Laser Molecular Spectroscopy CHE466 Fall 2009 David L. Cedeño, Ph.D. Illinois State University...

Laser Molecular SpectroscopyCHE466Fall 2009

David L. Cedeño, Ph.D.Illinois State University

Department of Chemistry

Electronic Spectroscopy

Electron Motion: Orbital and Spin Angular Momentum

Consider one electron in the hydrogen atom. Depending on its energy, electron motion is restricted to certain regions of space in a probabilistic manner. The waveunction associated to such probability is called an orbital. The particular “position” of an electron is dependent on its orbital angular momentum, which is a quantized property (with quantum number l, which depends on principal quantum number n).

Magnitude of orbital angular momentum: for l = 0, 1, 2, …(n-1)

Since an electron is charged, then its motion also generate a magnetic momentum called the spin angular momentum. The spin angular momentum is also quantized.

Magnitude of spin angular momentum: for s = ±1/2

Thus the total angular momentum of an electron is the sum of both orbital and spin angular momentum.

Magnitude of total angular momentum: for j = l+s, l+s-1, …|l-s|

Electronic Motion

)]1([ ll

)]1([ ss

)]1([ jj

Electron Motion: Orbital and Spin Angular Momentum

For multiple electrons in an atom, the angular momentum is defined by the sum of the angular momentum of the individual electrons, but bear in mind that angular momentum are vectors.

Example: Find the angular momentum of an excited state of Helium corresponding to electron configuration 1s1 2p1

The total orbital momentum L = l1 + l2, l1 + l2 – 1, …, |l1 – l2| = 1 + 0 = 1

The total spin momentum S = s1 + s2, s1 + s2 – 1, …, |s1 – s2| = ½ + ½, ½ - ½ = 1, 0

The total angular momentum J = L + S, L + S – 1, …, |L – S| = 2, 1, 0 for L = 1 and S = 1; and 1 for

L = 1, S = 0.

Electron Motion: Spectroscopic Term Symbols

A term symbol is used to identify a given electronic state with a given electron configuration in terms of the angular momentum. For an atom it corresponds to:

2S+1LJ

For the example above there are 4 electronic states associated to 1s1 2p1 configuration: 3P2, 3P1, 3P0

corresponding to the S = 1 (called triplet states) and 1P1 for that with S = 0 (a singlet state)

Electronic Motion

Electron Motion: Angular Momentum and Energy

Hund rules determine the energy ordering of the states of a given electron configuration. These state that for an atom:

1. If the states have the same orbital angular momentum (L), then the one with the greatest spin angular momentum (S) has the lowest energy

2. If two states have the same spin angular momentum (S), then the one with the largest orbital angular momentum (L) has the lowest energy

3. The ordering in terms of J for a given term depends on whether an electron shell is less or more than half-full. For a less than half-full, the one with the lowest J is the lowest energy, for more than half-full, the one with the highest J is the lowest energy.

In the previous example: 3P0 < 3P1 < 3P2 < 1P1

Electronic Motion

Diatomic Molecules: Molecular Orbital Theory

The calculation of the energy of a molecule using the Schrodinger equation is complicated by the fact that a molecule contains more than one electron that is somehow shared at different extents by a set of nuclei. In addition, the interaction between electrons is difficult to represent. These problems translate in our inability to solve the Schrodinger equation exactly for a molecule. Furthermore, suitable molecular wavefunctions are not easy to find.

Molecular orbital theory is a successful approach to obtain the energy of molecular electronic states which utilizes the variational principle to obtain the energy of molecular wavefunctions (i.e. orbitals) that are built as linear combinations of atomic orbitals (i.e. hydrogen-like wavefunctions).

Let us illustrate this methodology with the H2 molecule. Each atom contributes one 1s orbitals to the molecular wavefunctions. As a rule of thumb, the number of molecular orbitals (MOs) obtained is equal to the number of atomic orbitals (AOs) combined. These AOs are combined algebraically:

For H2: = ci(1s)i Therefore there will be two MOs resulting from the combination of 2 1s

AOs.

