Chiroptical Spectroscopy

Transcription

1 Chiroptical Spectroscopy Theory and Applications in Organic Chemistry Lecture 3: (Crash course in) Theory of optical activity Masters Level Class ( ) Mondays, am, NC 02/99 Wednesdays, am, NC 02/99 62

2 Crossed polarizers Chiroptical spectroscopy Dr. C. Merten 63

3 Crossed polarizers and optically active sample Chiroptical spectroscopy Dr. C. Merten 64

4 Optical rotation: Difference of refractive indices Linear polarized light can be expressed as superposition of left and right circular polarized light: with cos sin cos sin respectively cos sin x, y: unit vectors perpendicular to propagation direction Chiroptical spectroscopy Dr. C. Merten 65

5 Optical rotation: Difference of refractive indices with cos sin Taking into account that LCP and RCP light can have different velocities in a medium: Because 2f, we can rewrite the equation: Δ 2 / 1 2 Δ Chiroptical spectroscopy Dr. C. Merten 66

6 Optical rotation: Difference of refractive indices If medium is not circularly birefringant (optically active), then n=0 and cos sin Hence, the superposition is 2 cos i.e. it still oscillates in the plane of the direction of propagation and the unit vector x. Chiroptical spectroscopy Dr. C. Merten 67

7 Optical rotation: Difference of refractive indices If the medium shows circular birefringence, we have cos sin and the superposition becomes cos cos sin sin After some rearrangements, we get 2 cos z Δ 2 sin z Δ 2 cos Chiroptical spectroscopy Dr. C. Merten 68

8 Optical rotation: Difference of refractive indices The exiting ray is still linearly polarized, but its plane is now rotated by the angle : z Δ 2 Consequently, we have a dextrorotatory sample if n + > n - and a levorotatory sample if n - > n + Details on the following equations can be found in Atkins, Friedman, Molecular Quantum Mechanics, 5th Edition, p. 417 Chiroptical spectroscopy Dr. C. Merten 69

9 How does the refractive index depend on a molecular properties? From Maxwell equations, we find a relationship with the relative permittivity r, aka the dielectric constant: Relative permittivity r directly relates to electric susceptibility e, which measures how easily a dielectric is polarized in response to an electic field: 1 Ok, then: How does the dielectric constant depend on molecular properties? For non-polar molecules in a static electric field, it can be shown that with / polarizability 2 3 Chiroptical spectroscopy Dr. C. Merten 70

10 How does the refractive index depend on a molecular properties? From Maxwell equations, we find a relationship with the relative permittivity r, aka the dielectric constant: But the polarizability is not chiral!? We made a mistake when deriving r from the polarizability! We assumed polarizability electric field This is fully correct if there is a constant electric field. Chiroptical spectroscopy Dr. C. Merten 71

11 Contribution of the magnetic field Mistake: We ignored the contribution of the magnetic field to the polarization! Rewriting our definition of polarization as with being another molecular characteristic and not to be mixed up with the hyperpolarizability! Chiroptical spectroscopy Dr. C. Merten 72

12 The problem of static polarization From this definition of the polarization P, we get after solving a few equations again and thus the difference of the refractive indices becomes Δn For the optical rotation angle, it follows that after passing through a sample of length l, and taking into account that 0 µ v = c 2 with µ v being the vacuum permeability: Δ z Δ Chiroptical spectroscopy Dr. C. Merten 73

13 Polarization and quantum mechanics Polarization P = dipole moment density Quantum mechanical calculation of the induced average dipole moment should lead to expression for! Chiroptical spectroscopy Dr. C. Merten 74

14 Schrödinger equation The time-independent Schrödinger equation E while the full wavefunction has the form Ψ, It gives us access to observables by evaluating the observables expectation value Ω Ψ ΩΨ d Ψ Ω Ψ Chiroptical spectroscopy Dr. C. Merten 75

15 The basic idea of perturbation theory In perturbation theory, we define a (time-dependent) perturbation in the Hamilton operator: e.g. perturbation by electric field 2μ cos In presence of a perturbation, the wavefunction is a linear at combination of states: H 1 t Ψ For a n (t) it can be shown that 1 d with Chiroptical spectroscopy Dr. C. Merten 76

16 Evaluating The expectation value for the average dipole moment (its z-component) is Ψ Ψ Taking into account that is a perturbed wavefunction with ground state 0 and perturbed states, we get ,, Chiroptical spectroscopy Dr. C. Merten 77

17 Perturbation theory: Magnetically induced polarization In our case, the perturbation to the Hamiltonian is with : : : : electric dipole moment operator magnetic dipole moment operator electric field magnetic field More precisely, we also need to take into account the circular polarization states of the electric and magnetic field: with cos sin and cos sin Chiroptical spectroscopy Dr. C. Merten 78

18 Perturbation theory: Magnetically induced polarization cos sin cos sin cos sin cos sin As and cos sin cos sin cos 1 2 sin 2 i Chiroptical spectroscopy Dr. C. Merten 79

19 Perturbation theory: Magnetically induced polarization Based on this perturbation Hamiltonian, the coefficients of the perturbed wavefunctions can be determined: with 1, d, i Then, the induced electric dipole moment is the expectation value of the operator using the perturbed wavefunctions: Ψ Ψ Chiroptical spectroscopy Dr. C. Merten 80

20 Perturbation theory: Magnetically induced polarization Solving the equation for the expectation value Ψ Ψ leads to the final expression 2 Re 2 Im Comparison with the definition of polarization in an electromagnetic field shows that 2 Chiroptical spectroscopy Dr. C. Merten 81

21 The rotational strength Taking the rotational average ( xx, yy, zz ) we get 2 3 The Rosenfeld equation Δ l 3 rotational strength of transition n0 µ: electric dipole transition moment (EDTM) m: magnetic dipole transition moment (MDTM) Chiroptical spectroscopy Dr. C. Merten 82

22 Symmetry properties of the rotational strength With respect to reflection: EDTM transforms as translation MDTM transforms as rotation Requirement: Sign of R must be conserved upon action of any symmetry element of the molecule If a molecule has a mirror plane, reflection changes sign of EDTM but not of MDTM. Consequence: Rotational strength changes sign R = 0 Chiroptical spectroscopy Dr. C. Merten 83

23 Optical rotatory dispersion (ORD) The Rosenfeld equation Δ l 3 2 / At very high frequencies ( ) and in the opposite case of, the rotational angle is Δ 0 due to the sum rule 0 OR is close to zero far away from absorption bands, but becomes very large when approaching them: 0 Chiroptical spectroscopy Dr. C. Merten 84

24 Optical rotatory dispersion (ORD) When 0, i.e. if the excitation frequency is close to an absorption band, the contribution of that particular transition dominates the OR value: Δ Observation of Cotton effect Normal ORD curve: First resonances in the UV range Appears monosignate in usual range of measurement Chiroptical spectroscopy Dr. C. Merten 85

25 Outline of the lecture Dates topics Introduction Polarization of light Theoretical basis of optical activity Optical rotation Circular dichroism Circular dichroism Mondays am Wednesdays am Vibrational optical activity Vibrational optical activity applications applications your part Chiroptical spectroscopy Dr. C. Merten 86