Orientation Zx
Symmetry Operations
In the D3h Point Group, with orientation Zx there are the following symmetry operations
Operator Orientation
E { 0 , 0 , 0 } ,
C 3 { 0 , 0 , 1 } , { 0 , 0 , − 1 } ,
C 2 { 1 , 0 , 0 } , { 1 , √ 3 , 0 } , { 1 , − √ 3 , 0 } ,
σ h { 0 , 0 , 1 } ,
S 3 { 0 , 0 , 1 } , { 0 , 0 , − 1 } ,
σ v { 0 , 1 , 0 } , { √ 3 , 1 , 0 } , { − √ 3 , 1 , 0 } ,
Different Settings
Character Table
E (1) C 3 (2) C 2 (3) σ h (1) S 3 (2) σ v (3)
A ′ 1 1 1 1 1 1 1
A ′ 2 1 1 − 1 1 1 − 1
E' 2 − 1 0 2 − 1 0
A ″ 1 1 1 1 − 1 − 1 − 1
A ″ 2 1 1 − 1 − 1 − 1 1
E'' 2 − 1 0 − 2 1 0
Product Table
A ′ 1 A ′ 2 E' A ″ 1 A ″ 2 E''
A ′ 1 A ′ 1 A ′ 2 E' A ″ 1 A ″ 2 E''
A ′ 2 A ′ 2 A ′ 1 E' A ″ 2 A ″ 1 E''
E' E' E' A ′ 1 + A ′ 2 + E' E'' E'' A ″ 1 + A ″ 2 + E''
A ″ 1 A ″ 1 A ″ 2 E'' A ′ 1 A ′ 2 E'
A ″ 2 A ″ 2 A ″ 1 E'' A ′ 2 A ′ 1 E'
E'' E'' E'' A ″ 1 + A ″ 2 + E'' E' E' A ′ 1 + A ′ 2 + 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 = 0 k ∑ m = − k A k , m ( r ) C ( m ) k ( θ , ϕ )
Here A k , m ( r ) is a radial function and C ( m ) k ( θ , ϕ ) a renormalised spherical harmonics. C ( m ) k ( θ , ϕ ) = √ 4 π 2 k + 1 Y ( 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
A k , m = { 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
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]
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 ′ , m ′ ⟨ ψ n ″ , l ″ , m ″ ( r , θ , ϕ ) | V ( r , θ , ϕ ) | ψ n ′ , l ′ , m ′ ( r , θ , ϕ ) ⟩ a † n ″ , l ″ , m ″ a † n ′ , l ′ , m ′
For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. ψ n , l , m ( r , θ , ϕ ) = R n , 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.
A n ″ l ″ , n ′ l ′ ( k , m ) = ⟨ R n ″ , l ″ | A k , m ( r ) | R n ′ , l ′ ⟩
Note the difference between the function A k , m and the parameter A n ″ l ″ , n ′ l ′ ( k , m )
we can express the operator as
O = ∑ n ″ , l ″ , m ″ , n ′ , l ′ , m ′ , k , m A n ″ l ″ , n ′ l ′ ( k , m ) ⟨ Y ( m ″ ) l ″ ( θ , ϕ ) | C ( m ) k ( θ , ϕ ) | Y ( m ′ ) l ′ ( θ , ϕ ) ⟩ a † n ″ , l ″ , m ″ a † n ′ , l ′ , m ′
The table below shows the expectation value of O on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle A l ″ , l ′ ( k , m ) can be complex. Instead of allowing complex parameters we took A l ″ , l ′ ( k , m ) + I B l ″ , 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 ) 0 Ass ( 0 , 0 ) 0 0 0 0 0 Asd ( 2 , 0 ) √ 5 0 0 − Asf ( 3 , 3 ) √ 7 0 0 0 0 0 Asf ( 3 , 3 ) √ 7
