7.2.2 Classification of Electronic States For all diatomic molecules the coupling approximation which best describes electronic states is analogous to the Russell- Saunders approximation in atoms The orbital angular momenta of all the electrons in the molecule are coupled to give a resultant L and all the electron spin momenta to give a resultant S. If there is no highly charged nucleus in the molecule, the spin-orbit coupling between L and S is sufficiently weak, and they couple to the electrostatic field produced by the two nuclear charges. This situation is referred to as Hund s case (a). As the vector L is coupled to the electrostatic field, only the component Λħ of the orbital angular momentum along the internuclear axis is defined, where the quantum number Λ can take the values wbt Λ = 0, 1, 2, 3, Eq. (7.63) 1
All electronic states with Λ > 0 are doublet degenerate. Classically, this degeneracy can be thought of as being due to the electrons orbiting clockwise or anticlockwise around the internuclear axis, the enregy being the same in both cases. If Λ = 0 there is no orbiting motion and no degenercy. The Λ is value indicated for an electronic state, which is designated Σ, Π, Δ, Φ, Γ, corresponding toλ = 0, 1, 2, 3, 4, The coupling of S to the internuclear axis is caused not by the electrostatic field, but by the magnetic field along the axis due to the orbital motion of the electrons. The component of S along the internuclear axis is Σħ. The quantum number Σ is analogous to Ms in an atom and can take the values Σ = S, S-1,, -S Eq. (7.64) wbt 2
There are 2S+1 components corresponding to the number of values that Σ can take. The multiplicity of the state is the value of 2S+1 for example in 3 Π. The components of the total (orbital plus electron spin) angular momentum along the internuclear axis is Ωħ, where the quantum number Ω is given by. Ω = Λ + Σ Eq. (7.65) Since Λ = 1, and Σ = 1, 0, -1 the three components of 3 Π have Ω = 2, 1, 0 and are symbolized by 3 Π 2, 3 Π 1, and 3 Π 0. Hund s case (a) is the most commonly encountered case in diatomics. However, for molecule contains at least one highly charged nucleus, spin-orbital coupling may be sufficiently large. As in Hund s case (c), L and S couple to give J, and J couples to the internuclear axis along which the component is Ωħ. wbt 3
(a) Hund s cased (a), (b) Hund s case (c) coupling of orbital and electron spin angular momentum in a diatomic molecule. wbt 4
In diatomic molecules the quantum numbers Λ, S, Ω are not quite sufficient. We must use one (for heteronuclear) or two (for homoneuclear) symmetry properties of the electronic wave function ψ e. The g or u symmetry property which indicates that ψ e is symmetric or antisymmetric respectively to inversion through the center of the molecule, e.g. 4 Π g. The second symmetry property concerns the symmetry of ψ e with respect to reflect across any (σ v ) plane containing the internuclear axis. If ψ e is symmetric to this reflection the state is labeled + and if antisymmetric to this reflection the state is labeled -, as in 3 Σ g+ or 3 Σ g-. wbt 5
Electronic Selection Rules 1. ΔΛ = 0, ±1 Eq. (7.67) For example, Σ Σ, Π Σ, Δ Π transitions are allowed but not Δ Σ or Φ Π. 2. ΔS = 0 Eq. (7.68) As in atoms, this selection rule breaks down as the nuclear charge increases. For example, triplet singlet transitions are strictly forbidden in H 2, but in CO, the a 3 Π X 1 Σ +, is observed weakly. 3. Δ Σ = 0, ΔΩ = 0, ±1 Eq. (7.69) for transitions between multiplet components. 4. + -, + +, - - Eq. (7.70) For example, only Σ + Σ + and Σ - Σ - transitions are allowed. wbt 6
5. g u, g g, u u Eq. (7.71) For example, Σ u+ Σ g+ and Π u Σ g+ transitions are allowed, but Σ g+ Σ g+ and Π u Σ u+ transitions are forbidden. For Hund s case (c) the selection rules are slightly different. Eqs. (7.68) and (7.71) still apply, but because Λand Σ are not good quantum numbers Eqs. (7.68) and (7.69) do not. Concerning Eq. (7.70), the + and - labels refer to the symmetry of ψ e to reflection in a plane containing the internuclear axis only when Ω = 0 and the selection rule is 0 + 0 +, 0-0 -, 0 + 0 - Eq. (7.72) wbt 7
