Chapter 16 Exercises: Multi-Electron Atoms and the Periodic Table

Section A: The Multi-Electron Hamiltonian and Central Field (Problems 1--7)

Problem 1 (Basic)

Write the full non-relativistic Hamiltonian for (a) helium ($Z = 2$, $N = 2$), (b) lithium ($Z = 3$, $N = 3$), and (c) beryllium ($Z = 4$, $N = 4$). In each case, count the number of electron--electron repulsion terms and verify that it equals $\binom{N}{2}$.

Problem 2 (Basic)

For helium, the ground state energy ignoring electron--electron repulsion is $E^{(0)} = 2 \times (-Z^2 \times 13.6\;\text{eV}) = -108.8\;\text{eV}$ (with $Z = 2$, $n = 1$).

(a) The experimental ground state energy is $-79.0\;\text{eV}$. What is the electron--electron repulsion energy?

(b) Compute the ratio of the repulsion energy to the total nuclear attraction energy. Explain why this means electron--electron repulsion cannot be treated as a small perturbation.

(c) The first ionization energy of helium (energy to remove one electron) is $24.6\;\text{eV}$. The ground state energy of He$^+$ is $-54.4\;\text{eV}$. Verify that these are consistent with the total ground state energy of $-79.0\;\text{eV}$.

Problem 3 (Basic)

In the central field approximation, each electron moves in an effective potential $V_{\text{eff}}(r)$.

(a) At very small $r$ (much closer to the nucleus than any other electron), argue that $V_{\text{eff}}(r) \approx -Ze^2/(4\pi\epsilon_0 r)$.

(b) At very large $r$ (far outside all other electrons), argue that $V_{\text{eff}}(r) \approx -e^2/(4\pi\epsilon_0 r)$ for a neutral atom.

(c) Sketch $V_{\text{eff}}(r)$ qualitatively, comparing it with the bare Coulomb potential $-Ze^2/(4\pi\epsilon_0 r)$ and the fully screened potential $-e^2/(4\pi\epsilon_0 r)$.

Problem 4 (Intermediate)

Consider the hydrogen-like wavefunctions $R_{ns}(r)$ and $R_{np}(r)$ from Chapter 5.

(a) For $n = 2$, evaluate $|R_{20}(0)|^2$ and $|R_{21}(0)|^2$. Which is nonzero?

(b) More generally, show that $R_{nl}(r) \propto r^l$ near $r = 0$. (This follows from the behavior of the associated Laguerre polynomials.) Use this to explain why orbitals with lower $l$ penetrate more deeply toward the nucleus.

(c) Compute $\langle 1/r \rangle$ for the hydrogen $2s$ and $2p$ states. (Use the formula from Chapter 5: $\langle 1/r \rangle_{nl} = 1/(n^2 a_0)$.) Explain why this formula, which shows no $l$-dependence, is specific to the $1/r$ potential and does not hold for multi-electron atoms.

Problem 5 (Intermediate)

The screening function $\sigma(r)$ counts the average number of electrons inside radius $r$. For a filled $1s^2$ shell with hydrogen-like wavefunctions $\psi_{1s}(r) = (1/\sqrt{\pi})(Z/a_0)^{3/2}e^{-Zr/a_0}$:

(a) Compute the electron density $\rho(r) = 2|\psi_{1s}(r)|^2$ (factor of 2 for two electrons).

(b) Compute $\sigma(r) = \int_0^r 4\pi r'^2 \rho(r')\,dr'$.

(c) For $Z = 3$ (lithium), evaluate $\sigma(r)$ at $r = a_0, 2a_0, 5a_0$. At what radius does the valence $2s$ electron experience approximately $Z_{\text{eff}} = 1$?

