Chemistry 126 – Molecular Spectra & Molecular Structure

Week # 6 – Electronic Spectroscopy and Non-Radiative Processes

As was noted briefly last week, the ultraviolet spectroscopy of formaldehyde(specifically the π → π∗ and n → σ∗ transitions) is complicated by the fact that at 3eV or more of excitation the spectral lines are broadened by processes which limit thelifetime of the excited state. The energetics for the photochemical processes in H2CO areoutlined in the figure below:

2H COCO + H 2

H + HCO

H + HCO

?

?

?

A32

1A2

A11

( S) + ( )Π22

( S) + ( )2 2Σ

( Σ) + ( Σ )1 1 +g

n π∗

Figure 20.1– The photochemical energetics for formaldehyde photodissociation.

A major question in how fast the various photodissociation processes can occur is the natureof the barriers involved. That is, what are the barriers to the photochemical productionof H2 + CO or H + HCO from formaldehyde? The energetics are easy to lay out, butif large activation energies are involved then it may take photons of considerably greaterenergy to drive the photochemistry. Other questions concern the nature of singlet-tripletcoupling (i.e. is the 3A2 state involved in the decay of 1A2?), etc.

Studies of the isolated benzene molecule in the late 1960’s also raised a number ofinteresting questions. It was found, for example, that if the fluorescence yield (that is, thenumber of photons emitted/number of molecules excited) from the excitation into the firstexcited singlet state was examined as a function of pressure in a static cell, even at verylow pressures only about 20% of the molecules excited actually emitted photons. Strangerstill, if the fluorescence yield was examined as a function of vibrational energy contentin the A state, the yield started at 20% and then dropped sharply toward zero above anexcitation energy of 39,682 cm−1. These results are summarized in Figure 20.2 below. Thefact that the fluorescence yield was well below zero, and changed rapidly with excitation,was cited as a “breakdown of quantum mechanics” by the research teams involved!

142

0.2 10torr Ar

0.2 0.2

39,682 cm-1

Frequency

Φ Φf f

6 6C H

Figure 20.2– (Left) The fluorescence yield from the S1 state of benzene as a function ofbackground gas (in this case, Ar) pressure. Even at very low pressures the quantum yieldis ≪ 1 [J. Chem. Phys. 51, 1982(1969)]. (Right) The fluorescence yield from the S1 stateof benzene as a function of internal energy in the S1 state. The rapid drop in emissionabove a certain threshold energy was called the “channel three problem,” and working thisproblem out contributed greatly to our understanding of energy flow in isolated molecules[J. Chem. Phys. 46, 674(1967)].

This drop in fluorescence yield is now understood to arise from non-radiative processeswhich operate even on isolated molecules once their density of states becomes sufficientlyhigh. Pictorially, we use plots of the electronic state energies shown below, called Yablonskidiagrams, to illustrate the competition between radiative and non-radiative processes inthe energy flow within molecules after electronic excitation. Figure 20.3 below outlinesthe “kinetic rates” at which processes such as fluorescence, internal conversion, etc. occur,while Figure 17.4 stresses the important role that high lying vibrational states play inmany of these processes, especially those of internal conversion and intersystem crossing.

s-1kisc , 10 - 104 12

kic , 10 - 10 6 12

k , 10 - 10 sp -2 4 -1

s-1

, 10 - 10 6 9k s-1 s-1kisc , 10 - 104 12Fluorescence

Excitation

InternalConversion

Intersystem crossing

Phosphorescence

f

Intersystem crossing

Figure 20.3 A summary of the radiative (→) and non-radiative processes – along withtheir rates – in electronic spectroscopy.

143

Figure 20.4 The role of vibrational excitation and state density in the balance betweenradiative (→) and non-radiative processes after electronic excitation.

The quantum yield for fluorescence can thus be written as

Φf = kf/∑

i

ki ,

where i runs over all the possible radiative and non-radiative processes that can occur.Those include:

kf = fluorescence (spin, dipole allowed, relatively fast)kp = phosphorescence (spin, forbidden, typically slow for low z)kic = internal conversion (non− radiative transfer in the same spin manifold)kisc = intersystem crossing (non− radiative transfer to different spin manifolds)

The lifetime of the upper state, τf , is determined by all of the processes,

τf = 1/∑

i

ki ,

and so the quantum yield may be re-expressed in the form

kf = Φf/τf .

