Lecture 7: Electronic Spectra of Diatomics
Fortrat parabola, 1 ∆←1∆ (Symmetric Top) 1. Term symbols for diatomic molecules 2. Common molecular models for diatomics 3. Improved treatments 4. Quantitative absorption 1. Term symbols for diatomic molecules
Term symbols characterize key features of electron spin and orbital angular momentum For an atom: 2S 1 LJ Important terms 2S 1 For a diatomic: projection of orbital angular momentum onto the internuclear axis. Magnitude: | | L012 Symbols: Λ 012 Atoms Symbol ΣΠ∆ Symbol S P D total electronic spin angular momentum (the sum of electron spin in unfilled shells) S Magnitude:| S | S , S will have 1/2-integer values projection of S onto the internuclear axis (only defined when Λ≠0). Magnitude: | | Allowed values: S, S 1,...,S 2S 1 values sum of projections along the internuclear axis of electron spin and orbital angular momentum S, S 1,..., S 2S 1 values for S 2 1. Term symbols for diatomic molecules
Examples 2S 1 NO The ground state for NO is X2Π For a diatomic: S = 1/2, Λ = 1, Ω = 3/2, 1/2 2 2 There are two spin-split sub-states: Π1/2, Π3/2 Separation: 121cm-1
CO The ground state for CO is X1Σ+ S = 0 and Λ = 0, therefore Ω is unnecessary. This is a rigid rotor molecule. Easiest case!
3 – O2 The ground state for O2 is X Σg S = 1, Λ = 0 – The and g are notations about symmetry properties of wave functions. This is an example of a molecule that is modeled by Hund’s case b
3 2. Common molecular models for diatomics
Four common molecular models
Rigid Rotor Λ = 0, S = 0 2S+1 = 1 “singlets” no influence of electron spin Symmetric Top Λ≠ 0, S = 0 on spectra Hund’s a Λ≠ 0, S ≠ 0 Spin important through Hund’s b Λ = 0, S ≠ 0 interaction of Λ and Σ
. This lecture: Rigid Rotor Symmetric Top Followed by: Hund’s a Hund’s b
4 2. Common molecular models for diatomics
1 Rigid rotor ( Σ)
Axis of B IA ≈ 0 rotation C IB = IC J
m1 m2 A-axis
Center of mass
. Λ = 0, S = 0 1Σ type, Ω is not defined . Λ = 0 means the projection of the orbital angular momentum onto the A-axis is zero, and rotation must thus be around the B-axis
5 2. Common molecular models for diatomics
1 Rigid rotor ( Σ)
2 2 Rotational Energy FJ Bv J J 1 Dv J J 1
Total Energy E Te G F
ETe , v, J Te G v F J
Selection Rules Rotational spectra: J J 'J" 1 Rovibrational spectra: v v'v" 1 J 1 Rovibronic spectra: v determined by Frank- Condon factors J 1 Note: an alternate form is sometimes used
final initial J or v 6 2. Common molecular models for diatomics
1 Rigid rotor ( Σ) Intensity Distribution Within each band (v', v"), the intensity distribution follows the Boltzmann distribution for J modified by a J-dependent branching ratio (i.e., for the P and R branch), known as the Hönl-London factor. The relative intensities among all the vibrational bands
originating from a single initial level vinitial to all possible final levels vfinal are given by Franck-Condon factors.
