Lecture 7: Electronic Spectra of Diatomics

Transcription

1 Lecture 7: Electronic Spectra of Diatomics. Term symbols for diatomic molecules Fortrat parabola, (Symmetric Top). Common molecular models for diatomics 3. Improved treatments 4. Quantitative absorption

2 . Term symbols for diatomic molecules Term symbols characterize key features of electron spin and orbital angular momentum S For an atom: L Important terms S projection of orbital angular momentum onto the internuclear axis. Magnitude: Symbols: Λ 0 Symbol Σ Π For a diatomic: Atoms S total electronic spin angular momentum (the sum of electron spin in unfilled shells) Magnitude: S S, S will have /-integer values S projection of onto the internuclear axis (only defined when Λ 0). Magnitude: Allowed values: S, S,..., S S values sum of projections along the internuclear axis of electron spin and orbital momentum S, S,..., S S values for S L 0 Symbol S P D angular

3 . Term symbols for diatomic molecules Examples NO The ground state for NO is X Π S = /, Λ =, Ω = 3/, / For a diatomic: S There are two spin-split sub-states: Π /, Π 3/ Separation: cm - CO The ground state for CO is X Σ + S = 0 and Λ = 0, therefore Ω is unnecessary. This is a rigid rotor molecule. Easiest case! O The ground state for O is X 3 Σ g S =, 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

4 . Common molecular models for diatomics Four common molecular models Rigid Rotor Λ = 0, S = 0 Symmetric Top Λ 0, S = 0 Hund s a Λ 0, S 0 Hund s b Λ = 0, S 0 S+ = singlets no influence of electron spin on spectra Spin important through interaction of Λ and Σ This lecture: Followed by: Rigid Rotor Symmetric Top Hund s a Hund s b 4

5 . Common molecular models for diatomics Rigid rotor ( Σ) Axis of rotation C m m I A 0 I = I C A-axis Center of mass Λ = 0, S = 0 Σ 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 -axis 5

6 . Common molecular models for diatomics Rigid rotor ( Σ) Rotational Energy Total Energy Energy Change Selection Rules F E D v v T, v, T Gv F e E T e e G F Rotational spectra: Rovibrational spectra: v v v Rovibronic spectra: v determined by Frank- Condon factors final or v Note: an alternate form is sometimes used initial 6

7 . Common molecular models for diatomics Rigid rotor ( Σ) Intensity Distribution Within each band (v, v), the intensity distribution follows the oltzmann distribution for modified by a -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 v initial to all possible final levels v final are given by Franck-Condon factors. The relative total emission or absorption from v initial depends directly on the oltzmann fraction in that level, i.e., n v initial /n Examples Most stable diatomics: CO, Cl, r, N, H are rigid rotors Exceptions: NO (X Π), O (X 3 Σ) Note: no X states for diatomics all X states are Σ or Π! ~ XΣ ~XSome linear polyatomics: CO ( ), HCN and N O ( Σg) are rigid rotors with Σ ground states. Nuclear spin will have an impact on the statistics of homonuclear diatomic molecules 7

8 . Common molecular models for diatomics Symmetric top N I A 0 I A I = I C Λ 0, S = 0 (non-zero projection of orbital angular momentum on the internuclear axis and zero spin) ground states Π, Important components N angular momentum of nuclei A-axis projection of electron orbital angular momentum total angular momentum; 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

9 . Common molecular models for diatomics Symmetric top (Λ 0, S = 0) Rotational Energy F A A, h 8 ci A,,,,... Same spacing as the rigid rotor, but with a constant offset Since I A < I, A >, lines with < Λ are missing, as = Λ, Λ+, Selection Rules 0,,,0,0 0 is weak As a result of having a Q branch (i.e., = 0), the bands for a symmetric top will be doubleheaded, in contrast to the single-headed character of rigid rotor bands 9

10 Symmetric top (Λ 0, S = 0) 0. Common molecular models for diatomics Spectra for Λ = 0 ( Π Π or ) constant 0 for lower 0 for upper v v T G A T T G A T e e = 0 for ground state R Q P m m m R Q P P and R branches: Q branch: where, b a bm bm bm am

11 . Common molecular models for diatomics Symmetric top (Λ 0, S = 0) P and R branches: Spectra for Λ = 0 am bm bm bm Q branch: where a, b Notes: and heads in the Q and R branches for the typical case of <. m P = -, m Q = +, m R = + min = for m min =3 for R branch m min = for Q branch m min =3 for P branch missing lines near the origin Intensity Distribution Relative intensities depend on n /n, and Fortrat parabola, P, Q, R Hönl-London factors ( S ) - relative intensity factors / line strengths breakdown of the principle of equal probability

12 Example : Hönl-London Factors for Symmetric Top (see Herzberg). Common molecular models for diatomics For Λ = 0 for large for large 0 S S S S P Q R Notes:. ΣS = +, the total degeneracy!. The R-branch line for a specific, is ~ +/ times as strong as the P-branch line 3. For Λ=±, >>Λ 4 4 S S S S P Q R Q branch lines are twice as strong as P and R lines! Λ value is important in determining the relative line and branch strengths of rovibronic spectra.

