chem516.03.intro to spectroscopy - University Of...

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© K. S. Suslick, 2013

Applications of Group Theory to Spectroscopy

1. Introduction to Spectroscopy

Selection RulesSymmetry and Allowedness

2. Vibrational Spectroscopy

Raman & IRSymmetry of Vibrational ModesAllowedness of Modes

(seeing bands in IR & Raman)

3. Electronic Spectroscopy


Origins of Spectroscopy

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Sir George Gabriel Stokes, 1st Baronet

b., 13 Aug. 1819, Skreen, County Sligo, Ireland. d., 1 Feb. 1903, age 83, Cambridge.

The Lucasian professor of mathematics at Cambridge, 1849 – 1903. 1st u.g. physics labs!

1st suggested spectroscopic analysis ~1850. (e.g., the application of the prismatic analysis of light to solar and stellar chemistry.)

Described and named phenomenon of fluorescence. “On the Change of Refrangibilityof Light” Phil. Trans. 1852.

Founded modern hydrodynamics, 1845. Navier-Stokes Equation: the motion of fluids.


Sir George Gabriel Stokes, 1st Baronet

In 1899, on the 50th anniversary of Stokes’ Lucasian professorship, Lord Kelvin (Stokes’ protégé) presented Pembroke College, Cambridge, with a marble bust of Stokes.

The sculpture: Hamo (Sir William) Thornycroft.

Sir (William) Hamo Thornycroft RA(1850–1925)

1918 Bust of Sir John Isaac

Thornycroft(1843–1928)

His brother, Sir John Isaac Thornycroft, was a successful naval engineer.

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Sir John Isaac Thornycroft

H.M.S. Daring, the first high speed torpedo boat destroyer (185 ft.), built in 1893 by Sir John Thornycroft.

Failure of initial propeller led to discovery and naming of “cavitation”, which is also the source of sonochemistry & sonoluminescence.

“Torpedo-boat Destroyers. (Including Appendix & Plate At Back Of Volume)”, J.I. Thornycroft and S.W. Barnaby, Min. Proc. Inst. Civil Engineers, 122, 51-69 (1885)


Survey of Spectroscopic Techniquescm-1 108 107 105 104 103 102 10 1 0.1 0.01 0.001

thz

1 a.u.=

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cm-1 108 107 105 104 103 102 10 1 0.1 0.01 0.001

thz

Survey of Spectroscopic Techniques


Survey of Spectroscopic Techniques

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Interaction of Light and Molecules

y

z x

=

i.e., an oscillating perturbation of the electric field in time and space;ditto, magnetic field.



mole of photons = 1 einstein (ein)

400 nm ≡ 25,000 cm-1

≡ 25 kK ≡ 300 kJ / ein

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Every spectroscopic technique has an intrinsic timescale,shorten than which dynamic changes cannot be resolved.

Uncertainty Principle limits temporal resolution:



Temporal resolution is determined BOTHby the wavelength of the technique, andby the separation energy between the peaks.

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Lifetime for Infrared Spectra

Consider Case I, with two peaks just broadening:


Spectroscopic Lifetime Resolution

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Lifetime vs. Spectroscopic Technique

Consider mixed valence compounds (e.g., FeII – FeIII )


Lifetime vs. Spectroscopic Technique

Consider mixed valence compounds (e.g., FeII – FeIII ):now using Mössbauer vs. temperature:

lifetime ~ 10-8 s

lifetime ~ 10-7 s

lifetime ~ 10-6 s

lifetime > 10-6 s

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UV-vis spectra of dn metal ions in water.

Where do all these bandscome from?!

Thus, originated spectroscopy:where do all the colors come from?!


States vs. Orbitals

Let us now begin to look at the interaction of light with molecules (i.e., the origins of “spectroscopy”).

How does one state (e.g., ground) get perturbed into another state (e.g., excited)

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Expressing States: Dirac Notation

Any state may be represented as a vector of quantum numbers.

Under Dirac notation, we denote a vector using a “ket”: |>

c |> is another vector where c is an operator

|a> + |b> is also some other vector

|> - |> is the null vector, |0> or just 0.

(Aside: Hilbert space = a vector space containingall possible states of a system

an operator, c, simple takes one vector into another vector in that space, i.e., taking one state to another state. )

^

^


Dirac Notation

Any vector Xn can be represented two ways

Ket

|n>

z

y

x

w

v

Bra

<n| = |n>t

***** zyxwv

*m is the complex conjugate of m.

and |n>t is the transposed vector.