Consistency checks for the presence of fast, non-radiative processes can be performedby checking the magnitudes of the Einstein A and B coefficients. For purely radiativeprocesses, the relationships derived in Lecture #8 are valid, but the effective value of A

144

will be much larger if processes such as internal conversion and/or intersystem crossingoccur. We’ll next look at simple two versus three level modes of radiationless processes.

The Douglas Effect

Consider the two cases outlined in Figure 20.5 below. On the left we have a simpletwo level system. For such a system, the fluorescence yield must be unity, that is Φf = 1,and all the photons one expect are radiated (there is no place else for the energy to go).Thus, c1(t)

2 + c2(t)2 = 1 .

ψ

µ

µ

a0 = 0

= 0l 0

ψ0

a ψl

1

2

V

vs.

ε

Figure 20.5 The simplest quantum mechanical model of radiationless processes. On theleft is a simple two-level system. Non-radiative processes are included by adding a thirdlevel, degenerate with the upper state, for which the electric dipole matrix element withthe lower state is zero (right).

Life gets more interesting when a third level is added, as is shown on the right of Figure20.5. Let us suppose that, for a given H◦, the upper states are degenerate, that is

H◦|ψa > = ǫ|ψa > and H◦|ψl > = ǫ|ψl > .

Suppose further it is possible to pump to ψa, that is the 0 → a transition is allowed, butthat the 0 → l transition is forbidden (the H atom is one such system). ψa is called thebright state, and ψl the dark state.

Now, turn on a perturbing, or interaction, Hamiltonian H ′ with strength V .Degenerate perturbation theory tells us the energies of the two states in the fullHamiltonian H◦ +H ′ are given by

V

ε V

ε= ε + V- 2V

|->

|+> ε

ε

+ V

- V

with wavefunctions of the form

|− > =

√

1

2[|ψa > − |ψl >]

|+ > =

√

1

2[|ψa > + |ψl >]

145

Clearly, both of the |0 >→ |− > and |0 >→ |+ > transitions are now allowed, at leastformally. Let’s consider two limiting cases for t = 0, one in which |ϕ(0) >= |ϕ(+) >, andone in which |ϕ(0) >= |ψa >.

(A) t = 0; let |ϕ(0)> = |ϕ+>. In a real system, this would mean using a laser whosebandwidth ∆ω was such that ∆ω < 2V . Since |ϕ(+)> is an eigenstate of the totalHamiltonian, we have

Hϕ± = ǫ±ϕ± = (ǫ± V )ϕ± .

The time dependence for ϕ+ is

Φ(t) = U(t)ϕ+(0) = e−iHt/hϕ+(0) = e−iω+t/hϕ+(0) ,

where ω+ = (ǫ+ V )/h. Thus, we see that ϕ+ is a stationary state of the system since

|Φ(t)|2 = |ϕ+(0)|2 .

The same is true, of course, for ϕ−

(B) t = 0; let |ϕ(0)> = |ϕa> = [|ϕ+> + |ϕ−>]/√

(2), that is, suppose we prepare onlythe bright state ϕa which is composed of equal parts ϕ+ and ϕ− (which requres a laserwith ∆ω > 2V ). In this case,

|Φ(t)|2 = |e−iHt/h[ϕ+(0) + ϕ−(0)]/√

(2)|2 =1

2[|ϕ+|

2 + |ϕ−|2 + ϕ∗

+ϕ−e−i∆ωt + c.c.] ,

where c.c.= complex conjugate (of the third term), and ∆ω = ω+ − ω− ∝ 2V . Thus,because ψa is not an eigenstate of the full Hamiltonian the probability for finding thesystem in the ψa bright state, which is the one that fluoresces and hence is observableexperimentally, oscillates; and the stronger the interaction matrix element V the fasterthe oscillation is. This in turn induces an uncertainty in the energy which is proportionalto the splitting. Since the fluorescence is related to the probability distribution, which isgiven by

|ψ|2

t

πP (t) = [1 + cos(4 Vt/h)]/2

P

P

l

aa

where the so-called recurrence time is found to be τ = πh/V .The fluorescence decay is simply

Fluorescence ∼dP

dt= | < ψex|µ|ψgnd > |2

4ω3

3c3h.

