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Table of Contents

Orientation Z(x-y)

The point group D3d is a subgroup of Oh. Many materials of relevance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the states in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irreducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irreducible representation and two that belong to the eg irreducible representation. We label the eg orbitals that descend from the eg irreducible representation in Oh symmetry egσ and the eg orbitals that descend from the t2g irreducible representation egπ orbitals. (The mixing is given by the parameter Meg.)

As one can see in the list of supergroups of D3d, there are two different orientations of Oh that are a supergroup of this orientation of D3d. The different orientations of Oh with respect to D3d do however change the definitions of the egπ and egσ orbitals. We include three different representations of the orbitals and potentials for each setting of D3d symmetry. The orientation without additional letter takes the tesseral harmonics as a basis. This basis does not relate to the states in Oh symmetry. The orientation with an additional A or B relate to the two different supergroup representations of the Oh point group.

Symmetry Operations

In the D3d Point Group, with orientation Z(x-y) there are the following symmetry operations

Operator Orientation
E {0,0,0} ,
C3 {0,0,1} , {0,0,1} ,
C2 {1,1,0} , {2+3,1,0} , {1,2+3,0} ,
i {0,0,0} ,
S6 {0,0,1} , {0,0,1} ,
σd {1,1,0} , {2+3,1,0} , {1,2+3,0} ,

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=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 D3d Point group with orientation Z(x-y) the form of the expansion coefficients is:

Expansion

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

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Z(x-y).Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {(-1 + I)*A[4, 3], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {(1 + I)*A[4, 3], k == 4 && m == 3}, {(-I)*B[6, 6], k == 6 && m == -6}, {(-1 + I)*A[6, 3], k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {(1 + I)*A[6, 3], k == 6 && m == 3}, {I*B[6, 6], k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_D3d_Z(x-y).Quanty
Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {4, 0, A(4,0)} , 
       {4,-3, (-1+1*I)*(A(4,3))} , 
       {4, 3, (1+1*I)*(A(4,3))} , 
       {6, 0, A(6,0)} , 
       {6,-3, (-1+1*I)*(A(6,3))} , 
       {6, 3, (1+1*I)*(A(6,3))} , 
       {6,-6, (-I)*(B(6,6))} , 
       {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)00000Asd(2,0)5000000000
Y(1)10App(0,0)15App(2,0)0000000003527Apf(2,0)1327Apf(4,0)00(13+i3)Apf(4,3)0
Y(1)000App(0,0)+25App(2,0)000000(13+i3)Apf(4,3)3003537Apf(2,0)+4Apf(4,0)32100(13i3)Apf(4,3)3
Y(1)1000App(0,0)15App(2,0)000000(13+i3)Apf(4,3)003527Apf(2,0)1327Apf(4,0)00
Y(2)20000Add(0,0)27Add(2,0)+121Add(4,0)00(13i3)57Add(4,3)00000000
Y(2)100000Add(0,0)+17Add(2,0)421Add(4,0)00(13+i3)57Add(4,3)0000000
Y(2)0Asd(2,0)500000Add(0,0)+27Add(2,0)+27Add(4,0)000000000
Y(2)10000(13+i3)57Add(4,3)00Add(0,0)+17Add(2,0)421Add(4,0)00000000
Y(2)200000(13i3)57Add(4,3)00Add(0,0)27Add(2,0)+121Add(4,0)0000000
Y(3)300(13i3)Apf(4,3)3000000Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)00(111i11)7Aff(4,3)(1014310i143)73Aff(6,3)001013i733Bff(6,6)
Y(3)2000(13i3)Apf(4,3)000000Aff(0,0)733Aff(4,0)+10143Aff(6,0)00(133i33)14Aff(4,3)+(51435i143)42Aff(6,3)00
Y(3)103527Apf(2,0)1327Apf(4,0)000000000Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)00(133+i33)14Aff(4,3)(51435i143)42Aff(6,3)0
Y(3)0003537Apf(2,0)+4Apf(4,0)321000000(111+i11)7Aff(4,3)(10143+10i143)73Aff(6,3)00Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)00(1014310i143)73Aff(6,3)(111i11)7Aff(4,3)
Y(3)10003527Apf(2,0)1327Apf(4,0)000000(133+i33)14Aff(4,3)+(5143+5i143)42Aff(6,3)00Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)00
Y(3)20(13i3)Apf(4,3)000000000(133i33)14Aff(4,3)(5143+5i143)42Aff(6,3)00Aff(0,0)733Aff(4,0)+10143Aff(6,0)0
Y(3)300(13+i3)Apf(4,3)30000001013i733Bff(6,6)00(10143+10i143)73Aff(6,3)(111+i11)7Aff(4,3)00Aff(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
py0i20i2000000000000
pz0010000000000000
px012012000000000000
dxy0000i2000i20000000
dyz00000i20i200000000
d3z2r20000001000000000
dxz000001201200000000
dx2y2000012000120000000
fy(3x2y2)000000000i200000i2
fxyz0000000000i2000i20
fy(5z2r2)00000000000i20i200
fz(5z23r2)0000000000001000
fx(5z2r2)000000000001201200
fz(x2y2)000000000012000120
fx(x23y2)000000000120000012

