Chemistry 126 Spectroscopy Week # 5 Diatomic Bonding/Electronic Structure & the Franck-Condon Approximation As well described in Chapter 11 of

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1 Chemistry 126 Spectroscopy Week # 5 Diatomic Bonding/Electronic Structure & the Franck-Condon Approximation As well described in Chapter 11 of McQuarrie, for diatomics heavier than 2, we start with the simplest implementation of LCAO-MO description outlined previously, and then fill in the orbitals as a function of energy. The procedure is straightforward for homonuclear diatomic, since the atomic orbital energies all line up and since there is inversion symmetry that can be used to classify the resulting molecular orbitals. For heteronuclear species, such as F or Cl, the interactions are only significant if the atomic orbital energies are close to each other (think about perturbation theory). Let s consider the first row diatomic species as cases in point (see Figure 11.5, McQuarrie). Since the electronegativity increases across the periodic table rows to the right, the overall orbital energy decreases as Z increases from Li to F. The 1s orbitals combine to yield bonding and anti-bonding orbitals and are filled for all species. Recall that it is the component of the orbital angular momentum along the bond that is a good quantum number, and for s orbitals the resulting M.O.s are always cylindrically symmetric and so are σ in character. Bonding σ orbitals have g inversion symmetry, anti-bonding u inversion symmetry. For the p orbitals, if the z axis is along the bond axis then the p z atomic orbitals combine to yield a σ M.O., while the p x, p y orbitals yield π M.O.s. For the latter, it is the u combination that is bonding while the g combination is anti-bonding. The σ g 2p z M.O. is much more sensitive to the nuclear charge than are the π u 2p x,y orbitals, and starts out at higher energy than the latter for Li 2, but drops below the π u 2p x,y M.O.s for O 2 and F 2. Thus, the orbital energy ordering up to N 2 is given by σ g 1s σ u 1s σ g 2s σ u 2s (π u 2p x 2p y ) σ g 2p z (π g 2p x 2p y ) σ g 2p z, and two electrons can be placed into each M.O. according to the Pauli Principle. For O 2 and F 2 the (π u 2p x 2p y )/σ g 2p z ordering is swapped; and for degenerate orbitals (the π orbitals here, for transition metal species the d orbitals need to be considered), if there are only two electrons we place one electron each into the M.O.s with parallel spins. Thus, B 2 and molecular oxygen are predicted to be paramagnetic with (σ g 1s) 2 (σ u 1s) 2 (σ g 2s) 2 (σ u 2s) 2 (π u 2p x ) 1 (π u 2p y ) 1 and (σ g 1s) 2 (σ u 1s) 2 (σ g 2s) 2 (σ u 2s) 2 (σ g 2p z ) 2 (π u 2p x ) 2 (π u 2p y ) 2 (π g 2p x ) 1 (π g 2p y ) 1 ground states. To calculate the bond order, divide the difference of the number of electrons in bonding and anti-bonding M.O.s by two. This simplest picture explains satisfactorily the bond order (single, double, no bond), magnetism, and general binding properties of the first row diatomic molecules; but just as for 2 a quantitatively accurate description of their electronic structure requires the large, flexible wavefunctions generated by configuration interaction calculations. For heteronuclear diatomic species in which the electronegativity of the elements is not too different (think CN, CO, etc.) the treatment is similar, but in cases such as F or Cl where the orbital energies are very different the bonding/anti-bonding interactions are more restrictive (so, as McQuarrie describes, in the case of F it is largely the atom 1s and F atom 2p z atomic orbitals that interact to create the chemical bond). ere, the g and u labels no longer (strictly) apply, and so another common notation is to denote the anti-bonding orbitals as σ, π, etc., a notation we ll follow in the polyatomic case. What about excited electronic states and the spectra that result? There are two 107

2 possibilities. For a molecule like C 2, where the (π u 2p x,y ) and (σ g 2p z ) molecular orbitals are close in energy, strong transitions at visible wavelengths can result (and that were first detected in comets in the late 19 th century). For such situations, these are states tied to the lowest dissociation limit (two 3 P carbon atoms in the case just cited). More generally, configurations that result from the interaction of one ground state atom and the second atom in an excited state term (say O( 3 P) and O( 1 D), which we ll look at in just a bit) must also be considered, and the number of potential energy curves that derived from each possible dissociation limit can be extensive (see Figure 16.5)! Molecular und s Coupling Cases Once the electronic state symmetry is derived (see Lecture 5 notes), for the complete coupling of the various angular momenta in molecules we must combine: L = electronic orbital any momentum S = electronic spin any momentum R = nuclear rotational any momentum (sometimes N) J = total any momentum of molecule = L+S+R Five cases (a) - (e) may be described in analogy to the L S and j j coupling. Cases (d) and (e) are rarely or never observed, and will not be discussed here. These notes are provided for completeness so you know what the various cases look like, the most important part of this lecture is the analysis of the Franck-Condon factors that follows. Case (a) This is like L S coupling in atoms, and as shown in the figure below the coupling goes like Λ+Σ Ω; Ω+R J: R L J Λ S Σ ere the electronic spin-orbit coupling A L S is large, but the nuclear rotationelectron coupling N L is small. Thus Ω remains a good quantum number, and the energy expression becomes: F(J) B v [J(J +1)] AΩ 2 ], (16.1) where A B v (and so the formula looks much like a symmetric top). Note that each Ω state has its own rotational ladder. Examples of states for which case (a) coupling is valid include states like 2 Π or 3 states of molecules with first- and second-row atoms. For the 2 Π case, there are 2 Π 1/2 and 2 Π 3/2 states possible, each of which has its own complement of rotational levels, each of which is split by Λ doubling that is analogous to the l type doubling we encountered earlier for degenerate bending vibrations of linear molecules. Case (b) Again, this case is like L S, but is for the case Λ = 0. Now, the appropriate coupling becomes R+Λ N; N+S J, as shown below: 108