Suppose the excited state is written in terms of the ψa and ψl states discussed above, or

ψex = α(t)ψa + β(t)ψl .

146

We can rewrite the above and find that

dP

dt= |α(t)|2|µ0a|

2 4ω3

3c3h,

since |µ0l| = 0. Clearly, the experimentally measured lifetime must be longer (that is, thedecay rate slower) since

|µ0a|2effective = |α(t)|2|µ0a|

2 < |µ0a|2 .

This is called the Douglas effect.

(C) Suppose there are many |l> dark levels. For a pure continuum the probabilityof recurrence into |a> is extremely small. Thus, the Douglas effect transforms intoan irreversible non-radiative process (much like photodissocation) when the density ofvibrational states becomes large. This is illustrated in Figure 20.6. The energy uncertaintyis defined by the spread of the quasi-continuous |l> states. In general, this process ofrandomization of vibrational energy from a bright initial state to a range of dark statesthat are more or less statistical in nature is called Intramolecular Vibrational EnergyRedistribution, or IVR, and its rate can be roughly calculated using Fermi’s Golden Rule.In small molecules at low vibrational state densities IVR is slow, but for molecules as largeas benzene or larger, the IVR timescales can be of order picoseconds or faster.

ψ

µ

µ

a0 = 0

= 0l 0

ψ0

a ψl{ }V

Figure 20.6 An outline of IVR, where the irreversible non-radiative process is driven bythe high rovibrational state density of the dark |l> levels.

For the process of InterSystem Crossing, or ISC, the coupling occurs between singlet(|a>) and triplet (|l>) levels (or in general, between any two different spin manifolds,singlet-triplet, doublet-quartet, etc.). How does this take place?

Spin-Orbit Coupling in Molecules

In the Born-Oppenheimer approximation, the full wavefunction is written as a productof spatial and spin terms. If σg and σT are the ground and excited triplet state spinwavefunctions, then

1S0 → 3T1 ⇒ < ϕgσg|µ|ϕTσT > = < ϕg|µ|ϕT >< σg|σT > = 0

147

due to the orthogonality of the spin wavefunctions. However, the spin-orbit term

H ′ ∝∑

i

Li · Si

can couple the excited state singlets and triplets (that is, the normal states we draw in theYablonski diagrams are not the true eigenstates of the total Hamiltonian, just as the |nl>states alone are not the true eigenstates of the hydrogen atom)!

In the perturbed basis set, perturbation theory tells us the the true tripletwavefunction should be written

Ψtriplet = αϕTσT + βϕsσs ,

and that the phosphorescence rate is thus related to

< Ψground|µ|Ψtriplet > = α < ϕgσg|µ|ϕTσT > + β < ϕgσg|µ|ϕsσs > .

The first term is zero, but the second is not, and so

Phosphorescence ∝ |µgs|2β2 ,

where β ∼ H ′s.o./∆E. In these formulae, |µgs| is the singlet-singlet dipole coupling matrix

element, and since it is (or can be) non-zero, the triplet lifetime is finite. The spin-orbitmatrix element, H ′

s.o. is often of order 0.1 cm−1 for compounds involving first- or second-raw atoms, while ∆E, the electronic state energy separation, is more typically 104 cm−1.Thus, triplet lifetimes are some (105)2 times longer than that for singlet-singlet fluorescencefor most molecules. However, if heavy atoms are involved (such as V- or Os-containingporphyrins that are found in the environment), the spin-orbit coupling is much larger andthe phosphorescence yield can be much larger.