Electronic Motion in Molecules


The variational principle is used to obtain the energy: in this case the expectation value of the

energy is obtained on the basis of the optimization of the coefficients (c i) in a way that the

minimum possible energy is obtained. This process is usually known as energy minimization. The

expectation value of the energy E is:

Since AO’s are normalized and the Hamiltonian is a Hermitian operator then:

And the quantity


dsscssccssc

dsHscsHsccsHsccsHscE

dscscscsc

dscscHscsc

d

dHE

])1()1()1()1(2)1()1([

])1(ˆ)1()1(ˆ)1()1(ˆ)1()1(ˆ)1([

)])1()1(())1()1([(

)])1()1((ˆ))1()1([(ˆ

2*2

222

*1211

*1

21

2*2

221

*2122

*1211

*1

21

2211*

2211

2211*

2211

*

*

121*22

*1 )1()1()1()1( Sdssdss

121*22

*1

2*21

*1

)1(ˆ)1()1(ˆ)1(

1)1()1()1()1(

HdsHsdsHs

dssdss


The expectation value of the energy E can be written as:

Using the variational principle:

It is possible to have a system of equations that allow us to obtain the energy and coefficients. For

this example:


221221

21

2222122111

21

2

2

cSccc

HcHccHcE

0

ijic

E

0

0)()(

0)()(

221212

121211

22212121

12122111

EHESH

ESHEH

or

EHcESHc

ESHcEHc


This determinant yields a quadratic equation on E with solutions:

is called the resonance integral and has a negative value. is the Coulomb integral. Thus, the state with

E– = ( – )/(1 – S) is higher in energy than the other one with energy E+ = ( + )/(1 + S). Note that For

the hydrogen molecule a is equal to the energy of a hydrogen atom ( = EH)

The coefficients can be obtained from the equations (called secular) and the fact that:

If we assume S = 0, the wavefunction of the MO with the lower energy has c1 = c2 = √2/2 and the

wavefunction for the highest energy has c1 = –c2 = √2/2.


ij

ij

ii

SS

H

HS

E

1

122

21 cc


The molecular orbital energy diagram for H2 is shown below:


g(1s)

u*(1s)

E

–

The molecular orbital energy diagram for valence shell of N2 is shown here.


g(2s)

u*(2s)

E

u(2p)

g*(2p)

g(2p)

u*(2p)


The order of MOs as a function of energy for Li2, Be2, C2 and N2 is:

g(2s) < u*(2s) < u(2p) < g(2p) < g*(2p) < u*(2p)

But this is changed for O2, F2 and Ne2:

g(2s) < u*(2s) < g(2p) < u(2p) < g*(2p) < u*(2p)

The bond order is related to the strength of the molecule and their relative stability:


2

electrons gantibondin# - electrons bonding#BO

Diatomic Molecules: States and Term Symbols

Consider the ground state configuration of N2:

He2g(2s)2 u*(2s)2 u(2p)4 g(2p)2

The total spin (S) is zero, so it is a singlet (2S + 1 = 1). The total orbital angular momentum is also zero.

In order to assign the term symbol we utilize the symmetry symbols of the HOMO via the cross product:

g x g = g. The term symbol is 1g+.

Consider the excited state: He2g(2s)2 u*(2s)2 u(2p)4 g(2p)1 g*(2p)1:

g x u = u. However, there are two possible spin multiplicities 1 and 3 (for S = 0 and 1 respectively).

There is also the possibility of adding the spin and orbital angular momentum, which carries out a split of

the energy of the possible states for such an electron configuration. The sum of these angular momentum

defines a total angular momentum J (called ). For this configuration J = L + S, L + S -1, …, |L – S|. = 1.

For the triplet state: J = 2, 1, and 0; for the singlet state: J = 1. The term symbols are: 1u, 3u,2, 3u,1, 3u,0. Thus four excited states are linked to the electron configuration above. Hund’s rule still follows.



Consider the following electron configuration of O2:

He2g(2s)2 u*(2s)2 g(2p)2 u(2p)4 u*(2p)2

In this case there are two electrons in two degenerate MOs. The cross product is g x g = g+ + g

- + g

The spin multiplicity is singlet or triplet, however Pauli’s exclusion principle forbids two of the possible

triplet state. The only allowed triplet state is the ground state (Hund’s rules) with a term symbol is 3g-.

The negative parity is a consequence of the antisymmetric nature of the electronic wavefuction.