One particle coupling on a basis of symmetry adapted functions

After rotation we find

s py pz px dxy dyz d3z2r2 dxz dx2y2 fy(3x2y2) fxyz fy(5z2r2) fz(5z23r2) fx(5z2r2) fz(x2y2) fx(x23y2)
sAss(0,0)00000Asd(2,0)5000000000
py0App(0,0)15App(2,0)0000000013Apf(4,3)3527Apf(2,0)1327Apf(4,0)0013Apf(4,3)0
pz00App(0,0)+25App(2,0)0000001323Apf(4,3)003537Apf(2,0)+4Apf(4,0)321001323Apf(4,3)
px000App(0,0)15App(2,0)00000013Apf(4,3)003527Apf(2,0)1327Apf(4,0)13Apf(4,3)0
dxy0000Add(0,0)27Add(2,0)+121Add(4,0)1357Add(4,3)01357Add(4,3)00000000
dyz00001357Add(4,3)Add(0,0)+17Add(2,0)421Add(4,0)001357Add(4,3)0000000
d3z2r2Asd(2,0)500000Add(0,0)+27Add(2,0)+27Add(4,0)000000000
dxz00001357Add(4,3)00Add(0,0)+17Add(2,0)421Add(4,0)1357Add(4,3)0000000
dx2y2000001357Add(4,3)01357Add(4,3)Add(0,0)27Add(2,0)+121Add(4,0)0000000
fy(3x2y2)001323Apf(4,3)000000Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)0010143143Aff(6,3)11114Aff(4,3)001013733Bff(6,6)
fxyz013Apf(4,3)013Apf(4,3)000000Aff(0,0)733Aff(4,0)+10143Aff(6,0)13314Aff(4,3)+514342Aff(6,3)013314Aff(4,3)+514342Aff(6,3)00
fy(5z2r2)03527Apf(2,0)1327Apf(4,0)0000000013314Aff(4,3)+514342Aff(6,3)Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)0013314Aff(4,3)+514342Aff(6,3)0
fz(5z23r2)003537Apf(2,0)+4Apf(4,0)32100000010143143Aff(6,3)11114Aff(4,3)00Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)0011114Aff(4,3)10143143Aff(6,3)
fx(5z2r2)0003527Apf(2,0)1327Apf(4,0)00000013314Aff(4,3)+514342Aff(6,3)00Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)13314Aff(4,3)514342Aff(6,3)0
fz(x2y2)013Apf(4,3)013Apf(4,3)000000013314Aff(4,3)+514342Aff(6,3)013314Aff(4,3)514342Aff(6,3)Aff(0,0)733Aff(4,0)+10143Aff(6,0)0
fx(x23y2)001323Apf(4,3)0000001013733Bff(6,6)0011114Aff(4,3)10143143Aff(6,3)00Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)

Coupling for a single shell

Although the parameters Al,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 Al,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

Ak,m={Ea1gk=0m=00True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Z(x-y).Quanty.nb
Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_Z(x-y).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)0Ea1g

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

s
sEa1g

Rotation matrix used

Rotation matrix used

Y(0)0
s1

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Ea1g
ψ(θ,ϕ)=11 12π
ψ(ˆx,ˆy,ˆz)=11 12π

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={13(Ea2u+2Eeu)k=0m=05(Ea2uEeu)3k=2m=0

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Z(x-y).Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(5*(Ea2u - Eeu))/3, k == 2 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_Z(x-y).Quanty
Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , 
       {2, 0, (5/3)*(Ea2u + (-1)*(Eeu))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