3 J N S R L In this case, the electronic spin-orbit coupling is small, so the spin couples to the axis of rotation of the molecule. As noted above, this situation almost always applies to Σ states where Λ=0; but it often applies to Π and Σ states in light molecules where the rotational constants are large. N is the total angular momentum, excluding spin, and the J = N +S,N +S 1,... N S levels with the same N are close together. For example, in a 2 Σ state, the energy levels are: F 1 (N) = B v N(N +1)+1/2γN F 2 (N) = B v N(N +1) 1/2γ(N +1) (16.2a) (16.2b) where γ is the called the spin-rotation constant. For example, in Mg, B 0 =5.7365cm 1 while γ 0 =0.0264cm 1. For higher spin systems, for example 3 Σ (the ground state of molecular oxygen), the spin-spin interaction must also be taken into account. Case (c) This coupling case is analogous to j j coupling in atoms. Thus, as the figure below shows, we first couple L+S J a ; then couple the J a, or a J a Ω; and finally couple Ω+R J, as shown below: J L J a S R Again, R is the nuclear rotation, and R = 0,1,2,... Λ and Σ are no longer good quantum numbers, rigorously speaking. Like the atomic case, und s case (c) is most often encountered in molecules with heavy atoms. All of these cases are idealizations, most molecules will fall somewhere in between! Cases (a) and (b) are typically the best basis sets for molecules containing first and second row atoms. The selection rules for the electronic states involved in an electric dipole transition can be derived using group theory, as explained previously. As before, we require that the product of the symmetry types of the lower and upper state wave functions, and that of the dipole moment operator be totally symmetric: Γ(Ψ el u) Γ(d el ) Γ(Ψ el l ) = A 1. (16.3) 109

4 For diatomic molecules, this results in the following selection rules for cases (a), (b): Λ = 0±1 S = 0. (16.4) For case (c) the rule is Ω = 0,±1. Also, for sigma states, we find Σ + Σ +, Σ Σ, but Σ + Σ. For homonuclear species, we have the additional rule g u, but g g and u u. For completeness, the selection rules for the higher order transitions are: Magnetic Dipole: Λ = 0,±1 S = 0 Σ + Σ +, Σ Σ, Σ + Σ g g, u u, g u Electric Quadrupole: Λ = 0,±1,±2 S = 0 Σ + Σ +, Σ Σ, Σ + Σ g g, u u, g u *In both cases the S=0 rule breaks down to the extent that spin-orbit coupling occurs. With the electric dipole selection rules in hand for diatomic molecules, it is now possible to examine what sort of transitions occur (i.e. P-branch, Q-branch, etc.). This is often accomplished with the aid of erzberg digrams, named after the Canadian spectroscopist (and Nobelist) who invented them. erzberg diagrams for 1 Σ + 1 Σ + 1 Σ 1 Π transitions are presented on the next page. In these diagrams, the individual J levels for each electronic state are written horizontally, and labeled according to their value of J and any symmetry properties they possess. The J-values are then connected with any electric dipole-allowed transitions. Also presented are spectra of Cu and Al as experimental illustrations of these transitions. Note that for the 1 Σ + 1 Σ + case, the symmetry selection rules automatically rule out the Q-branch, but for the 1 Σ 1 Π transition a Q-branch is allowed. Thus, if a Q-branch is observed in a diatomic molecule electronic spectrum from a 1 Σ ground state, you know the excited state must have Λ 0. Fitting of the P,Q,R branches here is done much as for the l type doubling constants for linear vibrational spectra, but now the splitting is called lambda-doubling. For even higher spin-/lambda-states, the spectra can become quite complex. For example, a 3 Π 3 Π transition has eight allowed branches. Labeled by the Ω component from which they arise, they are the R 1, R 2, R 3, P 1, P 2, P 3, Q 2, and Q 3 branches. 110

5 J = Σ P(1) P(2) P(3) P(4) R(0) R(1) R(2) R(3) 1 + Σ Π R(0) Q(1) P(4) 1 Σ J = Figure 16.1 (Top) An absorption spectrum of the A 1 Σ X 1 Σ transition of Cu. Just below the experimental spectrum is the erzberg diagram that illustrates the selection rules for a transition of this type. (Bottom) An emission spectrum of the A 1 Π X 1 Σ transition of Al. Note in this case the selection rules permit a Q-branch, as the erzberg diagram immediately above the spectrum shows. The formation of the R-branch band heads is discussed on the following pages. 111

6 Rovibronic energy levels In earlier Lectures, we considered transitions between vibration-rotation levels within the ground electronic state. owever, as should be obvious from the analogy with atoms presented above, a molecule also has an (infinite) number of excited electronic states. Consider for simplicity a diatomic molecule. For each excited electronic state, there is a potential energy curve, and for bound states, the curve appears qualitatively similar to that of the ground state. owever, as can be seen by examining the wavefunctions for the simplest diatomic molecule, 2, some potential curves can be unbound or repulsive. In spectroscopy, generally only transitions between a bound upper electronic state and a bound lower state are observed. Such a transition involves simultaneous changes in the vibrational and rotational energy levels. For each state, the energy or spectroscopic term can be written as E = T e +G v +F v (J). (16.5) The electronic term T e measures the energy of the minimum of the potential curve for a particular state above the minimum of the ground state curve. For the ground state itself, T e =0. If the energies were measured with respect to the separated atoms, T e = D e for the ground state, but it is not customary to do so. The vibrational terms G v are given by the expressions in earlier lectures; as are the rotational term values F v (J). The actual spectrum consists of a large number of lines with frequencies hν = (T e T e )+G v +F v (J) G v F v (J). (16.6) Vibrational transitions accompanying an electronic transition are called vibronic transitions. The vibronic transitions and their accompanying rotational, or so-called ro-vibronic transitions, are grouped into bands in the spectrum, and the set of bands associated with a single electronic transition is called an electronic band system. Selection rules; the Franck Condon Principle As for vibrational and rotational transitions, the strength of an electronic transition is proportional to the square of the matrix element between the upper and lower state: R =< Ψ d Ψ >= Ψ dψ d( r R) (16.7) where el nuc d = d + d = r i + α i Z α Rα (16.8 is the electric dipole operator, and the integration in (16.12) is over both the electronic and the nuclear coordinates. In the Born-Oppenheimer approximation, we obtain (neglecting rotation for simplicity): < e v d e v >=< v < e d el + d nuc e > v > 112