Y ( 1 ) − 1 0 App ( 0 , 0 ) − 1 5 App ( 2 , 0 ) 0 0 0 0 0 0 − 3 7 Apd ( 3 , 3 ) 0 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 0 0 0 0
Y ( 1 ) 0 0 0 App ( 0 , 0 ) + 2 5 App ( 2 , 0 ) 0 0 0 0 0 0 0 0 0 3 5 √ 3 7 Apf ( 2 , 0 ) + 4 Apf ( 4 , 0 ) 3 √ 21 0 0 0
Y ( 1 ) 1 0 0 0 App ( 0 , 0 ) − 1 5 App ( 2 , 0 ) 3 7 Apd ( 3 , 3 ) 0 0 0 0 0 0 0 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 0 0
Y ( 2 ) − 2 0 0 0 3 7 Apd ( 3 , 3 ) Add ( 0 , 0 ) − 2 7 Add ( 2 , 0 ) + 1 21 Add ( 4 , 0 ) 0 0 0 0 0 0 0 0 5 33 √ 2 Adf ( 5 , 3 ) − 1 3 √ 2 7 Adf ( 3 , 3 ) 0 0
Y ( 2 ) − 1 0 0 0 0 0 Add ( 0 , 0 ) + 1 7 Add ( 2 , 0 ) − 4 21 Add ( 4 , 0 ) 0 0 0 0 0 0 0 0 − 1 3 √ 5 7 Adf ( 3 , 3 ) − 4 33 √ 5 Adf ( 5 , 3 ) 0
Y ( 2 ) 0 Asd ( 2 , 0 ) √ 5 0 0 0 0 0 Add ( 0 , 0 ) + 2 7 Add ( 2 , 0 ) + 2 7 Add ( 4 , 0 ) 0 0 1 3 √ 5 7 Adf ( 3 , 3 ) − 2 33 √ 5 Adf ( 5 , 3 ) 0 0 0 0 0 2 33 √ 5 Adf ( 5 , 3 ) − 1 3 √ 5 7 Adf ( 3 , 3 )
Y ( 2 ) 1 0 0 0 0 0 0 0 Add ( 0 , 0 ) + 1 7 Add ( 2 , 0 ) − 4 21 Add ( 4 , 0 ) 0 0 1 3 √ 5 7 Adf ( 3 , 3 ) + 4 33 √ 5 Adf ( 5 , 3 ) 0 0 0 0 0
Y ( 2 ) 2 0 − 3 7 Apd ( 3 , 3 ) 0 0 0 0 0 0 Add ( 0 , 0 ) − 2 7 Add ( 2 , 0 ) + 1 21 Add ( 4 , 0 ) 0 0 1 3 √ 2 7 Adf ( 3 , 3 ) − 5 33 √ 2 Adf ( 5 , 3 ) 0 0 0 0
Y ( 3 ) − 3 − Asf ( 3 , 3 ) √ 7 0 0 0 0 0 1 3 √ 5 7 Adf ( 3 , 3 ) − 2 33 √ 5 Adf ( 5 , 3 ) 0 0 Aff ( 0 , 0 ) − 1 3 Aff ( 2 , 0 ) + 1 11 Aff ( 4 , 0 ) − 5 429 Aff ( 6 , 0 ) 0 0 0 0 0 − 10 13 √ 7 33 Aff ( 6 , 6 )
Y ( 3 ) − 2 0 0 0 0 0 0 0 1 3 √ 5 7 Adf ( 3 , 3 ) + 4 33 √ 5 Adf ( 5 , 3 ) 0 0 Aff ( 0 , 0 ) − 7 33 Aff ( 4 , 0 ) + 10 143 Aff ( 6 , 0 ) 0 0 0 0 0
Y ( 3 ) − 1 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 0 0 0 0 0 0 1 3 √ 2 7 Adf ( 3 , 3 ) − 5 33 √ 2 Adf ( 5 , 3 ) 0 0 Aff ( 0 , 0 ) + 1 5 Aff ( 2 , 0 ) + 1 33 Aff ( 4 , 0 ) − 25 143 Aff ( 6 , 0 ) 0 0 0 0
Y ( 3 ) 0 0 0 3 5 √ 3 7 Apf ( 2 , 0 ) + 4 Apf ( 4 , 0 ) 3 √ 21 0 0 0 0 0 0 0 0 0 Aff ( 0 , 0 ) + 4 15 Aff ( 2 , 0 ) + 2 11 Aff ( 4 , 0 ) + 100 429 Aff ( 6 , 0 ) 0 0 0
Y ( 3 ) 1 0 0 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 5 33 √ 2 Adf ( 5 , 3 ) − 1 3 √ 2 7 Adf ( 3 , 3 ) 0 0 0 0 0 0 0 0 Aff ( 0 , 0 ) + 1 5 Aff ( 2 , 0 ) + 1 33 Aff ( 4 , 0 ) − 25 143 Aff ( 6 , 0 ) 0 0
Y ( 3 ) 2 0 0 0 0 0 − 1 3 √ 5 7 Adf ( 3 , 3 ) − 4 33 √ 5 Adf ( 5 , 3 ) 0 0 0 0 0 0 0 0 Aff ( 0 , 0 ) − 7 33 Aff ( 4 , 0 ) + 10 143 Aff ( 6 , 0 ) 0