7.2.5 Potential energy curves in excited electronic states The ground state configuration of C 2 is (σ g 1s) 2 (σ u *1s) 2 (σ g 2s) 2 (σ u *2s) 2 (π u 2p) 4 Eq. (7.80) Giving the X 1 g+ ground state. The low-lying excited electronic states arise from configurations in which an electron is promoted form the π u 2p or σ u *2s orbital to the σ g 2p orbital. The lowest excited state is the a 3 Π u state which is only 716 cm -1 above the ground X 1 g+ state. It is interesting to note that the Swan bands of C 2 are important in astrophysics. They have been observed in the emission of comets and stellar atmospheres. The Swan band results from the d 3 Π g a 3 Π u transition in the 785 340 nm region. Another interesting transition is the Mulliken band system which results from the D 1 Π u+ X 1 g+ transition in the 242 231 nm region. wbt 8
Potential energy curves of C 2 wbt 9
Vibrational progressions and sequences in the electronic spectrum of a diatomic molecule wbt 10
In electronic spectra there is no restriction on the values that Δv can take, but the Franck-Condon principle imposes limitations on the intensities of the transitions. Vibrational transitions accompanying an electronic transition are referred to as vibronic transition. These vibronic transitions, with their accompanying rotational or, strictly rovibronic transitions, give rise to a set of bands called an electronic band system. Because of the relatively high population of the v = 0 level, the progression with v = 0 is likely prominent in the absorption spectrum. Concerning the emission spectra, if the emission is between states of the same multiplicity, it is called fluorescence. If it involves only one vibrational level of the upper electronic state, it is single vibronic level fluorescence. Emission between states of difference multiplicity is called phosphorescence. wbt 11
Illustration of the Franck-Condon principle for (a) r e > r e and (b) r e r e. vertical transition wbt 12
In the case that the electronic transition gives rise to r e > r e, it often involves an electron in a bonding orbital to an orbital which is less bonding or even antibonding. For instance, in N 2 promotion of an electron from the σ g 2p to the π g *2p orbital leads to two excited electronic states, a 1 Π g and B 3 Π g, in which r e are 1.2203 and 1.2126 Å, respectively, considerably incresed from r e = 1.0977 Å in the X 1 g+ ground state. In the r e > r e case of the previous figure, transition from A to B (at the classical turning point) means nuclei are stationary. The transition from A to C is highly improbable because there is a large change of r. The transition from A to D is also unlikely because the nuclei is in motion at the point D. In the r e r e case of the previous figure, the most probable transition is from A to B with no vibrational energy in the upper state. The transition from A to C maintains the same value of r, but the nuclear velocities are increased due to their having kinetic energy equivalent the distance BC. wbt 13
The intensity of a vibronic transition is proportional to the square of the transition moment R ev, which is given by R = Ψ ' µ Ψ " dτ ev ev ev ev Eq. (7.84) where μis the electric dipole moment operator and ψ ev and ψ ev are the wave functions of the upper and lower state respectively. Assuming that the Born-Oppenheimer approximation holds, ψ ev can be factorized into ψ e ψ v. Then, Eq. (7.84) becomes R ev = Ψ e'* Ψ v'* µ Ψ e" Ψ v" dτedr integrate over electron coordinate τ e giving ' Eq. (7.85) R = Ψ ReΨ dr ev v v Eq. (7.86) wbt 14 "
where r is the internuclear distance and R e is the electronic transition moment given by R = ' µψ " dτe Eq. (7.87) e e e Ψ With the BO approximation, Eq. (7.86) becomes R ev ' v " v = R Ψ Ψ dr Eq. (7.88) e The quantity ψ v *ψ v dr is called the vibrational overlap integral; its square is known as the Franck-Condon factor. wbt 15
Franck-Condon principle applied to a case in which r e > r e and the 4 0 transition is the most probable. wbt 16
Typical vibrational progression intensity distribution wbt 17
Dissociation energies D 0 and D 0 of I 2 wbt 18
(a) The repulsive ground state and a bound excited state of He 2. (b) Two bound states and one repulsive state of H 2. (σ g 1s) 1 (σ g 2s) 1 (σ g 1s) 2 (σ u *1s) 1 (σ g 2s) 1 (σ g 1s) 1 (σ u *1s) 1 (σ g 1s) 2 (σ g 1s) 2 (σ u *1s) 2 wbt 19