zeeman splitting and mo theory notes
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Zeeman Splitting and MO Theory NotesTRANSCRIPT
CHEM 209Introductory Spectroscopy & Structure
Dr. W. S. Hopkins30th September, 2015
Zeeman Splitting & MO Theory
Lecture 8
Class Announcements
1. Assignment #1 due Friday
““No amount of experimentation can ever prove me right; a single experiment can prove me wrong.” --- Albert Einstein
Notable Quote:
Lecture 7 RecapMicrostates & Zeeman Splitting
(A more) Complete picture of configurations, terms, levels, & states• Multiple electron systems = rich spectrum• Hund’s Rules help us determine energy ordering
3 | Slide
External fields lift state degeneracy• (2J+1) state degeneracy for each level (field free)• Magnetic Field = Zeeman Effect
Lecture 8 OutlookZeeman Splitting & MO Theory
External fields lift state degeneracy• (2J+1) state degeneracy for each level (field free)• Magnetic Field = Zeeman Effect
2 Atomic Orbitals Combine To Make 2 Molecular Orbitals • Constructive Addition between nuclei = Bonding MO• Destructive Addition between nuclei = Anti-bonding MO• MO occupation → Bond Order
MOs labelled based on symmetry• Linear molecule = axial symmetry• s ≡ s, p ≡ p, d ≡ d,…• u/g (only when centre of inversion)
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Pieter Zeeman1865 - 1943
Top Hat Question
Lecture 8 – Question 1
Atomic oxygen has a 1s2 2s2 2p4 configuration, which gives rise to 3P, 1D, and 1S terms. What is the term symbol for the ground electronic state?
4 | Slide
A. 1S0
B. 1D2
C. 3P2
D. 3P1
E. 3P0
Top Hat Question
Lecture 8 – Question 2
Under field free conditions, how many degenerate quantum states are associated with a 3P1 level?
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A. 1B. 2C. 3
D. 4E. 5
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Lifting State Degeneracy
L and S give rise to magnetic moments. These can interact with an externally applied magnetic field, B
In the absence of external fields, the 3 ML components of a 1P term are degenerate
Interaction with an external B field lifts this degeneracy
Pieter Zeeman1865 - 1943
(quantization axis)
This phenomenon is known as the Zeeman Effect
Where and - the Bohr magneton
Interaction Energy: BMBME LBLe
ee m
e2
eB
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The Normal Zeeman EffectThe Normal Zeeman Effect occurs for states where L > 0 and S = 0
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The Anomalous Zeeman Effect
When S ≠ 0, the spin angular momentum also gives rise to a magnetic moment
When both L and S are non-zero, splittings become more complex:
BMgBJgE Jjeje
Where the Landé g-factor,
)1(2
)1(111)(
JJLLSSJJLSg J
gJ = 1.5 for a 3P2 levelgJ = 2.0 for a 3S1 level
Note: when S = 0, J = L and gJ = 1, which yields the expression for the Normal Zeeman Effect
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The Anomalous Zeeman EffectEnergy levels are split into 2J+1 components dependent on J, L, and S
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ExampleA sodium atom is placed in a magnetic field with B = 5.0 T. How is its ground electronic state effected?
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Combining Atomic Orbitals (AOs)Molecular orbitals (MOs) may be thought of as
combinations of atomic orbitals (AOs)
Hydrogen 1s AO:
Radial Wavefunction
Density Plot
Schematic Representation
Spherical shapemaximum at nucleus
decaying exponentially
Constructive (in-phase) addition:increased electron
density between nuclei (i.e. bond formation)
Density Plot Schematic Representation
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Anti-bonding Orbitals
Destructive (out-of-phase) addition:decreased electron
density between nuclei (i.e. anti-bonding)
Schematic Representation
Rules for Atomic Orbital Combinations:
1. AOs must be physically close enough to interact interaction inversely proportional to distance
2. AOs must combine along an axis of mutual symmetry interaction proportional to cosine of the alignment angle
3. AOs must be similar in size and energy interaction inversely proportional to the difference in size or energy
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Relative Energies of MOs
anti-bonding MO(destabilization)
bonding MO(stabilization)
Out-of-phaseaddition
In-phaseaddition
AO basis set
For MOs of similar symmetry the orbital with more nodes is higher in energy
This method of generating MOs from AOs is called Linear Combination of Atomic Orbitals (LCAO)
Rule of Thumb: 2 AOs yield 2 MOs
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Non-spherical AOs (p-Orbitals)
4 unique combinations
1. in-phase2. out-of-phase
O (2p) O (2p)pg*
pu
sg
su*
pz axial alignment
3. in-phase4. out-of-phase
px & py off-axis alignment
Px, Py constructive overlap
Pz constructive overlap
Px, Py destructive overlap
Pz destructive overlap
sgpu pg* su*
But what do the labels mean?
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MOs of Molecular Oxygen
O (2p) O (2p)1pg*
1pu
3sg
3su*
O (2s) O (2s)
2sg
2su*
O (1s) O (1s)
1sg
1su*
1pg*
1pu
3sg
3su*
2sg
2su*
1sg
1su*
3sg 3su*
2sg
2su*
1sg 1su*
s-orbitals view along z-axis
p-orbitals view along z-axis
1pg* 1pu
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MOs of Molecular Oxygen
1pg*
1pu
3sg
3su*
2sg
2su*
1sg
1su*
1pg*
3su*
g vs. u inversion symmetry
Gerade, g, means even
Y unchanged when inverting through
center of symmetry
Ungerade, u, means uneven
Y changes sign when inverting through
center of symmetry
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2nd Row Homonuclear Diatomics
1pg*
1pu
3sg
3su*
2sg
2su*
1pg*
1pu
3sg
3su*
2sg
2su*
O2Li2 Be2 Ne2B2 C2 N2 F2
Bond Order: 1 0 1 2 3 2 1 0
Bond Order = (# bonding electrons) – (# anti-bonding electrons)2
Top Hat Question
Lecture 8 – Question 3What is the bond order of Li2
+?
18 | Slide
1pg*
1pu
3sg
3su*
2sg
2su*
Li2
A. 0B. 0.5C. 1D. 1.5E. 2
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s-p mixingMolecular orbitals can interact too!
MOs should have the same symmetry and be of similar energies
B (2p) B (2p)1pg*
1pu
3sg
3su*
B (2s)
2sg
2su*
B2
B (2s)
2p3sg
2s2sg
Mainly 2s some (2p) sg
Mainly 2p some (2s) sg
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The Effect of Molecular Orbital Mixing
2p 2p1pg* 1pu
3sg
3su*
2s
2sg
2su*
2s
2p 2p
1pg*
1pu
3sg
3su*
2s
2sg
2su*
2s
The degree to which s-p mixing takes place depends on the energetic proximity of the combining orbitals
Li2, Be2, B2, C2, N2 O2, F2, Ne2
s-p mixing significant s-p mixing not significant
s-p mixing →
21 | Slide
The Effect of Molecular Orbital Mixing
Li2, Be2, B2, C2, N2 O2, F2, Ne2
s-p mixing significant s-p mixing not significant