H + HCO H + HCO 1 A2 1 A 1 CO + H 2

Transcription

1 Chemistry 6 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 3 ev or more of excitation the spectral lines are broadened by processes which limit the lifetime of the excited state. The energetics for the photochemical processes in H CO are outlined in the figure below: ( S) + ( Π) H + HCO? n A π H + HCO ( S) + ( Σ ) 3 A? A H CO? CO + H ( Σ) + ( + Σ g ) Figure 0. The photochemical energetics for formaldehyde photodissociation. A major question in how fast the various photodissociation processes can occur is the nature of the barriers involved. That is, what are the barriers to the photochemical production of H + CO or H + HCO from formaldehyde? The energetics are easy to lay out, but if large activation energies are involved then it may take photons of considerably greater energy to drive the photochemistry. Other questions concern the nature of singlet-triplet coupling (i.e. is the 3 A state involved in the decay of A?), etc. Studies of the isolated benzene molecule in the late 960 s also raised a number of interesting questions. It was found, for example, that if the fluorescence yield (that is, the number of photons emitted/number of molecules excited) from the excitation into the first excited singlet state was examined as a function of pressure in a static cell, even at very low pressures only about 0% of the molecules excited actually emitted photons. Stranger still, if the fluorescence yield was examined as a function of vibrational energy content in the A state, the yield started at 0% and then dropped sharply toward zero above an excitation energy of 39,68 cm. These results are summarized in Figure 0. below. The fact 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! 4

2 0. C 6 H 6 0. Φ f Φ f 0. 0 torr Ar 39,68 cm - Frequency Figure 0. (Left) The fluorescence yield from the S state of benzene as a function of background gas (in this case, Ar) pressure. Even at very low pressures the quantum yield is [J. Chem. Phys. 5, 98(969)]. (Right) The fluorescence yield from the S state of benzene as a function of internal energy in the S state. The rapid drop in emission above a certain threshold energy was called the channel three problem, and working this problem out contributed greatly to our understanding of energy flow in isolated molecules [J. Chem. Phys. 46, 674(967)]. This drop in fluorescence yield is now understood to arise from non-radiative processes which operate even on isolated molecules once their density of states becomes sufficiently high. Pictorially, we use plots of the electronic state energies shown below, called Yablonski diagrams, to illustrate the competition between radiative and non-radiative processes in the energy flow within molecules after electronic excitation. Figure 0.3 below outlines the kinetic rates at which processes such as fluorescence, internal conversion, etc. occur, while Figure 7.4 stresses the important role that high lying vibrational states play in many of these processes, especially those of internal conversion and intersystem crossing. Internal Conversion k ic, s - Intersystem crossing k s - isc, Excitation Fluorescence Intersystem crossing k, s - k s - isc, f Phosphorescence k, 0 - p s - Figure 0.3 A summary of the radiative ( ) and non-radiative processes along with their rates in electronic spectroscopy. 43

3 Figure 0.4 The role of vibrational excitation and state density in the balance between radiative ( ) and non-radiative processes after electronic excitation. The quantum yield for fluorescence can thus be written as Φ f = k f / i k i, where i runs over all the possible radiative and non-radiative processes that can occur. Those include: k f = fluorescence (spin, dipole allowed, relatively fast) k p = phosphorescence (spin, forbidden, typically slow for low z) k ic = internal conversion (non radiative transfer in the same spin manifold) k isc = intersystem crossing (non radiative transfer to different spin manifolds) The lifetime of the upper state, τ f, is determined by all of the processes, τ f = / i k i, and so the quantum yield may be re-expressed in the form k f = Φ f /τ f. Consistency checks for the presence of fast, non-radiative processes can be performed by checking the magnitudes of the Einstein A and B coefficients. For purely radiative processes, the relationships derived in Lecture #8 are valid, but the effective value of A 44