7 =< v < e d el e > v > + < e e >< v d nuc v >. (16.9) The second term of (16.9) vanishes, because the set of electronic wave functions is orthonormal. If the electronic transition dipole moment is defined as D el (R) =< e d el e > (16.10) then R =< v D el (R) v >= Ψ v (R)D el (R)Ψ v dr (16.11) where the integration in (16.11) is over the nuclear coordinates only. If D el varies little with R in the vicinity of the equilibrium internuclear coordinates, then it can be taken out of the integral, so that we obtain D el (R) D el (R e ), (16.12) R = D el (R e ) < v v >. (16.13) Thso-called oscillator strength for the transition then becomes f v v = 2 3 g Del (R e ) 2 < v v > 2 E v v (16.14) if both the transition dipole moment and the energy difference E v v are expressed in atomic units. ere g is a degeneracy factor equal to g = 2 δ Λ +Λ 2 δ Λ (16.15) for a diatomic molecule. The corresponding Einstein A-coefficient A v v = δ Λ +Λ 2 δ Λ = δ Λ +Λ 2 δ Λ ν 2 f v v ν 3 D el (R e ) 2 < v v > 2 (16.16) if ν is the transition frequency in wavenumbers. Thus, the relative intensity of a transition between any two vibrational states is given by the square of the vibrational overlap integral q v v = < v v > 2 (16.17) which is known as the Franck-Condon factor. The following sum rule holds for the Franck-Condon factors q v v = < v v >< v v >=< v v >= 1 (16.18) v v 113

8 where we have used the completeness relation v v >< v = 1. (16.19) The physical interpretation of the Franck-Condon factor is consistent with the original basis of the Born-Oppenheimer separation, namely that the nuclei are moving much more slowly than the electrons. It says that in the time required for an electronic transition to occur, which is of order h/ E s, the nuclei do not move. Thus, the band with the highest transition probability is the one for which the transition is vertical, that is, the molecule finds itself in the excited electronic state with the same internuclear separation as it had in the ground electronic state. This is illustrated in Figure 16.2 for the case of a diatomic molecule. The only regions of the excited state potential that are accessible in the transition are those for which the vibrational wave function of the ground state has a finite value. An analogous argument holds for emission spectra. Note that if the vibrational wave functions Ψ v and Ψ v have several nodes, there will be interference effects, leading to irregular variations in the Franck- Condon factors. For higher v, the maximum contribution comes from the part of the wave function closest to the classical turning point, as Figure 16.3 shows. The solid line in this figure indicates the maximum contribution to the vibrational overlap integral, which occurs in this example for v =4. owever, clearly the overlap integrals for v close to 4 are also appreciable, and give an intensity distribution like that illustrated in Figure 16.4b. Such an intensity distribution is called a progression : it involves a series of vibronic transitions with a common lower or upper levels. In this example, all members of the progression have v =0 in common. A group of transitions with the same value of v is referred to as a sequence. The situation illustrated in Figure 16.3 and 16.4b arises when R e > R e, that is, the equilibrium internuclear distance in the upper state is larger than that in the ground state. This is usually the case, since the ground state has the strongest bonding. Figure 16.4a shows the expected spectrum in the case R e R e. The maximum intensity occurs for the (0,0) band, and it falls off very rapidly. Very occasionally, the situation occurs in which R e < R e, although it usually only arises in transitions between two excited electronic states. The result is again an intensity distribution like that in Figure 16.4b, so that an observation of a long v = 0 progression with an intensity maximum at v > 0 indicates qualitatively an appreciable change in R e from the lower to the upper state, but does not indicate the sign of the change. owever, if anharmonicity is considered, there will be some slight differences between the two cases. For R e > R e, the relatively steep part of the excited state potential curve above v =0 is sampled, giving rise to a broad maximum in the progression intensity. For R e < R e, the shallower part of the excited state potential curve is probed, resulting in a sharper intensity distribution. If R e >> R e, appreciable intensity may arise from absorption into the continuum of vibrational levels above the dissociation limit, as illustrated in Figure 16.4c. 114

9 Figure 16.2 The vertical transitions permitted by the Franck-Condon principle between two electronic states. Also shown schematically is the relation among the dissociation energy of the ground state, D e, that of the excited state, D e, and the electronic term T e. Figure 16.3 The Franck-Condon principle applied to a case in which R e > R e and the (4,0) transition is the most probable. 115

10 Figure 16.4 Typical vibrational progressions and intensity distributions for the cases R e R e, R e > R e, R e R e. Example: The UV Spectrum of Molecular Oxygen Figure 16.5 illustrates some of the potential energy curves of the O 2 molecule. It is clear that there are a large number of electronic states at relatively low energies. Most of these states dissociate into ground state atoms O( 3 P) + O( 3 P). Conversely, these atoms give rise to a whole suite of distinct molecular states. The symmetries of these states can be found by adding the angular momentum of the separated atoms, and projecting those onto the internuclear axis. Thus, it is clear that two triplet states can give rise to singlet, triplet and quintet states, and that two P atoms with L=1 can result in Λ=0, 1 and 2 (Σ, Π and ) states. Just as for atoms, the fact that the two 3 P states are equivalent excludes some combinations (such as the 5 u state), but quite a number of possibilities remain. Such considerations of the possible electronic states correlating with the separated atoms are very useful, because they can be used to predict the presence of electronic states that have not yet been observed spectroscopically, especially repulsive states. Such states are indicated with dashed lines in Figure Tables of electronic states correlating with separated atoms can be found in erzberg Vol. I. The ground electronic state of O 2 has 3 Σ g symmetry. Thus, electric dipole allowed transitions are possible to states of 3 Σ g and 3 Π u symmetry. The lowest electric-dipole allowed transition is the B 3 Σ u X 3 Σ g transition. As Figure 16.5 shows, the B 3 Σ u potential curve is displaced to larger internuclear distances compared with the X 3 Σ g curve, thus giving rise to a long progression from Å. These are the so-called Schumann-Runge bands. Above 1750 Å, the molecule can dissociate into O( 3 P) and O( 1 D), so that the absorption becomes continuous. This is called the Schumann-Runge continuum. Figure 16.6 illustrates the absorption spectrum of the molecule at low spectral resolution. Both the Schumann-Runge bands and the continuum play an important role in the Earth s atmosphere. At low energies (λ >2000 Å), a weak continuum is observed, which can be ascribed to the forbidden A 3 Σ + u X 3 Σ g transition, and is called the erzberg I system. The transition occurs by magnetic dipole radiation. Other forbidden transitions which are 116

