Electron Orbitals and Quantum Numbers: Comprehensive Notes

Orbitals and how electrons are distributed

• Orbitals (e.g., the s, p, d shells) overlap in space; electrons have many possible locations within the orbital region.
• The probability of finding an electron is not uniform across the orbital; electrons sample different locations within the overlapping space.
• Goal: predict where the electron is most likely to be found among multiple possible locations.

Quantum numbers and orbitals

• Orbitals are described by three quantum numbers from Schrödinger’s equation: $n,\ l,\ m_{l}$.
  • $n$: principal quantum number; describes the shell size (the onion layer). The closer to the nucleus, the lower the energy.
  • $l$: azimuthal (orbital angular momentum) quantum number; describes the shape of the orbital. $l=0,1,2,3,\dots$ corresponding to the s, p, d, f, … sublevels with energies that can differ due to non-ideal degeneracy.
  • $m<em>{l}$: magnetic quantum number; describes the orientation of the orbital in space. For a given $l$, $m</em>{l} = -l,-l+1,\dots,+l$.
• Spin quantum number $m<em>{s}$ is independent of those three and describes the electron’s spin orientation: $m</em>{s}=\pm\tfrac{1}{2}$.
• Schrödinger’s equation yields $n, l, m<em>{l}$; $m</em>{s}$ is a separate quantum number describing spin behavior, particularly in magnetic fields.
• Spin affects emission/absorption of light and energy levels due to interactions with neighbors and with fields.

Spin, Pauli exclusion, and occupancy

• The two possible spin states of an electron are: $m<em>{s}=+\tfrac{1}{2}$ and $m</em>{s}=-\tfrac{1}{2}$.
• Pauli exclusion principle: no two electrons can have the same set of quantum numbers ($n,\ l,\ m<em>{l},\ m</em>{s}$). This is like not sitting two people on the same chair.
• An orbital (a specific set of $n, l, m_{l}$) can hold at most two electrons, and they must have opposite spins (one up, one down) when filled to the limit.
• The spin degree of freedom effectively provides the “number of chairs” in an orbital: two chairs per orbital (one for each spin) before moving to the next orbital.

Degeneracy, sublevels, and orbital energies

• Degeneracy: orbitals with the same energy within a subshell; e.g., within a given $n$ and $l$, the different $m_{l}$ orbitals are degenerate (same energy) before more electrons are present.
  • For a given $l$, there are $2l+1$ degenerate orbitals (e.g., p orbitals: $l=1\Rightarrow 3$ orbitals: $m_{l}=-1,0,+1$).
• Sublevels within a shell: $s, p, d, f$ correspond to $l=0,1,2,3$ respectively; inside a given shell you have different sublevel energies that can be nondegenerate due to shielding and penetration.
• Degeneracy is exact only for an isolated hydrogen-like atom; in multi-electron atoms, electrons break perfect degeneracy through screening, penetration, and Coulomb interactions.

Why electrons occupy different sublevels (Coulomb, shielding, penetration)

• Three factors determine energy differences among orbitals in multi-electron atoms:
  1. Coulomb’s law (electrostatic attraction between the nucleus and electrons): closer electrons feel stronger attraction to the nucleus, lowering energy.
  2. Shielding (electrons in inner shells reduce the effective nuclear charge felt by outer electrons): outer electrons feel a reduced positive charge after inner electrons partially cancel nucleus charge, raising energy.
  3. Penetration (electrons in certain orbitals can overlap into inner regions and temporarily occupy lower-energy space): this can lower the energy of outer electrons via temporary access to regions of higher nuclear attraction.
• Coulomb’s law (electrostatic force between charges): $F = k<em>e\frac{q</em>1 q_2}{r^2}$
  • Closer distance (smaller $r$) leads to stronger attraction or repulsion, lowering the electron’s potential energy when attraction dominates.
• Effective nuclear charge: outer electrons experience the net nuclear attraction after shielding by inner electrons.
  • Define $Z_{\text{eff}} = Z - S$ where $Z$ is the actual nuclear charge and $S$ is the shielding constant from inner electrons.
  • Higher shielding reduces attraction, increasing energy for outer electrons.
• Penetration can cause some outer electrons (notably in the 2s sublevel) to have a nonzero probability density close to the nucleus, effectively lowering their energy relative to other sublevels.

