Processing math: 49%

Table of Contents

Orientation Z

Symmetry Operations

In the C2 Point Group, with orientation Z there are the following symmetry operations

Operator Orientation
E {0,0,0} ,
C2 {0,0,1} ,

Different Settings

Character Table

E(1) C2(1)
A 1 1
B 1 1

Product Table

A B
A A B
B B A

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=0km=kAk,m(r)C(m)k(θ,ϕ) Here Ak,m(r) is a radial function and C(m)k(θ,ϕ) a renormalised spherical harmonics. C(m)k(θ,ϕ)=4π2k+1Y(m)k(θ,ϕ) The presence of symmetry induces relations between the expansion coefficients such that V(r,θ,ϕ) is invariant under all symmetry operations. For the C2 Point group with orientation Z the form of the expansion coefficients is:

Expansion

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

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[1, 0], k == 1 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}, {A[3, 2] + I*B[3, 2], k == 3 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {A[5, 2] - I*B[5, 2], k == 5 && m == -2}, {A[5, 0], k == 5 && m == 0}, {A[5, 2] + I*B[5, 2], k == 5 && m == 2}, {A[5, 4] + I*B[5, 4], k == 5 && m == 4}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{0, 0, A(0,0)} , 
       {1, 0, A(1,0)} , 
       {2, 0, A(2,0)} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} , 
       {3, 0, A(3,0)} , 
       {3,-2, A(3,2) + (-I)*(B(3,2))} , 
       {3, 2, A(3,2) + (I)*(B(3,2))} , 
       {4, 0, A(4,0)} , 
       {4,-2, A(4,2) + (-I)*(B(4,2))} , 
       {4, 2, A(4,2) + (I)*(B(4,2))} , 
       {4,-4, A(4,4) + (-I)*(B(4,4))} , 
       {4, 4, A(4,4) + (I)*(B(4,4))} , 
       {5, 0, A(5,0)} , 
       {5,-2, A(5,2) + (-I)*(B(5,2))} , 
       {5, 2, A(5,2) + (I)*(B(5,2))} , 
       {5,-4, A(5,4) + (-I)*(B(5,4))} , 
       {5, 4, A(5,4) + (I)*(B(5,4))} , 
       {6, 0, A(6,0)} , 
       {6,-2, A(6,2) + (-I)*(B(6,2))} , 
       {6, 2, A(6,2) + (I)*(B(6,2))} , 
       {6,-4, A(6,4) + (-I)*(B(6,4))} , 
       {6, 4, A(6,4) + (I)*(B(6,4))} , 
       {6,-6, A(6,6) + (-I)*(B(6,6))} , 
       {6, 6, A(6,6) + (I)*(B(6,6))} }

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,mψn,l,m(r,θ,ϕ)|V(r,θ,ϕ)|ψn,l,m(r,θ,ϕ)an,l,man,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. Anl,nl(k,m)=Rn,l|Ak,m(r)|Rn,l Note the difference between the function Ak,m and the parameter Anl,nl(k,m)

we can express the operator as O=n,l,m,n,l,m,k,mAnl,nl(k,m)Y(m)l(θ,ϕ)|C(m)k(θ,ϕ)|Y(m)l(θ,ϕ)an,l,man,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)0Asp(1,0)30Asd(2,2)+iBsd(2,2)50Asd(2,0)50Asd(2,2)iBsd(2,2)50Asf(3,2)+iBsf(3,2)70Asf(3,0)70Asf(3,2)iBsf(3,2)70
Y(1)10App(0,0)15App(2,0)0156(App(2,2)iBpp(2,2))0Apd(1,0)53Apd(3,0)750176(Apd(3,2)iBpd(3,2))03(Apf(2,2)+iBpf(2,2))35Apf(4,2)+iBpf(4,2)32103527Apf(2,0)1327Apf(4,0)01537(Apf(2,2)iBpf(2,2))1357(Apf(4,2)iBpf(4,2))02(Apf(4,4)iBpf(4,4))33
Y(1)0Asp(1,0)30App(0,0)+25App(2,0)0173(Apd(3,2)+iBpd(3,2))02Apd(1,0)15+3735Apd(3,0)0173(Apd(3,2)iBpd(3,2))0335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3703537Apf(2,0)+4Apf(4,0)3210335(Apf(2,2)iBpf(2,2))+2(Apf(4,2)iBpf(4,2))370
Y(1)10156(App(2,2)+iBpp(2,2))0App(0,0)15App(2,0)0176(Apd(3,2)+iBpd(3,2))0Apd(1,0)53Apd(3,0)7502(Apf(4,4)+iBpf(4,4))3301537(Apf(2,2)+iBpf(2,2))1357(Apf(4,2)+iBpf(4,2))03527Apf(2,0)1327Apf(4,0)03(Apf(2,2)iBpf(2,2))35Apf(4,2)iBpf(4,2)321
Y(2)2Asd(2,2)iBsd(2,2)50173(Apd(3,2)iBpd(3,2))0Add(0,0)27Add(2,0)+121Add(4,0)01753(Add(4,2)iBdd(4,2))27(Add(2,2)iBdd(2,2))013107(Add(4,4)iBdd(4,4))0Adf(1,0)72Adf(3,0)37+5Adf(5,0)3370533(Adf(5,2)iBdf(5,2))2(Adf(3,2)iBdf(3,2))37011110(Adf(5,4)iBdf(5,4))0
Y(2)10Apd(1,0)53Apd(3,0)750176(Apd(3,2)iBpd(3,2))0Add(0,0)+17Add(2,0)421Add(4,0)0176(Add(2,2)iBdd(2,2))22110(Add(4,2)iBdd(4,2))01357(Adf(3,2)+iBdf(3,2))1335(Adf(5,2)+iBdf(5,2))02235Adf(1,0)+13235Adf(3,0)533107Adf(5,0)0Adf(3,2)iBdf(3,2)215(Adf(5,2)iBdf(5,2))113021153(Adf(5,4)iBdf(5,4))
Y(2)0Asd(2,0)502Apd(1,0)15+3735Apd(3,0)01753(Add(4,2)+iBdd(4,2))27(Add(2,2)+iBdd(2,2))0Add(0,0)+27Add(2,0)+27Add(4,0)01753(Add(4,2)iBdd(4,2))27(Add(2,2)iBdd(2,2))01115(Adf(5,2)+iBdf(5,2))03Adf(1,0)35+4Adf(3,0)335+103357Adf(5,0)01115(Adf(5,2)iBdf(5,2))0
Y(2)10176(Apd(3,2)+iBpd(3,2))0Apd(1,0)53Apd(3,0)750176(Add(2,2)+iBdd(2,2))22110(Add(4,2)+iBdd(4,2))0Add(0,0)+17Add(2,0)421Add(4,0)021153(Adf(5,4)+iBdf(5,4))0Adf(3,2)+iBdf(3,2)215(Adf(5,2)+iBdf(5,2))11302235Adf(1,0)+13235Adf(3,0)533107Adf(5,0)01357(Adf(3,2)iBdf(3,2))1335(Adf(5,2)iBdf(5,2))
Y(2)2Asd(2,2)+iBsd(2,2)50173(Apd(3,2)+iBpd(3,2))013107(Add(4,4)+iBdd(4,4))01753(Add(4,2)+iBdd(4,2))27(Add(2,2)+iBdd(2,2))0Add(0,0)27Add(2,0)+121Add(4,0)011110(Adf(5,4)+iBdf(5,4))0533(Adf(5,2)+iBdf(5,2))2(Adf(3,2)+iBdf(3,2))370Adf(1,0)72Adf(3,0)37+5Adf(5,0)3370
Y(3)303(Apf(2,2)iBpf(2,2))35Apf(4,2)iBpf(4,2)32102(Apf(4,4)iBpf(4,4))3301357(Adf(3,2)iBdf(3,2))1335(Adf(5,2)iBdf(5,2))021153(Adf(5,4)iBdf(5,4))0Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)01325(Aff(2,2)iBff(2,2))+1116(Aff(4,2)iBff(4,2))104297(Aff(6,2)iBff(6,2))0111143(Aff(4,4)iBff(4,4))5143703(Aff(6,4)iBff(6,4))01013733(Aff(6,6)iBff(6,6))
Y(3)2Asf(3,2)iBsf(3,2)70335(Apf(2,2)iBpf(2,2))+2(Apf(4,2)iBpf(4,2))370Adf(1,0)72Adf(3,0)37+5Adf(5,0)33701115(Adf(5,2)iBdf(5,2))011110(Adf(5,4)iBdf(5,4))0Aff(0,0)733Aff(4,0)+10143Aff(6,0)02(Aff(2,2)iBff(2,2))35Aff(4,2)iBff(4,2)113+2042914(Aff(6,2)iBff(6,2))013370(Aff(4,4)iBff(4,4))+1014314(Aff(6,4)iBff(6,4))0
Y(3)103527Apf(2,0)1327Apf(4,0)01537(Apf(2,2)iBpf(2,2))1357(Apf(4,2)iBpf(4,2))02235Adf(1,0)+13235Adf(3,0)533107Adf(5,0)0Adf(3,2)iBdf(3,2)215(Adf(5,2)iBdf(5,2))11301325(Aff(2,2)+iBff(2,2))+1116(Aff(4,2)+iBff(4,2))104297(Aff(6,2)+iBff(6,2))0Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)02523(Aff(2,2)iBff(2,2))23310(Aff(4,2)iBff(4,2))10143353(Aff(6,2)iBff(6,2))0111143(Aff(4,4)iBff(4,4))5143703(Aff(6,4)iBff(6,4))
Y(3)0Asf(3,0)703537Apf(2,0)+4Apf(4,0)3210533(Adf(5,2)+iBdf(5,2))2(Adf(3,2)+iBdf(3,2))3703Adf(1,0)35+4Adf(3,0)335+103357Adf(5,0)0533(Adf(5,2)iBdf(5,2))2(Adf(3,2)iBdf(3,2))3702(Aff(2,2)+iBff(2,2))35Aff(4,2)+iBff(4,2)113+2042914(Aff(6,2)+iBff(6,2))0Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)02(Aff(2,2)iBff(2,2))35Aff(4,2)iBff(4,2)113+2042914(Aff(6,2)iBff(6,2))0
Y(3)101537(Apf(2,2)+iBpf(2,2))1357(Apf(4,2)+iBpf(4,2))03527Apf(2,0)1327Apf(4,0)0Adf(3,2)+iBdf(3,2)215(Adf(5,2)+iBdf(5,2))11302235Adf(1,0)+13235Adf(3,0)533107Adf(5,0)0111143(Aff(4,4)+iBff(4,4))5143703(Aff(6,4)+iBff(6,4))02523(Aff(2,2)+iBff(2,2))23310(Aff(4,2)+iBff(4,2))10143353(Aff(6,2)+iBff(6,2))0Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)01325(Aff(2,2)iBff(2,2))+1116(Aff(4,2)iBff(4,2))104297(Aff(6,2)iBff(6,2))
Y(3)2Asf(3,2)+iBsf(3,2)70335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))37011110(Adf(5,4)+iBdf(5,4))01115(Adf(5,2)+iBdf(5,2))0Adf(1,0)72Adf(3,0)37+5Adf(5,0)337013370(Aff(4,4)+iBff(4,4))+1014314(Aff(6,4)+iBff(6,4))02(Aff(2,2)+iBff(2,2))35Aff(4,2)+iBff(4,2)113+2042914(Aff(6,2)+iBff(6,2))0Aff(0,0)733Aff(4,0)+10143Aff(6,0)0
Y(3)302(Apf(4,4)+iBpf(4,4))3303(Apf(2,2)+iBpf(2,2))35Apf(4,2)+iBpf(4,2)321021153(Adf(5,4)+iBdf(5,4))01357(Adf(3,2)+iBdf(3,2))1335(Adf(5,2)+iBdf(5,2))01013733(Aff(6,6)+iBff(6,6))0111143(Aff(4,4)+iBff(4,4))5143703(Aff(6,4)+iBff(6,4))01325(Aff(2,2)+iBff(2,2))+1116(Aff(4,2)+iBff(4,2))104297(Aff(6,2)+iBff(6,2))0Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)

