Orientation xyz

In the Ih Point Group, with orientation xyz there are the following symmetry operations

• Point Group Ih with orientation xyz

• Point Group Oh with orientation XYZ

Any potential (function) can be written as a sum over spherical harmonics. $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the Ih Point group with orientation xyz the form of the expansion coefficients is:

$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ \frac{1}{2} \sqrt{\frac{105}{11}} A(6,0) & k=6\land (m=-6\lor m=6) \\ -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\ -\frac{1}{2} \sqrt{21} A(6,0) & k=6\land (m=-2\lor m=2) \\ A(6,0) & k=6\land m=0 \end{cases}$$

Akm_Ih_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(Sqrt[105/11]*A[6, 0])/2, k == 6 && (m == -6 || m == 6)}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {-(Sqrt[21]*A[6, 0])/2, k == 6 && (m == -2 || m == 2)}, {A[6, 0], k == 6 && m == 0}}, 0]

Akm_Ih_xyz.Quanty

Akm = {{0, 0, A(0,0)} , 
       {6, 0, A(6,0)} , 
       {6,-2, (-1/2)*((sqrt(21))*(A(6,0)))} , 
       {6, 2, (-1/2)*((sqrt(21))*(A(6,0)))} , 
       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6,-6, (1/2)*((sqrt(105/11))*(A(6,0)))} , 
       {6, 6, (1/2)*((sqrt(105/11))*(A(6,0)))} }

The operator representing the potential in second quantisation is given as: $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ 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. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. 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. $$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ 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 $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$

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 $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter.

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

After rotation we find

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

Click on one of the subsections to expand it or expand all

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