Problem 6 (Intermediate)

The electron--electron repulsion energy for helium can be estimated using first-order perturbation theory (to be developed fully in Chapter 17). Using the unperturbed ground state $\psi_0(\mathbf{r}_1, \mathbf{r}_2) = \psi_{1s}(\mathbf{r}_1)\psi_{1s}(\mathbf{r}_2)$ with $Z = 2$:

$$\langle V_{ee} \rangle = \frac{e^2}{4\pi\epsilon_0}\int\int\frac{|\psi_{1s}(\mathbf{r}_1)|^2|\psi_{1s}(\mathbf{r}_2)|^2}{|\mathbf{r}_1 - \mathbf{r}_2|}\,d^3\mathbf{r}_1\,d^3\mathbf{r}_2$$

The result of this integral is $\langle V_{ee} \rangle = (5/4)Z \times 13.6\;\text{eV} = 34.0\;\text{eV}$.

(a) Using this result, compute the first-order estimate of the helium ground state energy: $E^{(1)} = E^{(0)} + \langle V_{ee} \rangle$.

(b) Compare with the experimental value of $-79.0\;\text{eV}$. What is the percentage error?

(c) Why does this perturbative estimate give a result that is too high (not negative enough)?

Problem 7 (Advanced)

The variational approach to helium. Instead of treating electron--electron repulsion as a perturbation, we can use a variational trial function $\psi(\mathbf{r}_1, \mathbf{r}_2) = \psi_{1s}^{(Z')}(\mathbf{r}_1)\psi_{1s}^{(Z')}(\mathbf{r}_2)$ where $Z'$ is a variational parameter (effective nuclear charge).

(a) Using the known results for hydrogen-like atoms, show that the expectation value of the full Hamiltonian is:

$$E(Z') = \left[(Z')^2 - 2Z \cdot Z' + \frac{5}{8}Z'\right] \times 27.2\;\text{eV}$$

(Hint: $\langle T \rangle = (Z')^2 \times 13.6\;\text{eV}$ per electron; $\langle V_{\text{nuc}} \rangle = -2Z \cdot Z' \times 13.6\;\text{eV}$ per electron; $\langle V_{ee} \rangle = (5/8)Z' \times 27.2\;\text{eV}$.)

(b) Minimize $E(Z')$ with respect to $Z'$ and find the optimal $Z_{\text{opt}}$.

(c) Compute $E(Z_{\text{opt}})$ and compare with the experimental value. What is the percentage error?

(d) Interpret $Z_{\text{opt}}$ physically: why is it less than $Z = 2$?

Section B: Electron Configurations (Problems 8--13)

Problem 8 (Basic)

Write the ground-state electron configuration for each of the following atoms: (a) B ($Z = 5$), (b) Si ($Z = 14$), (c) Ar ($Z = 18$), (d) Fe ($Z = 26$), (e) Br ($Z = 35$), (f) Xe ($Z = 54$).

Use noble gas core notation where appropriate.

Problem 9 (Basic)

Write the electron configuration for each of the following ions: (a) Na$^+$ ($Z = 11$), (b) O$^{2-}$ ($Z = 8$), (c) Fe$^{2+}$ ($Z = 26$), (d) Fe$^{3+}$ ($Z = 26$), (e) Cu$^+$ ($Z = 29$).

For (c) and (d): which electrons are removed first, $4s$ or $3d$? Explain why.

Problem 10 (Intermediate)

The Madelung rule predicts that orbitals fill in order of increasing $n + l$ (with lower $n$ first for equal $n + l$). The rule gives the standard filling order $1s, 2s, 2p, 3s, 3p, 4s, 3d, \ldots$.

(a) List all orbitals with $n + l = 5$, in the order they fill according to the Madelung rule.

(b) The Madelung rule has exceptions. Chromium ($Z = 24$) has configuration $[\text{Ar}]\,3d^5\,4s^1$ rather than $[\text{Ar}]\,3d^4\,4s^2$. Using the concept of exchange energy (which stabilizes half-filled subshells), explain this exception qualitatively.

(c) Copper ($Z = 29$) has configuration $[\text{Ar}]\,3d^{10}\,4s^1$ rather than $[\text{Ar}]\,3d^9\,4s^2$. Explain this exception.

