1. 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

2. 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:

3. H-atom wavefunctions (cont.) The Coulombic potential can be generalized: Z Z = atomic number (= 1 for hydrogen)

4. H-atom wavefunctions (cont.) The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates. p+p+ e-e- r = interparticle distance (0 ≤ r ≤  )  = angle from “xy plane” (  /2 ≤  ≤ -  /2)  = rotation in “xy plane” (0 ≤  ≤ 2  )

5. 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.

6. 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 l, the “z component” of orbital angular momentum. Ranges in value from -l to 0 to l.

7. 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).

8. Orbital Shapes Let’s take a look at the lowest energy orbital, the “1s” orbital (n = 1, l = 0, m = 0) a 0 is referred to as the Bohr radius, and = 0.529 Å 1 1

9. Orbital Shapes (cont.) Note that the “1s” wavefunction has no angular dependence (i.e.,  and  do not appear). Probability = Probability is spherical

10. 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 - m l (-l…0…l) is usually “dropped”

11. Orbital Shapes (cont.) Table 12.3: Quantum Numbers and Orbitals nlOrbital m l # of Orb. 10 1s 0 1 20 2s 0 1 1 2p -1, 0, 1 3 30 3s 0 1 1 3p -1, 0, 1 3 2 3d -2, -1, 0, 1, 2 5

12. 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 m l sublevel) 4p (3 m l sublevels) 4d (5 m l sublevels 4f (7 m l sublevels)

13. Orbital Shapes (cont.) s (l = 0) orbitals r dependence only as n increases, orbitals demonstrate n-1 nodes.

14. Orbital Shapes (cont.) 2p (l = 1) orbitals not spherical, but lobed. labeled with respect to orientation along x, y, and z.

15. Orbital Shapes (cont.) 3p orbitals more nodes as compared to 2p (expected.). still can be represented by a “dumbbell” contour.

16. Orbital Shapes (cont.) 3d (l = 2) orbitals labeled as d xz, d yz, d xy, d x2-y2 and d z2.

17. Orbital Shapes (cont.) 3d (l = 2) orbitals d xy d x2-y2

18. Orbital Shapes (cont.) 3d (l = 2) orbitals d z2

19. Orbital Shapes (cont.) 4f (l = 3) orbitals exceedingly complex probability distributions.

20. Orbital Energies energy increases as 1/n 2 orbitals of same n, but different l are considered to be of equal energy (“degenerage”). the “ground” or lowest energy orbital is the 1s.