ISC in H2CO

As a quantitative example of how intersystem crossing occurs, let’s look at therelatively simple case of formaldehyde. As we saw before, for the n → π∗ excitation,the 1nπ∗ configuration has A2 symmetry in its spatial wavefunction. For singlets, thespin function has a1 symmetry, while for the triplets the three spin wavefunctions havea2, b1, and b2 symmetry. Thus, the symmetry of the singlet and triplet manifolds offormaldehyde have the symmetries outlined in Figure 17.7. Let Sx, Ty be given tripletstates. As always, for an interaction to occur we demand that the interaction matrixelement be of A1 symmetry, that is

< Sx|Hs.o.|Ty > ≡ A1

for the case of intersystem crossing. Since the spin-orbit interaction is a scalar, it is clearlyof A1 symmetry. Thus, for formaldehyde (and by inference other C2v species as well), thesinglet and triplet states must have at least one shared symmetry type for the coupling tobe finite. For example:

148

Singlet Symm. Triplet Symm. Interaction?

n→ π∗ 1A2 n→ π∗ 3A1,3B1,

3B2 Noπ → π∗ 1A1 n→ π∗ 3A1,

3B1,3B2 Yes

n→ σ∗ 1B2 n→ π∗ 3A1,3B1,

3B2 Yesπ → π∗ 1A1 π → π∗ 3A2,

3B1,3B2 No

. .

. .

. .

1B2 3B2

a

bb1

2

2

3B1

A3 1

3A2=n 1

n

σ∗σ∗3

1A2 3A2

a

bb1

2

2

A3 13B23B1

=

π∗nπ∗n 1

3

A3 1

3B2

3A23B1

a

bb1

2

2A11

=

π∗π∗1

3ππ

Figure 20.7 An outline of the potential singlet-triplet coupling in formaldehyde drivenby the spin-orbit interaction. Arrows denote states that are coupled, those with crossescannot interact by symmetry.

The interactions outlined above can be generalized, in that for C2v molecules singletsand triplets of the same configuration (nπ∗, etc.) do not interact, those from differentconfigurations do. Again, the spin orbit matrix element is of order Hs.o ∼ 0.1 cm−1, orabout 3 GHz. In this case, the appropriate level spacing (the ∆E above) is of the sameorder since it is the rovibrational states that are coupled, not isolated vibrations. Thus,the time scale for ISC is similar to that for fluorescence or even faster, that is τISC ≤ 10−9

sec. So, we expect the coupling to be fast and irreversible even for isolated molecules!

149

Intermolecular, or Forster, Energy Transfer

In the liquid or solid state, molecules are sufficiently close together that it is possiblefor the excitation on one chromophore (called the donor) to be transferred to anotherchromophore (called the acceptor). The most long range versus of this phenomenon usesnon-radiative singlet energy transfer, and is called Forster transfer after the scientist whofirst described the mechanism nearly 50 years ago.

Phenomenologically, the efficiency of Forster energy transfer is found to vary asηForster = R6

0/(R60 + R6), where R is the donor-acceptor separation and R0 is the

characteristic, or Forster, radius. Thus, the efficiency can be essentially unity for distanceswithin the Forster radius, and falls to nearly zero for distances larger than ∼ 2R0. Whatgoes into determining R0?

Figure 17.8 presents the basic functional form of Forster energy transfer and stepsinvolved. Qualitatively, what occurs is that with sufficient overlap in the emission spectrumof the donor and the absorption spectrum of the acceptor, an energy transfer process is setup whereby it is possible to generate emission from the acceptor by absorbing photons atshorter wavelengths where only the donor chromophore absorbs light.

Figure 20.8 An overview of the important steps in Forster energy transfer.

In his original work, Forster showed that the value of R0 for a donor-acceptor pairdepends on properties that can be measured separately for the donor and the acceptor,and has the form

R60 = QDeAOn

−4 ,

where QD is the fluorescence quantum yield for the donor, eA is the extinction coefficientfor the acceptor, n is the refractive index of the medium, and O is an overlap integral ofthe emission from the donor and the excitation of the acceptor. In addition to the spectralproperties of the donor and acceptor, the overlap integral O depends upon the dipolar

150

coupling between the electronic dipole matrix elements (and so is sensitive to variations inthe geometrical angles between the donor and acceptor).

