← Part 1: Atomic Structure & Quantum Mechanics

Part 1, Topic 2 | Lectures 5–8

Quantum Numbers & Electron Configuration

From quantum numbers to the architecture of the periodic table

2.1 The Four Quantum Numbers

The quantum mechanical description of the hydrogen atom yields three quantum numbers from the Schrodinger equation, plus a fourth (spin) discovered experimentally. Together, these four numbers completely specify the state of an electron in an atom:

Principal Quantum Number ($n$)

Determines the energy level and overall size of the orbital. Values: $n = 1, 2, 3, \ldots$

Energy: $E_n \propto -1/n^2$ | Shell capacity: $2n^2$ electrons

Angular Momentum Quantum Number ($l$)

Determines the shape of the orbital and the orbital angular momentum. Values: $l = 0, 1, 2, \ldots, n-1$

$l = 0$ (s), $l = 1$ (p), $l = 2$ (d), $l = 3$ (f) | Angular momentum: $L = \hbar\sqrt{l(l+1)}$

Magnetic Quantum Number ($m_l$)

Determines the orientation of the orbital in space. Values: $m_l = -l, -l+1, \ldots, 0, \ldots, l-1, l$

Number of orientations per subshell: $2l + 1$ | z-component: $L_z = m_l \hbar$

Spin Quantum Number ($m_s$)

Intrinsic angular momentum of the electron. Values: $m_s = +\frac{1}{2}$ (spin up) or $m_s = -\frac{1}{2}$ (spin down)

Spin angular momentum: $S = \hbar\sqrt{s(s+1)}$ where $s = 1/2$

No two electrons in an atom can have the same set of four quantum numbers — Pauli Exclusion Principle

2.2 Schrodinger Equation for Hydrogen

The time-independent Schrodinger equation for the hydrogen atom in spherical coordinates ($r, \theta, \phi$) is:

$$-\frac{\hbar^2}{2m_e}\nabla^2\psi + V(r)\psi = E\psi$$

where $V(r) = -e^2/(4\pi\epsilon_0 r)$ is the Coulomb potential. The key insight is that the wavefunction separates into radial and angular parts:

$$\psi_{nlm}(r, \theta, \phi) = R_{nl}(r) \cdot Y_l^{m_l}(\theta, \phi)$$

Radial Part $R_{nl}(r)$

The radial function determines the probability of finding the electron at distance $r$from the nucleus. The radial probability density is $P(r) = r^2|R_{nl}(r)|^2$. Key features:

●Number of radial nodes = $n - l - 1$
●The most probable radius for the 1s orbital equals the Bohr radius $a_0$
●Higher $n$ orbitals extend farther from the nucleus
●The average radius is $\langle r \rangle = \frac{n^2 a_0}{Z}\left[1 + \frac{1}{2}\left(1 - \frac{l(l+1)}{n^2}\right)\right]$

Angular Part $Y_l^{m_l}(\theta, \phi)$

The spherical harmonics determine the angular distribution (shape) of the orbital:

●s orbitals ($l = 0$): Spherically symmetric — no angular dependence
●p orbitals ($l = 1$): Dumbbell-shaped along x, y, or z axes (3 orientations)
●d orbitals ($l = 2$): Cloverleaf patterns with 5 orientations
●f orbitals ($l = 3$): Complex multilobed shapes with 7 orientations

2.3 Electron Configuration Rules

For multi-electron atoms, we fill orbitals according to three fundamental rules:

Aufbau Principle

Electrons fill orbitals starting from the lowest energy. The filling order follows the ($n + l$) rule: orbitals with lower $n + l$ fill first; for equal$n + l$, the lower $n$ fills first.

Order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

Hund's Rule

When filling degenerate orbitals (same $n$ and $l$), electrons occupy them singly with parallel spins before pairing. This minimizes electron-electron repulsion and maximizes total spin. For example, nitrogen (Z=7) has configuration 1s$^2$2s$^2$2p$^3$with all three 2p electrons unpaired.

Pauli Exclusion Principle

No two electrons in an atom can share the same four quantum numbers. This limits each orbital to at most 2 electrons (one spin-up, one spin-down), giving maximum occupancies of 2 for s, 6 for p, 10 for d, and 14 for f subshells.