Pauli’s exclusion principle excludes the states with symbols: 3g+ and 3g. The antisymmetric nature of

the electronic wavefunction also excludes the singlet with term symbol 1g-.

This, there are three states for the configuration above: 3g- (ground state), 1g, and 1g

+. The 1g state is

7882.39 cm-1 above the ground state, while the 1g+ is 13120.91 cm-1 above.


g* g*

3g-

g* g*

1g+

g* g*

1g


Another example, NO:

Ground stateHe2(2s)2 *(2s)2 (2p)2 (2p)4 *(2p)1

In this case there is one unpaired electrons in two degenerate MOs. There is no need for cross product,

the ground state has a term symbol The total spin is ½, thus the state is a doublet. The total angular

momentum (spin + orbital) could be (1 + ½) or (1 – ½), that is = 3/2 and ½. The ground state of NO is

the one with the lowest W, thus it has the symbol: 21/2. The other state (23/2) is only 119.73 cm-1 above

the ground state, indicating a small coupling of the spin and orbital momentum. With such a small

separation both states are populated at room temperature (see selection rules later).


Diatomic Molecules: Selection Rules

Whether an electronic transition is allowed depends on the effect of changing the total angular

momentum of the molecule as the induced dipole couples with the photons:

1. = 0, ±1 is allowed

2. S = 0 is allowed (exception to the rule in heavy diatomics like I2, where S = 1 is allowed)

3. = 0, ±1 is allowed

4. For to transitions: + ↔ + and – ↔ – are allowed

5. For homonuclear diatomics: g ↔ u is allowed

6. For molecules containing heavy atoms: regarding the states with W = 0, 0+ ↔ 0+ and 0– ↔ 0– are

allowed.

Example: Would the transition from O2 ground state (3g-) to its first excited state (1g) be allowed?

No because = 2 ( to transition), and S = 1. This is a doubly forbidden transition. Also it violates

the inversion selection rule because g ↔ g.

Example: Would the transition from I2 ground state (1g+) to its second excited state (3u) be allowed?

Yes because = 1 ( to transition) and S = 1 is now allowed (strong spin-orbital coupling). The

inversion rule is also allowed g ↔ u. The last rule could also be considered here, the complete term

symbol for the excited state is 3u+, thus 0+ ↔ 0+ also holds.


Diatomic Molecules: Vibrational coupling

The following shows the visible spectrum of I2 at a resolution of 0.02 nm.

The transition corresponds to the (1g+ → 3u) transition. The bands observed are the result of

changes in the vibrational motion that are concomitant to the changes in electronic motion. Overall there are three band progressions, with transitions originating at v” = 0, 1, and 2.

Electronic Motion

Iodine Gas Absorption Spectrum

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

490 510 530 550 570 590 610 630 650

Wavelength (nm)

Ab

sorb

ance

0.1

0.15

0.2

0.25

540 550 560 570 580 590

Wavelength (nm)

Quantum mechanically, the probability is associated to:

The square of the quantityis the Franck-Condon factor of the transition

The Franck-Condon Principle

The probability of a vibrational transition is dictated by the relative overlap of the potential energy surfaces and the overlap of the vibrational wavefunctions at a particular internuclear distance in the lower electronic state. In essence, the electronic transition occurs before the nuclei move (in other words, from the perspective of electron motion, nuclei are stationary).

Electronic Motion

drRR

drRR

drdR

vveev

vevev

eveveev

"*'

"*'

""ˆ*'*'

drRR

drRR

drdR

vveev

vevev

eveveev

"*'

"*'

""ˆ*'*'

drvv "*'

Franck-Condon Principle and Band Shape

The effect of the electronic transition on the internuclear distance is important in terms of the probability of the vibrational transitions and band shapes.