11 Figure 16.5 (Left) Potential energy curves for the low-lying electronic states of O 2. (Right) Energy level diagram for O 2, showing transitions important in atmospheric airglow spectra. Figure 16.6 The UV absorption spectrum of O 2 at low resolution. observed in the Earth s atmosphere are given in Figure At high energies, λ <1300 Å, the absorption occurs into high-lying Rydberg states such as the 2 3 Σ u state. The 117

12 ionization potential of O 2 is ev, so that absorption at λ <1030 Å gives rise to the ionization continuum. The Rotational Structure of Electronic Transitions Associated with the upper and lower vibronic states are sets of rotational levels, which give rise to rotational fine structure in the observed spectra. This structure is very similar to that found in the infrared vibrational transitions, except that a wider range of symmetry types can be involved. For simplicity, we will consider here only the structure of a 1 Σ 1 Σ transition in a diatomic molecule. As before, the rotational energy levels associated with two 1 Σ + electronic states have integer quantum numbers. The spectroscopic term of each of the states is given by (16.5) where F v (J) = B v J(J +1) D v J 2 (J +1) 2. (16.20) Just as for vibration-rotation transitions, the selection rule is J = ±1, resulting in P-branch ( J= 1) and R-branch ( J=+1) structure. If one of the electronic states has Λ >0, a Q-branch ( J=0) occurs as well (see above discussion). Thus, we expect the band to look very similar to the v=1-0 infrared band of simple diatomic species such as Cl or of parallel transitions such as the C or C N stretch of CN, as is outlined in Figure 16.7 (see also Figure 15.4). Figure 16.7 An illustration of the effect of the numerical values of B and T on the structure and intensity distribution for electronic emission or absorption bands having B B (headless bands). All the diagrams are on the same scale, except for the case B = 60 which is scaled by 25%. In practice, however, the electronic ro-vibronic bands are very asymmetrical about the band center ν 0 (which is defined as always as the wave number at which the utterly forbidden J = 0 J = 0 transition would occur). The reason for the asymmetry is that the rotational constants B and B are typically very different in different electronic states, whereas they are very similar in different vibrational states within the same electronic state. Since most likely R e > R e, B < B. This means that the rotational levels diverge more slowly in the upper state than in the lower state. The spectrum at the top of Figure

13 has been drawn for this case. A quantitative example for the I 2 molecule is shown in Figure The result is a very asymmetric branch with a so-called band head in which the low J lines in the R branch run together. This is due to the fact that the R-branch shows a reversal, that is, for low J the lines lie to the blue of the band center, but for high J, they lie to the red. The P-branch lines lie to the red for all J in this example. Such a band is said to be degraded, or shaded, to the red, that is, to lower wave number. If B > B, the P-branch forms a head, and the band is degraded to the blue, as is illustrated in the bottom of Figure The J value at which the branch turns around can be found by treating the line frequency ν R (J) = ν 0 +2B +(3B B )J +(B B )J 2 (16.21) as a continuous variable, and differentiating it with respect to J: d ν R (J) dj = (3B B )+2(B B )J = 0 J R = 3B B 2(B B ) (16.22) where JR is the nearest integer value to this ratio of B-values. In general, the spontaneous transition probability for a single rotational line (that is, a transition connecting a single Λ sublevel spin-multiplet component for angular momentum quantum number J in a specific v-level of electronic state e with a sublevel of another electronic state): A e v J s p e v J s p = δ 0,Λ +Λ ν 3 < e v D el (R) e v > 2 S J J 2 δ 0,Λ 2J +1 (16.23) where S J J is a rotational line strength factor for the line component J s p J s p and s and p designate the spin-multiplet and Λ-parity sublevels. It is easy to come to grief (usually by exactly factors of 2) in normalizing these line-strength factors; the proper normalization is S J J = (2 δ 0,Λ +Λ )(2J +1)(2S +1) (16.24) s p J and similarly for summations over the double-primed quantum states. The absorption oscillator strength for a line is then given by: f J J = 3 ( 2 δ 0,Λ 2J ν 2 ) δ 0,Λ 2J A e v J s p e +1 v J s p. (16.25) A clear summary of many of these confusing points is given by Larsson 1983, Astr. Ap. 128, 291, which is based upon more extensive discussions by: Whiting and Nicholls 1974, Ap. J. Suppl. 27, 1. Schadee 1978, J. Q. S. R. T. 19, 451. Whiting et al. 1980, J. Mol. Spectrosc. 80,

14 Figure 16.8 (Top three) Three illustrations of bands for which B < B, with associated R-branch band head formation. Frequency increases to the right, and so these bands are shaded to the red. (Bottom two) ere B > B, and so a P-branch band head is formed. Such bands are said to shade to the blue (or violet). Figure 16.9 Band head formation in the 0-0 band of the iodine B X transition. The intensity distribution shown corresponds to room temperature, and frequency increases to the left. Thus the band is red shaded. The 7:5 nuclear spin intensity alteration is also included. 120

