {((13/40 + (13*I)/40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/Sqrt[21], k == 6 && m == 3}, {(13/40 - (13*I)/40)*Sqrt[11/7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == 5}, {((-13*I)/80)*Sqrt[11/21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu), k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_111.Quanty
Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1 + Eeu2))} , 
       {2,-1, (-5/28+-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {2, 1, (5/28+-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {2,-2, (-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {2, 2, (5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {4, 0, (1/4)*((-1)*(Ea1u) + (-6)*(Ea2u1) + (3)*(Ea2u2) + (6)*(Eeu1) + (-2)*(Eeu2))} , 
       {4,-1, (-1/28+-1/28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , 
       {4, 1, (1/28+-1/28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , 
       {4, 2, (-1/7*I)*((1/(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4,-2, (1/7*I)*((1/(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4,-3, (-1/4+1/4*I)*((1/(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4, 3, (1/4+1/4*I)*((1/(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4,-4, (-1/4)*((sqrt(5/14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , 
       {4, 4, (-1/4)*((sqrt(5/14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , 
       {6, 0, (-13/280)*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2))} , 
       {6, 1, (-13/20+13/20*I)*((1/(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , 
       {6,-1, (13/20+13/20*I)*((1/(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , 
       {6,-2, (-13/80*I)*((1/(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , 
       {6, 2, (13/80*I)*((1/(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , 
       {6,-3, (-13/40+13/40*I)*((1/(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , 
       {6, 3, (13/40+13/40*I)*((1/(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , 
       {6,-4, (13/40)*((1/(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , 
       {6, 4, (13/40)*((1/(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , 
       {6,-5, (-13/40+-13/40*I)*((sqrt(11/7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , 
       {6, 5, (13/40+-13/40*I)*((sqrt(11/7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , 
       {6, 6, (-13/80*I)*((sqrt(11/21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} , 
       {6,-6, (13/80*I)*((sqrt(11/21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-3}^{(3)}} \frac{1}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2}) \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Meu}-\text{Ma2u})\right)}{\sqrt{6}} \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} \left(\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} -\frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right)
{Y_{-2}^{(3)}} -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) -\frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) \frac{i \text{Ma2u}}{\sqrt{6}} \frac{1}{12} (-1)^{3/4} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}}
{Y_{-1}^{(3)}} -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} \frac{1}{24} i \left(5 \text{Ea1u}-3 \text{Ea2u2}+3 \text{Eeu1}-5 \text{Eeu2}-6 \sqrt{5} \text{Meu}\right) \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2})
{Y_{0}^{(3)}} \left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) -\frac{i \text{Ma2u}}{\sqrt{6}} \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} \frac{1}{3} (\text{Ea2u2}+2 \text{Eeu1}) \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea2u2}-\text{Eeu1}-\sqrt{5} \text{Meu}\right)}{\sqrt{3}} \frac{i \text{Ma2u}}{\sqrt{6}} \left(-\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right)
{Y_{1}^{(3)}} \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) \frac{1}{24} i \left(-5 \text{Ea1u}+3 \text{Ea2u2}-3 \text{Eeu1}+5 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}}
{Y_{2}^{(3)}} \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) \frac{1}{12} \sqrt[4]{-1} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) -\frac{i \text{Ma2u}}{\sqrt{6}} -\frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}}
{Y_{3}^{(3)}} \frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} \frac{1}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2})

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{x^3+y^3+z^3} f_{x^3-y^3} f_{2z^3-x^3-y^3} f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y}
f_{\text{xyz}} \text{Ea2u1} \text{Ma2u} 0 0 0 0 0
f_{x^3+y^3+z^3} \text{Ma2u} \text{Ea2u2} 0 0 0 0 0
f_{x^3-y^3} 0 0 \text{Eeu1} 0 0 \text{Meu} 0
f_{2z^3-x^3-y^3} 0 0 0 \text{Eeu1} 0 0 \text{Meu}
f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} 0 0 0 0 \text{Ea1u} 0 0
f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} 0 0 \text{Meu} 0 0 \text{Eeu2} 0
f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} 0 0 0 \text{Meu} 0 0 \text{Eeu2}

Rotation matrix used

Rotation matrix used

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
f_{\text{xyz}} 0 \frac{i}{\sqrt{2}} 0 0 0 -\frac{i}{\sqrt{2}} 0
f_{x^3+y^3+z^3} \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} 0 -\frac{1}{4}-\frac{i}{4} \frac{1}{\sqrt{3}} \frac{1}{4}-\frac{i}{4} 0 \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}}
f_{x^3-y^3} \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} 0 \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} 0 \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} 0 \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}}
f_{2z^3-x^3-y^3} \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} 0 \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} \sqrt{\frac{2}{3}} -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} 0 \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}}
f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} -\frac{1}{4}-\frac{i}{4} \frac{1}{\sqrt{6}} \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} 0 \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} \frac{1}{\sqrt{6}} \frac{1}{4}-\frac{i}{4}
f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} \frac{1}{\sqrt{3}} \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} 0 \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} \frac{1}{\sqrt{3}} -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}}
f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} 0 \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} 0 \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} 0 \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}}