The relative total emission or absorption from vinitial depends directly on the Boltzmann fraction in that level, i.e., nv initial/n
Examples Most stable diatomics: CO, Cl2, Br2, N2, H2 are rigid rotors 2 3 Exceptions: NO (X Π), O2(X Σ) Note: no X∆ states for diatomics – all X states are Σ or Π! ~X1 Σ ~X1 Σ g Some linear polyatomics: CO2( ), HCN and N2O ( ) are rigid rotors with 1Σ ground states. Nuclear spin will have an impact on the statistics of homonuclear diatomic molecules 7 2. Common molecular models for diatomics
Symmetric top N J IA ≠ 0 IA ≠ IB = IC
. Λ≠ 0, S = 0 (non-zero projection of orbital angular momentum on the internuclear axis and zero spin) ground states 1Π, 1∆ . Important components N angular momentum of nuclei A-axis projection of electron orbital angular momentum J total angular momentum; J N Only the axial component of orbital angular momentum is used, because only is a “good” quantum number, i.e., a constant of the motion 8 2. Common molecular models for diatomics
Symmetric top (Λ≠0, S = 0)
Rotational Energy FJ BJJ 1 A B 2 , J , 1,... h A, B Same spacing as the rigid 2 rotor, but with a constant 8 cI A,B offset
Since IA < IB, A > B, lines with J < Λ are missing, as J = Λ, Λ+1, …
Selection Rules 0,J 1,0 J 0 is weak 1,J 1,0
As a result of having a Q branch (i.e., ∆J = 0), the bands for a symmetric top will be double- headed, in contrast to the single-headed character of rigid rotor bands
9 2. Common molecular models for diatomics
Symmetric top (Λ≠0, S = 0) Spectra for ∆Λ = 0 (1Π←1Π or 1∆←1∆)
T ' B' J 'J '1 A'B'2 Gv'T ' e = 0 for ground state 2 T" B"J" J"1 A"B" G v" Te"
upperfor J ' 0 lower for J" 0 constant
2 PJ" B'B"J B'B"J 2 Q J" B'B" J B'B" J 2 R J" B'B" J 1 B'B" J 1
mP J 2 P and R branches: am bm m J Q Q branch: bm bm2 m J 1 R where a B'B",b B'B"
10 2. Common molecular models for diatomics
Symmetric top (Λ≠0, S = 0)
Spectra for ∆Λ = 0 Fortrat parabola, 1∆←1∆ 2 P and R branches: am bm 2 Q branch: bm bm where a B'B",b B'B" Notes: . Band heads in the Q and R branches for the typical case of B' . mP = -J, mQ = +J, mR = J+1 1 1 . Jmin = 2 for ∆← ∆ mmin=3 for R branch mmin=2 for Q branch |mmin|=3 for P branch missing lines near the origin Intensity Distribution Relative intensities depend on nJ/n, and P,Q,R Hönl-London factors (S J ) - “relative intensity factors / line strengths” breakdown of the principle of equal probability 11 2. Common molecular models for diatomics Example – 1: Hönl-London Factors for Symmetric Top (see Herzberg) J 1 J 1 For ∆Λ = 0 S R J 1 J J J 1 2J 1 2 2 2 Q SJ 0 for large J SJ 2J 1 J J 1 J P J J SJ J for large J Notes: J 1. ΣSJ = 2J+1, the total degeneracy! 2. The R-branch line for a specific J, is ~ J+1/J times as strong as the P-branch line 3. For ∆Λ=±1, J>>Λ 2J 1 S R Q branch lines are twice as J 4 strong as P and R lines! Q 2J 1 SJ SJ 2J 1 ∆Λ value is important in 2 determining the relative line P 2J 1 and branch strengths of SJ 4 rovibronic spectra. 