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Lecture 19: The Hydrogen Atom. Reading: Zuhdahl 12.7-12.9 Outline The wavefunction for the H atom Quantum numbers and nomenclature Orbital shapes and energies. H-atom wavefunctions.
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Lecture 19: The Hydrogen Atom • Reading: Zuhdahl 12.7-12.9 • Outline • The wavefunction for the H atom • Quantum numbers and nomenclature • Orbital shapes and energies
H-atom wavefunctions • Recall from the previous lecture that the Hamiltonian is composite of kinetic (KE) and potential (PE) energy. • The hydrogen atom potential energy is given by:
H-atom wavefunctions (cont.) • The Coulombic potential can be generalized: Z • Z = atomic number (= 1 for hydrogen)
H-atom wavefunctions (cont.) • The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates. r = interparticle distance (0 ≤ r ≤ ) e- • = angle from “xy plane” (/2 ≤ ≤ - /2) p+ = rotation in “xy plane” (0 ≤ ≤ 2)
H-atom wavefunctions (cont.) • If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized: • n is the principle quantum number, and ranges from 1 to infinity.
H-atom wavefunctions (cont.) • In solving the Schrodinger Equation, two other quantum numbers become evident: l, the orbital angular momentum quantum number. Ranges in value from 0 to (n-1). ml, the “z component” of orbital angular momentum. Ranges in value from -l to 0 to l.
H-atom wavefunctions (cont.) • In solving the Schrodinger Equation, two other quantum numbers become evident: l, the orbital angular momentum quantum number. Ranges in value from 0 to (n-1). m, the “z component” of orbital angular momentum. Ranges in value from -l to 0 to l. • We can then characterize the wavefunctions based on the quantum numbers (n, l, m).
Orbital Shapes • Let’s take a look at the lowest energy orbital, the “1s” orbital (n = 1, l = 0, m = 0) • a0 is referred to as the Bohr radius, and = 0.529 Å 1 1
Orbital Shapes (cont.) • Note that the “1s” wavefunction has no angular dependence (i.e., Q and F do not appear). Probability = • Probability is spherical
Orbital Shapes (cont.) • Naming orbitals is done as follows • n is simply referred to by the quantum number • l (0 to (n-1)) is given a letter value as follows: • 0 = s • 1 = p • 2 = d • 3 = f - ml (-l…0…l) is usually “dropped”
Orbital Shapes (cont.) • Table 12.3: Quantum Numbers and Orbitals n l Orbital ml # of Orb. 0 1s 0 1 0 2s 0 1 1 2p -1, 0, 1 3 0 3s 0 1 1 3p -1, 0, 1 3 2 3d -2, -1, 0, 1, 2 5
Orbital Shapes (cont.) • Example: Write down the orbitals associated with n = 4. Ans: n = 4 l = 0 to (n-1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f 4s (1 ml sublevel) 4p (3 ml sublevels) 4d (5 ml sublevels 4f (7 ml sublevels)
Orbital Shapes (cont.) s (l = 0) orbitals • r dependence only • as n increases, orbitals demonstrate n-1 nodes.
Orbital Shapes (cont.) 2p (l = 1) orbitals • not spherical, but lobed. • labeled with respect to orientation along x, y, and z.
Orbital Shapes (cont.) 3p orbitals • more nodes as compared to 2p (expected.). • still can be represented by a “dumbbell” contour.
Orbital Shapes (cont.) 3d (l = 2) orbitals • labeled as dxz, dyz, dxy, dx2-y2 and dz2.
Orbital Shapes (cont.) 3d (l = 2) orbitals • dxy • dx2-y2
Orbital Shapes (cont.) 3d (l = 2) orbitals • dz2
Orbital Shapes (cont.) 4f (l = 3) orbitals • exceedingly complex probability distributions.
Orbital Energies • energy increases as 1/n2 • orbitals of same n, but different l are considered to be of equal energy (“degenerage”). • the “ground” or lowest energy orbital is the 1s.