Processing math: 92%

Orientation Zx

Symmetry Operations

In the D3h Point Group, with orientation Zx there are the following symmetry operations

Operator Orientation
E {0,0,0} ,
C3 {0,0,1} , {0,0,1} ,
C2 {1,0,0} , {1,3,0} , {1,3,0} ,
σh {0,0,1} ,
S3 {0,0,1} , {0,0,1} ,
σv {0,1,0} , {3,1,0} , {3,1,0} ,

Different Settings

Character Table

E(1) C3(2) C2(3) σh(1) S3(2) σv(3)
A1 1 1 1 1 1 1
A2 1 1 1 1 1 1
E' 2 1 0 2 1 0
A1 1 1 1 1 1 1
A2 1 1 1 1 1 1
E'' 2 1 0 2 1 0

Product Table

A1 A2 E' A1 A2 E''
A1 A1 A2 E' A1 A2 E''
A2 A2 A1 E' A2 A1 E''
E' E' E' A1+A2+E' E'' E'' A1+A2+E''
A1 A1 A2 E'' A1 A2 E'
A2 A2 A1 E'' A2 A1 E'
E'' E'' E'' A1+A2+E'' E' E' A1+A2+E'

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 D3h Point group with orientation Zx the form of the expansion coefficients is:

Expansion

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

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {-A[3, 3], k == 3 && m == -3}, {A[3, 3], k == 3 && m == 3}, {A[4, 0], k == 4 && m == 0}, {-A[5, 3], k == 5 && m == -3}, {A[5, 3], k == 5 && m == 3}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {A[6, 0], k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} , 
       {4, 0, A(4,0)} , 
       {5,-3, (-1)*(A(5,3))} , 
       {5, 3, A(5,3)} , 
       {6, 0, A(6,0)} , 
       {6,-6, A(6,6)} , 
       {6, 6, A(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)500Asf(3,3)700000Asf(3,3)7
Y(1)10App(0,0)15App(2,0)00000037Apd(3,3)003527Apf(2,0)1327Apf(4,0)0000
Y(1)000App(0,0)+25App(2,0)0000000003537Apf(2,0)+4Apf(4,0)321000
Y(1)1000App(0,0)15App(2,0)37Apd(3,3)000000003527Apf(2,0)1327Apf(4,0)00
Y(2)200037Apd(3,3)Add(0,0)27Add(2,0)+121Add(4,0)000000005332Adf(5,3)1327Adf(3,3)00
Y(2)100000Add(0,0)+17Add(2,0)421Add(4,0)000000001357Adf(3,3)4335Adf(5,3)0
Y(2)0Asd(2,0)500000Add(0,0)+27Add(2,0)+27Add(4,0)001357Adf(3,3)2335Adf(5,3)000002335Adf(5,3)1357Adf(3,3)
Y(2)10000000Add(0,0)+17Add(2,0)421Add(4,0)001357Adf(3,3)+4335Adf(5,3)00000
Y(2)2037Apd(3,3)000000Add(0,0)27Add(2,0)+121Add(4,0)001327Adf(3,3)5332Adf(5,3)0000
Y(3)3Asf(3,3)7000001357Adf(3,3)2335Adf(5,3)00Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)000001013733Aff(6,6)
Y(3)200000001357Adf(3,3)+4335Adf(5,3)00Aff(0,0)733Aff(4,0)+10143Aff(6,0)00000
Y(3)103527Apf(2,0)1327Apf(4,0)0000001327Adf(3,3)5332Adf(5,3)00Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)0000
Y(3)0003537Apf(2,0)+4Apf(4,0)321000000000Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)000
Y(3)10003527Apf(2,0)1327Apf(4,0)5332Adf(5,3)1327Adf(3,3)00000000Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)00
Y(3)2000001357Adf(3,3)4335Adf(5,3)00000000Aff(0,0)733Aff(4,0)+10143Aff(6,0)0
Y(3)3Asf(3,3)7000002335Adf(5,3)1357Adf(3,3)001013733Aff(6,6)00000Aff(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)50000000027Asf(3,3)
py0App(0,0)15App(2,0)0037Apd(3,3)0000003527Apf(2,0)1327Apf(4,0)0000
pz00App(0,0)+25App(2,0)0000000003537Apf(2,0)+4Apf(4,0)321000
px000App(0,0)15App(2,0)000037Apd(3,3)00003527Apf(2,0)1327Apf(4,0)00
dxy037Apd(3,3)00Add(0,0)27Add(2,0)+121Add(4,0)0000005332Adf(5,3)1327Adf(3,3)0000
dyz00000Add(0,0)+17Add(2,0)421Add(4,0)00001357Adf(3,3)+4335Adf(5,3)00000
d3z2r2Asd(2,0)500000Add(0,0)+27Add(2,0)+27Add(4,0)0000000013107Adf(3,3)23310Adf(5,3)
dxz0000000Add(0,0)+17Add(2,0)421Add(4,0)0000001357Adf(3,3)4335Adf(5,3)0
dx2y200037Apd(3,3)0000Add(0,0)27Add(2,0)+121Add(4,0)00001327Adf(3,3)5332Adf(5,3)00
fy(3x2y2)000000000Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)1013733Aff(6,6)000000
fxyz000001357Adf(3,3)+4335Adf(5,3)0000Aff(0,0)733Aff(4,0)+10143Aff(6,0)00000
fy(5z2r2)03527Apf(2,0)1327Apf(4,0)005332Adf(5,3)1327Adf(3,3)000000Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)0000
fz(5z23r2)003537Apf(2,0)+4Apf(4,0)321000000000Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)000
fx(5z2r2)0003527Apf(2,0)1327Apf(4,0)00001327Adf(3,3)5332Adf(5,3)0000Aff(0,0)+15Aff(2,0)+133Aff(4,0)25143Aff(6,0)00
fz(x2y2)00000001357Adf(3,3)4335Adf(5,3)000000Aff(0,0)733Aff(4,0)+10143Aff(6,0)0
fx(x23y2)27Asf(3,3)0000013107Adf(3,3)23310Adf(5,3)00000000Aff(0,0)13Aff(2,0)+111Aff(4,0)5429Aff(6,0)+1013733Aff(6,6)

