Lecture 11. Hydrogen Atom References Engel, Ch. 9 Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.3 Introductory Quantum Mechanics, R.
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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
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