Quantum theory and mechanism of Raman scattering. The molecule \(\ce{NaH}\) undergoes a rotational transition from \(J=0\) to \(J=1\) when it absorbs a photon of frequency \(2.94 \times 10^{11} \ Hz\). E_{photon} = h_{\nu} = hc\widetilde{\nu} &= J_f(J_f+1)\frac{\hbar^2}{2I} - J_i(J_i+1)\frac{\hbar^2}{2I}\\ Usefulness of rotational spectra 11 2. As we have just seen, quantum theory successfully predicts the line spacing in a rotational spectrum. We predict level degeneracy of the rotational type in diatomic molecules described by means of a cotangent-hindered rigid rotator. Figure \(\PageIndex{3}\) shows the rotational spectrum as a series of nearly equally spaced lines. The diagram shows a portion of the potential diagram for a stable electronic state of a diatomic molecule. E_{r.rotor} &= J(J+1)\frac{\hbar^2}{2I}\\ 1.2 Rotational Spectra of Rigid diatomic molecules A diatomic molecule may be considered as a rigid rotator consisting of atomic masses m 1 andm 2 connected by a rigid bond of length r, (Fig.1.1) Fig.1.1 A rigid diatomic molecule Consider the rotation of this rigid rotator about an axis perpendicular to its molecular axis and The moment of inertia about the center of mass is, Determining the structure of a diatomic molecule, Determining the structure of a linear molecule, Example of the structure of a polyatomic molecule, The rotational spectrum of a diatomic molecule consists of a series of equally spaced absorption lines, typically in the. The rotation of a rigid object in space is very simple to visualize. Molecular Structure, Vol. Rigid rotator: explanation of rotational spectra iv. In what ways does the quantum mechanical description of a rotating molecule differ from the classical image of a rotating molecule? E_{photon} = h_{\nu} = hc\widetilde{\nu} &= (1+J_i)(2+J_i)\frac{\hbar^2}{2I} - J_i(J_i+1)\frac{\hbar^2}{2I} \\ The rotations of a diatomic molecule can be modeled as a rigid rotor. Rigid-Rotor model of diatomic molecule Equal probability assumption (crude but useful) Abs. Infrared spectroscopists use units of wavenumbers. Isotope effect vii. Use Equation \(\ref{5.9.8}\) to prove that the spacing of any two lines in a rotational spectrum is \(2B\), i.e. What properties of the molecule can be physically observed? J = 0 ! &= \frac{\hbar^2}{2I}2(J_i+1)\\ To analyze molecules for rotational spectroscopy, we can break molecules down into 5 categories based on their shapes and their moments of inertia around their 3 orthogonal rotational axes: Diatomic Molecules. ROTATIONAL SPECTROSCOPY: Microwave spectrum of a diatomic molecule. The rotational energies for rigid molecules can be found with the aid of the Shrodinger equation. The rotational constant for 79 Br 19 F is 0.35717cm-1. Quantum symmetry effects. The Non-Rigid Rotor When greater accuracy is desired, the departure of the molecular rotational spectrum from that of the rigid rotor model can be described in terms of centrifugal distortion and the vibration-rotation interaction. To convert to kilograms, we need the conversion factor \(1 \ au = 1.66\times 10^{-27} \ kg\). To second order in the relevant quantum numbers, the rotation can be described by the expression The electromagnetic field exerts a torque on the molecule. This stretching increases the moment of inertia and decreases the rotational constant (Figure \(\PageIndex{5}\)). \[\begin{align*} Example: CO B = 1.92118 cm-1 → r CO = 1.128227 Å 10-6 Å = 10-16 m Ic h 8 2 2 r e Intensities of spectral lines 12 2. Rigid-Rotor model of diatomic molecule Measured spectra Physical characteristics of molecule Line spacing =2B B I r e Accurately! In this case the rotational spectrum in the vibrational ground state is characterized by Δ J = 0, ±1, Δ k = ±3 selection rules for the overall rotational angular momentum and for its projection along the symmetry axis of the molecule. To second order in the relevant quantum numbers, the rotation can be described by the expression . The selection rules for the rotational transitions are derived from the transition moment integral by using the spherical harmonic functions and the appropriate dipole moment operator, \(\hat {\mu}\). Khemendra Shukla M.Sc. Since microwave spectroscopists use frequency units and infrared spectroscopists use wavenumber units when describing rotational spectra and energy levels, both \(\nu\) and \(\bar {\nu}\) are important to calculate. Incident electromagnetic waves can excite the rotational levels of molecules provided they have an electric dipole moment. J = 5 4 3 2 1 0 Transitions observed in absorption spectrum. 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