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

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

Y(0)0 Y(1)1 Y(1)0 Y(1)1 Y(2)2 Y(2)1 Y(2)0 Y(2)1 Y(2)2 Y(3)3 Y(3)2 Y(3)1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
s1000000000000000
px012012000000000000
py0i20i2000000000000
pz0010000000000000
dx2y2000012000120000000
d3z2r20000001000000000
dyz00000i20i200000000
dxz000001201200000000
dxy0000i2000i20000000
fxyz0000000000i2000i20
fx(5x2r2)00000000054034034054
fy(5y2r2)000000000i540i340i340i54
fz(5z2r2)0000000000001000
fx(y2z2)00000000034054054034
fy(z2x2)000000000i340i540i540i34
fz(x2y2)000000000012000120

One particle coupling on a basis of symmetry adapted functions

After rotation we find

s px py pz dx2y2 d3z2r2 dyz dxz dxy fxyz fx(5x2r2) fy(5y2r2) fz(5z2r2) fx(y2z2) fy(z2x2) fz(x2y2)
sAss(0,0)00Asp(1,0)325Asd(2,2)Asd(2,0)50025Bsd(2,2)27Bsf(3,2)00Asf(3,0)70027Asf(3,2)
px0App(0,0)15App(2,0)+156App(2,2)156Bpp(2,2)000176Bpd(3,2)Apd(1,0)53Apd(3,0)75+176Apd(3,2)0031037Apf(2,0)+9Apf(2,2)514+Apf(4,0)221131021Apf(4,2)+1356Apf(4,4)3527Bpf(2,2)+13542Bpf(4,2)+1356Bpf(4,4)03Apf(2,0)235370Apf(2,2)+1657Apf(4,0)1327Apf(4,2)Apf(4,4)32635Bpf(2,2)Bpf(4,2)14+Bpf(4,4)320
py0156Bpp(2,2)App(0,0)15App(2,0)156App(2,2)000Apd(1,0)53Apd(3,0)75176Apd(3,2)176Bpd(3,2)003527Bpf(2,2)+13542Bpf(4,2)1356Bpf(4,4)31037Apf(2,0)9Apf(2,2)514+Apf(4,0)221+131021Apf(4,2)+1356Apf(4,4)0635Bpf(2,2)+Bpf(4,2)14+Bpf(4,4)323Apf(2,0)235370Apf(2,2)1657Apf(4,0)1327Apf(4,2)+Apf(4,4)320
pzAsp(1,0)300App(0,0)+25App(2,0)176Apd(3,2)2Apd(1,0)15+3735Apd(3,0)00176Bpd(3,2)635Bpf(2,2)2327Bpf(4,2)003537Apf(2,0)+4Apf(4,0)32100635Apf(2,2)+2327Apf(4,2)
dx2y225Asd(2,2)00176Apd(3,2)Add(0,0)27Add(2,0)+121Add(4,0)+13107Add(4,4)17103Add(4,2)272Add(2,2)0013107Bdd(4,4)11110Bdf(5,4)005332Adf(5,2)2327Adf(3,2)00Adf(1,0)72Adf(3,0)37+5Adf(5,0)337+11110Adf(5,4)
d3z2r2Asd(2,0)5002Apd(1,0)15+3735Apd(3,0)17103Add(4,2)272Add(2,2)Add(0,0)+27Add(2,0)+27Add(4,0)00272Bdd(2,2)17103Bdd(4,2)11110Bdf(5,2)003Adf(1,0)35+4Adf(3,0)335+103357Adf(5,0)0011110Adf(5,2)
dyz0176Bpd(3,2)Apd(1,0)53Apd(3,0)75176Apd(3,2)000Add(0,0)+17Add(2,0)176Add(2,2)421Add(4,0)22110Add(4,2)176Bdd(2,2)22110Bdd(4,2)002327Bdf(3,2)+5Bdf(5,2)3325Bdf(5,4)116335Adf(1,0)Adf(3,0)2105Adf(3,2)314+522521Adf(5,0)+5332Adf(5,2)+5Adf(5,4)1160111152Bdf(5,2)+11152Bdf(5,4)Adf(1,0)7+Adf(3,0)67542Adf(3,2)25Adf(5,0)667111103Adf(5,2)+11152Adf(5,4)0
dxz0Apd(1,0)53Apd(3,0)75+176Apd(3,2)176Bpd(3,2)000176Bdd(2,2)22110Bdd(4,2)Add(0,0)+17Add(2,0)+176Add(2,2)421Add(4,0)+22110Add(4,2)00335Adf(1,0)Adf(3,0)2105+Adf(3,2)314+522521Adf(5,0)5332Adf(5,2)+5Adf(5,4)1162327Bdf(3,2)+5Bdf(5,2)332+5Bdf(5,4)1160Adf(1,0)7Adf(3,0)67542Adf(3,2)+25Adf(5,0)667111103Adf(5,2)11152Adf(5,4)11152Bdf(5,4)111152Bdf(5,2)0
dxy25Bsd(2,2)00176Bpd(3,2)13107Bdd(4,4)272Bdd(2,2)17103Bdd(4,2)00Add(0,0)27Add(2,0)+121Add(4,0)13107Add(4,4)Adf(1,0)72Adf(3,0)37+5Adf(5,0)33711110Adf(5,4)002327Bdf(3,2)5332Bdf(5,2)0011110Bdf(5,4)
fxyz27Bsf(3,2)00635Bpf(2,2)2327Bpf(4,2)11110Bdf(5,4)11110Bdf(5,2)00Adf(1,0)72Adf(3,0)37+5Adf(5,0)33711110Adf(5,4)Aff(0,0)733Aff(4,0)13370Aff(4,4)+10143Aff(6,0)1014314Aff(6,4)002325Bff(2,2)+11123Bff(4,2)404297Bff(6,2)0013370Bff(4,4)1014314Bff(6,4)
fx(5x2r2)031037Apf(2,0)+9Apf(2,2)514+Apf(4,0)221131021Apf(4,2)+1356Apf(4,4)3527Bpf(2,2)+13542Bpf(4,2)1356Bpf(4,4)0002327Bdf(3,2)+5Bdf(5,2)3325Bdf(5,4)116335Adf(1,0)Adf(3,0)2105+Adf(3,2)314+522521Adf(5,0)5332Adf(5,2)+5Adf(5,4)11600Aff(0,0)215Aff(2,0)+2523Aff(2,2)+344Aff(4,0)11152Aff(4,2)+122352Aff(4,4)125Aff(6,0)1716+25572353Aff(6,2)2528672Aff(6,4)+2552733Aff(6,6)Bff(2,2)5611110Bff(4,2)5572353Bff(6,2)+2552733Bff(6,6) 0 \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) 0
f_{y\left(5y^2-r^2\right)} \color{darkred}{ 0 } \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}+\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }\color{darkred}{ 0 } 0 \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) 0 -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) 0
f_{z\left(5z^2-r^2\right)} \color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} } 0 0 \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} \color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) } \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) 0 0 \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) 0 0 -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2)
f_{x\left(y^2-z^2\right)} \color{darkred}{ 0 } -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }\color{darkred}{ 0 } 0 \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) 0 \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) 0
f_{y\left(z^2-x^2\right)} \color{darkred}{ 0 } \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} 0 \color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }\color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }\color{darkred}{ 0 } 0 \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) 0 \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) 0
f_{z\left(x^2-y^2\right)} \color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) } 0 0 \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) \color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }\color{darkred}{ 0 }\color{darkred}{ 0 }\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) } -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) 0 0 -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) 0 0 \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4)