Electron density and probability maps (penetration visual)

• Probability distribution for s orbitals shows a peak near the nucleus for 1s, and for 2s there is a substantial probability near the nucleus due to penetration into the inner region.
• Compare 2s and 2p: 2s has a higher probability of being close to the nucleus (due to penetration) than 2p, which reduces its energy more (lower), making 2s lower in energy than 2p within the same shell.
• Visual intuition: the probability density maps show how often you’d find the electron at a given distance from the nucleus; higher curves indicate higher likelihood.

Aufbau principle, electron configurations, and notation

• Aufbau principle: electrons fill orbitals in order of increasing energy (lowest first) before filling higher-energy orbitals.
• Order of filling: generally, fill in the sequence s → p → d → f within and across shells, with s orbitals generally lower in energy within a given principal quantum number before d, f sublevels come into play as you move to higher n.
• How to write electron configurations:
  • Full (standard) configuration: write the orbitals in order with superscripts for electrons per orbital, e.g.,
  • Hydrogen: $1s^{1}$
  • Helium: $1s^{2}$
  • Lithium: $1s^{2} 2s^{1}$
  • Beryllium: $1s^{2} 2s^{2}$
  • Boron: $1s^{2} 2s^{2} 2p^{1}$
  • Carbon: $1s^{2} 2s^{2} 2p^{2}$
  • Orbital diagram (box diagram): use lines or boxes for orbitals and arrows for electrons; ↑ for $m<em>s=+\tfrac{1}{2}$ and ↓ for $m</em>s=-\tfrac{1}{2}$.
  • Noble gas configuration: a shorthand where the preceding noble gas core is written in brackets, followed by the remaining valence electrons (the instructor will specify which form to use on exams).
• Aufbau sequence for building configurations: start with the lowest-energy sublevel (often 1s), then move to 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, etc., noting that the exact order can be nontrivial for larger atoms due to shielding and penetration effects.
• Maximum electrons per orbital and per subshell:
  • An orbital can hold at most two electrons with opposite spins: two electrons per orbital.
  • For a given subshell with angular momentum $l$ there are $2l+1$ orbitals, so the maximum electrons in the subshell is $2(2l+1)$.
  • Example: 2p can hold up to $2(2\cdot1+1)=6$ electrons (three degenerate orbitals with two electrons each).
• Degenerate orbitals within a subshell: while the orbitals with the same $l$ are degenerate in energy in a simple model, in practice electrons fill to minimize repulsion (Hund’s rule) before pairing.
• Hund’s rule (maximizing unpaired spins): within a set of degenerate orbitals (e.g., the three 2p orbitals), electrons will occupy separate orbitals with parallel spins before pairing up in the same orbital.
  • If there is an available empty orbital of the same energy, place electrons with parallel spins in separate orbitals first (instead of pairing in a single orbital).
  • Analogy: hotel rooms — it’s better to occupy empty rooms with unpaired guests than to share a room with a stranger when a free room exists.
• Practical takeaway: when filling electrons, prioritize unpaired spins in degenerate orbitals before pairing, and fill orbitals in order of increasing energy (Aufbau principle).

Examples of electron configurations (applications of the rules)

• Hydrogen: $1s^{1}$ (period 1, s-block).
• Helium: $1s^{2}$ (both electrons in the 1s orbital).
• Lithium: $1s^{2} 2s^{1}$ (3rd electron goes to next energy level; after 1s is full, move to 2s).
• Beryllium: $1s^{2} 2s^{2}$.
• Boron: $1s^{2} 2s^{2} 2p^{1}$ (after filling 2s, begin filling 2p).
• Carbon: $1s^{2} 2s^{2} 2p^{2}$.
• Notation variants for practice:
  • Full electron configuration: as above (e.g., $1s^{2} 2s^{2} 2p^{2}$).
  • Orbital diagram: represent each orbital with lines and arrows to show spin orientation; use up arrows for $m<em>s=+\tfrac{1}{2}$ and down arrows for $m</em>s=-\tfrac{1}{2}$.
  • Noble gas shorthand: e.g., [He] for neon-like core, followed by the remaining valence electron configuration.

