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Table of Contents

Orientation XYZ

Symmetry Operations

In the O Point Group, with orientation XYZ there are the following symmetry operations

Operator Orientation
E {0,0,0} ,
C3 {1,1,1} , {1,1,1} , {1,1,1} , {1,1,1} , {1,1,1} , {1,1,1} , {1,1,1} , {1,1,1} ,
C2 {1,1,0} , {1,1,0} , {1,0,1} , {1,0,1} , {0,1,1} , {0,1,1} ,
C4 {0,0,1} , {0,1,0} , {1,0,0} , {0,0,1} , {0,1,0} , {1,0,0} ,
C2 {0,0,1} , {0,1,0} , {1,0,0} ,

Different Settings

Character Table

E(1) C3(8) C2(6) C4(6) C2(3)
A1 1 1 1 1 1
A2 1 1 1 1 1
E 2 1 0 0 2
T1 3 0 1 1 1
T2 3 0 1 1 1

Product Table

A1 A2 E T1 T2
A1 A1 A2 E T1 T2
A2 A2 A1 E T2 T1
E E E E+A1+A2 T1+T2 T1+T2
T1 T1 T2 T1+T2 E+A1+T1+T2 E+A2+T1+T2
T2 T2 T1 T1+T2 E+A2+T1+T2 E+A1+T1+T2

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 O Point group with orientation XYZ the form of the expansion coefficients is:

Expansion

Ak,m={A(0,0)k=0m=0514A(4,0)k=4(m=4m=4)A(4,0)k=4m=072A(6,0)k=6(m=4m=4)A(6,0)k=6m=0

Input format suitable for Mathematica (Quanty.nb)

Akm_O_XYZ.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}, {A[4, 0], k == 4 && m == 0}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {A[6, 0], k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_O_XYZ.Quanty
Akm = {{0, 0, A(0,0)} , 
       {4, 0, A(4,0)} , 
       {4,-4, (sqrt(5/14))*(A(4,0))} , 
       {4, 4, (sqrt(5/14))*(A(4,0))} , 
       {6, 0, A(6,0)} , 
       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} }

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)000000000000000
Y(1)10App(0,0)0000000001327Apf(4,0)000131021Apf(4,0)
Y(1)000App(0,0)0000000004Apf(4,0)321000
Y(1)1000App(0,0)00000131021Apf(4,0)0001327Apf(4,0)00
Y(2)20000Add(0,0)+121Add(4,0)000521Add(4,0)0000000
Y(2)100000Add(0,0)421Add(4,0)0000000000
Y(2)0000000Add(0,0)+27Add(4,0)000000000
Y(2)10000000Add(0,0)421Add(4,0)00000000
Y(2)20000521Add(4,0)000Add(0,0)+121Add(4,0)0000000
Y(3)3000131021Apf(4,0)00000Aff(0,0)+111Aff(4,0)5429Aff(6,0)00011153Aff(4,0)+3514353Aff(6,0)00
Y(3)20000000000Aff(0,0)733Aff(4,0)+10143Aff(6,0)000533Aff(4,0)70143Aff(6,0)0
Y(3)101327Apf(4,0)000000000Aff(0,0)+133Aff(4,0)25143Aff(6,0)00011153Aff(4,0)+3514353Aff(6,0)
Y(3)0004Apf(4,0)321000000000Aff(0,0)+211Aff(4,0)+100429Aff(6,0)000
Y(3)10001327Apf(4,0)0000011153Aff(4,0)+3514353Aff(6,0)000Aff(0,0)+133Aff(4,0)25143Aff(6,0)00
Y(3)20000000000533Aff(4,0)70143Aff(6,0)000Aff(0,0)733Aff(4,0)+10143Aff(6,0)0
Y(3)30131021Apf(4,0)00000000011153Aff(4,0)+3514353Aff(6,0)000Aff(0,0)+111Aff(4,0)5429Aff(6,0)

