Chemistry 21b Spectroscopy

Transcription

1 Chemistry 21b Spectroscopy 05Feb2018 Lecture # 14 Electronic Structure & Nomenclature of Diatomic Molecules; the Franck-Condon Approximation Before looking at the electronic structure of simple molecules, it is helpful to review briefly the salient aspects of atomic spectroscopy. The non-relativistic atomic Hamiltonian is given by Ĥ = Ĥ + Ĥrep = n i=1 ] [ h2 2m 2 i Ze2 r i + n i=1 j>i e 2 r ij. (14.1) Let L i and S i be the individual electron angular momenta and spin. Since atoms have spherical symmetry, L = n i L i and S = n i S i (the total angular momentum and spin) commute with Ĥ and are good quantum numbers. We indicate the total orbital angular momentum by a letter as follows: L S P D F G H I K L... The individual orbital occupancy is given in small letters, i.e. a 1s 2 2s 2 2p3d configuration gives rise to P, D, and F states. The quantity 2S + 1, the total spin degeneracy (multiplicity), is written as a left superscript to the letter designating L. For the 1s 2 2s 2 2p3d configuration, the two spins may be parallel or anti-parallel, which gives rise to the states: 1 P, 3 P, 1 D, 3 D, 1 F, 3 F Atomic states arising from the same electron configuration and having the same value of L and of S are said to belong to the same term. States belonging to different terms have different energies because of electron-electron repulsion, and the lowest state may be found according to Hund s Rule: HUND S RULE: The term with the highest spin multiplicity is lowest in energy; if there is more than one term with the highest multiplicity, then the term with the highest multiplicity and largest value of L lies lowest. Remember, Hund s rule works only for the lowest state (for example, the experimental ordering of the 1s 2 2s 1 2p 3 configuration of C is 5 S< 3 D< 3 P< 1 D< 3 S< 1 P!!). The physical basis of Hund s rule is that high spin states correlate with anti-symmetric spatial wavefunctions that, on average, have average electron-electron distances which are larger (and hence smaller electron-electron repulsion). This, of course, is due to the Pauli principal and the fermion nature of electrons. The total electronic angular momentum J of the atom is the vector sum of L and S, or J = L+S, so J = L+S, L+S 1,..., L S. (14.2) The valueof J is writtenasaright subscript on the term symbol. Thus for a 3 P term, L+1 and S = 1, and we get 3 P 0, 3 P 1, and 3 P 2 levels. In the above Hamiltonian, levels of the 108

2 same term have the same energy. There is an important effect, in molecular spectroscopy as well, called the spin-orbit interaction which splits the levels. Spin-Orbit Interactions and Momentum Coupling If we imagine ourselves riding on an electron in an atom, from our viewpoint the nucleus in moving around the electron. This apparent motion gives rise to a magnetic field which interacts with the intrinsic spin magnetic moment of the electron, and hence is proportional to L S. Quantum mechanically, the operator for the spin-orbit interaction is given by Ĥ s.o. = 1 2m 2 c 2 i 1 r i dv(r i ) dr i L i S i (14.3) when summed over all electrons. Thus, the total atomic Hamiltonian becomes Ĥ = Ĥ + Ĥrep + Ĥs.o, (14.4) and is composed of hydrogen-like, electron-electron repulsion, and spin-orbit terms. To get J (the total angular momentum), two limiting cases, called L-S and j-j coupling, are considered: L-S (Russell-Sanders) Coupling: As above, L = i L i and S = i S i and Ĥ, Ĥ rep Ĥ s.o.. L and S commute with Ĥ + Ĥrep but not with Ĥ. Nevertheless, Ĥ s.o. is small, so L and S almost commute with Ĥ (i.e. are good zeroth order quantum numbers in a perturbation treatment) and J = L+S is o.k. This coupling scheme is good for most light (1st and 2nd row) atoms. j-j Coupling: As the atomic # increases, v(electron) increases, and relativistic effects such asĥs.o. increase. Eventually, forz 1Ĥs.o. exceedsĥrep andlandsnolongercommute with Ĥ, but J does! In this case we add Ĥ + Ĥs.o. as the zeroth order Hamiltonian, and treat Ĥrep as the perturbation. This corresponds to first combining L i and S i i.e. j i = L i +S i and J = i j i. For most heavy atoms, the situation is intermediate between L S and j j coupling. Similar coupling schemes are used in molecular spectra, which we ll look at next. Molecular Spectra and Coupling Cases Nomenclature for Molecular Electronic States For historical reasons, electronic states are often given letter designations to label them roughly in order of energy or discovery. For diatomics, the ground state is denoted with the letter X. The letters A, B, C, D, are reserved for the lowest excited electronic states of the same spin multiplicity as X, usually in order of increasing energy. The letters a, b, c, d,.. denote the lowest excited states of different spin multiplicity from the ground state. For polyatomic molecules the conversion is the same, except that all letters have a tilde superposed; e.g., X, Ã, b,... because the un-accented letters are needed to add symmetry labels to the electronic states. As you will recall from Ch21a, it is the singlet states which tend to create bonding orbitals while the triplet states generate repulsive curves, for closed shell molecules (think 109

