Editing Raman spectroscopy (section)

=== Raman Excitation Profile Analysis ===
Resonance Raman [[selection rule]]s can be explained by the [[Kramers–Heisenberg formula|Kramers–Heisenberg equation]] using the Albrecht A and B terms, as demonstrated.<ref>{{Cite journal |last=Albrecht |first=Andreas C. |date=1961-05-01 |title=On the Theory of Raman Intensities |url=http://dx.doi.org/10.1063/1.1701032 |journal=The Journal of Chemical Physics |volume=34 |issue=5 |pages=1476–1484 |doi=10.1063/1.1701032 |bibcode=1961JChPh..34.1476A |issn=0021-9606}}</ref> The Kramers–Heisenberg expression is conveniently linked to the polarizability of the molecule within its frame of reference:<ref name=":0">{{Cite book |last=McHale |first=Jeanne L. |date=2017-07-06 |title=Molecular Spectroscopy |url=http://dx.doi.org/10.1201/9781315115214 |doi=10.1201/9781315115214|isbn=978-1-4665-8659-8 }}</ref>

<math display="block">(\alpha_{\rho\sigma})_{if} = \frac{1}{\hbar} \sum_n \left[ \frac{\langle i \vert \mu_\rho \vert n \rangle \langle n \vert \mu_\sigma \vert f \rangle}{\omega_0 + \omega_{nf} + i \Gamma_n} - \frac{\langle i \vert \mu_\sigma \vert n \rangle \langle n \vert \mu_\rho \vert f \rangle}{\omega_0 - \omega_{nf} - i \Gamma_n} \right] \equiv \langle i \vert \hat{\alpha}_{\rho\sigma} \vert f \rangle</math>

The [[polarizability]] operator connecting the initial and final states expresses the transition polarizability as a [[Matrix element (physics)|matrix element]], as a function of the incidence frequency ω<sub>0</sub>.<ref name=":0" /> The directions x, y, and z in the molecular frame are represented by the [[Cartesian tensor]] ρ and σ here. Analyzing Raman excitation patterns requires the use of this equation, which is a sum-over-states expression for polarizability. This series of profiles illustrates the connection between a Raman active vibration's excitation [[frequency]] and [[Intensity (physics)|intensity]].<ref name=":0" />

This method takes into account sums over [[Franck–Condon principle|Franck-Condon's]] active vibrational states and provides insight into electronic [[Absorption spectroscopy|absorption]] and [[emission spectrum|emission spectra]]. Nevertheless, the work highlights a flaw in the sum-over-states method, especially for large molecules like visible [[chromophore]]s, which are commonly studied in Raman spectroscopy.<ref name=":0" /> The difficulty arises from the potentially infinite number of intermediary steps needed. While lowering the sum at higher vibrational states can help tiny molecules get over this issue, larger molecules find it more challenging when there are more terms in the sum, particularly in the condensed phase when individual [[Quantum state|eigenstates]] cannot be resolved spectrally.<ref name=":0" />

To overcome this, two substitute techniques that do not require adding eigenstates can be considered. Among these two methods are available: the transform method.<ref>{{Cite journal |last1=Hizhnyakov |first1=V.V. |last2=Tehver |first2=I.J. |date=March 1980 |title=Resonance Raman profile with consideration for quadratic vibronic coupling |url=http://dx.doi.org/10.1016/0030-4018(80)90274-6 |journal=Optics Communications |volume=32 |issue=3 |pages=419–421 |doi=10.1016/0030-4018(80)90274-6 |bibcode=1980OptCo..32..419H |issn=0030-4018}}</ref><ref>{{Cite journal |last1=Shreve |first1=Andrew P. |last2=Haroz |first2=Erik H. |last3=Bachilo |first3=Sergei M. |last4=Weisman |first4=R. Bruce |last5=Tretiak |first5=Sergei |last6=Kilina |first6=Svetlana |last7=Doorn |first7=Stephen K. |date=2007-01-19 |title=Determination of Exciton-Phonon Coupling Elements in Single-Walled Carbon Nanotubes by Raman Overtone Analysis |url=http://dx.doi.org/10.1103/physrevlett.98.037405 |journal=Physical Review Letters |volume=98 |issue=3 |page=037405 |doi=10.1103/physrevlett.98.037405 |pmid=17358727 |bibcode=2007PhRvL..98c7405S |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Blazej |first1=Daniel C. |last2=Peticolas |first2=Warner L. |date=1980-03-01 |title=Ultraviolet resonance Raman excitation profiles of pyrimidine nucleotides |url=http://dx.doi.org/10.1063/1.439547 |journal=The Journal of Chemical Physics |volume=72 |issue=5 |pages=3134–3142 |doi=10.1063/1.439547 |bibcode=1980JChPh..72.3134B |issn=0021-9606}}</ref> and Heller's time-dependent approach.<ref>{{Cite journal |last1=Lee |first1=Soo-Y. |last2=Heller |first2=E. J. |date=1979-12-15 |title=Time-dependent theory of Raman scattering |url=http://dx.doi.org/10.1063/1.438316 |journal=The Journal of Chemical Physics |volume=71 |issue=12 |pages=4777–4788 |doi=10.1063/1.438316 |bibcode=1979JChPh..71.4777L |issn=0021-9606}}</ref><ref>{{Cite journal |last1=Heller |first1=Eric J. |last2=Sundberg |first2=Robert |last3=Tannor |first3=David |date=May 1982 |title=Simple aspects of Raman scattering |url=http://dx.doi.org/10.1021/j100207a018 |journal=The Journal of Physical Chemistry |volume=86 |issue=10 |pages=1822–1833 |doi=10.1021/j100207a018 |issn=0022-3654}}</ref><ref>{{Cite journal |last=Heller |first=Eric J. |date=1981-12-01 |title=The semiclassical way to molecular spectroscopy |url=http://dx.doi.org/10.1021/ar00072a002 |journal=Accounts of Chemical Research |volume=14 |issue=12 |pages=368–375 |doi=10.1021/ar00072a002 |issn=0001-4842}}</ref><ref>{{Cite journal |last1=Tannor |first1=David J. |last2=Heller |first2=Eric J. |date=1982-07-01 |title=Polyatomic Raman scattering for general harmonic potentials |url=http://dx.doi.org/10.1063/1.443643 |journal=The Journal of Chemical Physics |volume=77 |issue=1 |pages=202–218 |doi=10.1063/1.443643 |bibcode=1982JChPh..77..202T |issn=0021-9606}}</ref> The goal of both approaches is to take into consideration the frequency-dependent Raman cross-section σ<sub>R</sub>(ω<sub>0</sub>) of a particular normal mode.<ref name=":0" />