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{Ea} & 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_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Ea, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{0, 0, Ea} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

\text{s}
\text{s} \text{Ea}

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{Ea}
\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{Ea}+\text{Ebx}+\text{Eby}) & k=0\land m=0 \\ 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ \frac{5 (\text{Ebx}-\text{Eby}+2 i \text{Mb})}{2 \sqrt{6}} & k=2\land m=-2 \\ \frac{5}{6} (2 \text{Ea}-\text{Ebx}-\text{Eby}) & k=2\land m=0 \\ \frac{5 (\text{Ebx}-\text{Eby}-2 i \text{Mb})}{2 \sqrt{6}} & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea + Ebx + Eby)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != 0 && m != 2)}, {(5*(Ebx - Eby + (2*I)*Mb))/(2*Sqrt[6]), k == 2 && m == -2}, {(5*(2*Ea - Ebx - Eby))/6, k == 2 && m == 0}}, (5*(Ebx - Eby - (2*I)*Mb))/(2*Sqrt[6])]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{0, 0, (1/3)*(Ea + Ebx + Eby)} , 
       {2, 0, (5/6)*((2)*(Ea) + (-1)*(Ebx) + (-1)*(Eby))} , 
       {2, 2, (5/2)*((1/(sqrt(6)))*(Ebx + (-1)*(Eby) + (-2*I)*(Mb)))} , 
       {2,-2, (5/2)*((1/(sqrt(6)))*(Ebx + (-1)*(Eby) + (2*I)*(Mb)))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-1}^{(1)}} {Y_{0}^{(1)}} {Y_{1}^{(1)}}
{Y_{-1}^{(1)}} \frac{\text{Ebx}+\text{Eby}}{2} 0 \frac{1}{2} (-\text{Ebx}+\text{Eby}-2 i \text{Mb})
{Y_{0}^{(1)}} 0 \text{Ea} 0
{Y_{1}^{(1)}} \frac{1}{2} (-\text{Ebx}+\text{Eby}+2 i \text{Mb}) 0 \frac{\text{Ebx}+\text{Eby}}{2}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

p_x p_y p_z
p_x \text{Ebx} \text{Mb} 0
p_y \text{Mb} \text{Eby} 0
p_z 0 0 \text{Ea}

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Ebx}
\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{Eby}
\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{Ea}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{3}{\pi }} z

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{5} (\text{Eax2y2}+\text{Eaxy}+\text{Eaz2}+\text{Ebxz}+\text{Ebyz}) & 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) \\ \frac{\sqrt{3} \text{Ebxz}-\sqrt{3} \text{Ebyz}-4 \text{Maxy2yz2}-4 i \text{Maz2xy}+2 i \sqrt{3} \text{Mb}}{2 \sqrt{2}} & k=2\land m=-2 \\ \frac{1}{2} (-2 \text{Eax2y2}-2 \text{Eaxy}+2 \text{Eaz2}+\text{Ebxz}+\text{Ebyz}) & k=2\land m=0 \\ \frac{\sqrt{3} \text{Ebxz}-\sqrt{3} \text{Ebyz}-4 \text{Maxy2yz2}+4 i \text{Maz2xy}-2 i \sqrt{3} \text{Mb}}{2 \sqrt{2}} & k=2\land m=2 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eax2y2}-\text{Eaxy}+2 i \text{Max2y2xy}) & k=4\land m=-4 \\ \frac{3 \left(\text{Ebxz}-\text{Ebyz}+\sqrt{3} \text{Maxy2yz2}+i \sqrt{3} \text{Maz2xy}+2 i \text{Mb}\right)}{\sqrt{10}} & k=4\land m=-2 \\ \frac{3}{10} (\text{Eax2y2}+\text{Eaxy}+6 \text{Eaz2}-4 \text{Ebxz}-4 \text{Ebyz}) & k=4\land m=0 \\ \frac{3 \left(\text{Ebxz}-\text{Ebyz}+\sqrt{3} \text{Maxy2yz2}-i \sqrt{3} \text{Maz2xy}-2 i \text{Mb}\right)}{\sqrt{10}} & k=4\land m=2 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eax2y2}-\text{Eaxy}-2 i \text{Max2y2xy}) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eax2y2 + Eaxy + Eaz2 + Ebxz + Ebyz)/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]*Ebxz - Sqrt[3]*Ebyz - 4*Maxy2yz2 - (4*I)*Maz2xy + (2*I)*Sqrt[3]*Mb)/(2*Sqrt[2]), k == 2 && m == -2}, {(-2*Eax2y2 - 2*Eaxy + 2*Eaz2 + Ebxz + Ebyz)/2, k == 2 && m == 0}, {(Sqrt[3]*Ebxz - Sqrt[3]*Ebyz - 4*Maxy2yz2 + (4*I)*Maz2xy - (2*I)*Sqrt[3]*Mb)/(2*Sqrt[2]), k == 2 && m == 2}, {(3*Sqrt[7/10]*(Eax2y2 - Eaxy + (2*I)*Max2y2xy))/2, k == 4 && m == -4}, {(3*(Ebxz - Ebyz + Sqrt[3]*Maxy2yz2 + I*Sqrt[3]*Maz2xy + (2*I)*Mb))/Sqrt[10], k == 4 && m == -2}, {(3*(Eax2y2 + Eaxy + 6*Eaz2 - 4*Ebxz - 4*Ebyz))/10, k == 4 && m == 0}, {(3*(Ebxz - Ebyz + Sqrt[3]*Maxy2yz2 - I*Sqrt[3]*Maz2xy - (2*I)*Mb))/Sqrt[10], k == 4 && m == 2}}, (3*Sqrt[7/10]*(Eax2y2 - Eaxy - (2*I)*Max2y2xy))/2]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{0, 0, (1/5)*(Eax2y2 + Eaxy + Eaz2 + Ebxz + Ebyz)} , 
       {2, 0, (1/2)*((-2)*(Eax2y2) + (-2)*(Eaxy) + (2)*(Eaz2) + Ebxz + Ebyz)} , 
       {2,-2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Ebxz) + (-1)*((sqrt(3))*(Ebyz)) + (-4)*(Maxy2yz2) + (-4*I)*(Maz2xy) + (2*I)*((sqrt(3))*(Mb))))} , 
       {2, 2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Ebxz) + (-1)*((sqrt(3))*(Ebyz)) + (-4)*(Maxy2yz2) + (4*I)*(Maz2xy) + (-2*I)*((sqrt(3))*(Mb))))} , 
       {4, 0, (3/10)*(Eax2y2 + Eaxy + (6)*(Eaz2) + (-4)*(Ebxz) + (-4)*(Ebyz))} , 
       {4, 2, (3)*((1/(sqrt(10)))*(Ebxz + (-1)*(Ebyz) + (sqrt(3))*(Maxy2yz2) + (-I)*((sqrt(3))*(Maz2xy)) + (-2*I)*(Mb)))} , 
       {4,-2, (3)*((1/(sqrt(10)))*(Ebxz + (-1)*(Ebyz) + (sqrt(3))*(Maxy2yz2) + (I)*((sqrt(3))*(Maz2xy)) + (2*I)*(Mb)))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Eax2y2 + (-1)*(Eaxy) + (-2*I)*(Max2y2xy)))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Eax2y2 + (-1)*(Eaxy) + (2*I)*(Max2y2xy)))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{-2}^{(2)}} \frac{\text{Eax2y2}+\text{Eaxy}}{2} 0 \frac{\text{Maxy2yz2}+i \text{Maz2xy}}{\sqrt{2}} 0 \frac{1}{2} (\text{Eax2y2}-\text{Eaxy}+2 i \text{Max2y2xy})
{Y_{-1}^{(2)}} 0 \frac{\text{Ebxz}+\text{Ebyz}}{2} 0 \frac{1}{2} (-\text{Ebxz}+\text{Ebyz}-2 i \text{Mb}) 0
{Y_{0}^{(2)}} \frac{\text{Maxy2yz2}-i \text{Maz2xy}}{\sqrt{2}} 0 \text{Eaz2} 0 \frac{\text{Maxy2yz2}+i \text{Maz2xy}}{\sqrt{2}}
{Y_{1}^{(2)}} 0 \frac{1}{2} (-\text{Ebxz}+\text{Ebyz}+2 i \text{Mb}) 0 \frac{\text{Ebxz}+\text{Ebyz}}{2} 0
{Y_{2}^{(2)}} \frac{1}{2} (\text{Eax2y2}-\text{Eaxy}-2 i \text{Max2y2xy}) 0 \frac{\text{Maxy2yz2}-i \text{Maz2xy}}{\sqrt{2}} 0 \frac{\text{Eax2y2}+\text{Eaxy}}{2}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{x^2-y^2} d_{3z^2-r^2} d_{\text{yz}} d_{\text{xz}} d_{\text{xy}}
d_{x^2-y^2} \text{Eax2y2} \text{Maxy2yz2} 0 0 \text{Max2y2xy}
d_{3z^2-r^2} \text{Maxy2yz2} \text{Eaz2} 0 0 \text{Maz2xy}
d_{\text{yz}} 0 0 \text{Ebyz} \text{Mb} 0
d_{\text{xz}} 0 0 \text{Mb} \text{Ebxz} 0
d_{\text{xy}} \text{Max2y2xy} \text{Maz2xy} 0 0 \text{Eaxy}