Periodic table structure and electron configurations

• Periodic table blocks reflect which subshells are being filled:
  • s-block: the first two groups; corresponds to filling the $n s$ orbitals (e.g., 1s, 2s, 3s, …).
  • p-block: the next six groups (after the s-block), corresponds to filling the $n p$ orbitals (e.g., 2p, 3p, etc.).
  • d-block: transition metals; corresponds to filling the $n-1 d$ orbitals (e.g., 3d, 4d, etc.).
  • f-block: lanthanides and actinides (bottom two rows in many tables).
• Ordering nuance with the table: as you move down and across, the actual orbital being filled can change; e.g., the d-block lags behind the period number by one (the 3d block corresponds to period 4, etc.). The speaker notes this as a general guide rather than a strict rule.
• The table helps track electron configurations by groups (or families) and periods (rows):
  • Group/family concept is commonly used to describe elements with similar valence electron configurations.
  • Hydrogen and helium can be treated conceptually near group 1/2 or the noble gas column for teaching purposes to illustrate electron filling, even though helium is often placed apart due to its full outer shell.
• Practical use: you will be asked to write electron configurations, identify the orbital into which the next electron goes, and read configurations off the periodic table.

Different representations of electron configurations

• Full electron configuration: explicit write-out of every occupied orbital with its electron count, e.g., $1s^{2} 2s^{2} 2p^{2}$ for carbon.
• Orbital diagram (box/line diagram): depicts orbitals as lines or boxes and populating them with arrows representing electrons and their spins (up for $m<em>s=+\tfrac{1}{2}$, down for $m</em>s=-\tfrac{1}{2}$).
• Noble gas configuration: use the noble gas in brackets to denote a completed inner shell core, then continue with the valence electrons, e.g., for sodium: [Ne] $3s^{1}$.
• In practice, instructors may specify which representation to use on exams; the content here covers all three formats.

Practical rules recap and key takeaways

• Electron filling order is driven by energy considerations: lower-energy orbitals fill first (Aufbau principle).
• Not all orbitals in a given shell have the same energy; degenerate orbitals exist within a subshell (e.g., the three 2p orbitals are degenerate in energy in a simple model).
• When a subshell can accommodate more electrons, fill according to Hund’s rule to maximize unpaired spins before pairing.
• The energy ordering of orbitals is influenced by: Coulomb attraction to the nucleus, shielding by inner electrons, and penetration of outer electrons into inner regions.
• The s orbitals tend to be lower in energy within a given shell due to greater penetration, which is why s orbitals are filled before p orbitals in many cases.
• In multi-electron atoms, the energy of the d and f subshells is influenced by the same principles but can lead to deviations from simple orderings (e.g., occasional crossing of energy levels).
• The Pauli principle governs occupancy: at most two electrons per orbital, with opposite spins.
• The noble gas configuration is a useful shorthand to simplify writing configurations for heavier elements.
• The periodic table’s blocks (s, p, d, f) reflect the subshell being filled in the typical order of electron configuration.
• Understanding the electron configuration helps explain chemical properties and periodic trends, since electrons determine bonding, reactivity, and overall chemistry of elements.

Quick memory aids and analogies from the lecture

• Mario analogy for energy steps: electrons move from lower to higher platforms (orbitals) in discrete steps; sometimes finer sub-steps exist within a platform, reflecting sublevel energy differences.
• Cookie bakery analogy for shielding: as you wait in line, the smells (nuclei’s positive charge) are muffled by people (inner electrons) in front, reducing your effective access (effective nuclear charge).
• Hotel room analogy for Hund’s rule: if there’s an empty room (an empty degenerate orbital), you’d rather not share (pair) with a stranger; keep rooms single-occupant until you run out of rooms.

Summary key equations and concepts (for quick reference)

• Coulomb's law (electrostatic interaction): $F = k<em>e\frac{q</em>1 q_2}{r^2}$
  • Closer proximity to nucleus (smaller $r$) increases attraction, lowering energy for bound electrons.
• Effective nuclear charge: $Z_{\text{eff}} = Z - S$ where $S$ depends on shielding by inner electrons.
• Spin quantum numbers: $m_s = \pm \tfrac{1}{2}$.
• Magnetic quantum numbers: for a given $l$, $m_l = -l, -l+1, \dots, +l$ with degeneracy $2l+1$.
• Degeneracy of a subshell: number of orbitals in a subshell is $2l+1$ and each orbital can hold two electrons.
• Maximum electrons in a subshell: $2(2l+1)$.
• Hund’s rule: maximize the number of unpaired electrons in degenerate orbitals before pairing.
• Aufbau principle: electrons fill in order of increasing energy (lowest first).
• Orbital energy order (simplified, typical for light elements): 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p …
• Orbital labels and angular momentum: $l=0\rightarrow s, l=1\rightarrow p, l=2\rightarrow d, l=3\rightarrow f$.
• Orbital penetration explains why some orbitals (notably 2s) can lower energy by overlapping inner regions, affecting the ordering of orbitals.