[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:
                   
t¹_1u(x): x₂+x₃+x₆+x₈
t¹_1u(y): y₁+y₃+x₅+y₇
t¹_1u(z): z₉+z₁₀+z₁₁+z₁₂
t²_1u(x): -(x₁+x₃+x₅+x₇+x₉+x₁₀ +x₁₁+x₁₂)
t²_1u(y): -(y₂+y₄+y₆+y₈+y₉+y₁₀+y₁₁+y₁₂)
t²_1u(z): -(z₁+z₂+z₃+z₄+z₅+z₆+z₇+z₈)
t³_1u(x): z₁-z₃-z₅+z₇+y₉-y₁₀+y₁₁+y₁₂
t³_1u(y): -z₂+z₄+z₆-z₈+x₉-x₁₀+x₁₁-x₁₂
t³_1u(z): x₁-x₃-x₅+x₇-y₂+y₄+y₆-y₈

The LCAO-MOs are then:

phi(x) = c₁p^M_x+c₂t¹_1u(x)+c₃t²_1u(x)+c₄t³_1u(z)

phi(y) = c₁p^M_y+c₂t¹_1u(y)+c₃t²_1u(y)+c₄t³_1u(y)

phi(z) = c₁p^M_z+c₂t¹_1u(z)+c₃t²_1u(z)+c₄t³_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:

<phi(i)a|H_SO|phi(j)a'>

where

a and a' are the spin functions.

H_SO=  E_Ml_M.s + SIGMA E_Xl_k.s

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

A*[Matrix of 1s and zeros]

where A = h²(c²₁E_M-4c₁c₂E_XS₁ -4c₁c₂E_XS_pi+4c₁c₃E_XS₁-4c₁c₄E_XS₂+4c₁c₂E_MS_pi+8c₁c₃ E_MS₁+8c₁c₃ E_MS₂)/8*pi

S₁ =  <p^M_z|-z₁>
S₂ =  <p^M_z|y₄>
S_pi =  <p^M_z|z₉>

The following article (J. Chem. Phys. 90(10) 1989) 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. I basically need to know how to evaluate the matrix elements: <phi(i)a|H_SO|phi(j)a'> . 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