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Eax2y2}
\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{Eaz2}
\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{Ebyz}
\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{Ebxz}
\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{Eaxy}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{2} \sqrt{\frac{15}{\pi }} x y

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} \frac{1}{7} (\text{Eax3}+\text{Eaxy2z2}+\text{Eay3}+\text{Eayz2x2}+\text{Ebxyz}+\text{Ebz3}+\text{Ebzx2y2}) & k=0\land m=0 \\ 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) \\ \frac{5}{56} \left(2 \left(\sqrt{6} \text{Eax3}-\sqrt{6} \text{Eay3}+\sqrt{10} (\text{Max3xy2z2}+\text{May3yz2x2}-2 \text{Mbz3zx2y2})\right)-i \left(\sqrt{6} \text{Max3y3}+\sqrt{10} \text{Max3yz2x2}+5 \sqrt{6} \text{Maxy2z2yz2x2}-\sqrt{10} \text{May3xy2z2}+4 \sqrt{10} \text{Mbxyzz3}\right)\right) & k=2\land m=-2 \\ -\frac{5}{14} \left(\text{Eax3}+\text{Eay3}-2 \text{Ebz3}-\sqrt{15} \text{Max3xy2z2}+\sqrt{15} \text{May3yz2x2}\right) & k=2\land m=0 \\ \frac{5}{56} \left(2 \left(\sqrt{6} \text{Eax3}-\sqrt{6} \text{Eay3}+\sqrt{10} (\text{Max3xy2z2}+\text{May3yz2x2}-2 \text{Mbz3zx2y2})\right)+i \left(\sqrt{6} \text{Max3y3}+\sqrt{10} \text{Max3yz2x2}+5 \sqrt{6} \text{Maxy2z2yz2x2}-\sqrt{10} \text{May3xy2z2}+4 \sqrt{10} \text{Mbxyzz3}\right)\right) & k=2\land m=2 \\ \frac{3 \left(3 \sqrt{5} \text{Eax3}-3 \sqrt{5} \text{Eaxy2z2}+3 \sqrt{5} \text{Eay3}-3 \sqrt{5} \text{Eayz2x2}-4 \sqrt{5} \text{Ebxyz}+4 \sqrt{5} \text{Ebzx2y2}+2 \sqrt{3} \text{Max3xy2z2}-8 i \sqrt{3} \text{Max3yz2x2}-8 i \sqrt{3} \text{May3xy2z2}-2 \sqrt{3} \text{May3yz2x2}+8 i \sqrt{5} \text{Mbxyzzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=-4 \\ \frac{3}{56} \left(-3 \sqrt{10} \text{Eax3}+7 \sqrt{10} \text{Eaxy2z2}+3 \sqrt{10} \text{Eay3}-7 \sqrt{10} \text{Eayz2x2}+2 \sqrt{6} \text{Max3xy2z2}+4 i \left(3 \sqrt{10} \text{Max3y3}-2 \sqrt{6} \text{Max3yz2x2}+\sqrt{10} \text{Maxy2z2yz2x2}+2 \sqrt{6} \text{May3xy2z2}-\sqrt{6} \text{Mbxyzz3}\right)+2 \sqrt{6} \text{May3yz2x2}-4 \sqrt{6} \text{Mbz3zx2y2}\right) & k=4\land m=-2 \\ \frac{3}{56} \left(9 \text{Eax3}+7 \text{Eaxy2z2}+9 \text{Eay3}+7 \text{Eayz2x2}-28 \text{Ebxyz}+24 \text{Ebz3}-28 \text{Ebzx2y2}-2 \sqrt{15} \text{Max3xy2z2}+2 \sqrt{15} \text{May3yz2x2}\right) & k=4\land m=0 \\ \frac{3}{56} \left(-3 \sqrt{10} \text{Eax3}+7 \sqrt{10} \text{Eaxy2z2}+3 \sqrt{10} \text{Eay3}-7 \sqrt{10} \text{Eayz2x2}+2 \sqrt{6} \text{Max3xy2z2}-4 i \left(3 \sqrt{10} \text{Max3y3}-2 \sqrt{6} \text{Max3yz2x2}+\sqrt{10} \text{Maxy2z2yz2x2}+2 \sqrt{6} \text{May3xy2z2}-\sqrt{6} \text{Mbxyzz3}\right)+2 \sqrt{6} \text{May3yz2x2}-4 \sqrt{6} \text{Mbz3zx2y2}\right) & k=4\land m=2 \\ \frac{3 \left(3 \sqrt{5} \text{Eax3}-3 \sqrt{5} \text{Eaxy2z2}+3 \sqrt{5} \text{Eay3}-3 \sqrt{5} \text{Eayz2x2}-4 \sqrt{5} \text{Ebxyz}+4 \sqrt{5} \text{Ebzx2y2}+2 \sqrt{3} \text{Max3xy2z2}+8 i \sqrt{3} \text{Max3yz2x2}+8 i \sqrt{3} \text{May3xy2z2}-2 \sqrt{3} \text{May3yz2x2}-8 i \sqrt{5} \text{Mbxyzzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=4 \\ \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eax3}+3 \sqrt{3} \text{Eaxy2z2}-5 \sqrt{3} \text{Eay3}-3 \sqrt{3} \text{Eayz2x2}-6 \sqrt{5} \text{Max3xy2z2}-10 i \sqrt{3} \text{Max3y3}-6 i \sqrt{5} \text{Max3yz2x2}+6 i \sqrt{3} \text{Maxy2z2yz2x2}+6 i \sqrt{5} \text{May3xy2z2}-6 \sqrt{5} \text{May3yz2x2}\right) & k=6\land m=-6 \\ -\frac{13 \left(15 \text{Eax3}-15 \text{Eaxy2z2}+15 \text{Eay3}-15 \text{Eayz2x2}+24 \text{Ebxyz}-24 \text{Ebzx2y2}+2 \sqrt{15} \text{Max3xy2z2}-8 i \sqrt{15} \text{Max3yz2x2}-8 i \sqrt{15} \text{May3xy2z2}-2 \sqrt{15} \text{May3yz2x2}-48 i \text{Mbxyzzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=-4 \\ \frac{13 \left(5 \sqrt{15} \text{Eax3}+3 \sqrt{15} \text{Eaxy2z2}-5 \sqrt{15} \text{Eay3}-3 \sqrt{15} \text{Eayz2x2}+34 \text{Max3xy2z2}+2 i \sqrt{15} \text{Max3y3}-26 i \text{Max3yz2x2}-14 i \sqrt{15} \text{Maxy2z2yz2x2}+26 i \text{May3xy2z2}+34 \text{May3yz2x2}+64 i \text{Mbxyzz3}+64 \text{Mbz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=-2 \\ -\frac{13}{560} \left(25 \text{Eax3}+39 \text{Eaxy2z2}+25 \text{Eay3}+39 \text{Eayz2x2}-24 \text{Ebxyz}-80 \text{Ebz3}-24 \text{Ebzx2y2}+14 \sqrt{15} \text{Max3xy2z2}-14 \sqrt{15} \text{May3yz2x2}\right) & k=6\land m=0 \\ \frac{13 \left(5 \sqrt{15} \text{Eax3}+3 \sqrt{15} \text{Eaxy2z2}-5 \sqrt{15} \text{Eay3}-3 \sqrt{15} \text{Eayz2x2}+34 \text{Max3xy2z2}-2 i \sqrt{15} \text{Max3y3}+26 i \text{Max3yz2x2}+14 i \sqrt{15} \text{Maxy2z2yz2x2}-26 i \text{May3xy2z2}+34 \text{May3yz2x2}-64 i \text{Mbxyzz3}+64 \text{Mbz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=2 \\ -\frac{13 \left(15 \text{Eax3}-15 \text{Eaxy2z2}+15 \text{Eay3}-15 \text{Eayz2x2}+24 \text{Ebxyz}-24 \text{Ebzx2y2}+2 \sqrt{15} \text{Max3xy2z2}+8 i \sqrt{15} \text{Max3yz2x2}+8 i \sqrt{15} \text{May3xy2z2}-2 \sqrt{15} \text{May3yz2x2}+48 i \text{Mbxyzzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=4 \\ \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eax3}+3 \sqrt{3} \text{Eaxy2z2}-5 \sqrt{3} \text{Eay3}-3 \sqrt{3} \text{Eayz2x2}-6 \sqrt{5} \text{Max3xy2z2}+10 i \sqrt{3} \text{Max3y3}+6 i \sqrt{5} \text{Max3yz2x2}-6 i \sqrt{3} \text{Maxy2z2yz2x2}-6 i \sqrt{5} \text{May3xy2z2}-6 \sqrt{5} \text{May3yz2x2}\right) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eax3 + Eaxy2z2 + Eay3 + Eayz2x2 + Ebxyz + Ebz3 + Ebzx2y2)/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*((-I)*(Sqrt[6]*Max3y3 + Sqrt[10]*Max3yz2x2 + 5*Sqrt[6]*Maxy2z2yz2x2 - Sqrt[10]*May3xy2z2 + 4*Sqrt[10]*Mbxyzz3) + 2*(Sqrt[6]*Eax3 - Sqrt[6]*Eay3 + Sqrt[10]*(Max3xy2z2 + May3yz2x2 - 2*Mbz3zx2y2))))/56, k == 2 && m == -2}, {(-5*(Eax3 + Eay3 - 2*Ebz3 - Sqrt[15]*Max3xy2z2 + Sqrt[15]*May3yz2x2))/14, k == 2 && m == 0}, {(5*(I*(Sqrt[6]*Max3y3 + Sqrt[10]*Max3yz2x2 + 5*Sqrt[6]*Maxy2z2yz2x2 - Sqrt[10]*May3xy2z2 + 4*Sqrt[10]*Mbxyzz3) + 2*(Sqrt[6]*Eax3 - Sqrt[6]*Eay3 + Sqrt[10]*(Max3xy2z2 + May3yz2x2 - 2*Mbz3zx2y2))))/56, k == 2 && m == 2}, {(3*(3*Sqrt[5]*Eax3 - 3*Sqrt[5]*Eaxy2z2 + 3*Sqrt[5]*Eay3 - 3*Sqrt[5]*Eayz2x2 - 4*Sqrt[5]*Ebxyz + 4*Sqrt[5]*Ebzx2y2 + 2*Sqrt[3]*Max3xy2z2 - (8*I)*Sqrt[3]*Max3yz2x2 - (8*I)*Sqrt[3]*May3xy2z2 - 2*Sqrt[3]*May3yz2x2 + (8*I)*Sqrt[5]*Mbxyzzx2y2))/(8*Sqrt[14]), k == 4 && m == -4}, {(3*(-3*Sqrt[10]*Eax3 + 7*Sqrt[10]*Eaxy2z2 + 3*Sqrt[10]*Eay3 - 7*Sqrt[10]*Eayz2x2 + 2*Sqrt[6]*Max3xy2z2 + 2*Sqrt[6]*May3yz2x2 + (4*I)*(3*Sqrt[10]*Max3y3 - 2*Sqrt[6]*Max3yz2x2 + Sqrt[10]*Maxy2z2yz2x2 + 2*Sqrt[6]*May3xy2z2 - Sqrt[6]*Mbxyzz3) - 4*Sqrt[6]*Mbz3zx2y2))/56, k == 4 && m == -2}, {(3*(9*Eax3 + 7*Eaxy2z2 + 9*Eay3 + 7*Eayz2x2 - 28*Ebxyz + 24*Ebz3 - 28*Ebzx2y2 - 2*Sqrt[15]*Max3xy2z2 + 2*Sqrt[15]*May3yz2x2))/56, k == 4 && m == 0}, {(3*(-3*Sqrt[10]*Eax3 + 7*Sqrt[10]*Eaxy2z2 + 3*Sqrt[10]*Eay3 - 7*Sqrt[10]*Eayz2x2 + 2*Sqrt[6]*Max3xy2z2 + 2*Sqrt[6]*May3yz2x2 - (4*I)*(3*Sqrt[10]*Max3y3 - 2*Sqrt[6]*Max3yz2x2 + Sqrt[10]*Maxy2z2yz2x2 + 2*Sqrt[6]*May3xy2z2 - Sqrt[6]*Mbxyzz3) - 4*Sqrt[6]*Mbz3zx2y2))/56, k == 4 && m == 2}, {(3*(3*Sqrt[5]*Eax3 - 3*Sqrt[5]*Eaxy2z2 + 3*Sqrt[5]*Eay3 - 3*Sqrt[5]*Eayz2x2 - 4*Sqrt[5]*Ebxyz + 4*Sqrt[5]*Ebzx2y2 + 2*Sqrt[3]*Max3xy2z2 + (8*I)*Sqrt[3]*Max3yz2x2 + (8*I)*Sqrt[3]*May3xy2z2 - 2*Sqrt[3]*May3yz2x2 - (8*I)*Sqrt[5]*Mbxyzzx2y2))/(8*Sqrt[14]), k == 4 && m == 4}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eax3 + 3*Sqrt[3]*Eaxy2z2 - 5*Sqrt[3]*Eay3 - 3*Sqrt[3]*Eayz2x2 - 6*Sqrt[5]*Max3xy2z2 - (10*I)*Sqrt[3]*Max3y3 - (6*I)*Sqrt[5]*Max3yz2x2 + (6*I)*Sqrt[3]*Maxy2z2yz2x2 + (6*I)*Sqrt[5]*May3xy2z2 - 6*Sqrt[5]*May3yz2x2))/160, k == 6 && m == -6}, {(-13*(15*Eax3 - 15*Eaxy2z2 + 15*Eay3 - 15*Eayz2x2 + 24*Ebxyz - 24*Ebzx2y2 + 2*Sqrt[15]*Max3xy2z2 - (8*I)*Sqrt[15]*Max3yz2x2 - (8*I)*Sqrt[15]*May3xy2z2 - 2*Sqrt[15]*May3yz2x2 - (48*I)*Mbxyzzx2y2))/(80*Sqrt[14]), k == 6 && m == -4}, {(13*(5*Sqrt[15]*Eax3 + 3*Sqrt[15]*Eaxy2z2 - 5*Sqrt[15]*Eay3 - 3*Sqrt[15]*Eayz2x2 + 34*Max3xy2z2 + (2*I)*Sqrt[15]*Max3y3 - (26*I)*Max3yz2x2 - (14*I)*Sqrt[15]*Maxy2z2yz2x2 + (26*I)*May3xy2z2 + 34*May3yz2x2 + (64*I)*Mbxyzz3 + 64*Mbz3zx2y2))/(160*Sqrt[7]), k == 6 && m == -2}, {(-13*(25*Eax3 + 39*Eaxy2z2 + 25*Eay3 + 39*Eayz2x2 - 24*Ebxyz - 80*Ebz3 - 24*Ebzx2y2 + 14*Sqrt[15]*Max3xy2z2 - 14*Sqrt[15]*May3yz2x2))/560, k == 6 && m == 0}, {(13*(5*Sqrt[15]*Eax3 + 3*Sqrt[15]*Eaxy2z2 - 5*Sqrt[15]*Eay3 - 3*Sqrt[15]*Eayz2x2 + 34*Max3xy2z2 - (2*I)*Sqrt[15]*Max3y3 + (26*I)*Max3yz2x2 + (14*I)*Sqrt[15]*Maxy2z2yz2x2 - (26*I)*May3xy2z2 + 34*May3yz2x2 - (64*I)*Mbxyzz3 + 64*Mbz3zx2y2))/(160*Sqrt[7]), k == 6 && m == 2}, {(-13*(15*Eax3 - 15*Eaxy2z2 + 15*Eay3 - 15*Eayz2x2 + 24*Ebxyz - 24*Ebzx2y2 + 2*Sqrt[15]*Max3xy2z2 + (8*I)*Sqrt[15]*Max3yz2x2 + (8*I)*Sqrt[15]*May3xy2z2 - 2*Sqrt[15]*May3yz2x2 + (48*I)*Mbxyzzx2y2))/(80*Sqrt[14]), k == 6 && m == 4}}, (13*Sqrt[11/7]*(5*Sqrt[3]*Eax3 + 3*Sqrt[3]*Eaxy2z2 - 5*Sqrt[3]*Eay3 - 3*Sqrt[3]*Eayz2x2 - 6*Sqrt[5]*Max3xy2z2 + (10*I)*Sqrt[3]*Max3y3 + (6*I)*Sqrt[5]*Max3yz2x2 - (6*I)*Sqrt[3]*Maxy2z2yz2x2 - (6*I)*Sqrt[5]*May3xy2z2 - 6*Sqrt[5]*May3yz2x2))/160]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{0, 0, (1/7)*(Eax3 + Eaxy2z2 + Eay3 + Eayz2x2 + Ebxyz + Ebz3 + Ebzx2y2)} , 
