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

Multiple Choice (Questions 1--14)

1. The fundamental difficulty in solving multi-electron atoms exactly is:

(a) The Schrodinger equation becomes a partial differential equation in more than three dimensions (b) The electron--electron repulsion terms couple the coordinates of different electrons, preventing separation (c) Electrons are indistinguishable, so we cannot label them (d) Spin introduces complex numbers into the wavefunction

2. In the central field approximation, the energy of an orbital depends on:

(a) $n$ only (as in hydrogen) (b) $n$ and $l$ (c) $n$, $l$, and $m_l$ (d) $n$, $l$, $m_l$, and $m_s$

3. In a multi-electron atom, the $2s$ orbital has lower energy than the $2p$ orbital because:

(a) The $2s$ orbital has a smaller value of $l$, and energy increases with $l$ in hydrogen (b) The $2s$ orbital penetrates closer to the nucleus, experiencing less screening and a larger effective nuclear charge (c) The $2p$ orbital has angular nodes that increase its kinetic energy (d) The Pauli principle forces $2s$ to fill before $2p$

4. According to the Madelung rule, which orbital fills after $4p$?

(a) $4d$ (b) $5s$ (c) $3d$ (d) $5p$

5. The ground state electron configuration of chromium ($Z = 24$) is $[\text{Ar}]\,3d^5\,4s^1$ rather than the expected $[\text{Ar}]\,3d^4\,4s^2$. This is due to:

(a) The Pauli exclusion principle (b) The Madelung rule requiring lower $n$ to fill first (c) Extra stability from the exchange interaction in a half-filled $3d$ subshell (d) Relativistic effects on the $4s$ orbital

6. For a filled subshell, the term symbol is always:

(a) ${}^1S_0$ (b) ${}^3P_0$ (c) ${}^1S_1$ (d) Depends on the value of $l$

7. The term symbol ${}^3D_2$ describes a state with:

(a) $S = 3$, $L = 2$, $J = 2$ (b) $S = 1$, $L = 2$, $J = 2$ (c) $S = 1$, $L = 3$, $J = 2$ (d) $S = 3/2$, $L = 2$, $J = 2$

8. For two equivalent $p$-electrons ($p^2$), which of the following terms is NOT allowed?

(a) ${}^1S$ (b) ${}^3P$ (c) ${}^3D$ (d) ${}^1D$

9. Hund's first rule states that the ground state has the maximum value of:

(a) $L$ (orbital angular momentum) (b) $J$ (total angular momentum) (c) $S$ (total spin) (d) $n$ (principal quantum number)

10. The ground state term of nitrogen ($2p^3$) is ${}^4S_{3/2}$. The subscript $J = 3/2$ arises because:

(a) Hund's third rule selects minimum $J = |L - S| = |0 - 3/2| = 3/2$ (less than half-filled) (b) Hund's third rule selects maximum $J = L + S = 0 + 3/2 = 3/2$ (half-filled) (c) When $L = 0$, the only possible value is $J = S = 3/2$ (d) Both (a) and (c) --- the rules coincide when $L = 0$

11. In the Hartree--Fock method, the exchange potential:

(a) Is a local potential that depends only on $\mathbf{r}$ (b) Is a nonlocal operator with no classical analogue, arising from the antisymmetry of the wavefunction (c) Is identical to the direct Coulomb potential (d) Can be ignored for atoms with $Z > 10$

12. The correlation energy is defined as $E_{\text{corr}} = E_{\text{exact}} - E_{\text{HF}}$. This quantity is:

(a) Always positive (HF underestimates the energy) (b) Always negative (HF overestimates the energy) (c) Zero for atoms with $Z \leq 2$ (d) Equal to the exchange energy

13. The period lengths of the periodic table are $2, 8, 8, 18, 18, 32, 32$. The fact that each length appears twice is due to:

(a) Electron spin doubling the number of states (b) The Madelung $(n + l)$ rule interleaving orbitals from different principal quantum numbers (c) The Pauli exclusion principle (d) The Heisenberg uncertainty principle

14. The first ionization energy of oxygen ($13.62\;\text{eV}$) is lower than that of nitrogen ($14.53\;\text{eV}$) because:

(a) Oxygen has a smaller nuclear charge (b) Oxygen's fourth $2p$ electron must pair with an existing electron, increasing electron--electron repulsion (c) Nitrogen's half-filled $2p^3$ subshell has extra exchange stabilization (d) Both (b) and (c)

Short Answer (Questions 15--20)

15. Write the ground state electron configuration and term symbol for sulfur ($Z = 16$). Show your work for the term symbol determination using Hund's rules.

16. Using Slater's rules, compute the effective nuclear charge $Z_{\text{eff}}$ experienced by a $3p$ electron in silicon ($Z = 14$, configuration $1s^2\,2s^2\,2p^6\,3s^2\,3p^2$). Then estimate the energy of this electron using $\varepsilon \approx -13.6\,Z_{\text{eff}}^2/n^2$ eV.

17. Explain in 3--4 sentences why the lanthanide contraction occurs and give one chemical consequence.

18. A student claims that the ground state of carbon ($2p^2$) is ${}^1D_2$ because it has the largest $L$ value among the allowed terms. Identify the error in the student's reasoning.

19. The Hartree--Fock method produces self-consistent orbitals. Describe the iterative procedure in 4--5 steps. What does "self-consistent" mean in this context?

20. Two equivalent $d$-electrons ($d^2$) give rise to the terms ${}^1S$, ${}^3P$, ${}^1D$, ${}^3F$, ${}^1G$. Determine the ground state term symbol ${}^{2S+1}L_J$ using Hund's rules and state the physical basis for each rule.