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={Ea1pk=0m=00True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Ea1p, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{0, 0, Ea1p} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

Y(0)0
Y(0)0Ea1p

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

s
sEa1p

Rotation matrix used

Rotation matrix used

Y(0)0
s1

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Ea1p
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(Ea2pp+2Eep)k=0m=05(Ea2ppEep)3k=2m=0

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea2pp + 2*Eep)/3, k == 0 && m == 0}, {(5*(Ea2pp - Eep))/3, k == 2 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{0, 0, (1/3)*(Ea2pp + (2)*(Eep))} , 
       {2, 0, (5/3)*(Ea2pp + (-1)*(Eep))} }

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)1Eep00
Y(1)00Ea2pp0
Y(1)100Eep

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

py pz px
pyEep00
pz0Ea2pp0
px00Eep

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

Eep
ψ(θ,ϕ)=11 123πsin(θ)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 123πy
Ea2pp
ψ(θ,ϕ)=11 123πcos(θ)
ψ(ˆx,ˆy,ˆz)=11 123πz
Eep
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(Ea1p+2(Eep+Eepp))k=0m=0Ea1p2Eep+Eeppk=2m=00k4m035(3Ea1p+Eep4Eepp)True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea1p + 2*(Eep + Eepp))/5, k == 0 && m == 0}, {Ea1p - 2*Eep + Eepp, k == 2 && m == 0}, {0, k != 4 || m != 0}}, (3*(3*Ea1p + Eep - 4*Eepp))/5]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{0, 0, (1/5)*(Ea1p + (2)*(Eep + Eepp))} , 
       {2, 0, Ea1p + (-2)*(Eep) + Eepp} , 
       {4, 0, (3/5)*((3)*(Ea1p) + Eep + (-4)*(Eepp))} }

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)2Eep0000
Y(2)10Eepp000
Y(2)000Ea1p00
Y(2)1000Eepp0
Y(2)20000Eep