Notable Exceptions

Some elements deviate from the simple Aufbau prediction due to extra stability from half-filled or fully-filled subshells:

●Chromium (Z=24): [Ar] 3d$^5$4s$^1$ instead of [Ar] 3d$^4$4s$^2$ (half-filled d subshell)
●Copper (Z=29): [Ar] 3d$^{10}$4s$^1$ instead of [Ar] 3d$^9$4s$^2$ (fully-filled d subshell)

2.4 Photoelectron Spectroscopy (PES)

Photoelectron spectroscopy provides direct experimental evidence for electron configuration by measuring the binding energies of electrons. High-energy photons (UV or X-ray) eject electrons from atoms, and the kinetic energy of the ejected electrons reveals the binding energy:

$$BE = h\nu - KE$$

A PES spectrum shows peaks at different binding energies, with heights proportional to the number of electrons in each subshell. For example, nitrogen shows three peaks:

●1s: highest BE (~400 eV), relative height 2
●2s: intermediate BE (~37 eV), relative height 2
●2p: lowest BE (~15 eV), relative height 3

2.5 Ionization Energies Across the Periodic Table

The first ionization energy (IE$_1$) is the energy required to remove the highest-energy electron from a neutral atom in the gas phase. General trends:

Across a Period (left to right)

IE generally increases due to increasing nuclear charge with same shielding

Down a Group (top to bottom)

IE generally decreases due to increasing distance and shielding

Notable exceptions to the trend across a period:

●B < Be: Boron removes a 2p electron (less stable) vs beryllium's 2s electron
●O < N: Oxygen has a paired 2p electron (higher repulsion) vs nitrogen's half-filled 2p subshell

The concept of effective nuclear charge ($Z_{\text{eff}}$) helps rationalize these trends. Slater's rules provide an approximate method:

$$Z_{\text{eff}} = Z - \sigma$$

where $\sigma$ is the shielding constant from inner electrons

Interactive Simulation: Orbital Probability Densities

This simulation visualizes the radial probability distribution $P(r) = r^2|R_{nl}(r)|^2$ for six hydrogen orbitals (1s through 3d), showing how the probability of finding the electron varies with distance from the nucleus.

Hydrogen Orbital Radial Probability Densities

Python

Plots radial probability distributions P(r) = r^2|R(r)|^2 for 1s, 2s, 2p, 3s, 3p, and 3d orbitals showing nodes and most probable radii.

script.py77 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import genlaguerre, factorial

# Radial wave function for hydrogen
def R_nl(n, l, r):
    """Radial wave function R_nl(r) for hydrogen atom.
    r is in units of a0 (Bohr radius)."""
    rho = 2.0 * r / n
    norm = np.sqrt((2.0/n)**3 * factorial(n-l-1) / (2.0*n*factorial(n+l)))
    L = genlaguerre(n-l-1, 2*l+1)(rho)
    return norm * np.exp(-rho/2) * rho**l * L

# Radial probability density: P(r) = r^2 |R_nl|^2
r = np.linspace(0, 35, 500)

# Plot for different (n, l) combinations
orbitals = [
    (1, 0, '1s', '#f472b6'),
    (2, 0, '2s', '#fb923c'),
    (2, 1, '2p', '#facc15'),
    (3, 0, '3s', '#4ade80'),
    (3, 1, '3p', '#38bdf8'),
    (3, 2, '3d', '#a78bfa'),
]

fig, axes = plt.subplots(2, 3, figsize=(15, 9))
fig.patch.set_facecolor('#0f172a')
axes = axes.flatten()

for idx, (n, l, label, color) in enumerate(orbitals):
    ax = axes[idx]
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    ax.xaxis.label.set_color('#cbd5e1')
    ax.yaxis.label.set_color('#cbd5e1')
    ax.title.set_color(color)
    for sp in ax.spines.values():
        sp.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

R = R_nl(n, l, r)
    P = r**2 * R**2

ax.fill_between(r, P, alpha=0.3, color=color)
    ax.plot(r, P, color=color, lw=2.5)
    ax.set_title(f'{label} orbital (n={n}, l={l})', fontsize=13, fontweight='bold')
    ax.set_xlabel('r / a\u2080', fontsize=10)
    ax.set_ylabel('r\u00b2|R(r)|\u00b2', fontsize=10)

# Mark the most probable radius
    i_max = np.argmax(P)
    r_max = r[i_max]
    ax.axvline(r_max, color=color, ls='--', alpha=0.6, lw=1)
    ax.text(r_max + 0.5, P[i_max]*0.9, f'r_mp={r_max:.1f} a\u2080',
            color=color, fontsize=8)

# Number of radial nodes = n - l - 1
    nodes = n - l - 1
    ax.text(0.95, 0.95, f'Nodes: {nodes}', transform=ax.transAxes,
            color='#94a3b8', fontsize=9, va='top', ha='right',
            bbox=dict(boxstyle='round,pad=0.3', facecolor='#0f172a', edgecolor='#334155'))

plt.tight_layout(pad=2)
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
plt.close()

print("Radial probability distributions plotted for 6 hydrogen orbitals.")
print("Key observations:")
print("  1s: Maximum at r = a0, no radial nodes")
print("  2s: Two peaks, one radial node")
print("  2p: Single peak at r = 4a0, no radial nodes")
print("  3s: Three peaks, two radial nodes")
print("  3p: Two peaks, one radial node")
print("  3d: Single peak, no radial nodes")
print(f"\nRadial nodes = n - l - 1")
print(f"Most probable radius for 1s: a0 = 52.9 pm")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Electron Configurations & Ionization Energies

This Fortran program generates electron configurations for elements Z=1–20 using the Aufbau principle and tabulates first ionization energies.