       {2, 0, (-5/14)*(Eax3 + Eay3 + (-2)*(Ebz3) + (-1)*((sqrt(15))*(Max3xy2z2)) + (sqrt(15))*(May3yz2x2))} , 
       {2,-2, (5/56)*((-I)*((sqrt(6))*(Max3y3) + (sqrt(10))*(Max3yz2x2) + (5)*((sqrt(6))*(Maxy2z2yz2x2)) + (-1)*((sqrt(10))*(May3xy2z2)) + (4)*((sqrt(10))*(Mbxyzz3))) + (2)*((sqrt(6))*(Eax3) + (-1)*((sqrt(6))*(Eay3)) + (sqrt(10))*(Max3xy2z2 + May3yz2x2 + (-2)*(Mbz3zx2y2))))} , 
       {2, 2, (5/56)*((I)*((sqrt(6))*(Max3y3) + (sqrt(10))*(Max3yz2x2) + (5)*((sqrt(6))*(Maxy2z2yz2x2)) + (-1)*((sqrt(10))*(May3xy2z2)) + (4)*((sqrt(10))*(Mbxyzz3))) + (2)*((sqrt(6))*(Eax3) + (-1)*((sqrt(6))*(Eay3)) + (sqrt(10))*(Max3xy2z2 + May3yz2x2 + (-2)*(Mbz3zx2y2))))} , 
       {4, 0, (3/56)*((9)*(Eax3) + (7)*(Eaxy2z2) + (9)*(Eay3) + (7)*(Eayz2x2) + (-28)*(Ebxyz) + (24)*(Ebz3) + (-28)*(Ebzx2y2) + (-2)*((sqrt(15))*(Max3xy2z2)) + (2)*((sqrt(15))*(May3yz2x2)))} , 
       {4, 2, (3/56)*((-3)*((sqrt(10))*(Eax3)) + (7)*((sqrt(10))*(Eaxy2z2)) + (3)*((sqrt(10))*(Eay3)) + (-7)*((sqrt(10))*(Eayz2x2)) + (2)*((sqrt(6))*(Max3xy2z2)) + (2)*((sqrt(6))*(May3yz2x2)) + (-4*I)*((3)*((sqrt(10))*(Max3y3)) + (-2)*((sqrt(6))*(Max3yz2x2)) + (sqrt(10))*(Maxy2z2yz2x2) + (2)*((sqrt(6))*(May3xy2z2)) + (-1)*((sqrt(6))*(Mbxyzz3))) + (-4)*((sqrt(6))*(Mbz3zx2y2)))} , 
       {4,-2, (3/56)*((-3)*((sqrt(10))*(Eax3)) + (7)*((sqrt(10))*(Eaxy2z2)) + (3)*((sqrt(10))*(Eay3)) + (-7)*((sqrt(10))*(Eayz2x2)) + (2)*((sqrt(6))*(Max3xy2z2)) + (2)*((sqrt(6))*(May3yz2x2)) + (4*I)*((3)*((sqrt(10))*(Max3y3)) + (-2)*((sqrt(6))*(Max3yz2x2)) + (sqrt(10))*(Maxy2z2yz2x2) + (2)*((sqrt(6))*(May3xy2z2)) + (-1)*((sqrt(6))*(Mbxyzz3))) + (-4)*((sqrt(6))*(Mbz3zx2y2)))} , 
       {4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eax3)) + (-3)*((sqrt(5))*(Eaxy2z2)) + (3)*((sqrt(5))*(Eay3)) + (-3)*((sqrt(5))*(Eayz2x2)) + (-4)*((sqrt(5))*(Ebxyz)) + (4)*((sqrt(5))*(Ebzx2y2)) + (2)*((sqrt(3))*(Max3xy2z2)) + (-8*I)*((sqrt(3))*(Max3yz2x2)) + (-8*I)*((sqrt(3))*(May3xy2z2)) + (-2)*((sqrt(3))*(May3yz2x2)) + (8*I)*((sqrt(5))*(Mbxyzzx2y2))))} , 
       {4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eax3)) + (-3)*((sqrt(5))*(Eaxy2z2)) + (3)*((sqrt(5))*(Eay3)) + (-3)*((sqrt(5))*(Eayz2x2)) + (-4)*((sqrt(5))*(Ebxyz)) + (4)*((sqrt(5))*(Ebzx2y2)) + (2)*((sqrt(3))*(Max3xy2z2)) + (8*I)*((sqrt(3))*(Max3yz2x2)) + (8*I)*((sqrt(3))*(May3xy2z2)) + (-2)*((sqrt(3))*(May3yz2x2)) + (-8*I)*((sqrt(5))*(Mbxyzzx2y2))))} , 
       {6, 0, (-13/560)*((25)*(Eax3) + (39)*(Eaxy2z2) + (25)*(Eay3) + (39)*(Eayz2x2) + (-24)*(Ebxyz) + (-80)*(Ebz3) + (-24)*(Ebzx2y2) + (14)*((sqrt(15))*(Max3xy2z2)) + (-14)*((sqrt(15))*(May3yz2x2)))} , 
       {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eax3)) + (3)*((sqrt(15))*(Eaxy2z2)) + (-5)*((sqrt(15))*(Eay3)) + (-3)*((sqrt(15))*(Eayz2x2)) + (34)*(Max3xy2z2) + (-2*I)*((sqrt(15))*(Max3y3)) + (26*I)*(Max3yz2x2) + (14*I)*((sqrt(15))*(Maxy2z2yz2x2)) + (-26*I)*(May3xy2z2) + (34)*(May3yz2x2) + (-64*I)*(Mbxyzz3) + (64)*(Mbz3zx2y2)))} , 
       {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eax3)) + (3)*((sqrt(15))*(Eaxy2z2)) + (-5)*((sqrt(15))*(Eay3)) + (-3)*((sqrt(15))*(Eayz2x2)) + (34)*(Max3xy2z2) + (2*I)*((sqrt(15))*(Max3y3)) + (-26*I)*(Max3yz2x2) + (-14*I)*((sqrt(15))*(Maxy2z2yz2x2)) + (26*I)*(May3xy2z2) + (34)*(May3yz2x2) + (64*I)*(Mbxyzz3) + (64)*(Mbz3zx2y2)))} , 
       {6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Eax3) + (-15)*(Eaxy2z2) + (15)*(Eay3) + (-15)*(Eayz2x2) + (24)*(Ebxyz) + (-24)*(Ebzx2y2) + (2)*((sqrt(15))*(Max3xy2z2)) + (-8*I)*((sqrt(15))*(Max3yz2x2)) + (-8*I)*((sqrt(15))*(May3xy2z2)) + (-2)*((sqrt(15))*(May3yz2x2)) + (-48*I)*(Mbxyzzx2y2)))} , 
       {6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Eax3) + (-15)*(Eaxy2z2) + (15)*(Eay3) + (-15)*(Eayz2x2) + (24)*(Ebxyz) + (-24)*(Ebzx2y2) + (2)*((sqrt(15))*(Max3xy2z2)) + (8*I)*((sqrt(15))*(Max3yz2x2)) + (8*I)*((sqrt(15))*(May3xy2z2)) + (-2)*((sqrt(15))*(May3yz2x2)) + (48*I)*(Mbxyzzx2y2)))} , 
       {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eax3)) + (3)*((sqrt(3))*(Eaxy2z2)) + (-5)*((sqrt(3))*(Eay3)) + (-3)*((sqrt(3))*(Eayz2x2)) + (-6)*((sqrt(5))*(Max3xy2z2)) + (-10*I)*((sqrt(3))*(Max3y3)) + (-6*I)*((sqrt(5))*(Max3yz2x2)) + (6*I)*((sqrt(3))*(Maxy2z2yz2x2)) + (6*I)*((sqrt(5))*(May3xy2z2)) + (-6)*((sqrt(5))*(May3yz2x2))))} , 
       {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eax3)) + (3)*((sqrt(3))*(Eaxy2z2)) + (-5)*((sqrt(3))*(Eay3)) + (-3)*((sqrt(3))*(Eayz2x2)) + (-6)*((sqrt(5))*(Max3xy2z2)) + (10*I)*((sqrt(3))*(Max3y3)) + (6*I)*((sqrt(5))*(Max3yz2x2)) + (-6*I)*((sqrt(3))*(Maxy2z2yz2x2)) + (-6*I)*((sqrt(5))*(May3xy2z2)) + (-6)*((sqrt(5))*(May3yz2x2))))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-3}^{(3)}} \frac{1}{16} \left(5 \text{Eax3}+3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \sqrt{15} (\text{May3yz2x2}-\text{Max3xy2z2})\right) 0 \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}+2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}+i \text{May3yz2x2}\right)\right) 0 \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}-4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right) 0 \frac{1}{16} \left(-5 \text{Eax3}-3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \left(\sqrt{15} \text{Max3xy2z2}+5 i \text{Max3y3}+i \sqrt{15} \text{Max3yz2x2}-3 i \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3yz2x2}-i \text{May3xy2z2})\right)\right)
{Y_{-2}^{(3)}} 0 \frac{\text{Ebxyz}+\text{Ebzx2y2}}{2} 0 \frac{\text{Mbz3zx2y2}+i \text{Mbxyzz3}}{\sqrt{2}} 0 \frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}+2 i \text{Mbxyzzx2y2}) 0
{Y_{-1}^{(3)}} \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}-2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}-i \text{May3yz2x2}\right)\right) 0 \frac{1}{16} \left(3 \text{Eax3}+5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}+2 \sqrt{15} (\text{Max3xy2z2}-\text{May3yz2x2})\right) 0 \frac{1}{16} \left(-3 \text{Eax3}-5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}-2 \left(\sqrt{15} \text{Max3xy2z2}+3 i \text{Max3y3}-i \sqrt{15} \text{Max3yz2x2}-5 i \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3yz2x2}+i \text{May3xy2z2})\right)\right) 0 \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}-4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right)
{Y_{0}^{(3)}} 0 \frac{\text{Mbz3zx2y2}-i \text{Mbxyzz3}}{\sqrt{2}} 0 \text{Ebz3} 0 \frac{\text{Mbz3zx2y2}+i \text{Mbxyzz3}}{\sqrt{2}} 0
{Y_{1}^{(3)}} \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}+4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right) 0 \frac{1}{16} \left(-3 \text{Eax3}-5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}-2 \sqrt{15} \text{Max3xy2z2}+2 i \left(3 \text{Max3y3}-\sqrt{15} \text{Max3yz2x2}-5 \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3xy2z2}+i \text{May3yz2x2})\right)\right) 0 \frac{1}{16} \left(3 \text{Eax3}+5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}+2 \sqrt{15} (\text{Max3xy2z2}-\text{May3yz2x2})\right) 0 \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}+2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}+i \text{May3yz2x2}\right)\right)
{Y_{2}^{(3)}} 0 \frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}-2 i \text{Mbxyzzx2y2}) 0 \frac{\text{Mbz3zx2y2}-i \text{Mbxyzz3}}{\sqrt{2}} 0 \frac{\text{Ebxyz}+\text{Ebzx2y2}}{2} 0
{Y_{3}^{(3)}} \frac{1}{16} \left(-5 \text{Eax3}-3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \left(\sqrt{15} \text{Max3xy2z2}-5 i \text{Max3y3}-i \sqrt{15} \text{Max3yz2x2}+3 i \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3yz2x2}+i \text{May3xy2z2})\right)\right) 0 \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}+4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right) 0 \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}-2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}-i \text{May3yz2x2}\right)\right) 0 \frac{1}{16} \left(5 \text{Eax3}+3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \sqrt{15} (\text{May3yz2x2}-\text{Max3xy2z2})\right)