15 Chemistry 126 Spectroscopy Week # 5 Photodissociation Processes & the Reflection Approximation In any environment or experiment where ultraviolet photons are present, photodissociation, typically depicted as XY + hν X + Y, (17.1) must be considered in addition to the bound state-bound state electronic transitions outlined in Lecture #18. Photodissociation plays a dominant role in the chemistry of diffuse interstellar clouds, in the outer parts of dense molecular clouds, in the atmospheres of planets, and is responsible for many of the molecules and radicals observed in circumstellar envelopes, and in cometary and planetary atmospheres. Photodissociation Mechanisms Photodissociation of a small molecule can proceed in various ways, which are illustrated in Figure 17.1 for the case of a diatomic molecule. For small polyatomic molecules, the processes are similar, but more complicated to illustrate because the potential surfaces are multi-dimensional. The simplest dissociation process is through direct absorption into a repulsive upper state as shown in Figure 17.1a. This absorption may also take place into the repulsive wall of a bound excited electronic state (not shown). As spontaneous emission back to the ground state is relatively slow compared to the time frame for movement along the nuclear coordinate, which occurs on the picosecond time scale, all absorptions therefore lead to dissociation of the molecule. The photodissociation cross section is continuous as a function of photon energy, and its energy dependence is governed to first order by the Franck-Condon principle in that its maximum value is at the vertical excitation energy indicated by the arrow in Figure 17.1a. This is the predominant photodissociation pathway of diatomics such as C +, N, Cl, and typically dominates for polyatomics (C 4, N 2 O, O 3,...). In contrast to direct photodissociation, which involves continuous absorption and can therefore occur over a range of wavelengths, the indirect photodissociation mechanisms each involve discrete transitions to bound vibrational levels of an excited electronic state as a first step. This has profound implications for the transfer of radiation, because line absorption can be saturated much more readily than continuous absorption. In the case of predissociation, illustrated in Figure 17.1b, the bound levels of the excited electronic state are coupled to the vibrational continuum of a third state of different symmetry. The third state usually crosses the excited electronic state within the adiabatic Born-Oppenheimer approximation. The transition to the dissociating state occurs without the emission of radiation, and can in most cases be described by first order perturbation theory. This mechanism is thought to be the predominant way of photodissociating CO. The spectral signature of this process appears as a broadening of the discrete peaks (corresponding to absorption into the bound excited state), due to the interaction with the third state. In the process of coupled states photodissociation, illustrated in Figure 17.1c, the bound levels of the excited state couple with the continuum of a dissociative state of the 121

16 Figure 17.1 Potential energy curves and characteristic cross sections for the processes of (a) direct photodissociation; (b) predissociation; (c) coupled states photodissociation; (d) spontaneous radiative dissociation. 122

17 same symmetry which, in the Born-Oppenheimer approximation, does not cross the bound state. If the states have a close avoided-crossing, the interaction is strong, and requires a coupled states description of the process. The spectral features in this case vary depending on the strength of the coupling and the relative sizes of the transition dipole moments involved. For example, they may consist of a broad continuous absorption background on which is superposed a series of sharper resonances. This mechanism appears to play an important role in the photodissociation of small, light molecules such as C and O. Finally, in the process of spontaneous radiative dissociation, illustrated in Figure 17.1d, the bound excited states simply decay by spontaneous emission into the vibrational continuum of either the ground electronic state, or a lower-lying repulsive state. The absorption cross section consists in this case of a series os sharp, discrete peaks. The photodissociation of 2 takes place through this mechanism. All of these processes may contribute to the photodissociation of a particular molecule, although in some cases one specific process may dominate. Let us look at some examples of diatomic molecule photodissociation in more detail. (i) Direct Photodissociation The clearest example of a molecule for which direct photodissociation dominates is that of + 2, for which the lowest potentials are reproduced in Figure In any reasonably cool environment, the molecule is almost entirely in the lowest vibrational and rotational eigenstates. Photodissociation can occur by direct absorption into the repulsive 2p σ u state. The continuum vibrational wave function is also illustrated in Figure In the simplest treatments of photodissociation, the continuum wavefunctions are approximated by delta functions at the classical turning points.in this limit, the cross section for photodissociation at an energy E = E k v is given by σ(e) = 2 3 πe 2 m e c g E k v < k D el (R) v > 2, (17.2) which is very similar to the formula for the oscillator strength for a bound-bound transition. The wavenumber k corresponds to the energy E k = E k v - (E f(r ) - E v ) of the nuclei above the dissociation limit. If the matrix element and the transition energy E k ν are in atomic units, the factor 2πe 2 /3m e c has the value 2.69 x cm 2. Figure 17.2(right) illustrates that the cross section reflects the shape of the vibrational wavefunction in the lower state. This is called the reflection approximation. The maximum usually occurs close to the vertical excitation energy. Note that the cross section is often negligibly small at the threshold wavelength, corresponding to the dissociation energy of the molecule. Thus, the dissociation energy of a molecule is not a good measure of how rapid a molecule is photodissociated; this depends entirely on the availability of excited electronic states through which photodissociation can occur. For example, the dissociation energy of + 3 is only 5 ev, but the photodissociation of interstellar + 3 is negligible since its first excited state occurs at 18 ev. A molecule with a more complicated electronic structure such as O generally has several low-lying photodissociation channels, as illustrated in Figure In this case, direct photodissociation can occur by absorption into the 1 2 Σ, 1 2, and B 2 Σ + states. 123

18 Figure 17.2 Potential energy curves for + 2 (left). Right: The reflection principle. Figure 17.3 Potential energy curves for the O molecule. 124

