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Chapter 6: Problem 12

Show group-theoretically that when a perturbation of the form \(H^{(1)}=a z\) is applied to a hydrogen atom, the 1 s-orbital is contaminated by the admixture of \(n \mathrm{p}_{z^{-}}\) orbitals. Deduce which orbitals mix into (a) \(2 p_{x}\) -orbitals, (b) \(2 \mathrm{p}_{z}\) -orbitals (c) \(3 d_{x y}\) -orbitals.

Short Answer

Expert verified
After applying az perturbation to a hydrogen atom, the 1s orbital is mixed with \(n p_{z}\) orbitals due to symmetry match. For (a) \(2 p_{x}\), there is no symmetry match so no mixing occurs. For (b) \(2 p_{z}\), there is a symmetry match with the perturbation so they do mix. For (c) \(3 d_{x y}\), there is no symmetry match hence, no mixing will occur.

Step by step solution

01

Identify the Symmetry Adapted Linear Combinations (SALC)

The first step is to identify the Symmetry Adapted Linear Combinations (SALCs) for the hydrogen atom, and for 1s, npz, 2px, 2pz, and 3dxy orbitals. The SALCs of the 1s orbital is totally symmetric, so is A1. This can be found in the character table. For the npz orbitals, it will also have A1 symmetry due to the z-direction.
02

Apply Perturbation Theory

Next, you must apply the principles of group theory and perturbation theory to the case where the hydrogen atom is subject to the az perturbation. In quantum mechanics, first order perturbation theory asserts that if there is a higher energy orbital (npz) with the same symmetry (A1) as a lower energy orbital (1s), an admixture of the higher energy orbital will contaminate the lower energy one. \(n p_{z}\) orbitals have the same symmetry as \(a z\) perturbation for any n, so they will mix into the 1s orbital.
03

Determine the Mixing of the 2px, 2pz, and 3dxy Orbitals

Applying the same principle as in step 2, the \(2 p_{x}\) has a symmetry of B2, \(2 p_{z}\) -orbitals have A1 symmetry and the \(3 d_{x y}\) has a symmetry of B1 or B2. Therefore, due to the az perturbation, the \(2 p_{x}\) -orbitals would not be contaminated since there is no symmetry match. However, the \(2 p_{z}\) -orbitals will be contaminated due to the A1 symmetry match. The \(3 d_{x y}\) orbitals, will not mix into the 1s orbital for the az perturbation, because there is no symmetry match.

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Most popular questions from this chapter

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.

Consider the hypothetical linear \(\mathrm{H}_{3}\) molecule. The wavefunctions may be modelled by expressing them as \(\psi=c_{\Lambda} s_{A}+c_{B} s_{B}+c_{C} s_{C}\) the \(s_{i}\) denoting hydrogen 1 s-orbitals of the relevant atom. Use the Rayleigh-Ritz method to find the optimum values of the coefficients and the energies of the orbital. Make the approximations \(H_{\mathrm{ss}}=\alpha, H_{\mathrm{ss}^{\prime}}=\beta\) for neighbours but 0 for non-neighbours, \(S_{\mathrm{ss}}=1,\) and \(S_{\mathrm{ss}^{\prime}}=0\) Hint. Although the basis can be used as it stands, it leads to a \(3 \times 3\) determinant and hence to a cubic equation for the energies. A better procedure is to set up symmetry-adapted combinations, and then to use the vanishing of \(H_{i j}\) unless \(\Gamma^{(i)}=\Gamma^{(j)}\).

\(\mathrm{A}\) biradical is prepared with its two electrons in a singlet state. A magnetic field is present, and because the two electrons are in different environments their interaction with the field is \(\left(\mu_{\mathrm{B}} / \hbar\right) \mathscr{B}\left(g_{1} s_{1 z}+g_{2} s_{2 z}\right)\) with \(g_{1} \neq g_{2} .\) Evaluate the time-dependence of the probability that the electron spins will acquire a triplet configuration (that is, the probability that the \(S=1, M_{s}=0\) state will be populated). Examine the role of the energy separation \(h J\) of the singlet state and the \(M_{S}=0\) state of the triplet. Suppose \(g_{1}-g_{2}\) \(\approx 1 \times 10^{-3}\) and \(J \approx 0 ;\) how long does it take for the triplet state to emerge when \(\mathscr{B}=1.0 \mathrm{T}\) ? Hint. Use eqn 6.63 ; take \(|0,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta-\beta \alpha)\) and \(|1,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta+\beta \alpha) .\) See Problem 4.26 for the significance of \(\mu_{\mathrm{B}}\) and \(g\)

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.

Calculate the second-order energy correction to the ground state of a particle in a one-dimensional box for a perturbation of the form \(H^{(1)}=-\varepsilon \sin (\pi x / L)\) by using the closure approximation. Infer a value of \(\Delta E\) by comparison with the numerical calculation in Example \(6.4 .\) These two problems \((6.14 \text { and } 6.15)\) show that the parameter \(\Delta E\) depends on the perturbation and is not simply a characteristic of the system itself.
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