Spin-orbit Coupling
- Page ID
- 5558
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.