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Lecture 32 General issues of spectroscopies. II (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies. General issues of spectroscopies We will learn two types of spectroscopies: absorption/emission and scattering. We will learn their relationship to dipole moment and polarizability as well as perturbation theories. Absorption/emission versus scattering spectroscopy Absorption/emission spectroscopy: microwave, IR, and UV/vis absorption as well as fluorescence Scattering spectroscopy: Raman IR Raman Absorption spectroscopy One-photon process 1st order perturbation theory I µ òY (0)* i ˆ xY dt (0) f 2 Scattering (Raman) spectroscopy C. V. Raman Public domain image from Wikipedia Two-photon process 2nd order perturbation theory Y ò Iµå k (0)* i ˆ xY (0) k dt ò Y (0)* k yˆ Y Ek(0) - Ei(0) - hn (0) f dt Y ò +å k (0)* i ˆ xY (0) k dt ò Y (0)* k yˆ Y Ek(0) - E (0) + hn f (0) f dt 2 Scattering (Raman) spectroscopy Stokes Raman Rayleigh Anti-Stokes Raman (a hot band) Scattering (Raman) spectroscopy Resonant Rayleigh Raman Y ò Iµå k (0)* i ˆ xY (0) k dt ò Y (0)* k yˆ Y Ek(0) - Ei(0) - hn (0) f dt Y ò +å k (0)* i Resonant Raman ˆ xY (0) k dt ò Y (0)* k yˆ Y Ek(0) - E (0) + hn f (0) f dt 2 Quantum in nature Resonanc e Rayleigh scattering Why is sky blue? Quantum in nature Resonanc e Rayleigh scattering Why is the Sun yellow? Dipole moment and polarizability Absorption spectroscopies (such as IR spectroscopy) are related to the dipole moment and its changes. Scattering spectroscopies (such as Raman spectroscopy) are related to the polarizability and its changes. Polarizability – softness of wave function; the larger the polarizability, the more easily the wave function is distorted by external electric field to create an induced dipole. Dipole moment ++++++++++++ Electric field Perturbation E0 E0 + µE – – – – – – – – – E (1) = òY x (0)* 0 (0) ˆ ( Ex ) Y0 dt Think of ( 0 )* 0 ˆx (00 ) d molecular length! First-order perturbation theory E (1) Polarizability ++++++++++++ E (2) E0 + µE + αE2 =E0 + (µ+ αE)E E0 – – – – – – – – – Induced dipole (0)* (0) (0)* (0) ˆ ˆ Y E x Y d t Y E x Y ò 0 ( ) k ò k ( ) 0 dt Think of E = å (0) (0) E E k 0 k molecular (0)* (0) (0)* (0) ˆ ˆ Y xY d t Y xY dt 0 k k 0 ò ò volume! a xx = å (0) (0) (2) k E0 - Ek Second-order perturbation theory Polarizability 0 ( 0 )* ++++++++++++ xy k x y – – – – – – – – – (0) ( 0 )* (0) xˆ k d k yˆ 0 d (0) E0 Ek (0) æ a a xy a xz xx ç ç a yx a yy a yz ç çè a zx a zy a zz ö ÷ ÷ ÷ ÷ø Classical (Smekal) theory of Rayleigh and Raman scattering An oscillating electric field (incident photon) causes the molecule to have an induced dipole: mind µ a cos w t Polarizability varies with molecular vibration; so does induced dipole: 0 cos 0 t mind µ éëa 0 + ( Da ) cos w 0t ùû cos w t = a 0 cos w t + 12 ( Da ) cos (w + w 0 ) t + 12 ( Da ) cos (w - w 0 ) t Rayleigh Anti-Stokes Stokes Absorption/emission versus Raman spectroscopies Absorption/emission occurs when molecular vibration, rotation, etc. alter dipole moment. The transition tends to transform as x, y, z. Raman occurs when molecular vibration, rotation, etc. alter polarizability. The transition tends to transform as xx, yy, zz, xy, yz, zx. Summary We have the general theories of absorption/emission spectroscopy and scattering (Raman) spectroscopy. We have learned the relationship between absorption/emission spectroscopies to dipole moment and that between scattering (Raman) spectroscopy and polarizability. We have made references to first- and second-order perturbation theories.