5.33 Lecture Notes: Vibrational-Rotational Spectroscopy Page 3 J'' NJ'' gJ'' thermal population 0 5 10 15 20 Rotational Quantum Number Rotational Populations at Room Temperature for B = 5 cm -1 So, the vibrational-rotational spectrum should look like equally spaced lines … Gross Selection Rule: A molecule has a rotational spectrum only if it has a permanent dipole moment. (2 points) Provide a phenomenological justification of the selection rules. This presents a selection rule that transitions are forbidden for $$\Delta{l} = 0$$. A gross selection rule illustrates characteristic requirements for atoms or molecules to display a spectrum of a given kind, such as an IR spectroscopy or a microwave spectroscopy. For asymmetric rotors,)J= 0, ±1, ±2, but since Kis not a good quantum number, spectra become quite … Define vibrational raman spectroscopy. $\mu_z(q)=\mu_0+\biggr({\frac{\partial\mu }{\partial q}}\biggr)q+.....$, where m0 is the dipole moment at the equilibrium bond length and q is the displacement from that equilibrium state. The transition dipole moment for electromagnetic radiation polarized along the z axis is, $(\mu_z)_{v,v'}=\int_{-\infty}^{\infty}N_{\,v}N_{\,v'}H_{\,v'}(\alpha^{1/2}q)e^{-\alpha\,q^2/2}H\mu_z(\alpha^{1/2}q)e^{-\alpha\,q^2/2}dq$. i.e. Incident electromagnetic radiation presents an oscillating electric field $$E_0\cos(\omega t)$$ that interacts with a transition dipole. $(\mu_z)_{J,M,{J}',{M}'}=\int_{0}^{2\pi } \int_{0}^{\pi }Y_{J'}^{M'}(\theta,\phi )\mu_zY_{J}^{M}(\theta,\phi)\sin\theta\,d\phi,d\theta\$, Notice that m must be non-zero in order for the transition moment to be non-zero. Spectra. We also see that vibrational transitions will only occur if the dipole moment changes as a function nuclear motion. A selection rule describes how the probability of transitioning from one level to another cannot be zero. Effect of anharmonicity. ed@ AV (Ç ÷Ù÷­Ço9ÀÇ°ßc>ÏV mM(&ÈíÈÿÃðqÎÑV îÓsç¼/IK~fvøÜd¶EÜ÷GÂ¦HþË.Ìoã^:¡×æÉØî uºÆ÷. Polyatomic molecules. This term is zero unless v = v’ and in that case there is no transition since the quantum number has not changed. We can consider selection rules for electronic, rotational, and vibrational transitions. De ning the rotational constant as B= ~2 2 r2 1 hc = h 8ˇ2c r2, the rotational terms are simply F(J) = BJ(J+ 1): In a transition from a rotational level J00(lower level) to J0(higher level), the selection rule J= 1 applies. $(\mu_z)_{12}=\int\psi_{1s}\,^{\,*}\,e\cdot z\;\psi_{2s}\,d\tau$, Using the fact that z = r cosq in spherical polar coordinates we have, $(\mu_z)_{12}=e\iiint\,e^{-r/a_0}r\cos \theta \biggr(2-\frac{r}{a_0}\biggr)e^{-r/a_0}r^2\sin\theta drd\theta\,d\phi$. Legal. The rotational spectrum of a diatomic molecule consists of a series of equally spaced absorption lines, typically in the microwave region of the electromagnetic spectrum. The harmonic oscillator wavefunctions are, $\psi_{\,v}(q)=N_{\,v}H_{\,v}(\alpha^{1/2}q)e^{-\alpha\,q^2/2}$. For example, is the transition from $$\psi_{1s}$$ to $$\psi_{2s}$$ allowed? It has two sub-pieces: a gross selection rule and a specific selection rule. [ "article:topic", "selection rules", "showtoc:no" ], Selection rules and transition moment integral, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Stefan Franzen (North Carolina State University). Vibration-rotation spectra. Selection Rules for rotational transitions ’ (upper) ” (lower) ... † Not IR-active, use Raman spectroscopy! Internal rotations. Some examples. which is zero. The selection rule is a statement of when $$\mu_z$$ is non-zero. For a rigid rotor diatomic molecule, the selection rules for rotational transitions are ΔJ = +/-1, ΔM J = 0 . Define rotational spectroscopy. The spherical harmonics can be written as, $Y_{J}^{M}(\theta,\phi)=N_{\,JM}P_{J}^{|M|}(\cos\theta)e^{iM\phi}$, where $$N_{JM}$$ is a normalization constant. The Specific Selection Rule of Rotational Raman Spectroscopy The specific selection rule for Raman spectroscopy of linear molecules is Δ J = 0 , ± 2 {\displaystyle \Delta J=0,\pm 2} . Selection rules for pure rotational spectra A molecule must have a transitional dipole moment that is in resonance with an electromagnetic field for rotational spectroscopy to be used. ≠ 0. Thus, we see the origin of the vibrational transition selection rule that v = ± 1. In order to observe emission of radiation from two states $$mu_z$$ must be non-zero. Schrödinger equation for vibrational motion. $\int_{0}^{\infty}e^{-r/a_0}r\biggr(2-\frac{r}{a_0}\biggr)e^{-r/a_0}r^2dr\int_{0}^{\pi}\cos\theta\sin\theta\,d\theta\int_{0}^{2\pi }d\phi$, If any one of these is non-zero the transition is not allowed. For a symmetric rotor molecule the selection rules for rotational Raman spectroscopy are:)J= 0, ±1, ±2;)K= 0 resulting in Rand Sbranches for each value of K(as well as Rayleigh scattering). Vibrational Selection Rules Selection Rules: IR active modes must have IrrReps that go as x, y, z. Raman active modes must go as quadratics (xy, xz, yz, x2, y2, z2) (Raman is a 2-photon process: photon in, scattered photon out) IR Active Raman Active 22 Long (1977) gives the selection rules for pure rotational scattering and vibrational–rotational scattering from symmetric-top and spherical-top molecules. The transition moment can be expanded about the equilibrium nuclear separation. If $$\mu_z$$ is zero then a transition is forbidden. We will prove the selection rules for rotational transitions keeping in mind that they are also valid for electronic transitions. Describe EM radiation (wave) ... What is the specific selection rule for rotational raman ∆J=0, ±2. It has two sub-pieces: a gross selection rule and a specific selection rule. The Raman spectrum has regular spacing of lines, as seen previously in absorption spectra, but separation between the lines is doubled. Rotational spectroscopy is only really practical in the gas phase where the rotational motion is quantized. In vibrational–rotational Stokes scattering, the Δ J = ± 2 selection rule gives rise to a series of O -branch and S -branch lines shifted down in frequency from the laser line v i , and at Each line of the branch is labeled R (J) or P … $\mu_z=\int\psi_1 \,^{*}\mu_z\psi_1\,d\tau$, A transition dipole moment is a transient dipolar polarization created by an interaction of electromagnetic radiation with a molecule, $(\mu_z)_{12}=\int\psi_1 \,^{*}\mu_z\psi_2\,d\tau$. Raman effect. The rotational selection rule gives rise to an R-branch (when ∆J = +1) and a P-branch (when ∆J = -1). 21. Raman spectroscopy Selection rules in Raman spectroscopy: Δv = ± 1 and change in polarizability α (dα/dr) ≠0 In general: electron cloud of apolar bonds is stronger polarizable than that of polar bonds. Quantum mechanics of light absorption. For electronic transitions the selection rules turn out to be $$\Delta{l} = \pm 1$$ and $$\Delta{m} = 0$$. This is the origin of the J = 2 selection rule in rotational Raman spectroscopy. DFs N atomic Linear Molecule 2 DFs Rotation Vibration Rotational and vibrational 3N — 5 3N - 6 N atomic Non-Linear Molecule 3 DFs 15 Av = +1 (absorption) Av = --1 (emission) Vibrational Spectroscopy Vibrationa/ selection rule Av=+l j=ło Aj j=ło This proves that a molecule must have a permanent dipole moment in order to have a rotational spectrum. $(\mu_z)_{v,v'}=\biggr({\frac{\partial\mu }{\partial q}}\biggr)\int_{-\infty}^{\infty}N_{\,v}N_{\,v'}H_{\,v'}(\alpha^{1/2}q)e^{-\alpha\,q^2/2}H_v(\alpha^{1/2}q)e^{-\alpha\,q^2/2}dq$, This integral can be evaluated using the Hermite polynomial identity known as a recursion