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

dxy dyz d3z2r2 dxz dx2y2
dxyEep0000
dyz0Eepp000
d3z2r200Ea1p00
dxz000Eepp0
dx2y20000Eep

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

Eep
ψ(θ,ϕ)=11 1415πsin2(θ)sin(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πxy
Eepp
ψ(θ,ϕ)=11 1415πsin(2θ)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πyz
Ea1p
ψ(θ,ϕ)=11 185π(3cos(2θ)+1)
ψ(ˆx,ˆy,ˆz)=11 145π(3z21)
Eepp
ψ(θ,ϕ)=11 1415πsin(2θ)cos(ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πxz
Eep
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(Ea1p+Ea2p+Ea2pp+2Eep+2Eepp)k=0m=0528(5Ea1p+5Ea2p4Ea2pp6Eep)k=2m=00(k6(k4m0))(m6m0m6)314(3Ea1p+3Ea2p+2(3Ea2pp+Eep7Eepp))k=4m=01320337(Ea1pEa2p)k=6(m=6m=6)13140(Ea1p+Ea2p20Ea2pp+30Eep12Eepp)True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea1p + Ea2p + Ea2pp + 2*Eep + 2*Eepp)/7, k == 0 && m == 0}, {(-5*(5*Ea1p + 5*Ea2p - 4*Ea2pp - 6*Eep))/28, k == 2 && m == 0}, {0, (k != 6 && (k != 4 || m != 0)) || (m != -6 && m != 0 && m != 6)}, {(3*(3*Ea1p + 3*Ea2p + 2*(3*Ea2pp + Eep - 7*Eepp)))/14, k == 4 && m == 0}, {(13*Sqrt[33/7]*(Ea1p - Ea2p))/20, k == 6 && (m == -6 || m == 6)}}, (-13*(Ea1p + Ea2p - 20*Ea2pp + 30*Eep - 12*Eepp))/140]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{0, 0, (1/7)*(Ea1p + Ea2p + Ea2pp + (2)*(Eep) + (2)*(Eepp))} , 
       {2, 0, (-5/28)*((5)*(Ea1p) + (5)*(Ea2p) + (-4)*(Ea2pp) + (-6)*(Eep))} , 
       {4, 0, (3/14)*((3)*(Ea1p) + (3)*(Ea2p) + (2)*((3)*(Ea2pp) + Eep + (-7)*(Eepp)))} , 
       {6, 0, (-13/140)*(Ea1p + Ea2p + (-20)*(Ea2pp) + (30)*(Eep) + (-12)*(Eepp))} , 
       {6,-6, (13/20)*((sqrt(33/7))*(Ea1p + (-1)*(Ea2p)))} , 
       {6, 6, (13/20)*((sqrt(33/7))*(Ea1p + (-1)*(Ea2p)))} }

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)3Ea1p+Ea2p200000Ea2pEa1p2
Y(3)20Eepp00000
Y(3)100Eep0000
Y(3)0000Ea2pp000
Y(3)10000Eep00
Y(3)200000Eepp0
Y(3)3Ea2pEa1p200000Ea1p+Ea2p2

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)Ea2p000000
fxyz0Eepp00000
fy(5z2r2)00Eep0000
fz(5z23r2)000Ea2pp000
fx(5z2r2)0000Eep00
fz(x2y2)00000Eepp0
fx(x23y2)000000Ea1p

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

Ea2p
ψ(θ,ϕ)=11 14352πsin3(θ)sin(3ϕ)
ψ(ˆx,ˆy,ˆz)=11 14352πy(y23x2)
Eepp
ψ(θ,ϕ)=11 14105πsin2(θ)cos(θ)sin(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 12105πxyz
Eep
ψ(θ,ϕ)=11 18212πsin(θ)(5cos(2θ)+3)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 14212πy(5z21)
Ea2pp
ψ(θ,ϕ)=11 1167π(3cos(θ)+5cos(3θ))
ψ(ˆx,ˆy,ˆz)=11 147πz(5z23)
Eep
ψ(θ,ϕ)=11 116212π(sin(θ)+5sin(3θ))cos(ϕ)
ψ(ˆx,ˆy,ˆz)=11 14212πx(5z21)
Eepp
ψ(θ,ϕ)=11 14105πsin2(θ)cos(θ)cos(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 14105πz(x2y2)
Ea1p
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={0k2m0A(2,0)True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, A[2, 0]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{2, 0, A(2,0)} }