19 A well-known example of a molecule for which direct photodissociation can occur by direct absorption into the repulsive part of a potential curve which exhibits a bound well at larger distances, is provided by the O 2 molecule. As shown in Lecture #19, absorption into the B 3 Σ u - X 3 Σ g Schumann-Runge continuum will lead to photodissociation of the molecule. In this case, the relative amounts of discrete and continuous absorption are very sensitive to small uncertainties in the relative positions of the upper and lower potential curves. An instructive example, illustrated in Figure 17.4, is provided by the photodissociation of the isovalent C + and Si + ions through the A 1 Π state. For C +, nearly 100% of the absorption cross section occurs into discrete levels. For Si +, however, the curves are slightly displaced, and most of the absorption takes place into the continuum of the A state. Thus, the photodissociation of Si + through the A state will be very rapid, whereas for C + it will be negligible. Figure 17.4 Comparison of the photodissociation of C + and Si + through the A 1 Π state. In many texts, the reflection approximation is presented as shown in Figure owever, this version is, rigorously, incorrect. The Franck-Condon approximation in fact states that both the position and momentum of the nuclei are conserved during the electronic transition. For low-lying states such as that shown in Figure 17.2, the amount of potential (or kinetic) energy is small, and so the reflection approximation is often invoked with arrows emanating at the energy level of the initial state and terminating on the repulsive potential. In Figure 17.5, the more correct treatment is outlined, in this case for emission from a bound upper state to a repulsive lower state. If V e and V g are the excited and ground state potentials, and T(R) is the kinetic energy of the nuclei, the Mulliken difference potential is defined as V Mulliken = V g (R) + T(R). The nature of the Mulliken difference potential is illustrated in Figure 17.5 for a series of repulsive potentials that range from very steep to very flat. The turning points remain the same, but clearly there can be major differences in regions where there are considerable amounts of kinetic energy. 125

20 Figure 17.5 An illustration of the correct reflection approximation. Shown are the Franck-Condon factors and wavefunctions for an emission from a bound state with considerable vibrational excitation onto a purely repulsive lower state potential. The dotted curves on the lower states are the Mulliken difference potential, and a variety of repulsive curves from very steep to very shallow are shown. The arrows labeled A, B, and C indicate the transitions from positions which correspond to the classical turning points and the maximum of the Mulliken difference potential. 126

21 The wavelength of emission (or absorption) is given by hν = E [V g (R)+T(R)] = V e (R) V g (R), and so we see the correct reflection approximation either must draw arrows from the bound state energy level to the Mulliken difference potential, or from the initial potential energy curve to the final potential energy curve. In either case, the turning points define the (approximate) limits. As Figure 17.5 shows, when the repulsive potential is fairly flat and/or the vibrational energy content of the bound state is high, the difference potential goes through a maximum. This maximum creates the possibility for emission at one frequency at two internuclear distances, and results in interference. The net result is oscillatory behavior in the emission or absorption. In Figure 17.5, the narrow, or fast, oscillations arise from the structure in the upper and lower state wavefunction, that it is arises from the nodal character of the vibrational state wavefunctions. Thus, the top plots in Figure 17.5 look very much like vibrational wavefunctions at high v. As the Mulliken difference potential becomes more important, broad oscillations are superposed on the nodal nature of the vibrational wavefunctions, as is illustrated by the dotted profiles in the lower sections of Figure The overall result is a very complex emission (or absorption) spectrum, particularly for high vibrational bands which have a great deal of kinetic energy and lots of nodal structure in their wavefunctions. (ii) Predissociation In addition to the direct photodissociation channel, the O molecule also has predissociation channels. For example, the bound A 2 Σ + state is crossed by the repulsive 1 4 Σ, 1 2 Σ, and 1 4 Π states (see Figure 17.3), which leads to rapid predissociation of all vibrational levels of the A state with ν 2. The strength of the predissociation process can be described by first-order perturbation theory. The rate for predissociation of level i, v, J by a final electronic state f, k, J is then given by the Fermi-Wentzel Golden Rule formula k pr ν J = 2π h < iv J int fkj > 2 s 1, (17.3) where int stands for the interaction amiltonian, which may, for example, be the spinorbit interaction. In the case of direct photodissociation, the cross section is continuous as a function of photon energy, and is typically cm 2, so the optical depth τ (where I trans = I e τ ) for absorption by a slab of molecules of column density N in the initial state is τ = Nσ N. (17.4) For comparison, consider a UV transition of a molecule with ν = 10 5 cm 1 or λ = 1000 Å, and a fairly large oscillator strength, f 0.1. The peak absorption cross section at line center in this case is σdν σ. (17.5) ν 127

22 From the oscillator strength sum rule, σdν = (πe 2 /m e c)f. For a strongly allowed transition, the line width may only be as small as that for thermal Doppler broadening, i.e. so that σ ν c 1 3 x 10 5 ν ν, (17.6) x c/3 x x cm 2, (17.7) which is orders of magnitude larger than a typical continuous cross section, although effective over only a narrow frequency interval. As a consequence, line absorption will become optically thick or saturated (i.e. τ >1) more readily than will continuous absorption. When this occurs, the absorption rate drops drastically with increasing depth into the absorbing slab because all of the photons have already been absorbed. If the slab is sufficiently thick, the molecules near the surface can thus effectively shield the molecules at greater optical depths from the dissociating photons. (iii) Coupled-States Photodissociation An example of a molecule for which the coupled states process is important is again the O molecule. As Figure 17.3 shows, the 2 2 Π and 3 2 Π states undergo a so-called avoided crossing in the neighborhood of R eq. The resulting cross section for absorption is similar to that in Figure 17.1c. Since it is mostly continuous, no self-shielding occurs. (iv) Spontaneous Radiative Dissociation For most molecules, this photodissociation mechanism plays only a minor role, but it is the dominant photodissociation process of the most important astrophysical molecule, 2. Figure 17.6 illustrates the potential energy curves involved. In the first step of the process, an 2 molecule in the ν = 0 vibrational level of the ground X 1 Σ + electronic state absorbs an ultraviolet photon. For photon energies less than 13.6 ev, only the B 1 Σ + u and C 1 Π u states can be accessed through electric dipole allowed transitions. The corresponding series of discrete absorptions into the various levels are called the Lyman and Werner systems, respectively. The electronically excited levels decay rapidly by spontaneous emission into the vibrational continuum of the ground state, leading to dissociation of the molecule. Detailed calculations of the radiative transition probabilities for this process show that on the average, 10% of the absorptions lead to dissociation in the interstellar radiation field. The remaining 90% of the fluorescent transitions populate various bound excited vibrationrotation levels of the ground X 1 Σ + state. These levels subsequently decay through slow quadrupole transitions at infrared wavelengths. The main reason for the large abundance of 2 in interstellar clouds is the fact that its photodissociation is initiated by line absorptions. These lines are not broadened due to any predissociation mechanism, but only due to the finite radiative lifetime of the state, so that they are very narrow. Thus, the lines rapidly become optically thick, and the 2 molecule shields itself quite effectively. 128