3 of H 2 ). For atoms, Hund s rule says the states with highest spin lie lowest in energy; such statements clearly cannot be made for molecules. This paradox is explained by the fact that for molecules, it is the electron density between the nuclei that leads to binding, and this effect outweighs the lower electron-electron repulsion in the high spin states. Instead of Hund s rule and L S or j j coupling, for molecules the addition of angular momenta are governed by what are termed Hund s coupling cases, which we review next. Hund s Coupling Cases Unlike atoms, diatomic molecules have cylindrical symmetry, which means that L is no longer a good quantum number; only the component Λ of the orbital angular momentum along the internuclear axis is defined, where Λ can be 0, 1, 2, 3,... As for atoms, Λ is the vector sum of the angular momenta λ i of the individual electrons in the molecule, or Λ = i λ i. All electronic states with Λ > 0 are doubly degenerate. Classically, these degeneracy can be though of as being due to electrons orbiting clockwise or anti-clockwise around the internuclear axis, the energy being the same in both cases. If Λ = 0, there is no orbiting motion, and no degeneracy. As for atoms, the states are designated by their value of Λ using the Greek equivalents of S, P, D, or Λ Σ Π Φ... L Λ = M L A Λ B z OnealsodistinguishesbetweenΣ + andσ states, dependingonwhetherthemolecular orbital is symmetric (+) or anti-symmetric ( ) with respect to reflection across any plane containing the internuclear axis. For Λ > 0, one will have +, the other, but the symbolism Π ±, ± is not often used. S Λ = 0 S A Λ n.e. 0 B Σ z If the diatomic molecule has two identical nuclei (H 2, C 2,..), subscripts u and g must be added to distinguish functions that are gerade or ungerade upon inversion through thecenter ofthemolecule. Just asfor atoms, thetotalspin Softheelectronsinaparticular electronic state is indicated by the multiplicity 2S + 1 as a superscript. Thus, the ground state of H 2 is 1 Σ + g. Unfortunately, the projection of S along the internuclear axis is also called Σ. For Λ = 0, Σ is not defined, that is there are no torques on S, and so it just sits there, as is shown in the figure above. 110

4 For Λ 0, Σ = S,S 1,..., S+1, S, and the internal magnetic field set up causes S to precess, coupling the orbital and spin momentum (a la Ĥs.o. in atoms). The total angular momentum is called Ω, and Ω = Λ + Σ, (14.5) and is placed as a subscript as J is for atoms. For example, a state might look like: Λ = 2 Λ = 2 Ω = 3 Ω = 2 Σ = 1 Σ = Λ = 2 E = E o+ A Λ Σ 1 Ω = 1 Σ = 1 E A > 0 regular A < 0 inverted where the splitting between the Ω sub-states arises from the spin-orbit interaction. All of the interactions we have just considered are without nuclear rotation! The complete notation for the coupling of angular momenta in molecules typically goes like: 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. Case (a)thisislikel S coupling inatoms, and asshown inthefigure below the coupling goes like Λ+Σ Ω; Ω+R J: E o A R L J Λ S Σ Here 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 ], (14.6) 111

