XAS dipole transition operator in D3d

Dear 蘇筠娟,

Indeed you need to consider the orientation of the polarisation with respect to the crystal orientation. For the full answer I would need to know your crystal orientation though.

In Quanty you have a global Carthesian coordinate system. You can define the polarization with respect to this coordinate system. For example for a 2p to 3d dipole transition you would get something like

-- next we define the dipole operator. The dipole operator is given as epsilon.r
-- with epsilon the polarization vector of the light and r the unit position vector
-- We can expand the position vector on (renormalized) spherical harmonics and use
-- the crystal-field operator to create the dipole operator. 
 
-- x polarized light is defined as x = Cos[phi]Sin[theta] = sqrt(1/2) ( C_1^{(-1)} - C_1^{(1)})
Akm = {{1,-1,sqrt(1/2)},{1, 1,-sqrt(1/2)}}
TXASx = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
-- y polarized light is defined as y = Sin[phi]Sin[theta] = sqrt(1/2) I ( C_1^{(-1)} + C_1^{(1)})
Akm = {{1,-1,sqrt(1/2)*I},{1, 1,sqrt(1/2)*I}}
TXASy = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
-- z polarized light is defined as z = Cos[theta] = C_1^{(0)}
Akm = {{1,0,1}}
TXASz = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)

For a specific experiment you can sum operate operators. If your polarisation vector is {ex,ey,ez} then you need the operator

TXASExperiment = ex * TXASx + ey * TXASy + ez * TXASz

Now to the crystal orientation. We need to express the crystal tight binding Hamiltonian. (crystal fields and hopping to ligands). On the same coordinate system. In $O_h$ point group we normally take the $C_4$ axis in the x, y and z direction. For $D_{3d}$ you normally would have the $C_{3d}$ direction in the $z$ direction. There is thus a rotation between the standard $O_h$ and $D_{3d}$ definitions. On our point group table page you find a list of point-groups and orientations. For the $D_{3d}$ point group you see several orientations. This are listed under the caption “Implemented Settings”. For example setting “D3d_111” is the setting were the $C_3$ axis is in the 111 direction. “D3d_Zx” has the $C_3$ rotation in the z direction and the $C_2$ in the x direction. You find pictures of all symmetry operations for each setting.

If you go to a specific setting you find the format of the crystal field. For $D_{3d}$ with the $C_3$ axis in the $z$ direction and the $C_2$ in the x direction you find the info here. The crystal field acting on a d-shell has in this setting the form:

Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg1 + Eeg2))} , 
       {2, 0, Ea1g + Eeg1 + (-2)*(Eeg2)} , 
       {4, 0, (3/5)*((3)*(Ea1g) + (-4)*(Eeg1) + Eeg2)} , 
       {4,-3, (3*I)*((sqrt(7/5))*(Meg))} , 
       {4, 3, (3*I)*((sqrt(7/5))*(Meg))} }

You find this under the section “Potentials for d orbitals - Input format suitable for Quanty”.

If you want to know how the axis relate to the cubic super group you find, on the top of the page listing the info for the $D_{3d}$ point group with the setting where the $C_3$ axis is in the $z$ direction and the $C_2$ in the $x$ a list of super and sub groups on the top of that page. Super groups have all symmetry operations of the group and some extra, sub-groups have less symmetry operations than the group.

I normally would work in the original crystal orientation. This is also why I needed all point-group settings and know how to expand the potential on spherical harmonics for the given point-group and setting. Important to know: A crystal can have multiple sites that are related by symmetry via a translation and rotation or mirror operation. These sites have the same point-group but a different setting. I would recommend to do the calculation for each setting and then sum the spectra or operator expectation values.

Hope this helped, Best wishes, Maurits