2.2.2: Quantum Numbers and Atomic Wave Functions

• A complete solution to the Schrodinger equation, both the three-dimensional wavefunction and energy, includes a set of three quantum numbers.
• The wavefunction describes an atomic orbital and defines the region in space where the electron is located.
• The quantum number n is the principle quantum number and represents the shell, including both the overall energy of the electron in that shell and the size of the shell.
• An allowed value for n is any non-zero, positive integer.
• The quantum number l is the angular momentum quantum number that corresponds to the subshell and its shape, and represents the angular dependence of the subshell.
• The allowed values of l for an electron in shell n are integer values between 0 to n-1.
• l = 0 is an s-orbital, l = 1 is an p-orbital, l = 2 is a d-orbital, and l = 3 is an f-orbital
• The quantum number mL is the magnetic quantum number and gives the number of orbitals within a subshell and its specific value gives the orbital's orientation in space.
• The value of mL is allowed to be any positive or negative integer between +L and -L.
• Quantum number mS accounts for the electron's spin and can be +1/2 or -1/2.
• The radial part of the wavefunction gives the radial variation of the wavefunction and defines how the wavefunction changes with distance from the nucleus.
• The square of the radial part of the wavefunction is called the radial distribution function and describes the probability of locating the electron at some distance, r, away from the nucleus.
• Radial nodes occur where the radial part of the wavefunction is zero.
• The number of radial nodes is n-l-1.
• Where there is a node, there is zero probability of finding an electron.
• The boundary surface represents the area around the nucleus where the electron exists most of the time.
• The angular contribution to the wavefunction describes the wavefunction's shape, or the angle with respect to a coordinate system.
• The angular part of the wavefunction becomes more "real" when you square it to get angular probability density, a more tangible concept described as the shape of orbitals.
• Angular nodes are planar and depend on the value of l. The number of angular nodes in any orbital is equal to l.
• S-orbitals have zero angular nodes, p-orbitals have 1 angular node, d-orbitals have 2 angular nodes, etc.
• Planar nodes can be flat planes or they can have a conical shape.
• Atomic orbitals result from a combination of both the radial and angular contributions of the wavefunction.