Y ( 3 ) 3 Asf ( 3 , 3 ) √ 7 0 0 0 0 0 2 33 √ 5 Adf ( 5 , 3 ) − 1 3 √ 5 7 Adf ( 3 , 3 ) 0 0 − 10 13 √ 7 33 Aff ( 6 , 6 ) 0 0 0 0 0 Aff ( 0 , 0 ) − 1 3 Aff ( 2 , 0 ) + 1 11 Aff ( 4 , 0 ) − 5 429 Aff ( 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
s 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
p y 0 i √ 2 0 i √ 2 0 0 0 0 0 0 0 0 0 0 0 0
p z 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
p x 0 1 √ 2 0 − 1 √ 2 0 0 0 0 0 0 0 0 0 0 0 0
d xy 0 0 0 0 i √ 2 0 0 0 − i √ 2 0 0 0 0 0 0 0
d yz 0 0 0 0 0 i √ 2 0 i √ 2 0 0 0 0 0 0 0 0
d 3 z 2 − r 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
d xz 0 0 0 0 0 1 √ 2 0 − 1 √ 2 0 0 0 0 0 0 0 0
d x 2 − y 2 0 0 0 0 1 √ 2 0 0 0 1 √ 2 0 0 0 0 0 0 0
f y ( 3 x 2 − y 2 ) 0 0 0 0 0 0 0 0 0 i √ 2 0 0 0 0 0 i √ 2
f xyz 0 0 0 0 0 0 0 0 0 0 i √ 2 0 0 0 − i √ 2 0
f y ( 5 z 2 − r 2 ) 0 0 0 0 0 0 0 0 0 0 0 i √ 2 0 i √ 2 0 0
f z ( 5 z 2 − 3 r 2 ) 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
f x ( 5 z 2 − r 2 ) 0 0 0 0 0 0 0 0 0 0 0 1 √ 2 0 − 1 √ 2 0 0
f z ( x 2 − y 2 ) 0 0 0 0 0 0 0 0 0 0 1 √ 2 0 0 0 1 √ 2 0
f x ( x 2 − 3 y 2 ) 0 0 0 0 0 0 0 0 0 1 √ 2 0 0 0 0 0 − 1 √ 2
One particle coupling on a basis of symmetry adapted functions
After rotation we find
s p y p z p x d xy d yz d 3 z 2 − r 2 d xz d x 2 − y 2 f y ( 3 x 2 − y 2 ) f xyz f y ( 5 z 2 − r 2 ) f z ( 5 z 2 − 3 r 2 ) f x ( 5 z 2 − r 2 ) f z ( x 2 − y 2 ) f x ( x 2 − 3 y 2 )
s Ass ( 0 , 0 ) 0 0 0 0 0 Asd ( 2 , 0 ) √ 5 0 0 0 0 0 0 0 0 − √ 2 7 Asf ( 3 , 3 )
p y 0 App ( 0 , 0 ) − 1 5 App ( 2 , 0 ) 0 0 3 7 Apd ( 3 , 3 ) 0 0 0 0 0 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 0 0 0 0
p z 0 0 App ( 0 , 0 ) + 2 5 App ( 2 , 0 ) 0 0 0 0 0 0 0 0 0 3 5 √ 3 7 Apf ( 2 , 0 ) + 4 Apf ( 4 , 0 ) 3 √ 21 0 0 0
p x 0 0 0 App ( 0 , 0 ) − 1 5 App ( 2 , 0 ) 0 0 0 0 − 3 7 Apd ( 3 , 3 ) 0 0 0 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 0 0
d xy 0 3 7 Apd ( 3 , 3 ) 0 0 Add ( 0 , 0 ) − 2 7 Add ( 2 , 0 ) + 1 21 Add ( 4 , 0 ) 0 0 0 0 0 0 5 33 √ 2 Adf ( 5 , 3 ) − 1 3 √ 2 7 Adf ( 3 , 3 ) 0 0 0 0
d yz 0 0 0 0 0 Add ( 0 , 0 ) + 1 7 Add ( 2 , 0 ) − 4 21 Add ( 4 , 0 ) 0 0 0 0 1 3 √ 5 7 Adf ( 3 , 3 ) + 4 33 √ 5 Adf ( 5 , 3 ) 0 0 0 0 0
d 3 z 2 − r 2 Asd ( 2 , 0 ) √ 5 0 0 0 0 0 Add ( 0 , 0 ) + 2 7 Add ( 2 , 0 ) + 2 7 Add ( 4 , 0 ) 0 0 0 0 0 0 0 0 1 3 √ 10 7 Adf ( 3 , 3 ) − 2 33 √ 10 Adf ( 5 , 3 )
d xz 0 0 0 0 0 0 0 Add ( 0 , 0 ) + 1 7 Add ( 2 , 0 ) − 4 21 Add ( 4 , 0 ) 0 0 0 0 0 0 − 1 3 √ 5 7 Adf ( 3 , 3 ) − 4 33 √ 5 Adf ( 5 , 3 ) 0
