Lecture 11. Hydrogen Atom

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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 Chemistry
• http//vergil.chemistry.gatech.edu/notes/quantrev
  /quantrev.html

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(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!
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in spherical coordinate
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Angular part (spherical harmonics)
Radial part (Radial equation)
principal quantum no.
, n-1
angular momentum quantum no.
(Laguerre polynom.)
magnetic quantum no.
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Radial Schrödinger Equation
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Wave Functions (Atomic Orbitals) Electronic
States
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Wave Functions (Atomic Orbitals) Electronic
States
Designated by three quantum numbers
Radial Wave Functions Rnl
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Radial Wave Functions Rnl
node
2 nodes
Bohr Radius
Reduced distance
Radial node (? 4, )
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Radial Wave Functions Rnl
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Radial Wave Functions (l 0, m 0) s Orbitals
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Radial Wave Functions (l ? 0)
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Probability Density
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Probability
Wave Function
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Radial Distribution Function
Probability density. Probability of finding an
electron at a point (r,?,f)
Integral over ? and f
Radial distribution function. Probability of
finding an electron at any radius r
Wave Function
Radial Distribution Function
Bohr radius
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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 )
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Energy Levels (Bound States)
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Ionization Energy
Minimum energy required to remove an electron
from the ground state
Energy of H atom at ground state (n1)
?
Rydberg Constant
Ionization energy of H atom
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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
m magnetic quantum number (m 0, ?1, ?2, , ?l)
Determines z-component of angular momentum of
the electron
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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
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Spectroscopic Transitions and Selection Rules
Transition (Change of State)
n1, l1,m1
n2, l2,m2

• All possible transitions are not permissible.
• Photon has intrinsic spin angular momentum s
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• d orbital (l2) ? s orbital (l0) (X) forbidden
  
• (Photon cannot carry away enough angular
  momentum.)

Selection rule for hydrogen atom
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Spectra of Hydrogen Atom (or Hydrogen-Like Atoms)
Balmer, Lyman and Paschen Series (J.
Rydberg) n1 1 (Lyman), 2 (Balmer), 3
(Paschen) n2 n11, n12, RH 109667 cm-1
(Rydberg constant)
Electric discharge is passed through gaseous
hydrogen. H2 molecules and H atoms emit lights of
discrete frequencies.
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