Spin orbit coupling for atom at centre of complex

[riclambo]

Hello Forum,
This is my first post. There is some preamble, but the problem itself is straightforward enough, even though I cannot solve it.
I'm trying to calculate the spin orbit matrix for an alkali metal atom at the centre of a complex of 12 noble gas atoms which form a tetradecahedron. The basis set consists of the metal atom excited state p orbitals plus the excited state rare gas (n + 1) p orbitals. The metal atom p orbitals (of $$t_{1u}$$ symmetry in $$O_h$$) mix with the $$t_{1u}$$ rare gas cage atom group orbitals. The cage group orbitals are then:

$$t^{1}_{1u}(x): x_{2}+x_{3}+x_{6}+x_{8}$$
$$t^{1}_{1u}(y): y_{1}+y_{3}+y_{5}+y_{7}$$
$$t^{1}_{1u}(z): z_{9}+z_{10}+z_{11}+z_{12}$$
$$t^{2}_{1u}(x): -(x_{1}+x_{3}+x_{5}+x_{7}+x_{9}+x_{10} +x_{11}+x_{12})$$
$$t^{2}_{1u}(y): -(y_{2}+y_{4}+y_{6}+y_{8}+y_{9}+y_{10}+y_{11}+y_{12})$$
$$t^{2}_{1u}(z): -(z_{1}+z_{2}+z_{3}+z_{4}+z_{5}+z_{6}+z_{7}+z_{8})$$
$$t^{3}_{1u}(x): z_{1}-z_{3}-z_{5}+z_{7}+y_{9}-y_{10}+y_{11}-y_{12}$$
$$t^{3}_{1u}(y): -z_{2}+z_{4}+z_{6}-z_{8}+x_{9}-x_{10}+x_{11}-x_{12}$$
$$t^{3}_{1u}(z): x_{1}-x_{3}-x_{5}+x_{7}-y_{2}+y_{4}+y_{6}-y_{8}$$

The LCAO-MOs are then:

$$\varphi(x) = c_{1}p^{M}_{x}+c_{2}t^{1}_{1u}(x)+c_{3}t^{2}_{1u}(x)+c_{4}t^{3}_{1u}(x)$$

$$\varphi(y) = c_{1}p^{M}_{y}+c_{2}t^{1}_{1u}(y)+c_{3}t^{2}_{1u}(y)+c_{4}t^{3}_{1u}(y)$$

$$\varphi(z) = c_{1}p^{M}_{z}+c_{2}t^{1}_{1u}(z)+c_{3}t^{2}_{1u}(z)+c_{4}t^{3}_{1u}(z)$$

Here the superscripts M, I, 2, and 3 refer to the metal and different rare gas group orbitals, respectively. The elements of the spin-orbit matrix have the form:

$$\left\langle \varphi(i)\eta \left|H_{SO}\right| \varphi(j)\eta'\right\rangle$$

where

$$\eta$$ and $$\eta'$$ are the spin functions.

$$H_{SO} = \zeta_{M}l_{M}.s + \zeta_{X}\sum l_{k}.s$$

where

$$\zeta_{M}$$

and

$$\zeta_{X}$$

are the metal and rare gas SO coupling constants. The sum is over all rare gas atom centers and makes use of the standard angular momentum operator relations. The final results looks like:

A*(Matrix of 1s and 0s)

where

$$A = h^{2}(c^{2}_1\zeta_{M} - 4c_1c_2\zeta_{X}S_1 -4c_1c_2\zeta_{X}S_{\pi}+ 4c_1c_3\zeta_{X}S_1 - 4c_1c_4\zeta_{X}S_2+4c_1c_2\zeta_{M}S_{\pi}+8c_1c_3\zeta_{M}S_1+8c_1c_3\zeta_{M}S_2)/8{\pi}$$

$$S_1 = \left\langle p^{M}_{z}\left|-z_1 \right\rangle$$
$$S_2 = \left\langle p^{M}_{z}\left|y_4 \right\rangle$$
$$S_{\pi} = \left\langle p^{M}_{z}\left|z_9 \right\rangle$$

The article I have attached gives more details and the result on page 3, which looks like a fairly simple expression. I have seen this kind of problem for tetrahedral and octahedral complexes but never for a tetradecahedral complex. Although I can derive some of its terms, I cannot arrive at the final form. If there are any physical chemists or chemical physicists out there, could you please give this 30 minutes of your time, even if it is just to point me to a more helpful reference.

Regards,
Ricardo

 

[Zefram]

Dear Ricardo,

Thank you for sharing your problem with the forum. As a fellow scientist, I understand the frustration that comes with trying to solve a complex problem. After reading through your post and the attached article, I believe I may be able to offer some insight into your calculations.

Firstly, I would like to clarify that I am not a physical chemist or chemical physicist, but I have some experience in the field of quantum mechanics and molecular orbital theory. From my understanding, you are trying to calculate the spin-orbit matrix for an alkali metal atom at the center of a complex of 12 noble gas atoms. This is a complicated problem, as it involves the interaction between the metal atom and the surrounding noble gas atoms, as well as the spin functions of the orbitals involved.

In order to arrive at the final expression, it seems that you have already derived some of the terms, but are struggling to put them together. My suggestion would be to break down the problem into smaller parts and focus on each term individually. For example, you could start by calculating the expectation values of the spin functions for each orbital involved. This will give you a better understanding of the spin interactions within the complex.

Additionally, I would recommend consulting with a colleague or a mentor who has experience in this specific area of research. They may be able to provide you with some helpful tips or point you to a more useful reference.

I hope this helps in some way and I wish you all the best in solving your problem. Keep persevering and don't hesitate to reach out for assistance.