12 2. Common molecular models for diatomics Example – 2: Symmetric Top Ground State If X = 1Π, possible transitions (Recall ∆Λ = 0, 1) 1Π←1Π 1∆←1Π 1Σ←1Π ∆Λ = 0 ∆Λ = 1 ∆Λ = –1 1. Three separate “systems” of bands possible from X1Π 2. Hönl-London factors for ∆Λ = 1 differ from for ∆Λ = 0 (see previous page) 13 3. Electronic Spectra of Diatomic Molecules: Improved Treatments (add Spin) Hund’s case a 1. Review of angular momentum 2. Interaction of Λ and Σ 3. Hund’s case a (Λ≠0, S ≠ 0) 4. Hund’s case b (Λ = 0, S ≠ 0) Hund’s case b 5. Λ-doubling 14 3.1. Review of angular momentum Review – then add spin Term symbol “Multiplicity” of state 2S 1 Term Symbol Sum of projections on A axis Λ 012 when Λ≠0 Projection of electron Symbol ΣΠ∆ orbital angular momentum on A axis S, S 1,..., S 4 models 1 Rigid Rotor Λ = S = 0 e.g., N2, H2: X Σ 1 Symmetric Top Λ≠ 0; S = 0 e.g., Π 2 Hund’s a Λ≠ 0; S ≠ 0 e.g., OH, NO (both X Π) Add spin 3 Hund’s b Λ = 0; S ≠ 0 e.g., O2: X Σ 15 3.1. Review of angular momentum Electronic angular momentum for molecules Orbital angular momentum of electrons 1. Separate from spin and nuclear rotation 2. Strong electrostatic field exists between nuclei. So L precesses about field direction (internuclear axis) with “allowed” components along axis ml = L, L-1, …, -L ≡ Λ L E 3. If we reverse direction of electron orbit in E field, we get the same energy but Λ→ –Λ (Λ doubling) 16 3.1. Review of angular momentum Electronic angular momentum for molecules Spin of electrons 1. To determine L and S for molecule, we usually sum l & s for all electrons. e.g., S si i So even number of electrons integral spin odd number of electrons half-integral spin 2. For Λ≠ 0, precession of L about internuclear axis magnetic field along axis. So ms is defined. ms ≡ Σ = S, S-1, … -S. Note for change of orbital direction, energy of electron spinning in magnetic field changes no degeneracy 2S+1 possibilities (multiplets) 3. For Λ = 0, no magnetic field exists and the projection of S on the nuclear axis is not conserved (Σ not defined) 17 3.1. Review of angular momentum Electronic angular momentum for molecules Total electronic angular momentum 1. Total electronic angular momentum along internuclear axis is But since all in same direction, use simple addition 2. For Λ≠ 0, magnetic field H ∝ Λ. Magnetic moment of “spinning” electron μH ∝ Σ. So interaction energy is proportional to E ~ μH ~ ΛΣ, or Te = T0 + AΛΣ (more on this later) For A > 0, “Regular” state For A < 0, “Inverted” state 2 2 1/ 2 3/ 2 18 3.2. Interaction of Λ and Σ This interaction is key to modeling the influence of spin on the electronic state structure. N J Nuclear Rotation When Λ≠0, S ≠ 0, they combine to form a net component of Ω. Λ≠0 an associated magnetic field due to net current about the axis. This field interacts with spinning electrons. Spin-orbit coupling (spin-splitting of energy levels) Comments: . Models are only approximations. . Coupling may change as J ranges from low to high values 19 3.2. Interaction of Λ and Σ Examples 3 ∆3 S = 1, Λ = 2, Ω = 3 (Σ = 1) 3 3 ∆ ∆2 S = 1, Λ = 2, Ω = 2 (Σ = 0) 3 ∆1 S = 1, Λ = 2, Ω = 1 (Σ = –1) Electronic energies Te T0 A Spin-orbit coupling constant, generally Energy without increases with molecular weight and interaction the number of electrons 3∆ S = 1, Λ = 2, Σ = 1,0,–1 3∆ 1 3 3∆ 2A For A>0 3 Te T0 A2 0 ∆2 3 1 ∆1 Sample constants Notes: -1 1. The parameter Y is often specified, ABeH ≈ 2 cm -1 where Y = A/B ANO ≈ 124 cm v -1 2. Values for A given in Herzberg, Vol.I AHgH ≈ 3600 cm A ≈ -140 cm-1 Negative! OH Now, consider Hund’s cases where S ≠ 0 20 3.3. Hund’s case a Λ≠0, S ≠ 0, Σ = S, S-1, …, -S FJ BJ J 1 A