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Ea2u1}
\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{Ea2u2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \left(\frac{1}{32}+\frac{i}{32}\right) \sqrt{\frac{7}{3 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{7}{3 \pi }} \left(5 x^3-15 x^2 y-3 x \left(5 y^2+5 z^2-1\right)+5 y^3+y \left(3-15 z^2\right)+4 z \left(5 z^2-3\right)\right)
\text{Eeu1}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{8} \sqrt{\frac{7}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(5 \sin ^2(\theta ) \sin (2 \phi )-5 \cos (2 \theta )-1\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{7}{2 \pi }} (x-y) \left(5 x^2+20 x y+5 y^2-15 z^2+3\right)
\text{Eeu1}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \left(-\frac{1}{32}-\frac{i}{32}\right) \sqrt{\frac{7}{6 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )-(4-4 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{7}{6 \pi }} \left(-5 x^3+15 x^2 y+3 x \left(5 y^2+5 z^2-1\right)-5 y^3+3 y \left(5 z^2-1\right)+8 z \left(5 z^2-3\right)\right)
\text{Ea1u}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{8} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(-\sin ^2(\theta ) \sin (2 \phi )+\sin (2 \theta ) (\sin (\phi )+\cos (\phi ))-2 \cos ^2(\theta )\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} -\frac{1}{16} \sqrt{\frac{35}{\pi }} (x-y) \left(x^2+4 x (y-z)+y^2-4 y z+5 z^2-1\right)
\text{Eeu2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{8} \sqrt{\frac{35}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )+2 \sin (2 \theta ) (\sin (\phi )+\cos (\phi ))+2 \cos ^2(\theta )\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{35}{2 \pi }} (x-y) \left(x^2+4 x (y+2 z)+y^2+8 y z+5 z^2-1\right)
\text{Eeu2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} -\frac{1}{8} \sqrt{\frac{105}{2 \pi }} \sin (\theta ) (\sin (\phi )+\cos (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )-2 \cos ^2(\theta )\right)
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{105}{2 \pi }} (x+y) \left(x^2-4 x y+y^2+5 z^2-1\right)

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & k=0\land m=0 \\ i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-2 \\ (1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-1 \\ (-1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=1 \\ -i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=2 \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {I*Sqrt[5/6]*Ma1g, k == 2 && m == -2}, {(1 + I)*Sqrt[5/6]*Ma1g, k == 2 && m == -1}, {(-1 + I)*Sqrt[5/6]*Ma1g, k == 2 && m == 1}, {(-I)*Sqrt[5/6]*Ma1g, k == 2 && m == 2}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_111.Quanty
Akm = {{2, 1, (-1+1*I)*((sqrt(5/6))*(Ma1g))} , 
       {2,-1, (1+1*I)*((sqrt(5/6))*(Ma1g))} , 
       {2, 2, (-I)*((sqrt(5/6))*(Ma1g))} , 
       {2,-2, (I)*((sqrt(5/6))*(Ma1g))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{0}^{(0)}} -\frac{i \text{Ma1g}}{\sqrt{6}} \frac{(1-i) \text{Ma1g}}{\sqrt{6}} 0 -\frac{(1+i) \text{Ma1g}}{\sqrt{6}} \frac{i \text{Ma1g}}{\sqrt{6}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{\text{yz}+\text{xz}+\text{xy}} d_{\text{yz}-\text{xz}} d_{2\text{xy}-\text{xz}-\text{yz}} d_{x^2-y^2} d_{3z^2-r^2}
\text{s} \text{Ma1g} 0 0 0 0

