Notes on Atomic Orbitals, Quantum Numbers, and Orbital Notation

Electromagnetic Waves and Atomic Orbitals

• The spectrum discussion places certain high-energy photons (X-rays, gamma rays) in the electromagnetic spectrum; these are wave phenomena that relate to how we probe atomic structure.

• Concept: Schrödinger equation solutions correspond to atomic orbitals; each solution describes a region of space associated with a particular orbital.

• An atomic orbital is a spatial region around the nucleus where the probability of finding the electron is nonzero; orbitals grow larger and their energies increase with the principal quantum number as you go to higher levels.

• For a one-electron atom, the azimuthal quantum number l starts at 0 for the s-like orbitals, and as n increases you obtain multiple orbital types.

• When n = 2, allowed l values are
  l \,\in\ {0,1}
  which correspond to the 2s and 2p orbitals.

• For a general n, l can take values from 0 up to n−1:
  l \in {0,1,\dots, n-1}

• Orbitals with l > 3 are possible for sufficiently large n (e.g., in the eighth period); such orbitals correspond to g, h, i… types with l = 4,5,6,7, etc.

• Example about the periodic table: element 119 would be placed under francium, and element 120 would be placed under radium in an extended periodic table.

• Quantum numbers and orbital identity

  • The state of an electron in an atom is labeled by four quantum numbers:
    n, \ l, m{l}, m{s}
  • The first three numbers (n, l, ml) identify the orbital; the fourth number (ms) distinguishes the two electrons that can occupy the same orbital.
  • If two electrons share the first three quantum numbers (n, l, ml), then they must have opposite spins to satisfy the Pauli exclusion principle: m{s} \in {+\tfrac{1}{2}, -\tfrac{1}{2}}
  • Thus, any given orbital can hold a maximum of two electrons with opposite spins.

• Principal and angular quantum numbers, and subshell notation

  • The principal quantum number n sets the energy level and radial extent; the azimuthal quantum number l determines the subshell type:
  • s subshell: l=0 (s orbitals are spherical)
  • p subshell: l=1 (three orbitals per subshell, oriented along axes)
  • The magnetic quantum number ml provides the orientation of the orbital in space; its allowed values are: m{l} \in {-l, -l+1, \dots, 0, \dots, l-1, l}
  • For p orbitals (l=1), the possible ml values are -1, 0, 1, corresponding to the three orbitals that orient along different spatial directions.
  • The three p orbitals are commonly labeled to reflect their spatial orientation: px, py, p_z; they correspond to ml values that differentiate their orientation.

• Shapes, nodes, and sign of the wave function

  • Orbitals are mathematical solutions to the wave equation; the wave function can take positive or negative values at different points in space.
  • The probability density is given by P(\mathbf{r}) = |\psi(\mathbf{r})|^{2}, which is always nonnegative, so the sign of the wave function does not affect the probability of finding the electron.
  • The sign is important for interference and bonding when combining orbitals; constructive or destructive interference depends on relative phases.
  • Nodes: regions where the probability density goes to zero; in general, orbitals have both angular nodes (due to angular dependence) and radial nodes (due to radial dependence).

• Node counts and orbital topology (general relations)

  • For a given principal quantum number n and azimuthal quantum number l:
  • Angular nodes (planes or cones where the wave function changes sign) = l
  • Radial nodes (spherical surfaces where the radial part goes to zero) = n - l - 1
  • Total number of nodes for an atomic orbital: \text{nodes} = n - 1
  • Number of orbitals in a given subshell (for fixed l) is 2l+1; hence:
  • s subshell (l = 0): 1 orbital
  • p subshell (l = 1): 3 orbitals
  • d subshell (l = 2): 5 orbitals
  • f subshell (l = 3): 7 orbitals
  • Total number of orbitals in shell n is \sum_{l=0}^{n-1} (2l+1) = n^{2}, so the maximum number of electrons in shell n is 2n^{2} given two electrons per orbital.

• Subshell notation and its relation to the quantum numbers

  • Subshell labels come from the azimuthal quantum number l, mapped to letters: 0→s, 1→p, 2→d, 3→f, 4→g, …
  • The notation combines n and the letter for l, e.g.,: 1s, 2s, 2p, 3s, 3p, 3d, 4f, etc.
  • The first three quantum numbers (n, l, ml) identify the specific orbital (e.g., 2px corresponds to a particular set of (n, l, ml)); the fourth quantum number ms specifies the electron's spin within that orbital.

• Connections to broader themes and real-world relevance

  • Orbital shapes and orientations explain chemical bonding, hybridization, and anisotropy in molecules.
  • The four quantum numbers underpin spectroscopy: transitions between orbitals yield photons with energies corresponding to differences in orbital energies.
  • The Pauli exclusion principle, via the four quantum numbers, explains electron configurations and periodic trends.
  • In extended periodic tables, high-n orbitals (e.g., with l up to 7 in very heavy elements) become relevant; the eighth period would accommodate orbitals with high l values beyond f (l=3).

• Quick recap of key formulas

  • Allowed azimuthal quantum numbers: l \in {0,1,\dots, n-1}
  • Magnetic quantum numbers: m_{l} \in {-l, -l+1, \dots, 0, \dots, l-1, l}
  • Spin quantum numbers: m_{s} \in { -\tfrac{1}{2}, +\tfrac{1}{2} }
  • Number of orbitals in shell n: \sum_{l=0}^{n-1} (2l+1) = n^{2}
  • Maximum electrons in shell n: 2n^{2}
  • Angular nodes: l
  • Radial nodes: n - l - 1
  • Total nodes: n - 1

• Notational summary (quick reference)

  • 1s, 2s, 2p, 3s, 3p, 3d, 4f, 5g, 6h, 7i, …
  • For a given n, orbitals exist for all l from 0 to n−1, with ml providing orientation and ms providing spin.

• Philosophical/epistemic note

  • These orbital concepts are models that approximate electron behavior; they capture essential features of atomic structure and spectroscopy, while acknowledging that electrons exhibit both wave-like and particle-like properties.

• Procedural takeaway for exams

  • Be able to determine the possible values of n, l, ml, and ms given an electron configuration or orbital description.
  • Remember the occupancy rule: no two electrons in the same orbital can have the same set of quantum numbers; they must have opposite spins.
  • Recognize s vs p orbital shapes, and that ml values differentiate p orbital orientations (px, py, p_z).