13 . Common molecular models for diatomics Example : Symmetric Top Ground State If X = Π, possible transitions (Recall Λ = 0, ) Π Π Π Σ Π Λ = 0 Λ = Λ =. Three separate systems of bands possible from X Π. Hönl-London factors for Λ = differ from for Λ = 0 (see previous page) 3

14 3. Electronic Spectra of Diatomic Molecules: Improved Treatments (add Spin). Review of angular momentum Hund s case a. 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 4

15 3.. Review of angular momentum Review then add spin Term symbol Multiplicity of state Λ 0 Symbol Σ Π 4 models Term Symbol S Projection of electron orbital angular momentum on A axis Sum of projections on A axis when Λ 0 S, S,..., S Rigid Rotor Λ = S = 0 e.g., N, H : X Σ Symmetric Top Λ 0; S = 0 e.g., Π Add spin Hund s a Λ 0; S 0 e.g., OH, NO (both X Π) Hund s b Λ = 0; S 0 e.g., O : X 3 Σ 5

16 3.. Review of angular momentum Electronic angular momentum for molecules Orbital angular momentum of electrons. Separate from spin and nuclear rotation. Strong electrostatic field exists between nuclei. So L precesses about field direction (internuclear axis) with allowed components along axis m l = L, L-,, -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) 6

17 3.. Review of angular momentum Electronic angular momentum for molecules Spin of electrons. To determine L and S for molecule, we usually sum l & s for all electrons. e.g., S s i i So even number of electrons integral spin odd number of electrons half-integral spin. For Λ 0, precession of L about internuclear axis magnetic field along axis. So m s is defined. m s Σ = S, S-, -S. Note for change of orbital direction, energy of electron spinning in magnetic field changes no degeneracy S+ 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) 7

18 3.. Review of angular momentum Electronic angular momentum for molecules Total electronic angular momentum. Total electronic angular momentum along internuclear axis is ut since all in same direction, use simple addition. For Λ 0, magnetic field H Λ. Magnetic moment of spinning electron μ H Σ. So interaction energy is proportional to E ~ μh ~ ΛΣ, or T e = T 0 + AΛΣ (more on this later) For A > 0, Regular state For A < 0, Inverted state / 3/ 8

19 3.. Interaction of Λ and Σ This interaction is key to modeling the influence of spin on the electronic state structure. Nuclear Rotation N 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 ranges from low to high values 9

20 3.. Interaction of Λ and Σ Examples S =, Λ =, Ω = 3 (Σ = ) 3 S =, Λ =, Ω = (Σ = 0) 3 S =, Λ =, Ω = (Σ = ) Electronic energies T e T 0 A Spin-orbit coupling constant, generally Energy without interaction 3 S =, Λ =, Σ =,0, increases with molecular weight and the number of electrons T e T 0 A Sample constants A eh cm - A NO 4 cm - A HgH 3600 cm - A OH -40 cm - 0 Negative! A For A>0 3 3 Notes:. The parameter Y is often specified, where Y = A/ v. Values for A given in Herzberg, Vol.I Now, consider Hund s cases where S 0 0

21 3.3. Hund s case a Λ 0, S 0, Σ = S, S-,, -S F A S, S,...,,,... P, Q, R branches for each value of Ω. Example: S Recall h h A, 8 I c 8 I A Not to be confused with spin-orbit constant c Π Ω = 3/ and /, two electronic sub-states a total of x 3 = 6 branches

22 3.4. Hund s case b Applies when spin is not coupled to the A-axis E.g.,. For Λ = 0, is not defined, must use S. At high, especially for hydrides, even with Λ 0 N S Allowed : = N+S, N+S-,, N-S, 0 only For this case, S and N couple directly,

23 3.4. Hund s case b Example O Ground state X 3 Σ has three s for each N! F (N) F (N) F 3 (N) = N + = N = N N = = 3 = Notes: Split rotational levels for N > 0 N = N = 0 = = = = 0 = Each level has a degeneracy of +, and a sum of Hönl- London factors of + Minimum is N-S In N = 0 level, only spin is active (S = ), this is the minimum value of 3

24 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 F() F F and F c y definition, F c ()>F d () (c,d replaced by e,f in some literature) Lambda doubling usually results in a very small change in energy, affecting oltzmann distribution only slightly. Change of parity between Λ-doubled states reduces the accessible fraction of molecules for a given transition (due to selection rules) d F c () F d () 4

25 4. Quantitative absorption Review of eer s law and spectral absorption as interpreted for molecules with multiplet structure eer s Law I I For a complex, multiple level system, we have quantities to specify: oltzmann fraction? 0 Oscillator strength for a specific transition? Integrated absorption intensity [cm - s - ] exp k n For two-level system n ni n i e k S mec, i initial, j final L n f exp h / kt f ij 5

26 4. Quantitative absorption oltzmann fraction Quantum numbers: n ni ni n i = the total number density of species I n /n i = the fraction of species i in state/level i n electronic v vibrational Σ spin Λ orbital total angular momentum N nuclear rotation c or d Λ-component We will illustrate this in the next lecture! n n Ni n, v,,,, N n N i 6

27 4. Quantitative absorption Oscillator strength Strength of a specific, single transition (i.e., from one of the substrates to a specific substrate), f Notes: f f m,v, n,v, f el system strength osc. f q vv Franck-Condon factor S normalized H-L factor or line strength q v v v S S δ = for Σ-Σ, otherwise δ = (Λ-doubling). el [(S+)δ] = 4 for OH s A Σ X Π system. f S f sum is f el for a single substate. v, 7

28 Oscillator strength 8 4. Quantitative absorption Remarks. and oscillator strength often is tabulated e.g., f 00 = 0.00 (OH A Σ X Π). e.g., if only P and R are allowed 3. In some cases, an additional correction term T is used, e.g., in OH 4. In terms of A-coefficient vv v v q f f el vv S f f, S S R P always near, vv T T S f f vv vv vv 8 f g g g g A e m c f e e e e e

29 Next: Case Study of Molecular Spectra Ultraviolet: OH