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

\[hcA\,\widehat{L}\cdot\widehat{S}=\dfrac{Z\alpha^2}{2}\dfrac{1}{r^3}\widehat{L}\cdot\widehat{S}\]

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

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

where

\[\biggr\langle\dfrac{1}{r^3}\biggr\rangle=\int\Psi^*\biggr(\dfrac{1}{r^3}\biggr)\Psi\,d\tau\]

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

For example for Y210 we have

\[\Psi_{210}=\dfrac{1}{4\sqrt{2\pi}}\biggr(\dfrac{Z}{a_0}\biggr)^\dfrac{3}{2}\dfrac{Zr}{a_0}e^{-Zr/2a_0}\cos\theta\]

so that the integral is

\[\biggr\langle\dfrac{1}{r^3}\biggr\rangle=\dfrac{1}{32\pi}\biggr
(\dfrac{Z}{a_0}\biggr)^5\int_{0}^{2z}d\phi\int_{0}^{z}\cos^2\theta\sin\theta\,d\theta cos\theta\int_{0}^{\infty}r^2e^{Zr/a_0}\biggr(\dfrac{1}{r^3}\biggr)r^2\,dr\]

which integrates to

\[\biggr\langle\dfrac{1}{r^3}\biggr\rangle=\dfrac{1}{32\pi}\biggr(\dfrac{Z}{a_0}\biggr)^52\pi\biggr(\dfrac{2}{3}\biggr)\biggr(\dfrac{a_0\,^2}{Z^2}\biggr)=\dfrac{1}{24}\biggr(\dfrac{Z}{a_0}\biggr)^3\]

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

Therefore in atomic units we have

\[\biggr\langle\dfrac{1}{r^3}\biggr\rangle=\dfrac{Z^3}{n^3l(l+1/2)(l+1)}\]

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

\[E_{so}=\dfrac{Z^4}{2(137)^2n^3}\Biggr(\dfrac{j(j+1)-l(l+1)-s(s+1)}{2l(l+1/2)(l+1)}\Biggr)\]

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.