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

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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?

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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

+?

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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 →

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The Effect of Molecular Orbital Mixing

Li2, Be2, B2, C2, N2 O2, F2, Ne2

s-p mixing significant s-p mixing not significant