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ \frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-2 \\ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-1 \\ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=1 \\ -\frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=2 \\ -\frac{1}{4} \sqrt{\frac{3}{10}} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land (m=-4\lor m=4) \\ \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-3 \\ i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-2 \\ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-1 \\ -\frac{1}{20} \sqrt{21} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land m=0 \\ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=1 \\ -i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=2 \\ \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -2}, {(1/3 + I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -1}, {(-1/3 + I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 1}, {(-I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 2}, {-(Sqrt[3/10]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/4, k == 4 && (m == -4 || m == 4)}, {(-1/2 + I/2)*Sqrt[3]*(Ma2u1 - Meu2), k == 4 && m == -3}, {I*Sqrt[6/7]*(Ma2u1 - Meu2), k == 4 && m == -2}, {(-1/2 - I/2)*Sqrt[3/7]*(Ma2u1 - Meu2), k == 4 && m == -1}, {-(Sqrt[21]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/20, k == 4 && m == 0}, {(1/2 - I/2)*Sqrt[3/7]*(Ma2u1 - Meu2), k == 4 && m == 1}, {(-I)*Sqrt[6/7]*(Ma2u1 - Meu2), k == 4 && m == 2}}, (1/2 + I/2)*Sqrt[3]*(Ma2u1 - Meu2)]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_111.Quanty
Akm = {{2, 1, (-1/3+1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {2,-1, (1/3+1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {2, 2, (-1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {2,-2, (1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {4, 0, (-1/20)*((sqrt(21))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , 
       {4,-1, (-1/2+-1/2*I)*((sqrt(3/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4, 1, (1/2+-1/2*I)*((sqrt(3/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4, 2, (-I)*((sqrt(6/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4,-2, (I)*((sqrt(6/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4,-3, (-1/2+1/2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4, 3, (1/2+1/2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4,-4, (-1/4)*((sqrt(3/10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , 
       {4, 4, (-1/4)*((sqrt(3/10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-1}^{(1)}} -\frac{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}} (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{\#$1}^4+1$,3\right] \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} (2 \text{Ma2u1}+\text{Meu2}) \text{Root}\left[2025 \text{\#$1}^4+1$,1\right] -\frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} \frac{(-1)^{3/4} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}}
{Y_{0}^{(1)}} \left(-\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2}) -\frac{i \text{Ma2u1}}{\sqrt{6}} (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{\#$1}^4+1$,3\right] -\frac{2 \text{Ma2u1}}{3 \sqrt{5}}+\text{Meu1}-\frac{\text{Meu2}}{3 \sqrt{5}} (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{\#$1}^4+1$,1\right] \frac{i \text{Ma2u1}}{\sqrt{6}} \left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2})
{Y_{1}^{(1)}} \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}} \frac{\sqrt[4]{-1} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} \frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} \frac{\left(\frac{1}{3}-\frac{i}{3}\right) (2 \text{Ma2u1}+\text{Meu2})}{\sqrt{10}} \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{\#$1}^4+1$,1\right] \frac{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{x^3+y^3+z^3} f_{x^3-y^3} f_{2z^3-x^3-y^3} f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y}
p_{x+y+z} \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(2-2 i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(-1+i)\right)\right) \left(-\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(-15 \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+15 i \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]-60 \sqrt{2} \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+(29+29 i) \sqrt{5}\right)+\text{Meu2} \left(-45 \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+45 i \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]-30 \sqrt{2} \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+(7+7 i) \sqrt{5}\right)-(90+90 i) \text{Meu1}\right) \frac{\left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)}{\sqrt{2}} \frac{\text{Ma2u1} \left(-(3-3 i) \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+(3+3 i) \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]+(24-24 i) \sqrt{2} \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+2 \sqrt{5}\right)+(1+i) \text{Meu2} \left(9 i \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+9 \sqrt{3} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]-12 i \sqrt{2} \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+(-1+i) \sqrt{5}\right)}{36 \sqrt{2}} \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+2 i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+2 i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)\right) -\frac{\left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(4 \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+4 i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)+\text{Meu2} \left(4 \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+4 i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)\right)}{\sqrt{2}} \frac{\left(\frac{1}{12}+\frac{i}{12}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(i \sqrt{15} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+\sqrt{15} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]+(-1+i)\right)}{\sqrt{2}}
p_{x-y} \frac{\left(\frac{1}{2}+\frac{i}{2}\right) (\text{Ma2u1}+\text{Meu2}) \left(\text{Root}\left[36 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]\right)}{\sqrt{2}} \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+i \sqrt{10}\right)}{30 \sqrt{3}} \text{Meu1} \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+i \sqrt{10}\right)}{15 \sqrt{6}} \frac{\left(\frac{1}{6}+\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(-1+i)\right)}{\sqrt{2}} \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(-1+i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(2-2 i)\right)\right) 0
p_{3z-r} \frac{\left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(\sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+(1+i)\right)}{\sqrt{2}} \left(\frac{1}{180}-\frac{i}{180}\right) \left(\text{Ma2u1} \left(-4 \text{Root}\left[\text{\#$1}^4+25$,1\right]+15 \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]-15 i \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]+(1+i) \sqrt{10}\right)+\text{Meu2} \left(-2 \text{Root}\left[\text{\#$1}^4+25$,1\right]+45 \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]-45 i \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]+(8+8 i) \sqrt{10}\right)\right) \left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right) \frac{\left(\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(15 i \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+15 \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]+120 i \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+(-7+7 i) \sqrt{10}\right)+\text{Meu2} \left(45 i \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+45 \sqrt{6} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]+60 i \text{Root}\left[2025 \text{\#$1}^4+1$,1\right]+(-11+11 i) \sqrt{10}\right)+(90-90 i) \sqrt{2} \text{Meu1}\right)}{\sqrt{2}} \frac{\left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(\text{Root}\left[36 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)+\text{Meu2} \left(\text{Root}\left[36 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)\right)}{\sqrt{2}} \left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+2 i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{\#$1}^4+1$,1\right]+2 i \text{Root}\left[36 \text{\#$1}^4+1$,3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{\#$1}^4+1$,1\right]+i \text{Root}\left[900 \text{\#$1}^4+1$,3\right]\right)\right)\right) \frac{1}{12} \left(\text{Ma2u1} \left(-(1-i) \sqrt{15} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+(1+i) \sqrt{15} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]-2\right)+(3+3 i) \text{Meu2} \left(i \sqrt{15} \text{Root}\left[900 \text{\#$1}^4+1$,1\right]+\sqrt{15} \text{Root}\left[900 \text{\#$1}^4+1$,3\right]+(1-i)\right)\right)

Table of several point groups

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

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