d x 2 − y 2 0 0 0 − 3 7 Apd ( 3 , 3 ) 0 0 0 0 Add ( 0 , 0 ) − 2 7 Add ( 2 , 0 ) + 1 21 Add ( 4 , 0 ) 0 0 0 0 1 3 √ 2 7 Adf ( 3 , 3 ) − 5 33 √ 2 Adf ( 5 , 3 ) 0 0
f y ( 3 x 2 − y 2 ) 0 0 0 0 0 0 0 0 0 Aff ( 0 , 0 ) − 1 3 Aff ( 2 , 0 ) + 1 11 Aff ( 4 , 0 ) − 5 429 Aff ( 6 , 0 ) − 10 13 √ 7 33 Aff ( 6 , 6 ) 0 0 0 0 0 0
f xyz 0 0 0 0 0 1 3 √ 5 7 Adf ( 3 , 3 ) + 4 33 √ 5 Adf ( 5 , 3 ) 0 0 0 0 Aff ( 0 , 0 ) − 7 33 Aff ( 4 , 0 ) + 10 143 Aff ( 6 , 0 ) 0 0 0 0 0
f y ( 5 z 2 − r 2 ) 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 0 0 5 33 √ 2 Adf ( 5 , 3 ) − 1 3 √ 2 7 Adf ( 3 , 3 ) 0 0 0 0 0 0 Aff ( 0 , 0 ) + 1 5 Aff ( 2 , 0 ) + 1 33 Aff ( 4 , 0 ) − 25 143 Aff ( 6 , 0 ) 0 0 0 0
f z ( 5 z 2 − 3 r 2 ) 0 0 3 5 √ 3 7 Apf ( 2 , 0 ) + 4 Apf ( 4 , 0 ) 3 √ 21 0 0 0 0 0 0 0 0 0 Aff ( 0 , 0 ) + 4 15 Aff ( 2 , 0 ) + 2 11 Aff ( 4 , 0 ) + 100 429 Aff ( 6 , 0 ) 0 0 0
f x ( 5 z 2 − r 2 ) 0 0 0 3 5 √ 2 7 Apf ( 2 , 0 ) − 1 3 √ 2 7 Apf ( 4 , 0 ) 0 0 0 0 1 3 √ 2 7 Adf ( 3 , 3 ) − 5 33 √ 2 Adf ( 5 , 3 ) 0 0 0 0 Aff ( 0 , 0 ) + 1 5 Aff ( 2 , 0 ) + 1 33 Aff ( 4 , 0 ) − 25 143 Aff ( 6 , 0 ) 0 0
f z ( x 2 − y 2 ) 0 0 0 0 0 0 0 − 1 3 √ 5 7 Adf ( 3 , 3 ) − 4 33 √ 5 Adf ( 5 , 3 ) 0 0 0 0 0 0 Aff ( 0 , 0 ) − 7 33 Aff ( 4 , 0 ) + 10 143 Aff ( 6 , 0 ) 0
f x ( x 2 − 3 y 2 ) − √ 2 7 Asf ( 3 , 3 ) 0 0 0 0 0 1 3 √ 10 7 Adf ( 3 , 3 ) − 2 33 √ 10 Adf ( 5 , 3 ) 0 0 0 0 0 0 0 0 Aff ( 0 , 0 ) − 1 3 Aff ( 2 , 0 ) + 1 11 Aff ( 4 , 0 ) − 5 429 Aff ( 6 , 0 ) + 10 13 √ 7 33 Aff ( 6 , 6 )
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 expand all
Potential for s orbitals
Potential parameterized with onsite energies of irriducible representations
Potential parameterized with onsite energies of irriducible representations
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
The Hamiltonian on a basis of symmetric functions
The Hamiltonian on a basis of symmetric functions
Irriducible representations and their onsite energy
Irriducible representations and their onsite energy
Ea1p
ψ ( θ , ϕ ) = √ 1 1 1 2 √ π
ψ ( ˆ x , ˆ y , ˆ z ) = √ 1 1 1 2 √ π
Potential for p orbitals
Potential parameterized with onsite energies of irriducible representations
Potential parameterized with onsite energies of irriducible representations
A k , m = { 1 3 ( Ea2pp + 2 Eep ) k = 0 ∧ m = 0 5 ( Ea2pp − Eep ) 3 k = 2 ∧ m = 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 ) − 1 Eep 0 0
Y ( 1 ) 0 0 Ea2pp 0
Y ( 1 ) 1 0 0 Eep
The Hamiltonian on a basis of symmetric functions
The Hamiltonian on a basis of symmetric functions