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(5z^2-r^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)} f_{z\left(x^2-y^2\right)}
f_{\text{xyz}} \text{Ebxyz} 0 0 \text{Mbxyzz3} 0 0 \text{Mbxyzzx2y2}
f_{x\left(5x^2-r^2\right)} 0 \text{Eax3} \text{Max3y3} 0 \text{Max3xy2z2} \text{Max3yz2x2} 0
f_{y\left(5y^2-r^2\right)} 0 \text{Max3y3} \text{Eay3} 0 \text{May3xy2z2} \text{May3yz2x2} 0
f_{z\left(5z^2-r^2\right)} \text{Mbxyzz3} 0 0 \text{Ebz3} 0 0 \text{Mbz3zx2y2}
f_{x\left(y^2-z^2\right)} 0 \text{Max3xy2z2} \text{May3xy2z2} 0 \text{Eaxy2z2} \text{Maxy2z2yz2x2} 0
f_{y\left(z^2-x^2\right)} 0 \text{Max3yz2x2} \text{May3yz2x2} 0 \text{Maxy2z2yz2x2} \text{Eayz2x2} 0
f_{z\left(x^2-y^2\right)} \text{Mbxyzzx2y2} 0 0 \text{Mbz3zx2y2} 0 0 \text{Ebzx2y2}

