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

(2-Body Problem)

r

Ze

mmVEEH N

Ne

enucleusKelectronK

0

22

22

2

,, 422ˆˆ

Electron coordinate

Nucleus coordinate

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!

r

ZeH

0

22

2

42

eNe mmm

1111

Separation of Internal Motion: Born-Oppenheimer Approximation

in spherical coordinate

angular momentum quantum no.

magnetic quantum no.

Angular part (spherical harmonics) Radial part (Radial equation)

...3,2,1 with 32 222

02

42

nne

eZEn

, n1principal quantum no.

nln

l

lnln eLn

NrR 2/,,, )()(

(Laguerre polynom.)

Radial Schrödinger Equation

2

20

0

4

ema

e

Wave Functions (Atomic Orbitals): Electronic States

nlm nl

Designated by three quantum numbers

Wave Functions (Atomic Orbitals): Electronic States

nlm nl

Radial Wave Functions Rnl

Radial Wave Functions Rnl

1s

2s

2p

3s

3p

3d

*Reduced distance

*Bohr Radius

2

20

0

4

ema

e

0

2

a

ZrRadial node

(ρ = 4, ) Zar /2 0

2 nodesnode

Radial Wave Functions Rnl

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

Radial Wave Functions (l 0)

2p

3p

3d

Probability Density

ProbabilityWave Function

Probability density. Probability of finding an electron at a point (r,θ,φ)

2

224)( rrP

0/2230

34)( aZrer

a

ZrP

Radial Distribution Function

Integral over θ and φ

Wave Function Radial Distribution Function

Bohr radius

Radial distribution function. Probability of finding an electron at any radius r

0/22 aZre

p orbital for n = 2, 3, 4, … ( l = 1; ml = 1, 0, 1 )

p Orbitals (l = 1) and d Orbitals (l = 2)

d orbital for n = 3, 4, 5, … (l = 2; ml = 2, 1, 0, 1, 2 )

Energy Levels (Bound States)

2

20

0

4

ema

e

32 22

02

4

1 e

ehcRE H

H

Energy of H atom at ground state (n=1)

HhcRI Ionization energy of H atom

Rydberg Constant

...3,2,1 with 32 222

02

42

nne

eZEn

32 22

02

4

e

ehcR H

H

Minimum energy required to remove

an electron from the ground state

Ionization Energy

2

20

0

4

ema

e

n: Principal quantum number (n = 1, 2, 3, …)Determines the energies of the electron

...3,2,1 with 32 222

02

42

nne

eZEn

Shells

Subshells

l ,...,2,1,0m with m llLz, m =

1,..,1,0 with 1)l(l 1/2 nlLl = (s, p, d, f,…)

Three Quantum Numbers

l: Angular momentum quantum number (l = 0, 1, 2, …, n1)

Determines the angular momentum of the electron

m: magnetic quantum number (m = 0, 1, 2, …, l) Determines z-component of angular momentum of the

electron

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), …, n1m = 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

Shells and Subshells...3,2,1 with

32 2220

2

42

nne

eZEn

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.)

n1, l1,m1

n2, l2,m2

PhotonhvE

Spectroscopic Transitions and Selection Rules

Selection rule for hydrogen atom 1,0 lm1l

Transition (Change of State)

22

21

11~nn

RH

hcRH

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)

Spectra of Hydrogen Atom (or Hydrogen-Like Atoms)

Electric discharge is passed through gaseous hydrogen.H2 molecules and H atoms emit lights of discrete frequencies.

22

21

11~nn

RH