Lecture 18: The Hydrogen Atom

1. Lecture 18: The Hydrogen Atom • Reading: Zumdahl 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. The Coulombic potential can be generalized: Z • Z = atomic number (= 1 for hydrogen)

4. H-atom wavefunctions The radial dependence of the potential invites us to switch from Cartesian to spherical polar coordinates to facilitate the q. m. solution (“Separation of variables”) r = interparticle distance (0 ≤ r ≤ ) e- • = angle from “xy plane” (/2 ≤  ≤ - /2) p+  = rotation in “xy plane” (0 ≤  ≤ 2)

5. H-atom wavefunctions 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 in value from 1 to infinity.

6. H-atom wavefunctions • In solving the Schrodinger Equation, two other quantum numbers become evident: l, the orbital angular momentum quantum number. Ranges in value 0, 1, 2, … (n-1). ml, 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).

7. 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

8. Note that the “1s” wavefunction has no angular dependence (i.e., Q and F do not appear). Probability = • Probability is spherical

9. Naming orbitals is done as follows: Principle q.n. ‘n’ is simply referred to as 1,2,3,… etc The quantum number l (0 to (n-1)) is given a letter value as follows: l 0 = s 1 = p 2 = d 3 = f ml(- l …,0,… l) subscript is usually dropped

10. 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

11. Which of the following sets of quantum numbers (n, l, m) is not allowed? (3, 2, 2). B. (0, 0, 0). C. (1, 0, 0). D. (2, 1, 0).

12. Orbital Shapes (cont.) • Example: Write down the orbitals associated with n = 4. Ans: n = 4 4s (1 ml sublevel) 4p (3 ml sublevels) 4d (5 ml sublevels 4f (7 ml sublevels) l = 0 to (n-1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f

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 dxz, dyz, dxy, dx2-y2 and dz2.

17. Orbital Shapes (cont.) 3d (l = 2) orbitals • dxy • dx2-y2

18. Orbital Shapes (cont.) 3d (l = 2) orbitals • dz2

19. What orbital is depicted in the following animation? A. 3dxy B. 3dz2 C. 3dxz D. 2s

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

21. Orbital Energies energy increases as 1/n2 orbitals of same n, but different l are considered to be of equal energy (“degenerate”). the “ground state” or lowest energy orbital is the 1s.

22. Which orbital is expected to have the greatest energy? 1s 2s 2p 3p