Invariant Potential expanded on renormalized spherical Harmonics
Any potential (function) can be written as a sum over spherical harmonics.
V(r,θ,ϕ)=∞∑k=0k∑m=−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:
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]
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′⟨ψn″,l″,m″(r,θ,ϕ)|V(r,θ,ϕ)|ψn′,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. 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.
An″l″,n′l′(k,m)=⟨Rn″,l″|Ak,m(r)|Rn′,l′⟩
Note the difference between the function Ak,m and the parameter An″l″,n′l′(k,m)
we can express the operator as
O=∑n″,l″,m″,n′,l′,m′,k,mAn″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 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)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)−15App(2,0)
0
0
0
0
0
0
−37Apd(3,3)
0
0
35√27Apf(2,0)−13√27Apf(4,0)
0
0
0
0
Y(1)0
0
0
App(0,0)+25App(2,0)
0
0
0
0
0
0
0
0
0
35√37Apf(2,0)+4Apf(4,0)3√21
0
0
0
Y(1)1
0
0
0
App(0,0)−15App(2,0)
37Apd(3,3)
0
0
0
0
0
0
0
0
35√27Apf(2,0)−13√27Apf(4,0)
0
0
Y(2)−2
0
0
0
37Apd(3,3)
Add(0,0)−27Add(2,0)+121Add(4,0)
0
0
0
0
0
0
0
0
533√2Adf(5,3)−13√27Adf(3,3)
0
0
Y(2)−1
0
0
0
0
0
Add(0,0)+17Add(2,0)−421Add(4,0)
0
0
0
0
0
0
0
0
−13√57Adf(3,3)−433√5Adf(5,3)
0
Y(2)0
Asd(2,0)√5
0
0
0
0
0
Add(0,0)+27Add(2,0)+27Add(4,0)
0
0
13√57Adf(3,3)−233√5Adf(5,3)
0
0
0
0
0
233√5Adf(5,3)−13√57Adf(3,3)
Y(2)1
0
0
0
0
0
0
0
Add(0,0)+17Add(2,0)−421Add(4,0)
0
0
13√57Adf(3,3)+433√5Adf(5,3)
0
0
0
0
0
Y(2)2
0
−37Apd(3,3)
0
0
0
0
0
0
Add(0,0)−27Add(2,0)+121Add(4,0)
0
0
13√27Adf(3,3)−533√2Adf(5,3)
0
0
0
0
Y(3)−3
−Asf(3,3)√7
0
0
0
0
0
13√57Adf(3,3)−233√5Adf(5,3)
0
0
Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0)
0
0
0
0
0
−1013√733Aff(6,6)
Y(3)−2
0
0
0
0
0
0
0
13√57Adf(3,3)+433√5Adf(5,3)
0
0
Aff(0,0)−733Aff(4,0)+10143Aff(6,0)
0
0
0
0
0
Y(3)−1
0
35√27Apf(2,0)−13√27Apf(4,0)
0
0
0
0
0
0
13√27Adf(3,3)−533√2Adf(5,3)
0
0
Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0)
0
0
0
0
Y(3)0
0
0
35√37Apf(2,0)+4Apf(4,0)3√21
0
0
0
0
0
0
0
0
0
Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)
0
0
0
Y(3)1
0
0
0
35√27Apf(2,0)−13√27Apf(4,0)
533√2Adf(5,3)−13√27Adf(3,3)
0
0
0
0
0
0
0
0
Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0)
0
0
Y(3)2
0
0
0
0
0
−13√57Adf(3,3)−433√5Adf(5,3)
0
0
0
0
0
0
0
0
Aff(0,0)−733Aff(4,0)+10143Aff(6,0)
0
Y(3)3
Asf(3,3)√7
0
0
0
0
0
233√5Adf(5,3)−13√57Adf(3,3)
0
0
−1013√733Aff(6,6)
0
0
0
0
0
Aff(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
s
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
py
0
i√2
0
i√2
0
0
0
0
0
0
0
0
0
0
0
0
pz
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
px
0
1√2
0
−1√2
0
0
0
0
0
0
0
0
0
0
0
0
dxy
0
0
0
0
i√2
0
0
0
−i√2
0
0
0
0
0
0
0
dyz
0
0
0
0
0
i√2
0
i√2
0
0
0
0
0
0
0
0
d3z2−r2
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
dxz
0
0
0
0
0
1√2
0
−1√2
0
0
0
0
0
0
0
0
dx2−y2
0
0
0
0
1√2
0
0
0
1√2
0
0
0
0
0
0
0
fy(3x2−y2)
0
0
0
0
0
0
0
0
0
i√2
0
0
0
0
0
i√2
fxyz
0
0
0
0
0
0
0
0
0
0
i√2
0
0
0
−i√2
0
fy(5z2−r2)
0
0
0
0
0
0
0
0
0
0
0
i√2
0
i√2
0
0
fz(5z2−3r2)
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
fx(5z2−r2)
0
0
0
0
0
0
0
0
0
0
0
1√2
0
−1√2
0
0
fz(x2−y2)
0
0
0
0
0
0
0
0
0
0
1√2
0
0
0
1√2
0
fx(x2−3y2)
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
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