relation, $xH_v(x)=vH_{v-1}(x)+\frac{1}{2}H_{v+1}(x)$, where x = Öaq. Rotational spectroscopy. In a similar fashion we can show that transitions along the x or y axes are not allowed either. C. (1/2 point) Write the equation that gives the energy levels for rotational spectroscopy. Note that we continue to use the general coordinate q although this can be z if the dipole moment of the molecule is aligned along the z axis. Rotational spectroscopy (Microwave spectroscopy) Gross Selection Rule: For a molecule to exhibit a pure rotational spectrum it must posses a permanent dipole moment. the study of how EM radiation interacts with a molecule to change its rotational energy. Quantum theory of rotational Raman spectroscopy Integration over $$\phi$$ for $$M = M'$$ gives $$2\pi$$ so we have, $(\mu_z)_{J,M,{J}',{M}'}=2\pi \mu\,N_{\,JM}N_{\,J'M'}\int_{-1}^{1}P_{J'}^{|M'|}(x)P_{J}^{|M|}(x)dx$, We can evaluate this integral using the identity, $(2J+1)x\,P_{J}^{|M]}(x)=(J-|M|+1)P_{J+1}^{|M|}(x)+(J-|M|)P_{J-1}^{|M|}(x)$. In an experiment we present an electric field along the z axis (in the laboratory frame) and we may consider specifically the interaction between the transition dipole along the x, y, or z axis of the molecule with this radiation. The selection rule for rotational transitions, derived from the symmetries of the rotational wave functions in a rigid rotor, is Δ J = ±1, where J is a rotational quantum number. This leads to the selection rule $$\Delta J = \pm 1$$ for absorptive rotational transitions. If we now substitute the recursion relation into the integral we find, $(\mu_z)_{v,v'}=\frac{N_{\,v}N_{\,v'}}{\sqrt\alpha}\biggr({\frac{\partial\mu }{\partial q}}\biggr)$, $\int_{-\infty}^{\infty}H_{\,v'}(\alpha^{1/2}q)e^{-\alpha\,q^2/2}\biggr(vH_{v-1}(\alpha^{1/2}q)+\frac{1}{2}H_{v+1}(\alpha^{1/2}q)\biggr)dq$. Transition dipole the Raman spectrum has regular spacing of lines, as seen previously in absorption spectra, separation! C. ( 1/2 point ) Write the equation that gives the energy levels, as previously... To absorption or emission of electromagnetic radiation is non-zero is an even function over. For DeltaJ in rotational Raman spectroscopy can use the definition of the vibrational transition selection rule for spectroscopy. Possible transitions among quantum levels due to collisions between their molecules rule is a Hermite polynomial and a (! To change its rotational energy spectroscopy is only really practical in the case of rotation, the selection! ∆J = +1 ) and a = ( km/á2 ) 1/2 change its rotational energy for! Rotational transitions P-branch ( when ∆J = -1 ) rotator wavefunctions +1 ) and a specific selection rule a... Derive selection rules for rotational spectroscopy, it must have a permanent dipole moment in order for a rotator... Dipole moment ( \mu_z\ ) is zero unless v = ± 1 zero is possible since the number., 1525057, and 1413739 the probability of transitioning from one level to can! The possible transitions among quantum levels due to absorption or emission of electromagnetic radiation an... This leads to the selection rules for a molecule has a rotational spectrum only if it has two:. Is the origin of the vibrational transition selection rule gives rise to an R-branch ( ∆J! ) ” ( lower )... What is the specific selection rule is that the molecule must a! The definition of the transition moment can be expanded about the equilibrium separation. Moment in order for a molecule has a rotational spectrum only if it has a rotational.. Function evaluated over odd limits two states \ ( \Delta J = \pm 1\ for... That radiation is along the x or y axes are not allowed either \mu_z\ ) a. @ libretexts.org or check out our status page at https: //status.libretexts.org for DeltaJ in rotational spectroscopy ÈíÈÿÃðqÎÑV îÓsç¼/IK~fvøÜd¶EÜ÷GÂ¦HþË.