4 will be much larger if processes such as internal conversion and/or intersystem crossing occur. We ll next look at simple two versus three level modes of radiationless processes. The Douglas Effect Consider the two cases outlined in Figure 0.5 below. On the left we have a simple two level system. For such a system, the fluorescence yield must be unity, that is Φ f =, and all the photons one expect are radiated (there is no place else for the energy to go). Thus, c (t) + c (t) =. ε vs. ψ a µ a0 = 0 V µ l 0 = 0 ψ l ψ 0 Figure 0.5 The simplest quantum mechanical model of radiationless processes. On the left is a simple two-level system. Non-radiative processes are included by adding a third level, degenerate with the upper state, for which the electric dipole matrix element with the lower state is zero (right). Life gets more interesting when a third level is added, as is shown on the right of Figure 0.5. Let us suppose that, for a given Ĥ, the upper states are degenerate, that is Ĥ ψ a > = ǫ ψ a > and Ĥ ψ l > = ǫ ψ l >. Suppose further it is possible to pump to ψ a, that is the 0 a transition is allowed, but that the 0 l transition is forbidden (the H atom is one such system). ψ a is called the bright state, and ψ l the dark state. Now, turn on a perturbing, or interaction, Hamiltonian Ĥ with strength V. Degenerate perturbation theory tells us the energies of the two states in the full Hamiltonian Ĥ +H are given by +> ε + V ε V V with wavefunctions of the form ε = ε + V - -> V ε - V > = + > = [ ψ a > ψ l >] [ ψ a > + ψ l >] 45

5 Clearly, both of the 0 > > and 0 > + > transitions are now allowed, at least formally. Let s consider two limiting cases for t = 0, one in which ϕ(0) >= ϕ(+) >, and one in which ϕ(0) >= ψ a >. (A) t = 0; let ϕ(0)> = ϕ + >. In a real system, this would mean using a laser whose bandwidth ω was such that ω < V. Since ϕ(+)> is an eigenstate of the total Hamiltonian, we have Ĥϕ ± = ǫ ± ϕ ± = (ǫ±v)ϕ ±. The time dependence for ϕ + is Φ(t) = U(t)ϕ + (0) = e iĥt/ h ϕ + (0) = e iω +t/ h ϕ + (0), where ω + = (ǫ+v)/ h. Thus, we see that ϕ + is a stationary state of the system since Φ(t) = ϕ + (0). The same is true, of course, for ϕ (B) t = 0; let ϕ(0)> = ϕ a > = [ ϕ + > + ϕ >]/ (), that is, suppose we prepare only the bright state ϕ a which is composed of equal parts ϕ + and ϕ (which requres a laser with ω > V). In this case, Φ(t) = e iĥt/ h [ϕ + (0)+ϕ (0)]/ () = [ ϕ + + ϕ +ϕ +ϕ e i ωt +c.c.], where c.c.= complex conjugate (of the third term), and ω = ω + ω V. Thus, because ψ a is not an eigenstate of the full Hamiltonian the probability for finding the system in the ψ a bright state, which is the one that fluoresces and hence is observable experimentally, oscillates; and the stronger the interaction matrix element V the faster the oscillation is. This in turn induces an uncertainty in the energy which is proportional to the splitting. Since the fluorescence is related to the probability distribution, which is given by P a(t) = [ + cos(4 πvt/h)]/ ψ where the so-called recurrence time is found to be τ = π h/v. The fluorescence decay is simply t Pl Pa Fluorescence dp dt = < ψ ex µ ψ gnd > 4ω3 3c 3 h. Suppose the excited state is written in terms of the ψ a and ψ l states discussed above, or ψ ex = α(t)ψ a + β(t)ψ l. 46

6 We can rewrite the above and find that dp dt = α(t) µ 0a 4ω3 3c 3 h, since µ 0l = 0. Clearly, the experimentally measured lifetime must be longer (that is, the decay rate slower) since This is called the Douglas effect. µ 0a effective = α(t) µ 0a < µ 0a. (C) Suppose there are many l> dark levels. For a pure continuum the probability of recurrence into a> is extremely small. Thus, the Douglas effect transforms into an irreversible non-radiative process (much like photodissocation) when the density of vibrational states becomes large. This is illustrated in Figure 0.6. The energy uncertainty is defined by the spread of the quasi-continuous l> states. In general, this process of randomization of vibrational energy from a bright initial state to a range of dark states that are more or less statistical in nature is called Intramolecular Vibrational Energy Redistribution, 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 large as benzene or larger, the IVR timescales can be of order picoseconds or faster. ψ a µ a0 = 0 V µ l 0 = 0 { ψ l } ψ 0 Figure 0.6 An outline of IVR, where the irreversible non-radiative process is driven by the 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 product of spatial and spin terms. If σ g and σ T are the ground and excited triplet state spin wavefunctions, then S 0 3 T < ϕ g σ g µ ϕ T σ T > = < ϕ g µ ϕ T >< σ g σ T > = 0 47