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

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

Y(0)0 Y(1)1 Y(1)0 Y(1)1 Y(2)2 Y(2)1 Y(2)0 Y(2)1 Y(2)2 Y(3)3 Y(3)2 Y(3)1 Y(3)0 Y(3)1 Y(3)2 Y(3)3
s1000000000000000
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)000000000000000
px0App(0,0)000000004Apf(4,0)32100000
py00App(0,0)000000004Apf(4,0)3210000
pz000App(0,0)000000004Apf(4,0)321000
dx2y20000Add(0,0)+27Add(4,0)00000000000
d3z2r200000Add(0,0)+27Add(4,0)0000000000
dyz000000Add(0,0)421Add(4,0)000000000
dxz0000000Add(0,0)421Add(4,0)00000000
dxy00000000Add(0,0)421Add(4,0)0000000
fxyz000000000Aff(0,0)411Aff(4,0)+80143Aff(6,0)000000
fx(5x2r2)04Apf(4,0)32100000000Aff(0,0)+211Aff(4,0)+100429Aff(6,0)00000
fy(5y2r2)004Apf(4,0)32100000000Aff(0,0)+211Aff(4,0)+100429Aff(6,0)0000
fz(5z2r2)0004Apf(4,0)32100000000Aff(0,0)+211Aff(4,0)+100429Aff(6,0)000
fx(y2z2)0000000000000Aff(0,0)233Aff(4,0)60143Aff(6,0)00
fy(z2x2)00000000000000Aff(0,0)233Aff(4,0)60143Aff(6,0)0
fz(x2y2)000000000000000Aff(0,0)233Aff(4,0)60143Aff(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={Ea1k=0m=00True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_O_XYZ.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Ea1, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_O_XYZ.Quanty
Akm = {{0, 0, Ea1} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

Y(0)0
Y(0)0Ea1

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

s
sEa1

Rotation matrix used

Rotation matrix used

Y(0)0
s1

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Ea1
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={Et1k=0m=00True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_O_XYZ.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Et1, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_O_XYZ.Quanty
Akm = {{0, 0, Et1} }

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)1Et100
Y(1)00Et10
Y(1)100Et1

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

px py pz
pxEt100
py0Et10
pz00Et1

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Et1
ψ(θ,ϕ)=11 123πsin(θ)cos(ϕ)
ψ(ˆx,ˆy,ˆz)=11 123πx
Et1
ψ(θ,ϕ)=11 123πsin(θ)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 123πy
Et1
ψ(θ,ϕ)=11 123πcos(θ)
ψ(ˆx,ˆy,ˆz)=11 123πz

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={2Ee5+3Et25k=0m=032710(EeEt2)k=4(m=4m=4)21(EeEt2)10k=4m=0

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_O_XYZ.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(2*Ee)/5 + (3*Et2)/5, k == 0 && m == 0}, {(3*Sqrt[7/10]*(Ee - Et2))/2, k == 4 && (m == -4 || m == 4)}, {(21*(Ee - Et2))/10, k == 4 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_O_XYZ.Quanty
Akm = {{0, 0, (2/5)*(Ee) + (3/5)*(Et2)} , 
       {4, 0, (21/10)*(Ee + (-1)*(Et2))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Ee + (-1)*(Et2)))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Ee + (-1)*(Et2)))} }

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)2Ee+Et22000EeEt22
Y(2)10Et2000
Y(2)000Ee00
Y(2)1000Et20
Y(2)2EeEt22000Ee+Et22

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

dx2y2 d3z2r2 dyz dxz dxy
dx2y2Ee0000
d3z2r20Ee000
dyz00Et200
dxz000Et20
dxy0000Et2

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Ee
ψ(θ,ϕ)=11 1415πsin2(θ)cos(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 1415π(x2y2)
Ee
ψ(θ,ϕ)=11 185π(3cos(2θ)+1)
ψ(ˆx,ˆy,ˆz)=11 145π(3z21)
Et2
ψ(θ,ϕ)=11 1415πsin(2θ)sin(ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πyz
Et2
ψ(θ,ϕ)=11 1415πsin(2θ)cos(ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πxz
Et2
ψ(θ,ϕ)=11 1415πsin2(θ)sin(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 1215πxy

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={17(Ea2+3(Et1+Et2))k=0m=034514(2Ea23Et1+Et2)k=4(m=4m=4)34(2Ea23Et1+Et2)k=4m=039(4Ea2+5Et19Et2)4014k=6(m=4m=4)39280(4Ea2+5Et19Et2)k=6m=0