23 Figure 17.6 Potential energy curves illustrating the photodissociation and excitation processes of 2. All of the above photodissociation processes are more than just mechanisms to destroy a molecule. Close to home, for example, it is the photodissociation of ozone (O 3 ) in the stratosphere that is responsible for shielding the near-surface regions of the Earth from ultraviolet radiation. O 3 absorbs strongly in the artley 1 B 2 1 A 1 transition from Å, and weakly in the visible through the uggins and Chappius bands. Figure 17.7 illustrates the relevant section through the O 3 potential surfaces, together with the measured absorption spectra. Figure 17.8 summarizes the most important sources of opacity in the Earth s atmosphere. Atoms resulting from photodissociation events can have significant kinetic energy, which can heat the ambient gas just as photoelectrons do. The photodissociation of O 2 through the Schumann-Runge continuum is the principal heating source of the thermosphere, for example. Note that photodissociation can also leave the product species in an excited electronic state, which can subsequently decay through characteristic emission-line radiation. 129

24 Figure 17.7 Absorption cross sections and potential curves for O 3. Figure 17.8 Depth of penetration of solar radiation in the ultraviolet as a function of λ. The line shows the altitude of unit optical depth, vertical arrows denote ionization limits. 130

25 Chemistry 126 Spectroscopy Week #5 Electronic Spectroscopy of Simple Polyatomic Molecules The electronic spectra of polyatomic molecules can become hopelessly congested at high resolution because of the very high density of eigenstates. Furthermore, a sometimes bewildering array of radiative and non-radiative processes become important. Before turningtothespectroscopyofformaldehydeasanexampleofofsomeofthemoreimportant rules of polyatomic electronic spectroscopy, we first present a brief tabulation of terms: I. A Brief Glossary of Terms in Electronic Spectroscopy A. Absorption and Emission Bound State-Bound State Transitions: Fluorescence Spin allowed, τ < 10 7 sec Phosphorescence Spin forbidden, τ > 10 6 sec Raman effects Spontaneous, resonant, stimulated Franck-Condon factors Bound-Free Transitions: Photoioinization resonant, non-resonant, pulsed field Photodissociation direct, predissociation, coupled states, excimers.. The Reflection approximation and the Mulliken difference potential B. Non-radiative Transitions Predissociation (curve crossing) Yablonski diagrams internal conversion (IC), intersystem crossing (ISC), intramolecular vibrational redistribution (IVR), inverse electronic relaxation (IER),... C. Popular Techniques Laser Induced Fluorescence: Excitation spectra (tune laser, measure total yield) Dispersed fluorescence (fix laser, disperse emission w/gratings, prisms,...) Stimulated emission pumping (SEP, including four-wave mixing) Photoelectron spectroscopy (lasers & mass spectrometers) Multi-photon dissociation (MPD) & ionization (REMPI) Picosecond/femtosecond pulses (pump-probe, CARS, ionization...) We ll now look at some of these processes using 2 CO as a test case. Formaldehyde: A Quick Rovibrational Review As the figure below reminds you, formaldehyde is a four atom non-linear molecule. ThepointgrouptowhichitbelongsisC 2v, anditisanearlyprolatetopwithanasymmetry parameter near -1. The permanent dipole moment is directed along the a-axis. 131

26 Since it is non-linear, there are 3(4)-6 = 6 non-degenerate normal modes of vibration. In the order derived by G. erzberg, these six modes are: σv x, c C 2 z, b O C σ v y, b -1 A = cm B = cm -1 C = cm -1 κ = O O O C C C ν 1 ν 2 ν 3 O O O + C C C - + ν 4 ν 5 ν 6 + Figure 18.1 The structure and vibrational normal modes of formaldehyde. What are the symmetries of the normal modes? Clearly, the C 2v symmetry of the molecule is maintained in the ν 1, ν 2, and ν 3 modes, and so they are all A 1. The ν 6, or out-of-plane bending, mode, is B 1, and the remaining ν 4 and ν 5 modes are B 2. The ν 1, ν 2, and ν 3 modes have dipole derivatives that change along the a-axis, and so these vibrational transitions are A-type bands with K p = 0 and K o = ±1. They thus look much like symmetric top perpendicular bands, with symmetric P QR profiles at low spectralresolution. Forν 4 andν 5, the µisalongtheb-axis, andtheyhave K p = ±1and K o = ±1. ν 6 has µ along the c-axis, and so is of C-type with K p = ±1, K o = 0. These latter B- and C-type bands are, as before, very messy with no easily noticeable structure by the untrained eye

27 Formaldehyde: Electronic Structure and Spectroscopy The molecular orbtials involved with the low-lying electronic states of formaldehyde are all associated with the valence electrons in the carbonyl group. If we ignore the core 1s orbitals on, C and O, then there are six 2p electrons left to account for. The molecular orbitals associated the carbonyl group are classified as σ and σ, π and π, and n (the are for anti-bonding orbitals, and the n is for non-bonding). A pictorial outline of these orbitals is presented below: C O C O σ σ C O C O + - π π C + O - n Figure 18.2 The molecular orbital structure of the valence electrons of formaldehyde. What are the C 2v symmetries of these orbitals? By examining the properties of each under the various symmetry operations [for example, under the C 2 operation the σ and σ orbitals are unchanged (and are hence of a symmetry), while the π, π, and n all change sign (and are hence of b symmetry)], we find the symmetry properties and orbital occupancy outlined next: a 1 σ a 1 σ b 1 π b 1 π b 2 n CO 2 b 2 n b 1 π b 1 π a 1 σ a 1 σ ow do we obtain the symmetry of the ground and excited electronic states? For the ground state, the configuration is σ 2 π 2 n 2, and so the symmetry is σ 2 π 2 n 2 = a 2 1b 2 1b 2 2 = a 1 a 1 a 1 = A

