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Spin-orbit Coupling

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  • Spin-orbit coupling refers to the interaction of a particle's "spin" motion with its "orbital" motion.

    The Spin-orbit coupling Hamiltonian

    The magnitude of spin-orbit coupling splitting is measured spectroscopically as

    \[\begin{align*} H_{so} &=\dfrac{1}{2} hcA \left( (l+s)(l+s+1)-l(l+1)-s(s+1)\right) \\[4pt] &= \dfrac{1}{2} hcA \left(l^2 +s^2 + ls + sl + l +s -l^2 -l -s^2 -s) \right) \\[4pt] &= hcA \textbf{l} \cdot \textbf{s} \end{align*}\]

    The expression can be modified by realizing that \(j = l + s\).

    \[H_{so}=\dfrac{1}{2} hcA \biggr(j(j+1)-l(l+1)-s(s+1)\biggr)\]​

    where \(A\) is the magnitude of the spin-orbit coupling in wave numbers. The magnitude of the spin orbit coupling can be calculated in terms of molecule parameters by the substitution


    where \(a\) is the fine structure constant (\(a = 1/137.037\)) and the carrots indicate that \(L\) and \(S\) are operators. The fine structure constant is a dimensionless constant, \(a = \dfrac{e^2}{ác}\). \(Z\) is an effective atomic number. The spin orbit coupling splitting can be calculated from

    \[E_{so}=\int \Psi^*H_{SO}\Psi\,d\tau=\dfrac{Z}{2(137)^2}\int\Psi^*\dfrac{\widehat{L}\cdot\widehat{S}}{r^3}\Psi\,d\tau\]

    This expression can be recast to give an spin-orbit coupling energy in terms of molecular parameters

    \[E_{so}=\dfrac{1}{2} \biggr(j(j+1)-l(l+1)-s(s+1)\biggr)=\dfrac{Z}{2(137)^2}\biggr\langle\dfrac{1}{r^3}\biggr\rangle\]



    We can evaluate this integral explicitly for a given atomic orbital.

    For example for Y210 we have


    so that the integral is

    (\dfrac{Z}{a_0}\biggr)^5\int_{0}^{2z}d\phi\int_{0}^{z}\cos^2\theta\sin\theta\,d\theta cos\theta\int_{0}^{\infty}r^2e^{Zr/a_0}\biggr(\dfrac{1}{r^3}\biggr)r^2\,dr\]

    which integrates to


    Or \(Z^3/24\) in atomic units.

    Therefore in atomic units we have


    Therefore, in general the spin-orbit splitting is given by


    Note that the spin-orbit coupling increases as the fourth power of the effective nuclear charge Z, but only as the third power of the principal quantum number n. This indicates that spin orbit-coupling interactions are significantly larger for atoms that are further down a particular column of the periodic table.

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