Electron Configurations & Ionization Energies

Fortran

Computes electron configurations using the Aufbau principle and tabulates experimental first ionization energies for elements 1-20.

electron_config.f9077 lines

program electron_config
  implicit none
  integer :: Z, n, l, ml, ms_idx, e_count
  integer :: max_n, max_electrons
  double precision :: IE_approx, Z_eff
  character(len=2) :: orbital_label
  character(len=1) :: l_labels(0:3)
  integer :: fill_order_n(20), fill_order_l(20)
  integer :: orbital_electrons(7, 4)  ! n=1..7, l=0..3
  integer :: i, j

l_labels(0) = 's'
  l_labels(1) = 'p'
  l_labels(2) = 'd'
  l_labels(3) = 'f'

! Aufbau filling order (by n+l, then n)
  fill_order_n = (/ 1,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7,7,7 /)
  fill_order_l = (/ 0,0,1,0,1,2,0,1,2,3,0,1,2,3,0,1,2,0,1,2 /)

write(*,'(A)') '=================================================='
  write(*,'(A)') '  Electron Configurations (Aufbau Principle)'
  write(*,'(A)') '=================================================='
  write(*,*)

write(*,'(A4, A6, A40, A15)') 'Z', 'Elem', 'Configuration', 'IE_1 (eV)'
  write(*,'(A)') '--------------------------------------------------'

! Element symbols for Z=1..20
  character(len=2) :: elements(20)
  elements = (/ 'H ', 'He', 'Li', 'Be', 'B ', 'C ', 'N ', 'O ', &
                'F ', 'Ne', 'Na', 'Mg', 'Al', 'Si', 'P ', 'S ', &
                'Cl', 'Ar', 'K ', 'Ca' /)

! Approximate first ionization energies (eV)
  double precision :: IE_data(20)
  IE_data = (/ 13.6d0, 24.6d0, 5.4d0, 9.3d0, 8.3d0, 11.3d0, 14.5d0, &
               13.6d0, 17.4d0, 21.6d0, 5.1d0, 7.6d0, 6.0d0, 8.2d0, &
               10.5d0, 10.4d0, 13.0d0, 15.8d0, 4.3d0, 6.1d0 /)

do Z = 1, 20
    ! Build configuration
    orbital_electrons = 0
    e_count = 0
    do i = 1, 20
      if (e_count >= Z) exit
      n = fill_order_n(i)
      l = fill_order_l(i)
      max_electrons = 2 * (2*l + 1)
      orbital_electrons(n, l+1) = min(max_electrons, Z - e_count)
      e_count = e_count + orbital_electrons(n, l+1)
    end do

! Build configuration string
    character(len=40) :: config
    config = ''
    do i = 1, 20
      if (fill_order_n(i) > 7) cycle
      n = fill_order_n(i)
      l = fill_order_l(i)
      if (orbital_electrons(n, l+1) > 0) then
        write(orbital_label, '(I1,A1)') n, l_labels(l)
        write(config, '(A,A2,I1)') trim(config), orbital_label, &
              orbital_electrons(n, l+1)
      end if
    end do

write(*,'(I4, A6, A40, F15.1)') Z, elements(Z), trim(config), IE_data(Z)
  end do

write(*,*)
  write(*,'(A)') '--- Periodic Trends in Ionization Energy ---'
  write(*,'(A)') 'General trend: IE increases across a period (left to right)'
  write(*,'(A)') 'General trend: IE decreases down a group (top to bottom)'
  write(*,'(A)') 'Exceptions: B < Be (2p vs 2s), O < N (electron pairing)'

end program electron_config

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

Lecture 5: Shell Models and Quantum Numbers

Goodie Bag 2: Electronic Transitions

Lecture 6: Electron Shell Model, Quantum Numbers, and PES

Lecture 7: Aufbau Principle and Atomic Orbitals

Goodie Bag 3: Ionic Solids

Lecture 8: Ionization Energy and PES

Key Takeaways

●Four quantum numbers ($n, l, m_l, m_s$) uniquely specify each electron state
●The Schrodinger equation separates into radial ($R_{nl}$) and angular ($Y_l^{m_l}$) parts
●Radial nodes = $n - l - 1$; angular nodes = $l$; total nodes = $n - 1$
●Aufbau, Hund's rule, and Pauli exclusion determine electron configurations
●PES provides direct experimental evidence for subshell structure
●Ionization energies reflect effective nuclear charge and electron-electron repulsion

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