1. Lecture 11. Hydrogen Atom References Engel, Ch. 9
Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.3
Introductory Quantum Mechanics, R. L. Liboff (4th ed, 2004), Ch.10
A Brief Review of Elementary Quantum Chemistry

2. Separation of Internal Motion: Born-Oppenheimer Approximation
(2-Body Problem)
Electron
coordinate
Nucleus
coordinate
Separation of Internal Motion: Born-Oppenheimer Approximation
Full Schrödinger equation can be separated into two equations:
1. Atom as a whole through the space;
2. Motion of electron around the nucleus.
“Electronic” structure (1-Body Problem): Forget about nucleus!

3. in spherical coordinate

4. angular momentum quantum no.
Angular part (spherical harmonics)
Radial part (Radial equation)
principal quantum no.
, n-1
angular momentum quantum no.
(Laguerre polynom.)
magnetic quantum no.

5. Radial Schrödinger Equation

7. Wave Functions (Atomic Orbitals): Electronic States
nlm nl

8. Wave Functions (Atomic Orbitals): Electronic States
Designated by three quantum numbers
nlm nl
Radial Wave Functions Rnl

10. Radial Wave Functions Rnl2p 3s 3p 3d
node
2 nodes
*Bohr Radius
*Reduced distance
Radial node
(ρ = 4, )

11. Radial Wave Functions Rnl

12. Radial Wave Functions (l = 0, m = 0): s Orbitals

14. Radial Wave Functions (l  0)2p 3p 3d

15. Probability Density

16. Wave Function
Probability

17. Radial Distribution Function
Probability density.
Probability of finding an electron at a point (r,θ,φ)
Integral over θ and φ
Radial distribution function.
Probability of finding an electron at any radius r
Wave Function
Radial Distribution Function
Bohr radius

22. p Orbitals (l = 1) and d Orbitals (l = 2)
p orbital for n = 2, 3, 4, …
( l = 1; ml = -1, 0, 1 )
d orbital for n = 3, 4, 5, …
(l = 2; ml = -2, -1, 0, 1, 2 )

24. Energy Levels (Bound States)

25. Ionization Energy Minimum energy required to remove
an electron from the ground state
Energy of H atom at ground state (n=1) 
Rydberg Constant
Ionization energy of H atom

26. Three Quantum Numbers n: Principal quantum number (n = 1, 2, 3, …)
Determines the energies of the electron
Shells
l: Angular momentum quantum number (l = 0, 1, 2, …, n1)
Determines the angular momentum of the electron
Subshells
Ll =
(s, p, d, f,…)
m: magnetic quantum number (m = 0, 1, 2, …, l)
Determines z-component of angular momentum of the electron
Lz, m =

27. Shells and Subshells Shell: n = 1 (K), 2 (L), 3 (M), 4(N), …
Sub-shell (for each n):
l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1
m = 0, 1, 2, …, l
Number of orbitals in the nth shell: n2
(n2 –fold degeneracy)
Examples : Number of subshells (orbitals)
n = 1 : l = 0 → only 1s (1) → 1
n = 2 : l = 0, 1 → 2s (1) , 2p (3) → 4
n = 3 : l = 0, 1, 2 → 3s (1), 3p (3), 3d (5) → 9

28. Spectroscopic Transitions and Selection Rules
Transition (Change of State)
hcRH
n1, l1,m1
Photon
n2, l2,m2
All possible transitions are not permissible.
Photon has intrinsic spin angular momentum : s = 1
d orbital (l=2)  s orbital (l=0) (X) forbidden
(Photon cannot carry away enough angular momentum.)
Selection rule for hydrogen atom

29. Spectra of Hydrogen Atom (or Hydrogen-Like Atoms)
Balmer, Lyman and Paschen Series (J. Rydberg)
n1 = 1 (Lyman), 2 (Balmer), 3 (Paschen)
n2 = n1+1, n1+2, …
RH = cm-1 (Rydberg constant)
Electric discharge is passed through gaseous hydrogen.
H2 molecules and H atoms emit lights of discrete frequencies.