(d) Are there analogous exceptions in the $4d$ series? Look up the configurations of Mo ($Z = 42$) and Ag ($Z = 47$) and compare with Cr and Cu.

Problem 11 (Intermediate)

Explain the following trends using the quantum mechanical model:

(a) The first ionization energy of Be ($9.32\;\text{eV}$) is higher than that of B ($8.30\;\text{eV}$), even though B has a larger nuclear charge.

(b) The first ionization energy of N ($14.53\;\text{eV}$) is higher than that of O ($13.62\;\text{eV}$).

(c) The first ionization energy of Na ($5.14\;\text{eV}$) is much lower than that of Ne ($21.56\;\text{eV}$), even though Na has one more proton.

Problem 12 (Intermediate)

For each of the following pairs, predict which element has the larger atomic radius and explain your reasoning: (a) Li vs. F (b) Na vs. Mg (c) Fe vs. Ru (d) Zr vs. Hf

For part (d), explain why the "lanthanide contraction" makes Zr and Hf nearly the same size.

Problem 13 (Advanced)

The aufbau principle breaks down for excited states. Consider the carbon atom in the configuration $1s^2\,2s^1\,2p^3$.

(a) Is this configuration an excited state of carbon? Explain.

(b) Find all possible term symbols for this configuration. (Hint: $2s^1$ contributes $l = 0$, $s = 1/2$. The $2p^3$ part gives terms ${}^4S$, ${}^2D$, ${}^2P$ as worked out in Section 16.4. Now couple the $2s$ electron with each of these terms.)

(c) The Aufbau principle says nothing about which excited configurations are accessible via photon absorption from the ground state. What selection rule(s) constrain these transitions?

Section C: Term Symbols and Hund's Rules (Problems 14--22)

Problem 14 (Basic)

Determine the ground state term symbol using Hund's rules for: (a) Al ($Z = 13$): $[\text{Ne}]\,3s^2\,3p^1$ (b) P ($Z = 15$): $[\text{Ne}]\,3s^2\,3p^3$ (c) Cl ($Z = 17$): $[\text{Ne}]\,3s^2\,3p^5$

Problem 15 (Intermediate)

Show explicitly that two equivalent $p$-electrons ($p^2$) give rise to the terms ${}^1S$, ${}^3P$, ${}^1D$.

(a) List all 15 microstates as pairs $(m_{l_1}, m_{s_1}; m_{l_2}, m_{s_2})$ with $m_{l_1} > m_{l_2}$ (or $m_{l_1} = m_{l_2}$ with $m_{s_1} > m_{s_2}$) to avoid double counting.

(b) Organize these microstates in a table of $M_L$ (columns) vs. $M_S$ (rows).

(c) Extract the terms by the standard "peeling" method: identify the largest $|M_L|$ and $|M_S|$, assign a term, remove its microstates, and repeat.

Problem 16 (Intermediate)

Find all allowed term symbols for the $d^2$ configuration (e.g., Ti).

(a) How many microstates are there? (Compute $\binom{10}{2}$.)

(b) The allowed terms are ${}^1S$, ${}^3P$, ${}^1D$, ${}^3F$, ${}^1G$. Verify that the total number of microstates (summing the degeneracies $(2L+1)(2S+1)$ for each term) equals your answer from part (a).

(c) Apply Hund's rules to determine the ground state term symbol ${}^{2S+1}L_J$.

Problem 17 (Intermediate)

For the $d^3$ configuration (e.g., V):

(a) The allowed terms are ${}^4F$, ${}^4P$, ${}^2H$, ${}^2G$, ${}^2F$, ${}^2D$ (two of these), ${}^2P$. Verify the microstate count: $\binom{10}{3} = 120$.

(b) Apply Hund's rules to determine the ground state. (Note: there are two terms with the same $S$; Rule 2 then selects the one with larger $L$.)