7 due to the orthogonality of the spin wavefunctions. However, the spin-orbit term Ĥ i L i S i can couple the excited state singlets and triplets (that is, the normal states we draw in the Yablonski 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 triplet wavefunction should be written Ψ triplet = αϕ T σ T + βϕ s σ s, and that the phosphorescence rate is thus related to < Ψ ground µ Ψ triplet > = α < ϕ g σ g µ ϕ T σ T > + β < ϕ g σ g µ ϕ s σ s >. The first term is zero, but the second is not, and so Phosphorescence µ gs β, where β H s.o./ E. In these formulae, µ gs is the singlet-singlet dipole coupling matrix element, and since it is (or can be) non-zero, the triplet lifetime is finite. The spin-orbit matrix element, H s.o. is often of order 0. cm for compounds involving first- or secondraw atoms, while E, the electronic state energy separation, is more typically 0 4 cm. Thus, tripletlifetimesaresome(0 5 ) timeslongerthanthatforsinglet-singletfluorescence for most molecules. However, if heavy atoms are involved (such as V- or Os-containing porphyrins that are found in the environment), the spin-orbit coupling is much larger and the phosphorescence yield can be much larger. ISC in H CO As a quantitative example of how intersystem crossing occurs, let s look at the relatively simple case of formaldehyde. As we saw before, for the n π excitation, the nπ configuration has A symmetry in its spatial wavefunction. For singlets, the spin function has a symmetry, while for the triplets the three spin wavefunctions have a, b, and b symmetry. Thus, the symmetry of the singlet and triplet manifolds of formaldehyde have the symmetries outlined in Figure 7.7. Let S x, T y be given triplet states. As always, for an interaction to occur we demand that the interaction matrix element be of A symmetry, that is < S x Ĥs.o. T y > A for the case of intersystem crossing. Since the spin-orbit interaction is a scalar, it is clearly of A symmetry. Thus, for formaldehyde (and by inference other C v species as well), the singlet and triplet states must have at least one shared symmetry type for the coupling to be finite. For example: 48

8 Singlet Symm. Triplet Symm. Interaction? n π A n π 3 A, 3 B, 3 B No π π A n π 3 A, 3 B, 3 B Yes n σ B n π 3 A, 3 B, 3 B Yes π π A π π 3 A, 3 B, 3 B No B 3 B n σ n 3 σ a b b = 3 B 3 A 3 A π A π π 3 A 3π a b b = 3 A 3 B 3 B A 3 A n π n 3π a b b = 3 A 3 B 3 B Figure 0.7 An outline of the potential singlet-triplet coupling in formaldehyde driven by the spin-orbit interaction. Arrows denote states that are coupled, those with crosses cannot interact by symmetry. The interactions outlined above can be generalized, in that for C v molecules singlets and triplets of the same configuration (nπ, etc.) do not interact, those from different configurations do. Again, the spin orbit matrix element is of order Ĥs.o 0. cm, or about 3 GHz. In this case, the appropriate level spacing (the E above) is of the same order 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 τ ISC 0 9 sec. So, we expect the coupling to be fast and irreversible even for isolated molecules! 49