28 (The individual orbitals are labeled with lower case letters since the capitals are needed for the overall wavefunctions). What is the lowest electronic transition? Clearly, from the orbital diagram above it is the n π transition. What is the symmetry of the upper state? Using the same arguments as above, the π state has a configuration σ 2 π 2 nπ and so σ 2 π 2 nπ = a 2 1b 2 1b 2 b 1 = A 2. Is this an allowed transition? Remember that the matrix element < ψ µ ψ > must be totally symmetric. From the C 2v character table we have that x,y,z are A 1,B 1,B 2. Since the ground electronic state is A 1, this means that ψ must be A 1,B 1, or B 2. Thus, for n π, < A 2 µ(a 1,B 1,B 2 ) A 1 > A 1 and so the transition is not electric dipole allowed. What about other low-lying transitions? π π a 2 1b 2 1b 1 b 1 so A 1 A 1 allowed, by z component of µ n σ a 2 1b 2 1b 2 a 1 so A 1 B 2 allowed, by y component of µ Although these latter two bands are electric dipole allowed, and intense, there are no discrete spectra because all the lines are lifetime broadened by photophysical and photochemical processes (more next time). The n π actually shows up weakly, and since it is the longest wavelength transition is actually diagnostic of carbonyl groups (just as the weak 2800 cm 1 C stretch in aldehydes is diagnostic of that group). Why? Basically, it is due to the breakdown of the Born-Oppenheimer approximation. In reality, the amount of vibrational excitation and its character does in fact slightly change the electronic wavefunction. Perturbation theory then tells you that other electronic states must be mixed in, with the degree of mixing being related to the matrix element between the states divided by their energy separation. Such electric dipole forbidden transitions that gain their intensity in this manner are called vibronically allowed. For example, in 2 CO the ground state electronic wavefunction is A 1, so if any single vibration is excited the overall wavefunction has that character (ignoring rotation). To some (small) extent, therefore, the overall wavefunction gains the character of the vibrational wavefunction. For the n π transition in formaldehyde, it is observed that the b symmetry vibrations create intensity, while the a symmetry bands do not. Further, the bands show a marked sensitivity to the polarization state of the radiation. Does this make sense? Let s look first at µ z in the limit where we can still use the symmetry products for the electronic and vibrational parts of the wavefunction, that is: < φ eφ v µ z φ eφ v > = < A 2 φ v a 1 A 1 a 1 > = 0 for all vibrations in the 1 A 2 excited electronic state and transitions from the ground electronic and vibrational state. owever, consider next µ y : < φ eφ v µ y φ eφ v > = < A 2 φ v b 2 A 1 a 1 > =? 134

29 For φ v = ν 6, the vibrational symmetry in the upper state is b 1, and b 1 A 2 b 2 = A 1, and so group theory says the transition is allowed. The strength of the transition depends on the magnitude of the breakdown of the Born-Oppenheimer approximation. We might expect the spectrum to contain a series of bands like 2 v , where the notation means ν 6 = 0 1 and ν 2 = 0 v. The different values of v are called a progression in the electronic transition (if v = 0, a sequence is defined by the different values of v that give rise to the bands). What governs the intensity distribution with v? Just as for the diatomic case, it is the Franck-Condon factors. Life gets more complicated because we are now dealing with a multi-dimensional potential energy surface, and so the change in geometry (hence the extent of the progressions in the Franck-Condon factors) can occur along several normal mode coordinates. In addition, the rotational sub-structure in the bands of the n π transition is complicated because a variety of selection rules apply since formaldehyde in the excited state is no longer planar. What does the n π transition in formaldehyde look like? The next page presents two kinds of spectra of this transition. The first is an absorption spectrum from the ground state that reveals long progressions in the ν 2 and ν 6 modes (in the figures, the mode labeled ν 4 is that called ν 6 by erzberg). From the diagrams at the beginning of the lecture, we see that these modes involve a carbonyl stretch and the out-of-plane bend. What does this tell us? The fact that the out-of-plane 0 3 bending transition is as intense as that for 0 1 means that the A 2 state is non-planar (it is, in fact, bent by approximately 32 ). The long progression in ν 2 means that the bond C O bond length changes, which is not too surprising since we are exciting to an anti-bonding orbital on the carbonyl chromophore. Not much else is lit-up, and so changes in the other degrees of freedom must be minimal. The lower spectra is that of the fluorescence after excitation of the transition in the π state. Whereas the absorption spectrum revealed long progressions of vibrational modes in the excited state, here the long progressions are to vibrational modes in the X 1 A 1 state. This is one of the real advantages of electronic spectroscopy. When sharp features are present, the rovibrational structure in absorption can be used to probe the upper state potential while that in emission can be used to look at the ground state rovibrational eigenvalues. It is therefore possible to acquire a great deal of information about the force field in the molecule from a single spectrum, when acquired with sufficient resolution. Crystal Field Theory Another interesting example is the optical spectroscopy of octahedrally coordinated transition metal complexes; in which the ligands split the metal-centered atomic-like d orbitals into a suite of molecular orbitals. The following pages are taken from Modern Spectroscopy, by J.M. ollas (Wiley), and outline this topic using a variety of systems. 135

30 Figure 18.3 (Top) The absorption spectrum of the n π transition of 2 CO at 6 cm 1 resolution. The various vibrational progressions in the 1 A 2 state are labeled for clarity. (Bottom) The dispersed fluorescence spectrum from 2 CO after excitation of the X 1 A A 1 A 2 transition at 3375 Å. The horizontal axis gives the displacement in cm 1 from the exciting line, and the vibrational assignments now correspond to the levels on which the fluorescence terminates in the ground electronic state. 136

31 137 ollas, Modern Spectroscopy

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