(c) What are the allowed $J$ values for the ground term? Which does Rule 3 select?

Problem 18 (Intermediate)

The concept of "holes" simplifies term symbol determination for subshells that are more than half-filled.

(a) Explain why $p^4$ (4 electrons) has the same set of terms as $p^2$ (2 electrons).

(b) Show that the ground state of $p^4$ is ${}^3P_2$ while the ground state of $p^2$ is ${}^3P_0$. What causes the difference in $J$?

(c) Use the hole formalism to determine the ground state term of $d^8$ (e.g., Ni) from the known terms of $d^2$.

Problem 19 (Advanced)

Consider the excited configuration of carbon $1s^2\,2s^2\,2p^1\,3d^1$. Since the two electrons are in different subshells (non-equivalent), all combinations of $L$ and $S$ are allowed (no Pauli restriction on $L + S$ being even).

(a) Find all allowed values of $L$ (couple $l_1 = 1$ with $l_2 = 2$).

(b) Find all allowed values of $S$ (couple two spin-$\frac{1}{2}$ particles).

(c) List all resulting term symbols ${}^{2S+1}L_J$ with all allowed $J$ values.

(d) How many total microstates are there? Verify that the sum of $(2J + 1)$ over all terms matches.

Problem 20 (Advanced)

The ground state configuration of oxygen is $[\text{He}]\,2s^2\,2p^4$.

(a) Derive all allowed terms for $p^4$, either by direct enumeration (35 microstates) or by the hole equivalence $p^4 \leftrightarrow p^2$.

(b) Determine the ground state term using Hund's rules.

(c) The three $J$ levels of the ${}^3P$ term have energies given by the Lande interval rule: $E(J) - E(J-1) \propto J$. If the splitting between ${}^3P_2$ and ${}^3P_1$ is $158.5\;\text{cm}^{-1}$, predict the splitting between ${}^3P_1$ and ${}^3P_0$. (The experimental value is $68.7\;\text{cm}^{-1}$ --- compare and comment.)

Problem 21 (Advanced)

Determine the ground state term symbol for the following transition metals: (a) Mn ($Z = 25$): $[\text{Ar}]\,3d^5\,4s^2$ (b) Co ($Z = 27$): $[\text{Ar}]\,3d^7\,4s^2$ (c) Ni ($Z = 28$): $[\text{Ar}]\,3d^8\,4s^2$

For each, state which Hund's rule determines $S$, $L$, and $J$.

Problem 22 (Advanced)

The term symbol notation extends to molecules. Consider a hypothetical atom with configuration $f^2$ (two equivalent $f$-electrons, $l = 3$).

(a) How many microstates are there? ($\binom{14}{2}$)

(b) The allowed terms are ${}^1S$, ${}^3P$, ${}^1D$, ${}^3F$, ${}^1G$, ${}^3H$, ${}^1I$. Verify the microstate count.

(c) Determine the ground state using Hund's rules.

(d) This is relevant to which series of elements in the periodic table?

Section D: Hartree--Fock and Screening (Problems 23--28)

Problem 23 (Basic)

Use Slater's rules to compute $Z_{\text{eff}}$ for: (a) The $1s$ electron in helium ($Z = 2$). (b) The $2p$ electron in nitrogen ($Z = 7$). (c) The $3d$ electron in iron ($Z = 26$, configuration $[\text{Ar}]\,3d^6\,4s^2$). (d) The $4s$ electron in iron.

Compare $Z_{\text{eff}}$ for (c) and (d). Which electron is more tightly bound? Is this consistent with the fact that Fe$^{2+}$ loses its $4s$ electrons first?

Problem 24 (Intermediate)

Using Slater's rules and the effective quantum number $n^*$, estimate the first ionization energy of (a) sodium ($Z = 11$) and (b) potassium ($Z = 19$). Compare with experimental values (5.14 eV and 4.34 eV, respectively).

