Chemistry

Write the time-independent Schrödinger equation for a particle in a specified potential.

Solve the time-independent Schrödinger equation for a particle in a one-dimensional infinite potential well.

Relate the nodes and wavelength of an infinite-well eigenfunction to its quantum number.

Calculate transition energies for a particle in a one-dimensional infinite potential well.

Construct the nonrelativistic Hamiltonian for an electron in the Coulomb field of a nucleus.

Explain how separation in spherical coordinates produces radial and angular quantum numbers for hydrogenic atoms.

Calculate the allowed energies of a hydrogenic ion from its principal quantum number and nuclear charge.

Determine the allowed quantum numbers for a specified hydrogenic state.

Interpret spherical harmonics as the angular components of atomic orbitals.

Calculate the number of radial and angular nodes in a hydrogenic orbital.

Use a radial probability distribution to identify the most probable electron–nucleus distance.

Sketch the qualitative shapes and nodal structures of s, p, and d orbitals.

Calculate the wavelength of a hydrogenic spectral transition using energy-level differences.

Interpret the Stern–Gerlach experiment as evidence for quantized electron spin.

Calculate the allowed magnitude and projection values of electron spin angular momentum.

Apply the Pauli exclusion principle to assign electrons to spin-orbitals.

Construct a Slater determinant for a simple two-electron state.

Distinguish singlet and triplet spin states for two electrons.

Apply Hund’s rules and orbital-energy ordering to construct ground-state electron configurations.

Identify common configuration exceptions caused by closely spaced subshell energies.

Distinguish electron shielding from orbital penetration in many-electron atoms.

Estimate effective nuclear charge using Slater’s rules.

Use effective nuclear charge and principal-shell structure to explain atomic-radius trends.

Use orbital energies and electron interactions to explain first-ionization-energy trends and their common exceptions.

Explain successive ionization-energy patterns using changes in electronic shell structure.

Distinguish exchange effects from classical electrostatic repulsion in many-electron states.

Use the variational principle to estimate an upper bound to a simple atomic ground-state energy.

Explain the purpose and iterative logic of the self-consistent-field method for atomic orbitals.

Generate allowed microstates for a specified open-subshell electron configuration.

Derive allowed atomic term symbols from a small set of electron microstates.

Apply Hund’s rules to identify the lowest-energy term of an atom or ion.

Calculate the allowed total angular momentum values arising from specified orbital and spin angular momenta.

Distinguish Russell–Saunders coupling from jj coupling and identify when each approximation is appropriate.

Use spin–orbit coupling to explain fine-structure splitting of an atomic term.

Apply the Landé interval rule to analyze an LS-coupled multiplet.

Relate absorption and emission lines to transitions between quantized atomic states.

Apply parity and angular-momentum selection rules to identify allowed electric-dipole transitions.

Use term symbols to label transitions in a simple atomic spectrum.

Calculate transition wavelengths from tabulated atomic term energies.

Use the Rydberg–Ritz combination principle to test assignments of spectral lines.

Distinguish line intensity, line position, and line width as sources of atomic information.

Explain natural, Doppler, and collisional broadening of atomic spectral lines.

Infer an unknown energy level from a consistent set of observed spectral transitions.

Calculate first-order Zeeman shifts for a simple atomic level in a weak magnetic field.

Distinguish the normal Zeeman effect from the anomalous Zeeman effect.

Calculate a Landé g-factor for an LS-coupled atomic term.

Explain qualitatively how an external electric field produces Stark splitting.

Distinguish fine structure, hyperfine structure, and isotope shifts by their physical origins.

Build an atomic energy-level diagram from configurations, term symbols, and measured transition energies.

Analyze a small atomic spectrum by assigning levels and checking selection rules.

Evaluate where nonrelativistic and independent-electron atomic models cease to be reliable.