Pairs of chromophores are known with R0 covering the range 1-10 nm. Experimentalverification for Forster’s formula for the efficiency of nonradiative singlet energy transferhas been achieved by attaching the donor and acceptor to a rigid molecule, so that thedistance R is known. Alternatively, polymers with a strong preference for the formation ofa helix of known geometry, with the donor attached at one end, and the acceptor attachedat the other end, can be used. This helix provides a particularly strong test, becausepolymers of different molecular weight will form helices of different lengths. One suchstudy used poly(L-proline) as the polymer (Stryer, L. & Haugland, R. P. 1967, “EnergyTransfer: A Spectroscopic Ruler” Proc. Natl. Acad. Sci., 58, 719-726). In the solventsystem used for this work, poly(L-proline) forms a left-handed helix with 3 units per turn,and a length of 0.30 nm per unit. The samples studied had degrees of polymerizationranging from 5 to 12, with the donor (naphthyl) at one end, and the acceptor (dansyl)at the other end. Forster’s equation yields an R0 of 2.72 nm for this donor-acceptorpair. The measured efficiencies were in good agreement with theoretical prediction. Morerecently, Forster energy transfer has become an extremely powerful means of examiningthe dynamical and folding properties of biopolymers such as DNA or proteins.

A spectacular natural example of Forster energy transfer is provided by thephotosynthetic reaction complex (PRC) shown in Figure 20.9. In this system, lightgathering chromophores transfer their electronic energy over very long distances to theprotein at the center of the PRC with nearly unit efficiency (apart from the loss incurringby the energy difference between the donor-acceptor states).

Figure 20.9 A schematic depiction of the photosynthetic reaction complex in purplebacteria. By transferring the energy over such long distances, the PRC can separatelyoptimize the light collection and chemical properties of the system.

151

Chemistry 126 – Molecular Spectra & Molecular Structure

Week # 6 – Electronic Spectroscopy of Periodic Solids

Previously we have investigated the vibrational modes of periodic solids using theharmonic potential approximation. Here we are interested in the electronic propertiesof such materials. From the LCAO-MO perspective, we could think about combininghydrogen orbitals in a periodic potential that mimics atomic structure. This can be done,but for the purposes of illustration we will look at a one-dimensional model that replacesthe long range potential associated with Coloumb attraction with a simpler (positive)square-well potential whose height and width are variable. This is called the Kronig-Penney model for the electronic structure of periodic solids. The Atkins & Friedman texton reserve has a nice summary of this model, from which the following notes are derived.

The Kronig-Penney Model of the Electronic Structure of Solids

When we thought about the vibrational spectroscopy of crystalline solids, theperiodicity of the lattice assumed paramount importance. Not surprisingly, the sameis true for the electronic behavior of such materials. Here we briefly outline the simplestone-dimensional model of the so-called band structure of solids, called the Kronig-Penneymodel. A pictorial representation of this approach is shown below.

Figure 21.1 – A schematic of the 1-D potential that defines the Kronig-Penney model ofthe electronic structure of crystalline solids.

Briefly, we consider an infinite array of square wells, with a potential barrier heightof V0, a barrier width of b, and a well spacing of a. Formally, the solutions we will presentbelow are valid for periodic potentials where V0b=constant. In this model, it is clear thatV (x+ a) = V (x). Under such conditions, the time independent Schrodinger equation forthe electrons in the solid has a solution of the form

ψq(x) = uq(x)eiqx uq(x+ a) = uq(x) . (21.1)

152

The periodic functions uq(x) are called Bloch functions. As in our analysis of the vibrationsof solids, the behavior of the energy with the wave vectors q will be pivotal.

If we insert eq. (21.1) into the Schrodinger equation, we find

d2uqdx2

+ 2iqduqdx

−

(

2me

h2{V (x)− E} + q2

)

uq = 0 . (21.2)

Here we will only think about solutions for which E < V , and to make the problem assimple as possible we will only consider the V0b=constant case as V0 → ∞. Why? In sucha case, the square well wave functions will have no change in wavelength across the verynarrow “teeth” of the periodic potential, making it only tedious (but straightforward) togenerate the solutions.

As always in quantum mechanics, the key it to utilize the boundary conditions and tomake the appropriate change of variables to yield differential equations with well knownsolutions (well known, at least, to mathematicians). In this case, parameters of the form

α2 =2meE

h2β2 =

2me(V − E)

h2(21.3)

will be helpful. There are two regions to think about, that where V = 0 and that whereV 6= 0, for which eq. (21.2) becomes:

d2uqdx2

+ 2iqduqdx

+(

α2 − q2)

uq = 0 V = 0. (21.4)

d2uqdx2

+ 2iqduqdx

−(

β2 + q2)

uq = 0 V 6= 0.