Y(1)1 Y(1)0 Y(1)1
Y(1)1Eeu00
Y(1)00Ea2u0
Y(1)100Eeu

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

py pz px
pyEeu00
pz0Ea2u0
px00Eeu

Rotation matrix used

Rotation matrix used

Y(1)1 Y(1)0 Y(1)1
pyi20i2
pz010
px12012

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Eeu
ψ(θ,ϕ)=11 123πsin(θ)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 123πy
Ea2u
ψ(θ,ϕ)=11 123πcos(θ)
ψ(ˆx,ˆy,ˆz)=11 123πz
Eeu
ψ(θ,ϕ)=11 123πsin(θ)cos(ϕ)
ψ(ˆx,ˆy,ˆz)=11 123πx

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={15(Ea1g+2(Eeg1+Eeg2))k=0m=0Ea1g+Eeg12Eeg2k=2m=0(3+3i)75Megk=4m=335(3Ea1g4Eeg1+Eeg2)k=4m=0(3+3i)75Megk=4m=3

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Z(x-y).Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea1g + 2*(Eeg1 + Eeg2))/5, k == 0 && m == 0}, {Ea1g + Eeg1 - 2*Eeg2, k == 2 && m == 0}, {(-3 + 3*I)*Sqrt[7/5]*Meg, k == 4 && m == -3}, {(3*(3*Ea1g - 4*Eeg1 + Eeg2))/5, k == 4 && m == 0}, {(3 + 3*I)*Sqrt[7/5]*Meg, k == 4 && m == 3}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_Z(x-y).Quanty
Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg1 + Eeg2))} , 
       {2, 0, Ea1g + Eeg1 + (-2)*(Eeg2)} , 
       {4, 0, (3/5)*((3)*(Ea1g) + (-4)*(Eeg1) + Eeg2)} , 
       {4,-3, (-3+3*I)*((sqrt(7/5))*(Meg))} , 
       {4, 3, (3+3*I)*((sqrt(7/5))*(Meg))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

Y(2)2 Y(2)1 Y(2)0 Y(2)1 Y(2)2
Y(2)2Eeg200(1i)Meg0
Y(2)10Eeg100(1+i)Meg
Y(2)000Ea1g00
Y(2)1(1+i)Meg00Eeg10
Y(2)20(1i)Meg00Eeg2

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

dxy dyz d3z2r2 dxz dx2y2
dxyEeg2Meg0Meg0
dyzMegEeg100Meg
d3z2r200Ea1g00
dxzMeg00Eeg1Meg
dx2y20Meg0MegEeg2

Rotation matrix used

Rotation matrix used

Y(2)2 Y(2)1 Y(2)0 Y(2)1 Y(2)2
dxyi2000i2
dyz0i20i20
d3z2r200100
dxz0120120
dx2y21200012

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Eeg2
ψ(θ,ϕ)=11 1415πsin2(θ)sin(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πxy
Eeg1
ψ(θ,ϕ)=11 1415πsin(2θ)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πyz
Ea1g
ψ(θ,ϕ)=11 185π(3cos(2θ)+1)
ψ(ˆx,ˆy,ˆz)=11 145π(3z21)
Eeg1
ψ(θ,ϕ)=11 1415πsin(2θ)cos(ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πxz
Eeg2
ψ(θ,ϕ)=11 1415πsin2(θ)cos(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 1415π(x2y2)

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={17(Ea1u+Ea2u1+Ea2u2+2Eeu1+2Eeu2)k=0m=0528(5Ea1u+5Ea2u14Ea2u26Eeu1)k=2m=0(323i2)(3Ma2u+22Meu)7k=4m=3314(3Ea1u+3Ea2u1+2(3Ea2u2+Eeu17Eeu2))k=4m=0(32+3i2)(3Ma2u+22Meu)7k=4m=31320i337(Ea1uEa2u1)k=6m=6(131013i10)37(Ma2u32Meu)k=6m=313140(Ea1u+Ea2u120Ea2u2+30Eeu112Eeu2)k=6m=0(131013i10)37(Ma2u32Meu)k=6m=31320i337(Ea1uEa2u1)k=6m=6