p y p z p x
p y Eep 0 0
p z 0 Ea2pp 0
p x 0 0 Eep
Y ( 1 ) − 1 Y ( 1 ) 0 Y ( 1 ) 1
p y i √ 2 0 i √ 2
p z 0 1 0
p x 1 √ 2 0 − 1 √ 2
Irriducible representations and their onsite energy
Irriducible representations and their onsite energy
Eep
ψ ( θ , ϕ ) = √ 1 1 1 2 √ 3 π sin ( θ ) sin ( ϕ )
ψ ( ˆ x , ˆ y , ˆ z ) = √ 1 1 1 2 √ 3 π y
Ea2pp
ψ ( θ , ϕ ) = √ 1 1 1 2 √ 3 π cos ( θ )
ψ ( ˆ x , ˆ y , ˆ z ) = √ 1 1 1 2 √ 3 π z
Eep
ψ ( θ , ϕ ) = √ 1 1 1 2 √ 3 π sin ( θ ) cos ( ϕ )
ψ ( ˆ x , ˆ y , ˆ z ) = √ 1 1 1 2 √ 3 π x
Potential for d orbitals
Potential parameterized with onsite energies of irriducible representations
Potential parameterized with onsite energies of irriducible representations
A k , m = { 1 5 ( Ea1p + 2 ( Eep + Eepp ) ) k = 0 ∧ m = 0 Ea1p − 2 Eep + Eepp k = 2 ∧ m = 0 0 k ≠ 4 ∨ m ≠ 0 3 5 ( 3 Ea1p + Eep − 4 Eepp ) 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 ) − 2 Eep 0 0 0 0
Y ( 2 ) − 1 0 Eepp 0 0 0
Y ( 2 ) 0 0 0 Ea1p 0 0
Y ( 2 ) 1 0 0 0 Eepp 0
Y ( 2 ) 2 0 0 0 0 Eep
The Hamiltonian on a basis of symmetric functions
The Hamiltonian on a basis of symmetric functions
d xy d yz d 3 z 2 − r 2 d xz d x 2 − y 2
d xy Eep 0 0 0 0
d yz 0 Eepp 0 0 0
d 3 z 2 − r 2 0 0 Ea1p 0 0
d xz 0 0 0 Eepp 0
d x 2 − y 2 0 0 0 0 Eep
Y ( 2 ) − 2 {Y_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
d_{\text{xy}} \frac{i}{\sqrt{2}} 0 0 0 -\frac{i}{\sqrt{2}}
d_{\text{yz}} 0 \frac{i}{\sqrt{2}} 0 \frac{i}{\sqrt{2}} 0
d_{3z^2-r^2} 0 0 1 0 0
d_{\text{xz}} 0 \frac{1}{\sqrt{2}} 0 -\frac{1}{\sqrt{2}} 0
d_{x^2-y^2} \frac{1}{\sqrt{2}} 0 0 0 \frac{1}{\sqrt{2}}
Irriducible representations and their onsite energy
Irriducible representations and their onsite energy
\text{Eep}
\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
\text{Eepp}
\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{Ea1p}
\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{Eepp}
\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{Eep}
\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)
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{Ea1p}+\text{Ea2p}+\text{Ea2pp}+2 \text{Eep}+2 \text{Eepp}) & k=0\land m=0 \\
-\frac{5}{28} (5 \text{Ea1p}+5 \text{Ea2p}-4 \text{Ea2pp}-6 \text{Eep}) & k=2\land m=0 \\
0 & (k\neq 6\land (k\neq 4\lor m\neq 0))\lor (m\neq -6\land m\neq 0\land m\neq 6) \\
\frac{3}{14} (3 \text{Ea1p}+3 \text{Ea2p}+2 (3 \text{Ea2pp}+\text{Eep}-7 \text{Eepp})) & k=4\land m=0 \\