Problem 25 (Intermediate)

The Hartree--Fock method is iterative. Consider a highly simplified model: suppose the helium atom has two electrons in a $1s$-like orbital $\psi(r) = (\alpha^3/\pi)^{1/2}e^{-\alpha r}$, where $\alpha$ is an effective decay constant.

(a) If we start with $\alpha = Z = 2$ (no screening), compute the electron density $\rho(r) = 2|\psi(r)|^2$.

(b) The average potential seen by one electron due to the other is $V_{\text{screen}}(r) = \frac{e^2}{4\pi\epsilon_0}\int\frac{\rho(r')/2}{|\mathbf{r} - \mathbf{r}'|}\,d^3\mathbf{r}'$. For the $1s$ density, this evaluates to:

$$V_{\text{screen}}(r) = \frac{e^2}{4\pi\epsilon_0}\left[\frac{1}{r}(1 - e^{-2\alpha r}) - \alpha e^{-2\alpha r}\right]$$

Sketch this potential and show that $V_{\text{screen}}(r) \to e^2/(4\pi\epsilon_0 r)$ for $r \gg 1/\alpha$ and $V_{\text{screen}}(r) \to e^2\alpha/(4\pi\epsilon_0)$ for $r \to 0$.

(c) Explain qualitatively why the next iteration would produce an orbital with a slightly smaller $\alpha$ (the electron expands slightly due to screening).

Problem 26 (Intermediate)

The correlation energy $E_{\text{corr}} = E_{\text{exact}} - E_{\text{HF}}$ is always negative (Hartree--Fock always overestimates the energy).

(a) Explain physically why $E_{\text{HF}} > E_{\text{exact}}$. (Hint: what type of electron--electron avoidance does HF miss?)

(b) For helium, $E_{\text{HF}} = -77.9\;\text{eV}$ and $E_{\text{exact}} = -79.0\;\text{eV}$, so $|E_{\text{corr}}| = 1.1\;\text{eV}$. What fraction of the total energy is this?

(c) A typical chemical bond has a dissociation energy of $2$--$5\;\text{eV}$. Explain why correlation energy, despite being only $\sim 1\%$ of the total energy, can be critical for predicting chemical bonding.

Problem 27 (Advanced)

The exchange interaction between electrons in the same subshell provides the quantum-mechanical basis for Hund's first rule.

(a) Consider two electrons in a triplet state (symmetric spin, antisymmetric spatial wavefunction). Show that the probability of finding both electrons at the same position vanishes: $|\Psi_{\text{spatial}}(\mathbf{r}, \mathbf{r})|^2 = 0$. This is the Fermi hole.

(b) For two electrons in a singlet state (antisymmetric spin, symmetric spatial wavefunction), show that there is no such constraint.

(c) Using the results from (a) and (b), explain why the triplet state has lower electron--electron repulsion energy and hence lower total energy (for the same orbital configuration).

Problem 28 (Advanced)

Relativistic effects on orbital energies. In heavy atoms, electrons in $s$ and $p$ orbitals (which penetrate close to the nucleus) experience relativistic speeds, causing their orbitals to contract. The relativistic mass increase $m \to \gamma m$ contracts the orbital by approximately:

$$\frac{a_{\text{rel}}}{a_{\text{nonrel}}} \approx \sqrt{1 - (Z\alpha)^2/n^2}$$

where $\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137$ is the fine structure constant.

(a) Compute this ratio for the $1s$ electron of gold ($Z = 79$). By what percentage has the orbital contracted?

(b) The contraction of inner $s$ and $p$ orbitals increases the screening of outer $d$ and $f$ orbitals, causing them to expand. Explain qualitatively why this makes gold yellow (the $5d \to 6s$ transition shifts into the visible range) while silver is white.

(c) The "inert pair effect" (reluctance of $6s^2$ electrons to participate in bonding) in elements like Pb and Bi is attributed to relativistic stabilization of the $6s$ orbital. Explain this connection.