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

\text{Ebxyz}
\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{Eax3}
\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{Eay3}
\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{Ebz3}
\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{Eaxy2z2}
\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{Eayz2x2}
\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{Ebzx2y2}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-p 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 1\lor m\neq 0 \\ A(1,0) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 1 || m != 0}}, A[1, 0]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{1, 0, A(1,0)} }

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_{0}^{(0)}} 0 \frac{A(1,0)}{\sqrt{3}} 0

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

p_x p_y p_z
\text{s} 0 0 \frac{A(1,0)}{\sqrt{3}}

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_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]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{2, 0, A(2,0)} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{0}^{(0)}} \frac{A(2,2)+i B(2,2)}{\sqrt{5}} 0 \frac{A(2,0)}{\sqrt{5}} 0 \frac{A(2,2)-i B(2,2)}{\sqrt{5}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

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

Potential for s-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 3\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ A(3,2)-i B(3,2) & k=3\land m=-2 \\ A(3,0) & k=3\land m=0 \\ A(3,2)+i B(3,2) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 0 && m != 2)}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}}, A[3, 2] + I*B[3, 2]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{3, 0, A(3,0)} , 
       {3,-2, A(3,2) + (-I)*(B(3,2))} , 
       {3, 2, A(3,2) + (I)*(B(3,2))} }

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_{0}^{(0)}} 0 \frac{A(3,2)+i B(3,2)}{\sqrt{7}} 0 \frac{A(3,0)}{\sqrt{7}} 0 \frac{A(3,2)-i B(3,2)}{\sqrt{7}} 0

