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÷G¦Hþ˜Ë“.Ìoã^:‘¡×æɕØî‘ 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. 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