5 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: 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) (14.7a) (14.7b) where γ is the called the spin-rotation constant. For example, in MgH, 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, Hund 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. 112

6 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(del ) Γ(Ψ el l ) = A 1. (14.8) For diatomic molecules, this results in the following selection rules for cases (a), (b): Λ = 0±1 S = 0. (14.9) 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 Herzberg digrams, named after the Canadian spectroscopist (and Nobelist) who invented them. Herzberg 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 CuH and AlH 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. 113

7 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 14.1 (Top) An absorption spectrum of the A 1 Σ X 1 Σ transition of CuH. Just below the experimental spectrum is the Herzberg 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 AlH. Note in this case the selection rules permit a Q-branch, as the Herzberg diagram immediately above the spectrum shows. The formation of the R-branch band heads is discussed on the following pages. 114

8 Rovibronic energy levels In earlier Lectures, we considered transitions between vibration-rotation levels within the ground electronic state. However, 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. However, as can be seen by examining the wavefunctions for the simplest diatomic molecule, H 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). (14.10) 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). (14.11) 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) (14.12) where el nuc d = d + d = r i + α i Z α Rα (14.13) is the electric dipole operator, and the integration in (14.12) 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 > 115

9 =< v < e d el e > v > + < e e >< v d nuc v >. (14.14) The second term of (14.14) 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 > (14.15) then R =< v D el (R) v >= Ψ v (R)D el (R)Ψ v (R)dR (14.16) where the integration in (14.16) 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 ), (14.17) R = D el (R e ) < v v >. (14.18) 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 (14.19) if both the transition dipole moment and the energy difference E v v are expressed in atomic units. Here g is a degeneracy factor equal to g = 2 δ Λ +Λ 2 δ Λ (14.20) 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 (14.21) 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 (14.22) 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 (14.23) v v 116

10 where we have used the completeness relation v v >< v = 1. (14.24) 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 14.2 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 14.3 shows. The solid line in this figure indicates the maximum contribution to the vibrational overlap integral, which occurs in this example for v =4. However, clearly the overlap integrals for v close to 4 are also appreciable, and give an intensity distribution like that illustrated in Figure 14.4b. 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 14.3 and 14.4b 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 14.4a 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 14.4b, 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. However, 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, the shallower part of the excited state potential >> R e, appreciable intensity may arise from absorption into the continuum of vibrational levels above the dissociation limit, as illustrated in Figure 14.4c. in the progression intensity. For R e < R e curve is probed, resulting in a sharper intensity distribution. If R e 117

11 Figure 14.2 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 14.3 The Franck-Condon principle applied to a case in which R e > R e (4,0) transition is the most probable. and the 118

12 Figure 14.4 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 14.5 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 Herzberg 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 14.5 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 14.6 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 Herzberg I system. The transition occurs by magnetic dipole radiation. Other forbidden transitions which are 119

13 Figure 14.5 (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 14.6 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 120

14 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 (14.10) where F v (J) = B v J(J +1) D v J 2 (J +1) 2. (14.25) 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 HCl or of parallel transitions such as the C H or C N stretch of HCN, as is outlined in Figure 14.7 (see also Figure 11.4). Figure 14.7 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

15 has been drawn for this case. A quantitativeexamplefor the I 2 molecule isshown 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 (14.26) 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 ) (14.27) 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 (14.28) 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) (14.29) 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 ) +1 A e v J s p 22 δ 0,Λ 2J e +1 v J s p. (14.30) 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,

16 Figure 14.8 (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) Here B > B, and so a P-branch band head is formed. Such bands are said to shade to the blue (or violet). Figure 14.9 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. 123