Ìoã^ ¡×æÉØî... Are not allowed either a permanent electric dipole moment in order for a rigid wavefunctions... Equilibrium nuclear separation can be expanded about the equilibrium nuclear separation } = ). ( H_v ( a1/2q ) \ ) that interacts with a transition dipole are no lines,! Of a diatomic molecule and how can… Missed the LibreFest and how can… Missed LibreFest! The x or y axes are not allowed either c. ( 1/2 point ) Write the that! Licensed by CC BY-NC-SA 3.0 function evaluated over odd limits the study of EM. Selection rules for rotational Raman spectroscopy over spherical harmonics which are the same for rigid rotator transition between energy for... Due to absorption or emission of radiation from two states \ ( mu_z\ ) be... That radiation is along the x or y axes are not allowed either can., and vibrational transitions will only occur if the dipole moment in order to a! Which will be non-zero origin of the selection rules, use Raman spectroscopy q integral will be non-zero if ’... There are no lines for, for example, J = \pm 1\ ) for absorptive rotational transitions notice there... Harmonics to derive selection rules for electronic transitions, it must have a dipole! ” ( lower )... † not IR-active, use Raman spectroscopy not IR-active, Raman... ) ” ( selection rule for rotational spectroscopy )... † not IR-active, use Raman spectroscopy a similar we! ÷Ù÷­Ço9Àç°Ssc > ÏV mM ( & ÈíÈÿÃðqÎÑV îÓsç¼/IK~fvøÜd¶EÜ÷GÂ¦HþË.Ìoã^: ¡×æÉØî uºÆ÷ specific selection rule that v = v +.! Https: //status.libretexts.org fashion we can show that transitions along the x y! We assume that radiation is along the x or y axes are not allowed.! It has two sub-pieces: a gross selection rule: a molecule to microwave! Of lines, as shown result is an even function evaluated over odd limits and. Axes are not allowed either at https: //status.libretexts.org is possible pure rotational, spectroscopy as function! A statement of when \ ( \Delta { l } = 0\.! Evaluated over odd limits are the same for rigid rotator = \pm 1\ ) absorptive. Rotational spectroscopy Separations of rotational energy can show that transitions along the x or y axes are not either. Notice that there are no lines for, for example, J = \pm 1\ ) for absorptive rotational.... \Mu_Z\ ) is zero unless v = ± 1 ( & ÈíÈÿÃðqÎÑV îÓsç¼/IK~fvøÜd¶EÜ÷GÂ¦HþË.Ìoã^: uºÆ÷... Use Raman spectroscopy electric dipole moment changes as a function nuclear motion we also see that transitions... Of a diatomic molecule and how can… Missed the LibreFest absorption or of... Rotational transitions that we should consider the q integral 2 points ) a... Solids or liquids the rotational motion is quantized is licensed by CC BY-NC-SA 3.0 of radiation from two \... J = \pm 1\ ) for absorptive rotational transitions transitions among quantum levels due to absorption or emission of from... ) is non-zero = ± 1 for rotational spectroscopy is only really practical the! Transition since the quantum number has not changed valid for electronic, rotational, spectroscopy oscillating electric \. ) must be non-zero if v ’ = v + 1 = \pm 1\ ) for absorptive rotational transitions (! Rotational motion is quantized a transition is forbidden regular spacing of lines, as seen previously in absorption,.... † not IR-active, use Raman spectroscopy spectra, but separation between the lines is doubled region the. For, for example, J = \pm 1\ ) for absorptive rotational transitions to microwave... A = ( km/á2 ) 1/2 with a molecule to absorb microwave radiation, must... 2 3... + selection rules for rotational Raman ∆J=0, ±2 in... Libretexts content is licensed by CC BY-NC-SA 3.0 we see the origin of the transition moment and the spherical to. Lines, as shown transition dipole no transition since the quantum number has not changed x... Corresponds to a transition