Electronic Motion

Diatomic Molecules: Molecular Constants and Dissociation Energies from SpectraIt is possible to obtain the dissociation energy from the spectrum by carrying out a combination of differences within a given band progression (i.e. constant v” or v’). In this case the difference between two consecutive bands (with vibrational numbers v’ and v’+1) belonging to a given progression (say v” = 0) is plotted against the value of v’+1. It can be shown that such difference is:

G(v’+1)-G(v’) = e’ – 2e’xe’(v’+1)

Similarly for two bands with vibrational number v” and v”+1 that belong to the same progression (same v’):

G(v”)-G(v”+1) = e” – 2e”xe”(v”+1)

The dissociation energy relative to thev’= 0 (zpe), D0, is the area underthe line from v’+1 = 1 to the max v’+1


Birge-Sponer Plot B state of I2

y = -1.9964x + 133.07

R2 = 0.9984

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80

v' + 1

G(v

'+1)

- G

(v')

Diatomic Molecules: Bond Dissociation Energies from Spectra


Polyatomic Molecules: Molecular Orbital Theory

It is possible to extend the treatment of the LCAO and variational principle to any molecule. Indeed most computational software in the market does this to provide the energy of the orbitals and the contribution of atomic orbitals.

Application: AH2 molecules: This can be bent (C2v) or linear (D∞h). In any case the valence MOs are built from the combination of the 1s orbitals of the H atoms and the ns and np orbitals of the A atom. If the molecule were linear the MOs would be labeled according to the symmetry representation (using the greek symbols), but once the molecule is bent the MOs are named using the a1, a2, b1 or b2 symmetry symbols from the C2v group.The MOs in both geometries are correlated via a Walsh diagram (see next page)



180o90o

2g

1u

u

3g

2u

2a1

3a1

1b2

1b1

4a1

2b2

[1s(H) + 2s(A) + 1s(H)]

[1s(H) + 2pz(A) - 1s(H)]

[1s(H) + 2py(A) + 1s(H)]

[1s(H) + 2px(A) + 1s(H)]

[1s(H) -2s(A) + 1s(H)]

[1s(H) - 2pz(A) - 1s(H)]

Walsh Diagram for HAH Molecule (not at scale)

Energ

y

CHE466 – D. Cedeno 180o90o

2g

1u

u

3g

2u

2a1

3a1

1b2

1b1

4a1

2b2

[1s(H) + 2s(A) + 1s(H)]

[1s(H) + 2pz(A) - 1s(H)]

[1s(H) + 2py(A) + 1s(H)]

[1s(H) + 2px(A) + 1s(H)]

[1s(H) -2s(A) + 1s(H)]

[1s(H) - 2pz(A) - 1s(H)]

Walsh Diagram for HAH Molecule (not at scale)

Energ

y

CHE466 – D. Cedeno

Example:BeH2 is though to have a linear ground state, with 4 valence electrons, the electron configuration is (2g)2(1u)2. Term symbol is 1g

+.The first excited state is bent with electron configuration (2a1)2(1b2)1(3a1)1. The possible term symbols are 3B2 and 1B2.


Example: Formaldehyde.A MO treatment yields the frontier MOs shown. The MOs are labeled according to the symmetry representation (C2v). The energy ordering is 1b1 < 2b2 < 2b1.The ground state is 1A1 (b2 x b2).The lowest energy transition corresponds to a HOMO to LUMO electronic motion. This transition is common in carbonyl containing compounds and is called a n → p* transition. This transition results in two excited states (triplet and singlet) with the same term symbol: A2, thus the 3A2 and 1A2 excited states.

1b1: HOMO-1(a MO)

2b2: HOMO(non bonding)

2b1: LUMO(non bonding)(A * MO)


The Huckel MO MethodFor conjugated molecules involving valence MOs that originate from pure p AOs, the

Huckel method provides a simplified, yet satisfactory modeling ot the MOs. The Huckel method considers a LCAO of pz orbitals from atoms that participate in the conjugation. The secular determinant is given by:

The following approximations apply:1. For any Smn, if m ≠ n, then Smn = 0, if m = n, then Snm = 12. Any Hnn = a (Coulomb integral)3. Any Hmn = b (resonance integral) only if m and n are adjacent (i.e. atoms are bonded),

while Hmn = 0 for atoms that are not bonded.

0

2211

22221221

11121211

EHESHESH

ESHEHESH

ESHESHEH

nnnnnn

nn

nn


Example: BenzeneThe secular determinant for benzene is:

Solutions are x = ±2, ±1, ±1, or E1 = + 2, E2 = E3 = + , E4 = E5 = – , E6 = – 2.

