Chapter 6

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 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.

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.

The symmetry of the ground electronic state of the water molecule is \(\mathrm{A}_{1}\). (a) An electric field, (b) a magnetic field is applied perpendicular to the molecular plane. What symmetry species of excited states may be mixed into the ground state by the perturbations? Hint. The electric interaction has the form \(H^{(1)}=a x ;\) the magnetic interaction has the form \(H^{(1)}=b l_{x^{*}}\)

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.

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 \(\Gamma^{(i)}=\Gamma^{(j)}\).

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 \(R=80 \mathrm{pm}, a_{0}=53 \mathrm{pm}\)

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}\\]

\(\mathrm{A}\) 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 \(g\)