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Dirac Notation

An Inner Product is a Bra multiplied by a Ket:

<a| ● |b> is written <a|b> and is a scalar (i.e., a number)

<a|b> =

p

o

n

m

l

***** zyxwv***** pzoynxmwlv


Dirac Notation

|a><b| =

p

o

n

m

l

=

*****

*****

*****

*****

*****

pzpypxpwpv

ozoyoxowov

nznynxnwnv

mzmymxmwmv

lzlylxlwlv

***** zyxwv

By Definition acbcba

An Outer Product is a Ket multiplied by a Bra

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Dirac Notation

1 11 12 1 1

2 21 22 2 2

1 2

matrix multiplicationˆTherefore the action of is simply equivalent to :

'

'

'

ând can then

n

n

n n n nn n

A

v A A A v

v A A A v

v A A A v

A

be represented by an matrix.n n

1 1 2 2 1 1 2 2

Â takes any vector in a linear vector space to another vector

ˆ în that space: ' and satisfies:

linear o

' '

perator A

A v v A c v c v c v c v


Dirac Notation

Within the context of quantum mechanics, we can

consider the Dirac Inner Product, <X|A>, to mean

“the probability amplitude that state A becomes/arrives-at/is-converted-to state X.”(The inner product is read right to left.)

<X|A> is equivalent to XA d integrated over all space.

If we ask, “what is the probability of an electron at A being scattered by atom i to location B” then we write

<B|i>S<i|A> where S is the scattering function/operator.

and BSi A d = <B|Si|A>

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Dirac Notation

DiracChemists QM


Dirac Notation

< Q >

and if Ψ is already normalized (as we usually assume):

< Q Ψ> = < Q > = <Ψ|Q|Ψ>

and if Q is the Hamiltonianthen H|Ψ> = E|Ψ>

^

ExpectationValue of Q

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Dirac Notation

Within the context of electronic structure,

inter-electron repulsion is so commonly wanted,

that you will also see a special “diracish” notation

to represent the Coulombic repulsion operator,

1/r, where r is the inter-electron distance operator:

( j|i ) = <j|r-1|i> = j (1/rij) i d = repulsion between

electrons j and i



ICBST,

HoΨ = EΨ˄

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t

So, a Selection Rule just asks when is

(i.e., rate of excitation)

Fermi’s Golden Rule

“transition moment integral”



ICBST: The interaction of light with matter comes

overwhelmingly from interaction of an electric dipole

(in the molecule) with the electric field component of light.

i.e., H’ is the electric dipole operator

Interactions between light and either molecular quadrupole moments

or magnetic dipoles are pretty much negligible.

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Light Absorption Selection Rule

So, all Selection Rules grow out of the Transition Moment Integral:


Oscillator Strength: The Allowedness Observable

where 0 ≤ f ≤ 1

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Extinction Coefficient and Oscillator Strength

Beer’s Law:



i.e., the Dipole Moment (M or µ) must change during excitation

ICBST

where µlm is the change in dipole moment from the ground to excited state.

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(i.e., transition is “allowed”)


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Selection Rules and Group Theory

For an allowed transition:

But that will be true iff:


Group Theory View of Selection Rules

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All f2(x) must include an A1g (totally symmetric) function.

Group Theory View of Selection Rules

or any non-E sym op


Absorption Selection Rule

Summary: During the transition, the dipole moment must changefrom the GS to the XS.

More later about the group theory consequences of this.

where Ψes = excited state wavefunction

Ψgs = ground state wavefunction

M = dipole moment operator (x, y, z)^ ^ ^ ^

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Absorption Selection Rule vs. Spin

Consider the wavefunction that includes spin.To a first approximation, the dipole moment operator doesn’t interact

with spin (when it does, we call it spin-orbit coupling).

So, we can separate spin from the rest of the wavefunction:

Ψ = Ψspin Ψorbital and

M(Ψspin Ψorbital) = Ψspin M(Ψorbital)

So we now have to worry, does Ψspin ES Ψspin GS = 0 integrated over all space.

During the transition, the dipole moment must changefrom the GS to the XS.

What's the role of spin in this?


Spin Selection Rule

This is the strictest of electronic selection rules (more later) (at least before 2nd row TMs).

Assumes (1) spin can be separated from orbital functions, and(2) dipole operator does notaffect spin.

(l = GS, m = XS)

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For the Total Spin of an atom or molecule the rules apply:

1. Doubly occupied orbitals do NOT contribute to the total Spin

2. Singly occupied orbitals can be occupied with either spin-up or spin-down e-

3. Unpaired e- can be coupled parallel or antiparallel, giving a total spin S

4. For a state with total spin S there are 2S+1 “components” with M = S,S-1,...,-S. Hence terms singlet, doublet, triplet, …

5. The MS quantum number is always the sum of all individual ms QNs.

Spin


Spin Names (aside, more later)

“Russel-Saunders” Term Symbols

Examples for dn configurations:

doublet

sextet

triplet

for atoms; Irr Rep Mulliken for molecules

Atoms Molecules

2H

6A

3l=1 or 3

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