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Z(x-y).Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea1u + Ea2u1 + Ea2u2 + 2*Eeu1 + 2*Eeu2)/7, k == 0 && m == 0}, {(-5*(5*Ea1u + 5*Ea2u1 - 4*Ea2u2 - 6*Eeu1))/28, k == 2 && m == 0}, {((-3/2 + (3*I)/2)*(3*Ma2u + 2*Sqrt[2]*Meu))/Sqrt[7], k == 4 && m == -3}, {(3*(3*Ea1u + 3*Ea2u1 + 2*(3*Ea2u2 + Eeu1 - 7*Eeu2)))/14, k == 4 && m == 0}, {((3/2 + (3*I)/2)*(3*Ma2u + 2*Sqrt[2]*Meu))/Sqrt[7], k == 4 && m == 3}, {((13*I)/20)*Sqrt[33/7]*(Ea1u - Ea2u1), k == 6 && m == -6}, {(13/10 - (13*I)/10)*Sqrt[3/7]*(Ma2u - 3*Sqrt[2]*Meu), k == 6 && m == -3}, {(-13*(Ea1u + Ea2u1 - 20*Ea2u2 + 30*Eeu1 - 12*Eeu2))/140, k == 6 && m == 0}, {(-13/10 - (13*I)/10)*Sqrt[3/7]*(Ma2u - 3*Sqrt[2]*Meu), k == 6 && m == 3}, {((-13*I)/20)*Sqrt[33/7]*(Ea1u - Ea2u1), k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_Z(x-y).Quanty
Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1) + (2)*(Eeu2))} , 
       {2, 0, (-5/28)*((5)*(Ea1u) + (5)*(Ea2u1) + (-4)*(Ea2u2) + (-6)*(Eeu1))} , 
       {4, 0, (3/14)*((3)*(Ea1u) + (3)*(Ea2u1) + (2)*((3)*(Ea2u2) + Eeu1 + (-7)*(Eeu2)))} , 
       {4,-3, (-3/2+3/2*I)*((1/(sqrt(7)))*((3)*(Ma2u) + (2)*((sqrt(2))*(Meu))))} , 
       {4, 3, (3/2+3/2*I)*((1/(sqrt(7)))*((3)*(Ma2u) + (2)*((sqrt(2))*(Meu))))} , 
       {6, 0, (-13/140)*(Ea1u + Ea2u1 + (-20)*(Ea2u2) + (30)*(Eeu1) + (-12)*(Eeu2))} , 
       {6, 3, (-13/10+-13/10*I)*((sqrt(3/7))*(Ma2u + (-3)*((sqrt(2))*(Meu))))} , 
       {6,-3, (13/10+-13/10*I)*((sqrt(3/7))*(Ma2u + (-3)*((sqrt(2))*(Meu))))} , 
       {6, 6, (-13/20*I)*((sqrt(33/7))*(Ea1u + (-1)*(Ea2u1)))} , 
       {6,-6, (13/20*I)*((sqrt(33/7))*(Ea1u + (-1)*(Ea2u1)))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

Y(3)3 Y(3)2 Y(3)1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
Y(3)3Ea1u+Ea2u1200(12i2)Ma2u0012i(Ea1uEa2u1)
Y(3)20Eeu200(1i)Meu00
Y(3)100Eeu100(1+i)Meu0
Y(3)0(12+i2)Ma2u00Ea2u200(12+i2)Ma2u
Y(3)10(1+i)Meu00Eeu100
Y(3)200(1i)Meu00Eeu20
Y(3)312i(Ea1uEa2u1)00(12i2)Ma2u00Ea1u+Ea2u12

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

fy(3x2y2) fxyz fy(5z2r2) fz(5z23r2) fx(5z2r2) fz(x2y2) fx(x23y2)
fy(3x2y2)Ea1u+Ea2u1200Ma2u200Ea1uEa2u12
fxyz0Eeu2Meu0Meu00
fy(5z2r2)0MeuEeu100Meu0
fz(5z23r2)Ma2u200Ea2u200Ma2u2
fx(5z2r2)0Meu00Eeu1Meu0
fz(x2y2)00Meu0MeuEeu20
fx(x23y2)Ea1uEa2u1200Ma2u200Ea1u+Ea2u12

Rotation matrix used

Rotation matrix used

Y(3)3 Y(3)2 Y(3)1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
fy(3x2y2)i200000i2
fxyz0i2000i20
fy(5z2r2)00i20i200
fz(5z23r2)0001000
fx(5z2r2)001201200
fz(x2y2)012000120
fx(x23y2)120000012

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Ea1u+Ea2u12
ψ(θ,ϕ)=11 14352πsin3(θ)sin(3ϕ)
ψ(ˆx,ˆy,ˆz)=11 14352πy(y23x2)
Eeu2
ψ(θ,ϕ)=11 14105πsin2(θ)cos(θ)sin(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 12105πxyz
Eeu1
ψ(θ,ϕ)=11 18212πsin(θ)(5cos(2θ)+3)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 14212πy(5z21)
Ea2u2
ψ(θ,ϕ)=11 1167π(3cos(θ)+5cos(3θ))
ψ(ˆx,ˆy,ˆz)=11 147πz(5z23)
Eeu1
ψ(θ,ϕ)=11 116212π(sin(θ)+5sin(3θ))cos(ϕ)
ψ(ˆx,ˆy,ˆz)=11 14212πx(5z21)
Eeu2
ψ(θ,ϕ)=11 14105πsin2(θ)cos(θ)cos(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 14105πz(x2y2)
Ea1u+Ea2u12
ψ(θ,ϕ)=11 14352πsin3(θ)cos(3ϕ)
ψ(ˆx,ˆy,ˆz)=11 14352πx(x23y2)

