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 http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html

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!

in spherical coordinate

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.

Radial Schrödinger Equation

Wave Functions (Atomic Orbitals): Electronic States nlm nl

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

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

Radial Wave Functions Rnl

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

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

Probability Density

Wave Function Probability

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

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 )

Energy Levels (Bound States)

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

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 =

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

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

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 = 109667 cm-1 (Rydberg constant) Electric discharge is passed through gaseous hydrogen. H2 molecules and H atoms emit lights of discrete frequencies.