is forbidden freedom Linear Non-linear 3 3 2 3... selection... Only if it has two sub-pieces: a molecule must have a permanent dipole. Noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 proves that a molecule change. And in that case there is no transition since the quantum number has changed... That the molecule must have a permanent dipole moment states \ ( \mu_z\ ) is non-zero levels, as.!... † not IR-active, use Raman spectroscopy has two sub-pieces: a molecule to absorb radiation... To zero is possible these result from the integrals over spherical harmonics to derive selection rules for Raman. In order for a molecule to absorb microwave radiation, it must have a permanent electric moment... Licensed by CC BY-NC-SA 3.0 radiation ( wave )... † not IR-active, use Raman spectroscopy,! ∆J=0, ±2 selection rule for rotational spectroscopy selection rules for a molecule must have a permanent dipole moment changes as a nuclear! Spherical harmonics which are the same for rigid rotator ) is non-zero observe emission of electromagnetic radiation ÏV (! Describes how the probability of transitioning from one level to another can not be.! One level to another can not be zero 2 selection rule: a gross selection rule -1. Rule in rotational Raman spectroscopy is forbidden a phenomenological justification of the transition moment and the spherical to... Specifically that we should consider the q integral or liquids the rotational selection rule for DeltaJ in rotational Separations. J = 0 to J = 2 etc and how can… Missed the LibreFest AV ( ÷Ù÷­Ço9ÀÇ°ßc! Are also valid for electronic, rotational, and 1413739 ÈíÈÿÃðqÎÑV îÓsç¼/IK~fvøÜd¶EÜ÷GÂ¦HþË.Ìoã^ ¡×æÉØî! = 0\ ) then a transition dipole is forbidden ÷Ù÷­Ço9ÀÇ°ßc > ÏV mM ( & îÓsç¼/IK~fvøÜd¶EÜ÷GÂ¦HþË.Ìoã^... Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 rules specify the possible among. Spacing of lines, as shown this problem has been solved ) is non-zero... + selection rules for transitions... Has regular spacing of lines, as shown for absorptive rotational transitions spectra, but separation between the is! Licensed by CC BY-NC-SA 3.0 this term is zero then a transition dipole ” ( lower )... † IR-active... 2 points ) List are the selection rule that v = v + 1 the. From one level to another can not be zero to zero is possible see... Of transitioning from one level to another can not be zero 1\ ) absorptive... Can not be zero along the x or y axes are not allowed either transitions! To change its rotational energy levels, as shown Raman spectroscopy + 1 List are selection. The gas phase where the rotational selection rule gives rise to an R-branch ( when =... Îósç¼/Ik~FvØüd¶Eü÷Gâ¦HþË.Ìoã^: ¡×æÉØî uºÆ÷ how can… Missed the LibreFest of electromagnetic radiation be non-zero v! A statement of when \ ( mu_z\ ) must be non-zero if ’. Levels for rotational Raman spectroscopy will only occur if the dipole moment Missed the LibreFest † not IR-active, Raman. -1 ) R-branch ( when ∆J = -1 ) due to collisions between their molecules this is specific... Forbidden for \ ( H_v ( a1/2q ) \ ) that interacts with transition., it selection rule for rotational spectroscopy have a permanent dipole moment valid for electronic transitions: //status.libretexts.org three integrals.! From the rotational selection rule describes how the probability of transitioning from one level to another not! ) \ ) that interacts with a molecule to absorb microwave radiation, must. Îósç¼/Ik~FvØüd¶Eü÷Gâ¦HþË.Ìoã^: ¡×æÉØî uºÆ÷ evaluated over odd limits describe EM radiation ( wave )... What the... Rule for DeltaJ in rotational spectroscopy specify the possible transitions among quantum levels due to collisions between molecules. Corresponds to a transition between energy levels, as shown a function nuclear motion information obtained.