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(5z^2-r^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)} f_{z\left(x^2-y^2\right)}
\text{s} -\sqrt{\frac{2}{7}} B(3,2) 0 0 \frac{A(3,0)}{\sqrt{7}} 0 0 \sqrt{\frac{2}{7}} A(3,2)

Potential for p-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\neq 3\land (k\neq 1\lor m\neq 0))\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ A(1,0) & k=1\land m=0 \\ A(3,2)-i B(3,2) & k=3\land m=-2 \\ A(3,0) & k=3\land m=0 \\ A(3,2)+i B(3,2) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || m != 0)) || (m != -2 && m != 0 && m != 2)}, {A[1, 0], k == 1 && m == 0}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}}, A[3, 2] + I*B[3, 2]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{1, 0, A(1,0)} , 
       {3, 0, A(3,0)} , 
       {3,-2, A(3,2) + (-I)*(B(3,2))} , 
       {3, 2, A(3,2) + (I)*(B(3,2))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-2}^{(2)}} {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{-1}^{(1)}} 0 \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} 0 -\frac{1}{7} \sqrt{6} (A(3,2)-i B(3,2)) 0
{Y_{0}^{(1)}} \frac{1}{7} \sqrt{3} (A(3,2)+i B(3,2)) 0 \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} 0 \frac{1}{7} \sqrt{3} (A(3,2)-i B(3,2))
{Y_{1}^{(1)}} 0 -\frac{1}{7} \sqrt{6} (A(3,2)+i B(3,2)) 0 \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} 0

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

d_{x^2-y^2} d_{3z^2-r^2} d_{\text{yz}} d_{\text{xz}} d_{\text{xy}}
p_x 0 0 -\frac{1}{7} \sqrt{6} B(3,2) \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)+5 \sqrt{6} A(3,2)\right) 0
p_y 0 0 \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) -\frac{1}{7} \sqrt{6} B(3,2) 0
p_z \frac{1}{7} \sqrt{6} A(3,2) \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} 0 0 -\frac{1}{7} \sqrt{6} B(3,2)

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

A_{k,m} = \begin{cases} 0 & (k\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(2,2)-i B(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i B(2,2) & k=2\land m=2 \\ A(4,4)-i B(4,4) & k=4\land m=-4 \\ A(4,2)-i B(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i B(4,2) & k=4\land m=2 \\ A(4,4)+i B(4,4) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_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]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{2, 0, A(2,0)} , 
       {2,-2, A(2,2) + (-I)*(B(2,2))} , 
       {2, 2, A(2,2) + (I)*(B(2,2))} , 
       {4, 0, A(4,0)} , 
       {4,-2, A(4,2) + (-I)*(B(4,2))} , 
       {4, 2, A(4,2) + (I)*(B(4,2))} , 
       {4,-4, A(4,4) + (-I)*(B(4,4))} , 
       {4, 4, A(4,4) + (I)*(B(4,4))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-1}^{(1)}} \frac{3 (A(2,2)+i B(2,2))}{\sqrt{35}}-\frac{A(4,2)+i B(4,2)}{3 \sqrt{21}} 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 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 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}}

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(5z^2-r^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)} f_{z\left(x^2-y^2\right)}
p_x 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} 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}} 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)

Potential for d-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 5\land (((k\neq 1\lor m\neq 0)\land k\neq 3)\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(1,0) & k=1\land m=0 \\ A(3,2)-i B(3,2) & k=3\land m=-2 \\ A(3,0) & k=3\land m=0 \\ A(3,2)+i B(3,2) & k=3\land m=2 \\ A(5,4)-i B(5,4) & k=5\land m=-4 \\ A(5,2)-i B(5,2) & k=5\land m=-2 \\ A(5,0) & k=5\land m=0 \\ A(5,2)+i B(5,2) & k=5\land m=2 \\ A(5,4)+i B(5,4) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_C2_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 5 && (((k != 1 || m != 0) && k != 3) || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {A[1, 0], k == 1 && m == 0}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}, {A[3, 2] + I*B[3, 2], k == 3 && m == 2}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {A[5, 2] - I*B[5, 2], k == 5 && m == -2}, {A[5, 0], k == 5 && m == 0}, {A[5, 2] + I*B[5, 2], k == 5 && m == 2}}, A[5, 4] + I*B[5, 4]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_C2_Z.Quanty
Akm = {{1, 0, A(1,0)} , 
       {3, 0, A(3,0)} , 
       {3,-2, A(3,2) + (-I)*(B(3,2))} , 
       {3, 2, A(3,2) + (I)*(B(3,2))} , 
       {5, 0, A(5,0)} , 
       {5,-2, A(5,2) + (-I)*(B(5,2))} , 
       {5, 2, A(5,2) + (I)*(B(5,2))} , 
       {5,-4, A(5,4) + (-I)*(B(5,4))} , 
       {5, 4, A(5,4) + (I)*(B(5,4))} }

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_{-2}^{(2)}} 0 \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} 0 \frac{5}{33} (A(5,2)-i B(5,2))-\frac{2 (A(3,2)-i B(3,2))}{3 \sqrt{7}} 0 \frac{1}{11} \sqrt{10} (A(5,4)-i B(5,4)) 0
{Y_{-1}^{(2)}} \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,2)+i B(3,2))-\frac{1}{33} \sqrt{5} (A(5,2)+i B(5,2)) 0 \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) 0 -\frac{A(3,2)-i B(3,2)}{\sqrt{21}}-\frac{5 (A(5,2)-i B(5,2))}{11 \sqrt{3}} 0 -\frac{2}{11} \sqrt{\frac{5}{3}} (A(5,4)-i B(5,4))
{Y_{0}^{(2)}} 0 \frac{1}{11} \sqrt{5} (A(5,2)+i B(5,2)) 0 \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} 0 \frac{1}{11} \sqrt{5} (A(5,2)-i B(5,2)) 0
{Y_{1}^{(2)}} -\frac{2}{11} \sqrt{\frac{5}{3}} (A(5,4)+i B(5,4)) 0 -\frac{A(3,2)+i B(3,2)}{\sqrt{21}}-\frac{5 (A(5,2)+i B(5,2))}{11 \sqrt{3}} 0 \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) 0 \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,2)-i B(3,2))-\frac{1}{33} \sqrt{5} (A(5,2)-i B(5,2))
{Y_{2}^{(2)}} 0 \frac{1}{11} \sqrt{10} (A(5,4)+i B(5,4)) 0 \frac{5}{33} (A(5,2)+i B(5,2))-\frac{2 (A(3,2)+i B(3,2))}{3 \sqrt{7}} 0 \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} 0

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{\text{xyz}} f_{x\left(5x^2-r^2\right)} f_{y\left(5y^2-r^2\right)} f_{z\left(5z^2-r^2\right)} f_{x\left(y^2-z^2\right)} f_{y\left(z^2-x^2\right)} f_{z\left(x^2-y^2\right)}
d_{x^2-y^2} -\frac{1}{11} \sqrt{10} B(5,4) 0 0 \frac{5}{33} \sqrt{2} A(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} A(3,2) 0 0 \frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)+21 \sqrt{10} A(5,4)\right)
d_{3z^2-r^2} -\frac{1}{11} \sqrt{10} B(5,2) 0 0 \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} 0 0 \frac{1}{11} \sqrt{10} A(5,2)
d_{\text{yz}} 0 \frac{44 \sqrt{7} B(3,2)+35 \left(B(5,2)-\sqrt{3} B(5,4)\right)}{231 \sqrt{2}} \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(-11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)+70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} 0 \frac{1}{11} \sqrt{\frac{5}{2}} \left(\sqrt{3} B(5,2)+B(5,4)\right) \frac{1}{462} \left(66 \sqrt{7} A(1,0)+11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)-25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)+21 \sqrt{10} A(5,4)\right) 0
d_{\text{xz}} 0 \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)-70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} \frac{44 \sqrt{7} B(3,2)+35 \left(B(5,2)+\sqrt{3} B(5,4)\right)}{231 \sqrt{2}} 0 -\frac{66 \sqrt{35} A(1,0)+11 \sqrt{35} A(3,0)+55 \sqrt{42} A(3,2)-25 \sqrt{35} A(5,0)+70 \sqrt{6} A(5,2)+105 \sqrt{2} A(5,4)}{462 \sqrt{5}} \frac{1}{11} \sqrt{\frac{5}{2}} \left(B(5,4)-\sqrt{3} B(5,2)\right) 0
d_{\text{xy}} \frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)-21 \sqrt{10} A(5,4)\right) 0 0 \frac{2}{3} \sqrt{\frac{2}{7}} B(3,2)-\frac{5}{33} \sqrt{2} B(5,2) 0 0 -\frac{1}{11} \sqrt{10} B(5,4)

Table of several point groups

Return to Main page on Point Groups

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