\frac{13}{20} \sqrt{\frac{33}{7}} (\text{Ea1p}-\text{Ea2p}) & k=6\land (m=-6\lor m=6) \\
-\frac{13}{140} (\text{Ea1p}+\text{Ea2p}-20 \text{Ea2pp}+30 \text{Eep}-12 \text{Eepp}) & \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[{{(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_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-3}^{(3)}} \frac{\text{Ea1p}+\text{Ea2p}}{2} 0 0 0 0 0 \frac{\text{Ea2p}-\text{Ea1p}}{2}
{Y_{-2}^{(3)}} 0 \text{Eepp} 0 0 0 0 0
{Y_{-1}^{(3)}} 0 0 \text{Eep} 0 0 0 0
{Y_{0}^{(3)}} 0 0 0 \text{Ea2pp} 0 0 0
{Y_{1}^{(3)}} 0 0 0 0 \text{Eep} 0 0
{Y_{2}^{(3)}} 0 0 0 0 0 \text{Eepp} 0
{Y_{3}^{(3)}} \frac{\text{Ea2p}-\text{Ea1p}}{2} 0 0 0 0 0 \frac{\text{Ea1p}+\text{Ea2p}}{2}
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)}
f_{y\left(3x^2-y^2\right)} \text{Ea2p} 0 0 0 0 0 0
f_{\text{xyz}} 0 \text{Eepp} 0 0 0 0 0
f_{y\left(5z^2-r^2\right)} 0 0 \text{Eep} 0 0 0 0
f_{z\left(5z^2-3r^2\right)} 0 0 0 \text{Ea2pp} 0 0 0
f_{x\left(5z^2-r^2\right)} 0 0 0 0 \text{Eep} 0 0
f_{z\left(x^2-y^2\right)} 0 0 0 0 0 \text{Eepp} 0
f_{x\left(x^2-3y^2\right)} 0 0 0 0 0 0 \text{Ea1p}
{Y_{-3}^{(3)}} {Y_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
f_{y\left(3x^2-y^2\right)} \frac{i}{\sqrt{2}} 0 0 0 0 0 \frac{i}{\sqrt{2}}
f_{\text{xyz}} 0 \frac{i}{\sqrt{2}} 0 0 0 -\frac{i}{\sqrt{2}} 0
f_{y\left(5z^2-r^2\right)} 0 0 \frac{i}{\sqrt{2}} 0 \frac{i}{\sqrt{2}} 0 0
f_{z\left(5z^2-3r^2\right)} 0 0 0 1 0 0 0
f_{x\left(5z^2-r^2\right)} 0 0 \frac{1}{\sqrt{2}} 0 -\frac{1}{\sqrt{2}} 0 0
f_{z\left(x^2-y^2\right)} 0 \frac{1}{\sqrt{2}} 0 0 0 \frac{1}{\sqrt{2}} 0
f_{x\left(x^2-3y^2\right)} \frac{1}{\sqrt{2}} 0 0 0 0 0 -\frac{1}{\sqrt{2}}
Irriducible representations and their onsite energy
Irriducible representations and their onsite energy
\text{Ea2p}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} -\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)
\text{Eepp}
\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{Eep}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)
\text{Ea2pp}
\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{Eep}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)
\text{Eepp}
\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)
\text{Ea1p}
\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )
\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} \frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)
Coupling between two shells
Click on one of the subsections to expand it or expand all