B2 . Recall h h S, S 1,..., S A 2 , B 2 8 I Ac 8 I Bc J , 1,... Not to be confused with spin-orbit constant P, Q, R branches for each value of Ω. Example: 2Π Ω = 3/2 and 1/2, two electronic sub-states a total of 2 x 3 = 6 branches 21 3.4. Hund’s case b Applies when spin is not coupled to the A-axis E.g., 1. For Λ = 0, is not defined, must use S 2. At high J, especially for hydrides, even with Λ≠ 0 J N S Allowed J: J = N+S, N+S-1, …, N-S, J ≥ 0 only For this case, S and N couple directly, 22 3.4. Hund’s case b Example – O2 3 Ground state X Σ has three J’s for each N! F1(N) F2(N) F3(N) J = N + 1 J = N J = N – 1 J = 3 Notes: N = 2 J = 2 . Split rotational levels for N > 0 J = 1 . Each level has a degeneracy J = 2 of 2J + 1, and a sum of Hönl- N = 1 London factors of 2J + 1 J = 1 . Minimum J is |N-S| J = 0 . In N = 0 level, only spin is J = 1 active (S = 1), this is the N = 0 minimum value of J 23 3.5. Λ – doubling Further complexity in the energy levels resulting from Λ-doubling Different coupling with nuclear rotation ( N and interaction) The two orientations of (± Λ along the A-axis) have slightly different energies FJ Fc J and Fd J Fc(J) F(J) Fd(J) . By definition, Fc(J)>Fd(J) (c,d replaced by e,f in some literature) . Lambda doubling usually results in a very small change in energy, affecting Boltzmann distribution only slightly. . Change of parity between Λ-doubled states – reduces the accessible fraction of molecules for a given transition (due to selection rules) 24 4. Quantitative absorption Review of Beer’s law and spectral absorption as interpreted for molecules with multiplet structure Beer’s Law I exp k L I 0 n1 For two-level system n1 ni ni e2 k S12 n1 f12 1 exp h / kT mec f ,i initial, j final Integrated absorption ij intensity [cm-1s-1] For a complex, multiple level system, we have 2 quantities to specify: . Boltzmann fraction? . Oscillator strength for a specific transition? 25 4. Quantitative absorption Boltzmann fraction n1 n1 ni ni ni = the total number density of species I n1/ni = the fraction of species i in state/level 1 n1 Ni n, v,,, J, N ni Ni Quantum numbers: n – electronic v – vibrational Σ –spin Λ –orbital J – total angular momentum N – nuclear rotation c or d – Λ-component We will illustrate this in the next lecture! 26 4. Quantitative absorption Oscillator strength Strength of a specific, single transition (i.e., from one of the J" substrates to a specific J' substrate), fJ"J‘ f12 fm,v",J " n,v',J ' f J "J ' S f q J "J ' el v"v' 2J"1 "system" osc. Franck-Condon strength factor normalized H-L factor Notes: or line strength . qv"v' 1 v' . SJ "J ' 2J"1 2S 1 δ = 1 for Σ-Σ, otherwise δ = 2 (Λ-doubling). J ' [(2S+1)δ] = 4 for OH’s A2Σ←X2Π system. . f J "J ' 2S 1 fel sum is fel for a single J" substate. v',J " 27 4. Quantitative absorption Oscillator strength Remarks 1. Band oscillator strength f v" v' f el q v" v' often is tabulated 2 2 e.g., f00 = 0.001 (OH A Σ←X Π) SJ "J ' 2. f J "J ' fv"v' 2J"1 P R e.g., if only P and R are allowed SJ "J ' J", SJ "J ' J"1 3. In some cases, an additional “correction term” TJ"J' is used, e.g., in OH SJ "J ' f J "J ' fv"v' TJ "J ' ,TJ "J ' always near 1 2J"1 m c 2 g f e A e' 4. In terms of A-coefficient v"v' 2 2 v'v" 8 e ge" ge' fv'v" ge" 28 Next: Case Study of Molecular Spectra Ultraviolet: OH