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_O_XYZ.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Ea2 + 3*(Et1 + Et2))/7, k == 0 && m == 0}, {(-3*Sqrt[5/14]*(2*Ea2 - 3*Et1 + Et2))/4, k == 4 && (m == -4 || m == 4)}, {(-3*(2*Ea2 - 3*Et1 + Et2))/4, k == 4 && m == 0}, {(-39*(4*Ea2 + 5*Et1 - 9*Et2))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(39*(4*Ea2 + 5*Et1 - 9*Et2))/280, k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_O_XYZ.Quanty
Akm = {{0, 0, (1/7)*(Ea2 + (3)*(Et1 + Et2))} , 
       {4, 0, (-3/4)*((2)*(Ea2) + (-3)*(Et1) + Et2)} , 
       {4,-4, (-3/4)*((sqrt(5/14))*((2)*(Ea2) + (-3)*(Et1) + Et2))} , 
       {4, 4, (-3/4)*((sqrt(5/14))*((2)*(Ea2) + (-3)*(Et1) + Et2))} , 
       {6, 0, (39/280)*((4)*(Ea2) + (5)*(Et1) + (-9)*(Et2))} , 
       {6,-4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2) + (5)*(Et1) + (-9)*(Et2)))} , 
       {6, 4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2) + (5)*(Et1) + (-9)*(Et2)))} }

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)318(5Et1+3Et2)0001815(Et1Et2)00
Y(3)20Ea2+Et22000Et2Ea220
Y(3)10018(3Et1+5Et2)0001815(Et1Et2)
Y(3)0000Et1000
Y(3)11815(Et1Et2)00018(3Et1+5Et2)00
Y(3)20Et2Ea22000Ea2+Et220
Y(3)3001815(Et1Et2)00018(5Et1+3Et2)

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

fxyz fx(5x2r2) fy(5y2r2) fz(5z2r2) fx(y2z2) fy(z2x2) fz(x2y2)
fxyzEa2000000
fx(5x2r2)0Et100000
fy(5y2r2)00Et10000
fz(5z2r2)000Et1000
fx(y2z2)0000Et200
fy(z2x2)00000Et20
fz(x2y2)000000Et2

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
fxyz0i2000i20
fx(5x2r2)54034034054
fy(5y2r2)i540i340i340i54
fz(5z2r2)0001000
fx(y2z2)34054054034
fy(z2x2)i340i540i540i34
fz(x2y2)012000120

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

Ea2
ψ(θ,ϕ)=11 14105πsin2(θ)cos(θ)sin(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 12105πxyz
Et1
ψ(θ,ϕ)=11 1167πsin(θ)cos(ϕ)(10sin2(θ)cos(2ϕ)5cos(2θ)7)
ψ(ˆx,ˆy,ˆz)=11 1167πx(5x215y215z2+3)
Et1
ψ(θ,ϕ)=11 1167πsin(θ)sin(ϕ)(10sin2(θ)cos(2ϕ)+5cos(2θ)+7)
ψ(ˆx,ˆy,ˆz)=11 1167πy(15x2+5y215z2+3)
Et1
ψ(θ,ϕ)=11 1167π(3cos(θ)+5cos(3θ))
ψ(ˆx,ˆy,ˆz)=11 147πz(5z23)
Et2
ψ(θ,ϕ)=11 116105πsin(θ)cos(ϕ)(2sin2(θ)cos(2ϕ)+3cos(2θ)+1)
ψ(ˆx,ˆy,ˆz)=11 116105πx(x23y2+5z21)
Et2
ψ(θ,ϕ)=11 132105πsin(θ)sin(ϕ)(4sin2(θ)cos(2ϕ)+6cos(2θ)+2)
ψ(ˆx,ˆy,ˆz)=11 116105πy(3x2+y2+5z21)
Et2
ψ(θ,ϕ)=11 14105πsin2(θ)cos(θ)cos(2ϕ)
ψ(ˆx,ˆy,ˆz)=11 14105πz(x2y2)

Coupling between two shells

Click on one of the subsections to expand it or

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

Ak,m={0k4(m4m0m4)514A(4,0)k=4(m=4m=4)A(4,0)True

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_O_XYZ.Quanty.nb
Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -4 && m != 0 && m != 4)}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}}, A[4, 0]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_O_XYZ.Quanty
Akm = {{4, 0, A(4,0)} , 
       {4,-4, (sqrt(5/14))*(A(4,0))} , 
       {4, 4, (sqrt(5/14))*(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)1001327A(4,0)000131021A(4,0)
Y(1)00004A(4,0)321000
Y(1)1131021A(4,0)0001327A(4,0)00

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

fxyz fx(5x2r2) fy(5y2r2) fz(5z2r2) fx(y2z2) fy(z2x2) fz(x2y2)
px04A(4,0)32100000
py004A(4,0)3210000
pz0004A(4,0)321000

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 Cv Dh