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 0 \\
A(2,0) & \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 != 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_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{0}^{(0)}} 0 0 \frac{A(2,0)}{\sqrt{5}} 0 0
The Hamiltonian on a basis of symmetric functions
The Hamiltonian on a basis of symmetric functions
d_{\text{xy}} d_{\text{yz}} d_{3z^2-r^2} d_{\text{xz}} d_{x^2-y^2}
\text{s} 0 0 \frac{A(2,0)}{\sqrt{5}} 0 0
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 -3\land m\neq 3) \\
-A(3,3) & k=3\land m=-3 \\
A(3,3) & \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 || (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_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{0}^{(0)}} -\frac{A(3,3)}{\sqrt{7}} 0 0 0 0 0 \frac{A(3,3)}{\sqrt{7}}
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)}
\text{s} 0 0 0 0 0 0 -\sqrt{\frac{2}{7}} A(3,3)
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\lor (m\neq -3\land m\neq 3) \\
-A(3,3) & k=3\land m=-3 \\
A(3,3) & \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 || (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_{-1}^{(2)}} {Y_{0}^{(2)}} {Y_{1}^{(2)}} {Y_{2}^{(2)}}
{Y_{-1}^{(1)}} 0 0 0 0 -\frac{3}{7} A(3,3)
{Y_{0}^{(1)}} 0 0 0 0 0
{Y_{1}^{(1)}} \frac{3}{7} A(3,3) 0 0 0 0
The Hamiltonian on a basis of symmetric functions
The Hamiltonian on a basis of symmetric functions
d_{\text{xy}} d_{\text{yz}} d_{3z^2-r^2} d_{\text{xz}} d_{x^2-y^2}
p_y \frac{3}{7} A(3,3) 0 0 0 0
p_z 0 0 0 0 0
p_x 0 0 0 0 -\frac{3}{7} A(3,3)
Potential for p-f orbital mixing
Potential parameterized with onsite energies of irriducible representations
Potential parameterized with onsite energies of irriducible representations
A_{k,m} = \begin{cases}
0 & (k\neq 2\land k\neq 4)\lor m\neq 0 \\
A(2,0) & k=2\land m=0 \\
A(4,0) & \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 != 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_{-2}^{(3)}} {Y_{-1}^{(3)}} {Y_{0}^{(3)}} {Y_{1}^{(3)}} {Y_{2}^{(3)}} {Y_{3}^{(3)}}
{Y_{-1}^{(1)}} 0 0 \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) 0 0 0 0
{Y_{0}^{(1)}} 0 0 0 \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} 0 0 0
{Y_{1}^{(1)}} 0 0 0 0 \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) 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)}
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