Chapter 8, Problem 8D.5ST

Interpretation Introduction

Interpretation:

The terms and levels that may arise from the given electronic configuration has to be identified.

Concept-Introduction:

Clebsch–Gordan series:

The plausible values of quantum number L can be calculated using orbital angular momentum quantum numbers of the electrons in outermost shell of the atom through a series called Clebsch–Gordan series. It is given as,

${\text{L = l}}_{\text{1}}{\text{ + l}}_{\text{2}}{\text{, l}}_{\text{1}}{\text{ + l}}_{\text{2}}\text{ - 1, }\mathrm{...}{\text{, |l}}_{\text{1}}{\text{ - l}}_{\text{2}}\text{|}$

Total angular momentum quantum number: The total angular momentum being the vector sum of the orbital and the spin angular momenta. The possible values of J (a positive number) are found by forming the series

$\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}$

Explanation of Solution

The Clebsch–Gordan series should be used to identify the L values. The given electronic configuration is ${\text{4p}}^{\text{1}}{\text{3d}}^{\text{1}}$ in which one p and one d electrons are coupled. For d and p electrons, only angular momenta should be considered.

For each electron $\text{l}\text{=}\text{1}\text{and}\text{2}{\text{(l}}_{\text{1}}\text{ = 1 }\text{for}{\text{p electron and l}}_{\text{2}}\text{ = 2 for d electron)}$. The L values are given as,

    $\begin{array}{l}{\text{L = l}}_{\text{1}}{\text{ + l}}_{\text{2}}{\text{, l}}_{\text{1}}{\text{ + l}}_{\text{2}}\text{ - 1, }\mathrm{...}{\text{, |l}}_{\text{1}}{\text{ - l}}_{\text{2}}\text{|}\\ \\ \text{L}\text{=}\text{1}\text{+}\text{2,}\text{1}\text{+}\text{2-1,}\text{1-2}\\ \\ \text{L}\text{=}\text{3}\text{,}\text{2,}\text{1}\end{array}$

Thus, the term symbols are P, D, and F. Now the total spin angular momentum quantum number S for two d electrons is calculated as,

     $\begin{array}{l}{\text{S = s}}_{\text{1}}{\text{ + s}}_{\text{2}}{\text{, s}}_{\text{1}}{\text{ + s}}_{\text{2}}\text{ - 1, }\mathrm{...}{\text{, |s}}_{\text{1}}{\text{ - s}}_{\text{2}}\text{|}\\ \\ \text{S}\text{=}\frac{\text{1}}{\text{2}}\text{+}\frac{\text{1}}{\text{2}}\text{,}\text{|}\frac{\text{1}}{\text{2}}\text{-}\frac{\text{1}}{\text{2}}\text{|}\\ \\ \text{S}\text{=}\text{1,}\text{0}\end{array}$

The multiplicities are calculated as,

    $\begin{array}{l}\text{S}\text{=}\text{1,}\text{0}\\ \\ \text{2S + 1 =}\text{2(1) + 1 =}\text{3}\\ \\ \text{2S + 1 =}\text{2(0) + 1 =}\text{1}\end{array}$

Therefore, the term symbols are ${}^{\text{1}}{\text{F, }}^{\text{1}}{\text{D}}_{\text{2}}{\text{, }}^{\text{1}}\text{P,}{}^3{\text{F, }}^3{\text{D, }}^3\text{P}$. Now, the total angular momentum quantum number, J, is calculated as,

    $\begin{array}{l}\text{J = L + S, L + S - 1, }\mathrm{...}\text{, |L - S|}\\ \\ \text{L}\text{=}\text{1,}\text{2,}\text{3}\text{(derived}\text{from}\text{symbol}\text{F,}\text{D,}\text{P)}\\ \\ \text{S}\text{=}\text{1,}\text{0}\\ \\ \text{For}{}^{\text{3}}\text{P term, J}\text{=}\text{2,}\text{1,}\text{0}\\ \text{For}{}^{\text{1}}\text{P term,}\text{J}\text{=}\text{1}\\ \text{For}{}^{\text{3}}\text{D term,}\text{J}\text{=}\text{3,}\text{2,}\text{1}\\ \text{For}{}^{\text{1}}\text{D term,}\text{J}\text{=}\text{2}\\ \text{For}{}^{\text{3}}\text{F term,}\text{J}\text{=}\text{4,}\text{3,}\text{2}\\ \text{For}{}^{\text{1}}\text{F term,}\text{J=}\text{3}\end{array}$

Therefore, term symbols with levels are ${}^{\text{1}}{\text{F}}_{\text{3}}{\text{, }}^{\text{1}}{\text{D}}_{\text{2}}{\text{, }}^{\text{1}}{\text{P}}_{\text{1}}\text{,}{}^3{\text{F}}_{4,3,2}{\text{, }}^3{\text{D}}_{3,2,1}{\text{, }}^3{\text{P}}_{2,1,0}$