0

10001

11000

01100

00110

00011

10001

makingafter as written becan which

0

000

000

000

000

000

000

x

x

x

x

x

x

β

Eα x

E

E

E

E

E

E

Example: BenzeneThe MOs are represented by a LCAO: i = cii(pz). Six pz AOs produce six MOs.


Images courtesy of http://chemical-quantum-images.blogspot.com/2007/01/benzenes-mos.html

1 (a2u)

2 (e1g) 3 (e1g)

5 (e2u)4 (e2u)

6 (b2g)

Selection Rules in Polyatomic Molecules

Electronic motion is highly allowed in polyatomic molecules. The one rule that hold is the one concerning to the change in the spin angular momentum:

S = 0 (for low spin-orbital coupling)

The selection rules of a polyatomic molecule depends on symmetry properties of the electronic wavefunctions. In terms of transition that depend on the coupling of light with the electric dipole moment (), the rules is:

Г(el’) x Г() x Г(el”) ⊇ A or

Г(el’) x Г(x) x Г(el”) ⊇ AГ(el’) x Г(y) x Г(el”) ⊇ AГ(el’) x Г(z) x Г(el”) ⊇ A

Which is then written as: Г(el’) x Г(el”) ⊇ Г(x) Г(el’) x Г(el”) ⊇ Г(y) Г(el’) x Г(el”) ⊇ Г(z)


Selection Rules in Polyatomic Molecules

Example: The ground state of benzene has a term symbol 1A1g. Determine if the transition to the excited state 1B1u is allowed.

Cross product: A1g x B1u = B1u. For D6h, Г(z) = A2u and Г(x,Ty) = E1u. Since the cross product of the symmetry labels of the excited state does not contain any of the symmetry representations for translation, the transition is NOT allowed.

Note that in order to have a transition from the ground state in benzene, the excited state must be a 1A2u, or 1E1u.


Diffuse Spectra and Photophysical Processes

The UV-Vis spectra of polyatomic molecules (even in the gas phase) is mostly composed of very broad bands. The reason being that many vibrational (and rotational) transitions concur in a relatively small range of frequencies that correspond to the electronic transition.


In cyclohexane solution, Du et al, Photochem. Photobiol. 1998, 68, 141

This spectrum of pyrene shows 3 distinctive bands, in which some vibrational progressions are evident. The ground state of pyrene (D2h) is 1Ag, with allowed transitions into 1B1u and 1B2u and 1B3u excited states.

Photophysical Processes


Ab

sorp

tio

n

IC

Flu

ore

scen

ce

Phosphorescence

ISC

ISC

1M 0

1M 1

3M 1

Ab

sorp

tio

n

IC

Flu

ore

scen

ce

Phosphorescence

ISC

ISC

1M 0

1M 1

3M 1

Radiative ProcessesFluorescence: Light emission S = 0Phosphorescence: Light emission S =1

Non-Radiative ProcessesInternal Conversion: heat release via vibrational relaxation induced by collisions (S = 0)Intersystem Crossing: heat release as a result of a spin-orbital change induced by internal or external factors (S = 1)

Photophysical Processes

Quantum Yields: In any process initiated by light, the quantum yield is the efficiency of the process per every 100 photons absorbed. For example, a 0.12 fluorescence quantum yield means that out of 100 photons absorbed by the compound, 12 of them are converted into emitted photons.

The quantum yield of a photophysical process depends on the relative rates of competing processes. For example the fluorescence quantum yield is defined as:

Where the kf, kISC and kIC are the rate constants for fluorescence, intersystem crossing and internal conversion respectively. Note that the sum of these rate constants represent the rate constant for decay of the excited singlet state.

The rates of photophysical processes and therefore the quantum yields are thus dependent on the structure of the compound and the molecular environment around it.


ISCICf

ff kkk

k

Photophysical Processes: Effects of excitation on molecular structure

From N. Turro, Modern Molecular Photochemistry, 1st Ed, 1978

Photophysical Processes: Chromophores and Orbital Configurations (Typical)


Photophysical Processes: Effect of Environmental Conditions


Laser Molecular Spectroscopy CHE466 Fall 2009 David L. Cedeño, Ph.D. Illinois State University...

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Transcript of Laser Molecular Spectroscopy CHE466 Fall 2009 David L. Cedeño, Ph.D. Illinois State University...