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Chapter 6: Problem 19

Repeat the last problem but set \(H_{\mathrm{s}, \mathrm{s}}=\gamma\) and \(S_{\mathrm{s}^{\prime}} \neq 0\) Evaluate the overlap integrals between 1 s-orbitals on centres separated by \(R ;\) use $$S=\left\\{1+\frac{R}{a_{0}}+\frac{1}{3}\left(\frac{R}{a_{0}}\right)^{2}\right\\} \mathrm{e}^{-R / a_{0}}$$ Suppose that \(\beta / \gamma=S_{\mathrm{s}, \mathrm{s}_{2}} / S_{\mathrm{s}, \mathrm{s}} .\) For a numerical result, take \(R=80 \mathrm{pm}, a_{0}=53 \mathrm{pm}\)

Short Answer

Expert verified
Follow the given steps to evaluate the overlap integral between 1 s-orbitals using the values of \(R\) and \(a_0\) provided, and then calculate the ratio of beta/gamma which is equal to \(S_{\mathrm{s}, \mathrm{s}2} / S_{\mathrm{s}, \mathrm{s}}\). The exact numerical result depends on the computations at each step.

Step by step solution

01

Substitution in Overlap Integral Function

The first step is to substitute the given values of \(R\) and \(a_0\) in the function for overlap integral \(S\). It is given as \(S=\left\{1+\frac{R}{a_{0}}+\frac{1}{3}\left(\frac{R}{a_{0}}\right)^{2}\right\}\mathrm{e}^{-R / a_{0}}\). Here, \(R = 80\) pm and \(a_0 = 53\) pm.
02

Evaluate the Overlap Integral

Once the values of \(R\) and \(a_0\) are substituted in the overlap integral function, the next step is to evaluate \(S\). For this, calculate the values inside the brackets and exponent first, and then multiply the resultant values as per the function.
03

Calculate Beta/Gamma Ratio

Finally, it is mentioned that the ratio of beta/gamma is equal to \(S_{\mathrm{s}, \mathrm{s}2} / S_{\mathrm{s}, \mathrm{s}}\). Since \(S_{\mathrm{s}, \mathrm{s}}\) is given as \(\gamma\), the ratio of beta/gamma can be obtained as \(S_{\mathrm{s}, \mathrm{s}2} / \gamma\). Substitute the value of \(S = S_{\mathrm{s}, \mathrm{s}2}\) evaluated in the previous step here to get the ratio.

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Most popular questions from this chapter

Calculate the first-order correction to the energy of a ground-state harmonic oscillator subject to an anharmonic potential of the form \(a x^{3}+b x^{4}\) where \(a\) and \(b\) are small (anharmonicity) constants. Consider the three cases in which the anharmonic perturbation is present (a) during bond expansion \((x \geq 0)\) and compression \((x \leq 0)\) (b) during expansion only, (c) during compression only.

A hydrogen atom in a \(2 \mathrm{s}^{1}\) configuration passes into a region where it experiences an electric field in the \(z\) -direction for a time \(\tau .\) What is its electric dipole moment during its exposure and after it emerges? Hint. Use eqn 6.62 with \(\omega_{21}=0 ;\) the dipole moment is the expectation value of \(-e z\) \\[\text { use } \int \psi_{2 \mathrm{s}} z \psi_{2 \mathrm{p}} \mathrm{d} \tau=3 a_{0}\\]

A simple calculation of the energy of the helium atom supposes that each electron occupies the same hydrogenic 1 s-orbital (but with \(Z=2\) ). The electronelectron interaction is regarded as a perturbation, and calculation gives $$\int \psi_{1 s}^{2}\left(r_{1}\right)\left(\frac{e^{2}}{4 \pi \varepsilon_{0} r_{12}}\right) \psi_{1 s}^{2}\left(r_{2}\right) \mathrm{d} \tau=\frac{5}{4}\left(\frac{e^{2}}{4 \pi \varepsilon_{0} a_{0}}\right)$$ (see Example 7.2 ). Estimate (a) the binding energy of helium, (b) its first ionization energy. Hint. Use eqn 6.15 with \(E_{1}=E_{2}=E_{1 \mathrm{s}} .\) Be careful not to count the electronelectron interaction energy twice.

Suppose that the potential energy of a particle on a ring depends on the angle \(\varphi\) as \(H^{(1)}=\varepsilon \sin ^{2} \varphi .\) Calculate the first- order corrections to the energy of the degenerate \(m_{l}=\pm 1\) states, and find the correct linear combinations for the perturbation calculation. Find the second-order correction to the energy. Hint. This is an example of degenerate-state perturbation theory, and so find the correct linear combinations by solving eqn 6.42 after deducing the energies from the roots of the secular determinant. For the matrix elements, express \(\sin \varphi\) as \((1 / 2 \mathrm{i})\left(\mathrm{e}^{\mathrm{i} \varphi}-\mathrm{e}^{-\mathrm{i} \varphi}\right)\) When evaluating eqn \(6.42,\) do not forget the \(m_{1}=0\) state lying beneath the degenerate pair. The energies are equal to \(m_{l}^{2} \hbar^{2} / 2 m r^{2} ;\) use \(\psi_{m_{l}}=(1 / 2 \pi)^{1 / 2} \mathrm{e}^{i m_{\mu} \varphi}\) for the unperturbed states.

Find the complete dependence of the \(A\) and \(B\) coefficients on atomic number for the \(2 \mathrm{p} \rightarrow 1\) s transitions of hydrogenic atoms. Calculate how the stimulated emission rate depends on \(Z\) when the atom is exposed to black-body radiation at \(1000 \mathrm{K} .\) Hint. The relevant density of states also depends on \(Z\).
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