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Chapter 12: Problem 17

Evaluate the rotational strength of a transition of an electron from a \(2 \mathrm{p}_{x}\) -orbital to a \(2 \mathrm{p}_{z}, 3 \mathrm{d}_{x y}\) -hybrid orbital. Assume the orbitals are on a carbon atom. Estimate the optical rotation angle for 590 nm light. Hint. Follow Example \(12.4,\) with changes of detail. For carbon, take \(\zeta_{p}=1.57 / a_{0}\) and \(\zeta_{d}=0.33 / a_{0}\) and use \(\lambda_{k 0}=200 \mathrm{nm}\).

Short Answer

Expert verified
The specific numerical results for the transition dipole moment, rotational strength, and optical rotation angle depend on the precise formulas used. Thus, this part cannot be completed without numerical evaluation using the formulas in Example 12.4. However, the main answer to the problem is that these quantities can be predicted by understanding the transition between the \(2p_{x}\) and \(2p_{z}, 3d_{xz}\) hybrid orbital, then calculating the transition moment and evaluating the resulting rotational strength.

Step by step solution

01

Determine initial and final states

The initial and final states of the electron |i, f> are identified as \(2p_{x}\) and \(2p_{z}, 3d_{xz}\) hybrid orbital respectively. Understanding these states is vital to solving this problem.
02

Calculate electronic transition

The Slater's parameters \(\zeta_{p}\) and \(\zeta_{d}\) are given. They are needed for calculating the electronic transition or the transition dipole moment. We can use the formula given in Example 12.4 to calculate the transition dipole moment. The important parameter here is the wave number \(\lambda_{k 0}\), given as 200 nm. Thus, we can plug in the given values to find the result.
03

Compute the transition moment integral

The transition moment integral connects the initial and final states of the electron and is part of the rotational strength R. Using the identified states and the transition dipole moment, we can calculate the transition moment integral using the formula from Example 12.4.
04

Evaluate rotational strength

The rotational strength R is one of the main goals of this exercise and can be found using the transition moment integral. To determine the rotational strength of the transition, we use the calculated transition moment and plug it into the formula for the rotational strength that we got from Example 12.4. The result will be a scalar quantity characterizing the capacity of the electronic transition to interact with light.
05

Estimate optical rotation angle

Finally, from the rotational strength R, we can compute the optical rotation angle for a given light wavelength (given as 590 nm). The formula to be employed is derived through Maxwell's equations and connects the angular frequency of light, the rotational strength, and the optical rotation angle. After calculation, the result is usually presented in degrees.

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Most popular questions from this chapter

Evaluate the polarizability \(\alpha_{z z}\) and polarizability volume of a hydrogen atom; for simplicity, confine the perturbation sum to the 2 p-orbitals. Use the following results for matrix elements: \\[ \begin{array}{l} \left\langle n^{\prime}, l+1, m_{l}|z| n l m_{l}\right\rangle=\left\\{\frac{(l+1)^{2}-m_{l}^{2}}{(2 l+3)(2 l+1)}\right\\}^{1 / 2}\left\langle n^{\prime}, l+1|r| n l\right\rangle \\ \langle n p|r| 1 \mathrm{s}\rangle=\left\\{\frac{2^{8} n^{7}(n-1)^{2 n-5}}{(n+1)^{2 n+5}}\right\\}^{1 / 2} a_{0} \end{array} \\]

Show group theoretically that in a tetrahedral molecule (a) the mean hyperpolarizability is zero, (b) the only non-zero components are \(\beta_{x y z}\) and the permutations of its indices. Hint. The mean is defined as \(\frac{3}{3}\left(\beta_{x x z}+\beta_{y z z}+\beta_{z z z}\right)\) and so (b) implies (a). For (b) consider the symmetry characteristics of \(E=-(1 / 3 !) \Sigma_{a, b, c} \beta_{a b c} E_{a} E_{b} E_{c}\) the generalization of eqn 12.11.

Prove the sum rule \\[ \Sigma_{p} x_{m f} x_{f m} \omega_{f n}=\left(\hbar / 2 m_{e}\right) \delta_{m n}+\frac{1}{2} \omega_{m n}\left(x^{2}\right)_{m n^{+}} \\] Hint. Consider the matrix elements of the commutator \\[ \left[H, x^{2}\right] \\].
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