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)000A(2,0)500

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

dxy dyz d3z2r2 dxz dx2y2
s00A(2,0)500

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={0k3(m3m3)A(3,3)k=3m=3A(3,3)True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != 3)}, {-A[3, 3], k == 3 && m == -3}}, A[3, 3]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} }

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(0)0A(3,3)700000A(3,3)7

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)
s00000027A(3,3)

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={0k3(m3m3)A(3,3)k=3m=3A(3,3)True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != 3)}, {-A[3, 3], k == 3 && m == -3}}, A[3, 3]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} }

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(1)1000037A(3,3)
Y(1)000000
Y(1)137A(3,3)0000

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

dxy dyz d3z2r2 dxz dx2y2
py37A(3,3)0000
pz00000
px000037A(3,3)

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={0(k2k4)m0A(2,0)k=2m=0A(4,0)True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || m != 0}, {A[2, 0], k == 2 && m == 0}}, A[4, 0]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{2, 0, A(2,0)} , 
       {4, 0, A(4,0)} }

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)10011527(9A(2,0)5A(4,0))0000
Y(1)000027A(2,0)+20A(4,0)1521000
Y(1)1000011527(9A(2,0)5A(4,0))00

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) f_{x\left(x^2-3y^2\right)}
p_y 0 0 \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) 0 0 0 0
p_z 0 0 0 \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} 0 0 0
p_x 0 0 0 0 \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) 0 0

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 3\land k\neq 5)\lor (m\neq -3\land m\neq 3) \\ -A(3,3) & k=3\land m=-3 \\ A(3,3) & k=3\land m=3 \\ -A(5,3) & k=5\land m=-3 \\ A(5,3) & \text{True} \end{cases}

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zx.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, (k != 3 && k != 5) || (m != -3 && m != 3)}, {-A[3, 3], k == 3 && m == -3}, {A[3, 3], k == 3 && m == 3}, {-A[5, 3], k == 5 && m == -3}}, A[5, 3]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_D3h_Zx.Quanty
Akm = {{3,-3, (-1)*(A(3,3))} , 
       {3, 3, A(3,3)} , 
       {5,-3, (-1)*(A(5,3))} , 
       {5, 3, A(5,3)} }

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 0 0 0 \frac{5}{33} \sqrt{2} A(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} A(3,3) 0 0
{Y_{-1}^{(2)}} 0 0 0 0 0 -\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,3)+28 A(5,3)\right) 0
{Y_{0}^{(2)}} \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)-\frac{2}{33} \sqrt{5} A(5,3) 0 0 0 0 0 \frac{2}{33} \sqrt{5} A(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)
{Y_{1}^{(2)}} 0 \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)+\frac{4}{33} \sqrt{5} A(5,3) 0 0 0 0 0
{Y_{2}^{(2)}} 0 0 \frac{1}{3} \sqrt{\frac{2}{7}} A(3,3)-\frac{5}{33} \sqrt{2} A(5,3) 0 0 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)}
d_{\text{xy}} 0 0 \frac{5}{33} \sqrt{2} A(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} A(3,3) 0 0 0 0
d_{\text{yz}} 0 \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)+\frac{4}{33} \sqrt{5} A(5,3) 0 0 0 0 0
d_{3z^2-r^2} 0 0 0 0 0 0 \frac{1}{231} \sqrt{10} \left(11 \sqrt{7} A(3,3)-14 A(5,3)\right)
d_{\text{xz}} 0 0 0 0 0 -\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,3)+28 A(5,3)\right) 0
d_{x^2-y^2} 0 0 0 0 \frac{1}{3} \sqrt{\frac{2}{7}} A(3,3)-\frac{5}{33} \sqrt{2} A(5,3) 0 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

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