These two differential equations have solutions of the form

uq(x) = Aei(α−q)x + Be−i(α+q)x

uq(x) = Ce(β−iq)x + De−(β+iq)x

respectively, as can be verified by substitution. The key now is to use the periodic boundaryconditions and to require that the full wave functions be both continuous and differentiable.

To make a long story short, these conditions lead to an equation of the form

P

αasin(αa) + cos(αa) = cos(qa) , (21.5)

where P = abβ2/2, the area under a potential ‘tooth’ but above the particle energy. Theform of this function is shown below.

153

Clearly, cos(qa) oscillates between -1 and +1, but the absolute value of the left handside of eq. (21.5) can exceed unity. Thus, only certain values of α, that is energy, arepermitted! Figure 21.2 presents the energy solutions for a particular value of P (that is, aparticular form of the Kronig-Penney potential). As before with vibrational spectroscopy,it is most convenient to plot the solutions versus wave vector, in which the various Brillouinzones that correspond to the solutions are visible.

For sufficiently small values of P , a continuous range of energies may be possible.More generally, however, there exist forbidden regions of q for which no solution to theSchrodinger equation is possible. These forbidden regions are called band gaps. The extentof these band gaps is sensitive to the height and width of the potential barrier, and a goodplace on the web to examine the behavior of the Kronig-Penney potential numerically ishttp://fermi.la.asu.edu/ccli/applets/kp/index.html. This applet will enable youto try various forms of the potential using sliders, and presents the energies and wavefunctions that are consistent with a given potential.

Actual solids, of course, are three dimensional, but the Kronig-Penney model doesillustrate the basics of the situation. For mono-atomic solids, it is easiest to think aboutthe bands that are allowed as arising from the combination of hydrogen-like atomic orbitalsspaced at regular intervals. This leads to the general behavior outlined in Figure 21.3.

The combination of orbitals to yield bands must still follow the Pauli exclusionprinciple. Thus, only a certain number of electrons can be placed into each band. Inmetals, the highest occupied band is only partially filled. Think about what this means.In a band with O(1024) wave functions, those that are close to each other in energy onlyhave a small number of nodes that are different from each other. Thus, the Franck-Condonoverlap is extremely good and the energy spacing with q is extremely small, << kT . Sincethe wave functions are extended over the solid, the electrons are free to move and thusmetals have high conductivity.

154

Figure 21.2 – Energies as a function of the magnitude of the wave vector, q, for theKronig-Penney model with P = 3π/2, in which band structure is clearly evident.

In insulators and semiconductors, on the other hand, the highest occupied band (calledthe valence band) is completely filled. Thus, in order for electrons to move, they must beexcited to the next available band (called the conduction band). For insulators, the bandgap energy is >> kT , thus they are poor conductors. For semiconductors, the bandgap is such that the conductivity is a strong function of temperature near 200-300 K (atcryogenic temperatures, nearly all semiconductors are insulators). Still, the conductivity isnowhere near that for metals at room temperature. How then, are semiconductors used inelectronics? The key is to use dopants whose energy levels create either holes or electronsin energy regions close to the valence band so that efficient conductivity is possible.

Another way to generate conductivity is to use photons. For indirect semiconductorsthe electronic matrix elements are dipole forbidden, but vibronically allowed; while fordirect semiconductors transitions between the valence and conduction band are electricdipole allowed. The resulting difference in skin depth is enormous, and can be used togreat advantage in integrated photonic/electronic devices. Si, for example, is an indirectbandgap semiconductor, while GaAs (or InGaAs) is a direct bandgap semiconductor. Byintegrating these materials, the GaAs/InGaAs portion of the circuit can be an efficientphotonic conduit, even with 1-2 µm thick films, while the Si can be doped to conductelectrons but not respond to optical excitation, even when the photons have sufficientenergy to excited Si valance electrons.

155

Figure 21.3 – A schematic diagram of the band structures that develop from hydrogenatom-like orbitals in mono-atomic solids. The major difference between metals, insulators,and semiconductors is the electron populations in the various bands along with the size ofthe band gaps with respect to the thermal energy available.

156