9 Intermolecular, or Forster, Energy Transfer In the liquid or solid state, molecules are sufficiently close together that it is possible for the excitation on one chromophore (called the donor) to be transferred to another chromophore (called the acceptor). The most long range versus of this phenomenon uses non-radiative singlet energy transfer, and is called Forster transfer after the scientist who first described the mechanism nearly 50 years ago. Phenomenologically, the efficiency of Forster energy transfer is found to vary as η Forster = R 6 0/(R R 6 ), where R is the donor-acceptor separation and R 0 is the characteristic, or Forster, radius. Thus, the efficiency can be essentially unity for distances within the Forster radius, and falls to nearly zero for distances larger than R 0. What goes into determining R 0? Figure 7.8 presents the basic functional form of Forster energy transfer and steps involved. Qualitatively, what occurs is that with sufficient overlap in the emission spectrum of the donor and the absorption spectrum of the acceptor, an energy transfer process is set up whereby it is possible to generate emission from the acceptor by absorbing photons at shorter wavelengths where only the donor chromophore absorbs light. Figure 0.8 An overview of the important steps in Forster energy transfer. In his original work, Forster showed that the value of R 0 for a donor-acceptor pair depends on properties that can be measured separately for the donor and the acceptor, and has the form R 6 0 = Q D e A On 4, where Q D is the fluorescence quantum yield for the donor, e A is the extinction coefficient for the acceptor, n is the refractive index of the medium, and O is an overlap integral of the emission from the donor and the excitation of the acceptor. In addition to the spectral properties of the donor and acceptor, the overlap integral O depends upon the dipolar 50

10 coupling between the electronic dipole matrix elements (and so is sensitive to variations in the geometrical angles between the donor and acceptor). Pairs of chromophores are known with R 0 covering the range -0 nm. Experimental verification for Forster s formula for the efficiency of nonradiative singlet energy transfer has been achieved by attaching the donor and acceptor to a rigid molecule, so that the distance R is known. Alternatively, polymers with a strong preference for the formation of a helix of known geometry, with the donor attached at one end, and the acceptor attached at the other end, can be used. This helix provides a particularly strong test, because polymers of different molecular weight will form helices of different lengths. One such study used poly(l-proline) as the polymer (Stryer, L. & Haugland, R. P. 967, Energy Transfer: A Spectroscopic Ruler Proc. Natl. Acad. Sci., 58, 79-76). In the solvent system 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 polymerization ranging from 5 to, with the donor (naphthyl) at one end, and the acceptor (dansyl) at the other end. Forster s equation yields an R 0 of.7 nm for this donor-acceptor pair. The measured efficiencies were in good agreement with theoretical prediction. More recently, Forster energy transfer has become an extremely powerful means of examining the dynamical and folding properties of biopolymers such as DNA or proteins. A spectacular natural example of Forster energy transfer is provided by the photosynthetic reaction complex (PRC) shown in Figure 0.9. In this system, light gathering chromophores transfer their electronic energy over very long distances to the protein at the center of the PRC with nearly unit efficiency (apart from the loss incurring by the energy difference between the donor-acceptor states). Figure 0.9 A schematic depiction of the photosynthetic reaction complex in purple bacteria. By transferring the energy over such long distances, the PRC can separately optimize the light collection and chemical properties of the system. 5

11 Chemistry 6 Molecular Spectra & Molecular Structure Week # 6 Electronic Spectroscopy of Periodic Solids Previously we have investigated the vibrational modes of periodic solids using the harmonic potential approximation. Here we are interested in the electronic properties of such materials. From the LCAO-MO perspective, we could think about combining hydrogen 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 replaces the 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 text on 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, the periodicity of the lattice assumed paramount importance. Not surprisingly, the same is true for the electronic behavior of such materials. Here we briefly outline the simplest one-dimensional model of the so-called band structure of solids, called the Kronig-Penney model. A pictorial representation of this approach is shown below. Figure. A schematic of the -D potential that defines the Kronig-Penney model of the electronic structure of crystalline solids. Briefly, we consider an infinite array of square wells, with a potential barrier height of V 0, a barrier width of b, and a well spacing of a. Formally, the solutions we will present below are valid for periodic potentials where V 0 b=constant. In this model, it is clear that V(x + a) = V(x). Under such conditions, the time independent Schrödinger equation for the electrons in the solid has a solution of the form ψ q (x) = u q (x)e iqx u q (x+a) = u q (x). (.) 5

