10.6 Atomic Spectra, the Hydrogen Atom, Many-Electron Atoms, and Term Symbols

10.6 Atomic Spectra, the Hydrogen Atom, Many-Electron Atoms, and Term Symbols

Quick Answer: The Physical Chemistry TOS quantum area lists three phenomena (blackbody radiation, the photoelectric effect, and line spectra), simple systems, term symbols, the polyelectronic wave equation, and diatomic energy levels. The Inorganic TOS adds electromagnetic radiation to its quantum-theory area. Section 10.5 covers blackbody radiation, the photoelectric effect, and particle-in-a-box and oscillator models. This section covers photon energy, hydrogen line spectra, the hydrogen-atom solution, many-electron atoms, and atomic term symbols.

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Electromagnetic Radiation

$c = \lambda\nu, \qquad E = h\nu = \frac{hc}{\lambda} = hc\tilde{\nu}$

with $h = 6.626 \times 10^{-34}$ J s and $c = 2.998 \times 10^8$ m/s. For a 500 nm photon, $E = \frac{(6.626 \times 10^{-34})(2.998 \times 10^8)}{500 \times 10^{-9}} = 3.97 \times 10^{-19}$ J (2.48 eV). Per mole of photons that is 239 kJ, comparable to bond energies, which is why visible and ultraviolet light can drive photochemistry.

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Hydrogen Line Spectra and the Bohr Model

Excited hydrogen emits only certain wavelengths. Rydberg fitted them with

$\tilde{\nu} = \frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \qquad R_H = 109{,}678\text{ cm}^{-1}$

Worked example (Balmer, n = 4 to 2): $\tilde{\nu} = 109{,}678\left(\frac{1}{4} - \frac{1}{16}\right) = 20{,}565\text{ cm}^{-1}$, so $\lambda = 486$ nm (blue-green). The n = 3 to 2 line is at 656 nm (red).

Bohr's model (1913) quantized angular momentum ($mvr = n\hbar$). It gave $E_n = -\frac{13.6\text{ eV}}{n^2}$ and orbit radii $r_n = n^2a_0$ with $a_0 = 52.9$ pm. It reproduces hydrogen-like spectra but fails for many-electron atoms and wrongly assigns the ground state definite orbits and nonzero angular momentum.

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The Schrodinger Solution for Hydrogen-like Atoms

The wavefunction separates into radial and angular parts, $\psi_{nlm_l} = R_{nl}(r)\,Y_{lm_l}(\theta,\phi)$, with energies

$E_n = -\frac{Z^2}{n^2}\,(13.6\text{ eV})$

• Energy depends only on $n$ in a one-electron atom, so each level is $n^2$-fold degenerate ($2n^2$ with spin). That degeneracy is lifted in many-electron atoms.
• Ionization energies scale with $Z^2$: H 13.6 eV, $He^+$ 54.4 eV, and $Li^{2+}$ 122.4 eV.
• The orbital has $n - l - 1$ radial nodes and $l$ angular nodes.
• The radial distribution function $P(r) = r^2R^2$ gives the probability of finding the electron at distance $r$. For the 1s orbital it peaks at $r = a_0/Z$, although $\psi^2$ itself is greatest at the nucleus. The mean radius of the 1s orbital is $1.5\,a_0$.

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Many-Electron Atoms

The helium Hamiltonian contains an electron-electron repulsion term, $e^2/4\pi\varepsilon_0 r_{12}$. Because of it, the equation cannot be solved exactly.

• Ignoring repulsion gives $E = 2(-54.4) = -108.8$ eV, far below the experimental $-79.0$ eV.
• Variation method: any trial function gives an energy at or above the true value. Letting the effective nuclear charge vary gives $Z_{eff} = 27/16 = 1.69$ and $E = -77.5$ eV; the electrons partly screen each other.
• Orbital approximation and the self-consistent field (Hartree-Fock): each electron moves in the average field of the others, and the orbitals are iterated until they stop changing.
• Pauli principle: the total wavefunction must be antisymmetric when any two electrons are exchanged. A Slater determinant builds this in automatically, and it vanishes if two electrons share all four quantum numbers.
• Shielding and penetration split $s$, $p$, $d$, and $f$ energies within a shell. That gives the filling order covered in Section 2.2.

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Russell-Saunders Term Symbols

For light atoms, combine orbital angular momenta into $L$ and spins into $S$, then couple them to $J$:

$^{2S+1}L_J, \qquad L = 0, 1, 2, 3 \rightarrow S, P, D, F; \qquad J = L+S, \ldots, \lvert L-S\rvert$

Hund's rules for the ground term: maximize $S$, then maximize $L$; then $J = \lvert L - S\rvert$ if the subshell is less than half full or $J = L + S$ if it is more than half full.

Selection rules for one-electron atomic transitions: $\Delta l = \pm 1$, $\Delta S = 0$, $\Delta L = 0, \pm 1$, and $\Delta J = 0, \pm 1$ (but not $J = 0$ to $J = 0$). The sodium D doublet shows spin-orbit coupling: $^2P_{3/2} \rightarrow {}^2S_{1/2}$ at 589.0 nm and $^2P_{1/2} \rightarrow {}^2S_{1/2}$ at 589.6 nm. Term symbols for $d^n$ ions in ligand fields are covered in Section 3.3.