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

Ak,m={0k2m05Ma1gTrue

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Z(x-y).Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, Sqrt[5]*Ma1g]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_Z(x-y).Quanty
Akm = {{2, 0, (sqrt(5))*(Ma1g)} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

Y(2)2 Y(2)1 Y(2)0 Y(2)1 Y(2)2
Y(0)000Ma1g00

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

dxy dyz d3z2r2 dxz dx2y2
s00Ma1g00

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={0k=0m=0521(21Ma2u2+214Meu1)k=2m=0(323i2)3Ma2u1k=4m=3314(221Ma2u2314Meu1)k=4m=0(323i2)3Ma2u1k=4m=3

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Z(x-y).Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {(5*(Sqrt[21]*Ma2u2 + 2*Sqrt[14]*Meu1))/21, k == 2 && m == 0}, {(3/2 - (3*I)/2)*Sqrt[3]*Ma2u1, k == 4 && m == -3}, {(3*(2*Sqrt[21]*Ma2u2 - 3*Sqrt[14]*Meu1))/14, k == 4 && m == 0}, {(-3/2 - (3*I)/2)*Sqrt[3]*Ma2u1, k == 4 && m == 3}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3d_Z(x-y).Quanty
Akm = {{2, 0, (5/21)*((sqrt(21))*(Ma2u2) + (2)*((sqrt(14))*(Meu1)))} , 
       {4, 0, (3/14)*((2)*((sqrt(21))*(Ma2u2)) + (-3)*((sqrt(14))*(Meu1)))} , 
       {4, 3, (-3/2+-3/2*I)*((sqrt(3))*(Ma2u1))} , 
       {4,-3, (3/2+-3/2*I)*((sqrt(3))*(Ma2u1))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

Y(3)3 Y(3)2 Y(3)1 Y(3)0 Y(3)1 Y(3)2 {Y_{3}^{(3)}}
{Y_{-1}^{(1)}} 0 0 \text{Meu1} 0 0 \text{Ma2u1} \text{Root}\left[4 \text{\#$1}^4+9$,3\right] 0
{Y_{0}^{(1)}} \left(\frac{1}{2}+\frac{i}{2}\right) \text{Ma2u1} 0 0 \text{Ma2u2} 0 0 \left(-\frac{1}{2}+\frac{i}{2}\right) \text{Ma2u1}
{Y_{1}^{(1)}} 0 \text{Ma2u1} \text{Root}\left[4 \text{\#$1}^4+9$,1\right] 0 0 \text{Meu1} 0 0

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

f_{y\left(3x^2-y^2\right)} f_{\text{xyz}} f_{y\left(5z^2-r^2\right)} f_{z\left(5z^2-3r^2\right)} f_{x\left(5z^2-r^2\right)} f_{z\left(x^2-y^2\right)} f_{x\left(x^2-3y^2\right)}
p_y 0 \frac{1}{2} \text{Ma2u1} \left(\text{Root}\left[4 \text{\#$1}^4+9$,1\right]-\text{Root}\left[4 \text{\#$1}^4+9$,3\right]\right) \text{Meu1} 0 0 -\frac{1}{2} i \text{Ma2u1} \left(\text{Root}\left[4 \text{\#$1}^4+9$,1\right]+\text{Root}\left[4 \text{\#$1}^4+9$,3\right]\right) 0
p_z -\frac{\text{Ma2u1}}{\sqrt{2}} 0 0 \text{Ma2u2} 0 0 \frac{\text{Ma2u1}}{\sqrt{2}}
p_x 0 -\frac{1}{2} i \text{Ma2u1} \left(\text{Root}\left[4 \text{\#$1}^4+9$,1\right]+\text{Root}\left[4 \text{\#$1}^4+9$,3\right]\right) 0 0 \text{Meu1} \frac{1}{2} \text{Ma2u1} \left(\text{Root}\left[4 \text{\#$1}^4+9$,3\right]-\text{Root}\left[4 \text{\#$1}^4+9$,1\right]\right) 0

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