Orientation Zx_A

The point group D3d is a subgroup of Oh. Many materials of relevance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the states in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irreducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irreducible representation and two that belong to the eg irreducible representation. We label the eg orbitals that descend from the eg irreducible representation in Oh symmetry eg$\sigma$ and the eg orbitals that descend from the t2g irreducible representation eg$\pi$ orbitals. (The mixing is given by the parameter Meg.) ###

As one can see in the list of supergroups of D3d, there are two different orientations of Oh that are a supergroup of this orientation of D3d. The different orientations of Oh with respect to D3d do however change the definitions of the eg$\pi$ and eg$\sigma$ orbitals. We include three different representations of the orbitals and potentials for each setting of D3d symmetry. The orientation without additional letter takes the tesseral harmonics as a basis. This basis does not relate to the states in Oh symmetry. The orientation with an additional A or B relate to the two different supergroup representations of the Oh point group.

The parameterisation A of the orientation Zx is related to the orientation Sqrt[2]1z of the Oh pointgroup.

===== Symmetry Operations =====

In the D3d Point Group, with orientation Zx_A there are the following symmetry operations

===== Different Settings =====

• Point Group D3d with orientation 111
• Point Group D3d with orientation Zx
• Point Group D3d with orientation Zx_A
• Point Group D3d with orientation Zx_B
• Point Group D3d with orientation Z(x-y)
• Point Group D3d with orientation Z(x-y)_A
• Point Group D3d with orientation Z(x-y)_B
• Point Group D3d with orientation Zy
• Point Group D3d with orientation Zy_A
• Point Group D3d with orientation Zy_B

===== Character Table =====

===== Product Table =====

===== Sub Groups with compatible settings =====

• Point Group C1 with orientation 1
• Point Group C2 with orientation X
• Point Group C3v with orientation Zx
• Point Group C3 with orientation Z
• Point Group Ci with orientation
• Point Group Cs with orientation X
• Point Group D3d with orientation Zx
• Point Group D3d with orientation Zx_B
• Point Group D3 with orientation Zx
• Point Group S6 with orientation Z

===== Super Groups with compatible settings =====

• Point Group D3d with orientation Zx
• Point Group D3d with orientation Zx_B
• Point Group Oh with orientation 0sqrt2-1z
• Point Group Oh with orientation 0sqrt21z

===== Invariant Potential expanded on renormalized spherical Harmonics =====

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 D3d Point group with orientation Zx_A the form of the expansion coefficients is:

==== Expansion ====

$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ A(2,0) & k=2\land m=0 \\ i B(4,3) & k=4\land (m=-3\lor m=3) \\ A(4,0) & k=4\land m=0 \\ A(6,6) & k=6\land (m=-6\lor m=6) \\ i B(6,3) & k=6\land (m=-3\lor m=3) \\ A(6,0) & k=6\land m=0 \end{cases}$$

==== Input format suitable for Mathematica (Quanty.nb) ====

Akm_D3d_Zx_A.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {I*B[4, 3], k == 4 && (m == -3 || m == 3)}, {A[4, 0], k == 4 && m == 0}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {I*B[6, 3], k == 6 && (m == -3 || m == 3)}, {A[6, 0], k == 6 && m == 0}}, 0]

==== Input format suitable for Quanty ====

Akm_D3d_Zx_A.Quanty

Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {4, 0, A(4,0)} , 
       {4,-3, (I)*(B(4,3))} , 
       {4, 3, (I)*(B(4,3))} , 
       {6, 0, A(6,0)} , 
       {6,-3, (I)*(B(6,3))} , 
       {6, 3, (I)*(B(6,3))} , 
       {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: $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ 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.

==== 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

==== One particle coupling on a basis of symmetry adapted functions ====

After rotation we find

===== 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 $l'$.

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==== Potential for s orbitals ====

==== Potential for p orbitals ====

==== Potential for d orbitals ====

==== Potential for f orbitals ====

===== Coupling between two shells =====

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==== Potential for s-d orbital mixing ====

==== Potential for p-f orbital mixing ====

===== Table of several point groups =====

Return to Main page on Point Groups

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