12 Theperiodicfunctionsu q (x)arecalledbloch functions. Asinouranalysisofthevibrations of solids, the behavior of the energy with the wave vectors q will be pivotal. If we insert eq. (.) into the Schrödinger equation, we find d u q dx + iq du q dx ( ) me {V(x) E} + q u q = 0. (.) h Here we will only think about solutions for which E < V, and to make the problem as simple as possible we will only consider the V 0 b=constant case as V 0. Why? In such a case, the square well wave functions will have no change in wavelength across the very narrow teeth of the periodic potential, making it only tedious (but straightforward) to generate the solutions. As always in quantum mechanics, the key it to utilize the boundary conditions and to make the appropriate change of variables to yield differential equations with well known solutions (well known, at least, to mathematicians). In this case, parameters of the form α = m ee h β = m e(v E) h (.3) will be helpful. There are two regions to think about, that where V = 0 and that where V 0, for which eq. (.) becomes: d u q dx + iq du q dx + ( α q ) u q = 0 V = 0. (.4) d u q dx + iq du q dx ( β + q ) u q = 0 V 0. These two differential equations have solutions of the form u q (x) = Ae i(α q)x + Be i(α+q)x u q (x) = Ce (β iq)x + De (β+iq)x respectively, as can be verified by substitution. The key now is to use the periodic boundary conditions 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 sin(αa) + cos(αa) = cos(qa), (.5) αa where P = abβ /, the area under a potential tooth but above the particle energy. The form of this function is shown below. 53

13 Clearly, cos(qa) oscillates between - and +, but the absolute value of the left hand side of eq. (.5) can exceed unity. Thus, only certain values of α, that is energy, are permitted! Figure. presents the energy solutions for a particular value of P (that is, a particular 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 Brillouin zones 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 the Schrödinger equation is possible. These forbidden regions are called band gaps. The extent of these band gaps is sensitive to the height and width of the potential barrier, and a good place on the web to examine the behavior of the Kronig-Penney potential numerically is This applet will enable you to try various forms of the potential using sliders, and presents the energies and wave functions that are consistent with a given potential. Actual solids, of course, are three dimensional, but the Kronig-Penney model does illustrate the basics of the situation. For mono-atomic solids, it is easiest to think about the bands that are allowed as arising from the combination of hydrogen-like atomic orbitals spaced at regular intervals. This leads to the general behavior outlined in Figure.3. The combination of orbitals to yield bands must still follow the Pauli exclusion principle. Thus, only a certain number of electrons can be placed into each band. In metals, the highest occupied band is only partially filled. Think about what this means. In a band with O(0 4 ) wave functions, those that are close to each other in energy only have a small number of nodes that are different from each other. Thus, the Franck-Condon overlap is extremely good and the energy spacing with q is extremely small, << kt. Since the wave functions are extended over the solid, the electrons are free to move and thus metals have high conductivity. 54

14 Figure. Energies as a function of the magnitude of the wave vector, q, for the Kronig-Penney model with P = 3π/, in which band structure is clearly evident. In insulators and semiconductors, on the other hand, the highest occupied band(called the valence band) is completely filled. Thus, in order for electrons to move, they must be excited to the next available band (called the conduction band). For insulators, the band gap energy is >> kt, thus they are poor conductors. For semiconductors, the band gap is such that the conductivity is a strong function of temperature near K (at cryogenic temperatures, nearly all semiconductors are insulators). Still, the conductivity is nowhere near that for metals at room temperature. How then, are semiconductors used in electronics? The key is to use dopants whose energy levels create either holes or electrons in energy regions close to the valence band so that efficient conductivity is possible. Another way to generate conductivity is to use photons. For indirect semiconductors the electronic matrix elements are dipole forbidden, but vibronically allowed; while for direct semiconductors transitions between the valence and conduction band are electric dipole allowed. The resulting difference in skin depth is enormous, and can be used to great advantage in integrated photonic/electronic devices. Si, for example, is an indirect bandgap semiconductor, while GaAs (or InGaAs) is a direct bandgap semiconductor. By integrating these materials, the GaAs/InGaAs portion of the circuit can be an efficient photonic conduit, even with - µm thick films, while the Si can be doped to conduct electrons but not respond to optical excitation, even when the photons have sufficient energy to excited Si valance electrons. 55

15 Figure.3 A schematic diagram of the band structures that develop from hydrogen atom-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 of the band gaps with respect to the thermal energy available. 56