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VIBRATIONAL SPECTROSCOPY • the Vibrational Energy V(R) Can Be Calculated Using the (Classical) Model of the Harmonic Oscillator

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VIBRATIONAL SPECTROSCOPY • the Vibrational Energy V(R) Can Be Calculated Using the (Classical) Model of the Harmonic Oscillator VIBRATIONAL SPECTROSCOPY • The vibrational energy V(r) can be calculated using the (classical) model of the harmonic oscillator: • Using this potential energy function in the Schrödinger equation, the vibrational frequency can be calculated: The vibrational frequency is increasing with: • increasing force constant f = increasing bond strength • decreasing atomic mass • Example: f cc > f c=c > f c-c Vibrational spectra (I): Harmonic oscillator model • Infrared radiation in the range from 10,000 – 100 cm –1 is absorbed and converted by an organic molecule into energy of molecular vibration –> this absorption is quantized: A simple harmonic oscillator is a mechanical system consisting of a point mass connected to a massless spring. The mass is under action of a restoring force proportional to the displacement of particle from its equilibrium position and the force constant f (also k in followings) of the spring. MOLECULES I: Vibrational We model the vibrational motion as a harmonic oscillator, two masses attached by a spring. nu and vee! Solving the Schrödinger equation for the 1 v h(v 2 ) harmonic oscillator you find the following quantized energy levels: v 0,1,2,... The energy levels The level are non-degenerate, that is gv=1 for all values of v. The energy levels are equally spaced by hn. The energy of the lowest state is NOT zero. This is called the zero-point energy. 1 R h Re 0 2 Vibrational spectra (III): Rotation-vibration transitions The vibrational spectra appear as bands rather than lines. When vibrational spectra of gaseous diatomic molecules are observed under high-resolution conditions, each band can be found to contain a large number of closely spaced components— band spectra. The structure observed is due to that a single vibrational energy change is accompanied by a number of rotational energy changes. The form of such a vibration-rotation spectrum can be predicted from the energy levels of a vibrating-rotating molecule. –> “vibrational-rotational bands” A vibrational absorption transition from to +1 gives rise to three sets of lines called branches: Lower-frequency P branch: =1, J=-1; Higher-frequency R branch: =1, J=+1; Q branch: branch: =1, J=0. EVIB ( En+1 – En ) =h× nosc • The energy difference between the transition from n to n+1 corresponds to the energy of the absorbed light quantum • The difference between two adjacent energy levels gets smaller with increasing n until dissociation of the molecule occurs (Dissociation energy ED ) Note: Weaker transitions called “overtones” are sometimes observed. These correspond to =2 or 3, and their frequencies are less than two or three times the fundamental frequency (=1) because of anharmonicity. Typical energy spacings for vibrational levels are on the order of 10-20 J. from the Bolzmann distribution, it can be shown that at room temperature typically 1% or less of the molecules are in excited states in the absence of external radiation. Thus most absorption transitions observed at room temperature are from the =0 to the =1 level. Total Energy The total energy is the energy of each degree of freedom: trans rot vib elec 2 2 2 2 h nx ny nz 1 nx ,ny ,nz 2 2 2 8m a b c v h(v 2 ) For each vib. DOF nx 1,2,3,... ny 1,2,3,... Look up values in a nz 1,2,3,... table (i.e., De). 2 J(J 1) J 2I J 0,1,2,... For linear molecules. Potential Energy for Harmonic Oscillator • The oscillator has total energy equal to kinetic energy + potential energy. • when the oscillator is at A, it is momentarily at rest, so has no U=0 kinetic energy 9 Morse Potential The dissociation energy De is larger than the true energy required for dissociation D0 due to the zero point energy of the lowest (v = 0) vibrational level. 10 Energy Levels for a Quantum Mechanical Harmonic Oscillator 11 Molecular Spectra - Vibrational States • For a simple harmonic oscillator, the classical frequency f of oscillation is given by 1 k k is the stiffness constant f 2 is the reduced mass Solution of the Schrodinger equation for the simple harmonic oscillator potential shows that the oscillator energy Evib is quantised. 1 Vibrational quantum number v = 0, 1, 2, 3, .... E (v )hf vib 2 f is the frequency Note that the lowest vibrational energy ( for v = 0) is not zero (as is the case for rotation) but hf/2. This is known as the zero point energy. Also note that the energy levels are equally spaced. Energy spacing is hf. Sept. 2002 Molecules Slide 12 Molecular Spectra - Vibrational Vibrational transitions are subjectStates to the following selection rule: v = 1 • The selection rule shows that allowed vibrational transitions can only occur between adjacent vibrational energy levels. • In the simple harmonic approximation, the energy E of emitted or absorbed photon is given by E hf Transition energies are 10 to 100 times those for rotational and wavelengths are in the infrared spectral region (l ~ 1m to 100m) Sept. 2002 Molecules Slide 13 Vibrational spectra (II): Anharmonic oscillator model The actual potential energy of vibrations fits the parabolic function fairly well only near the equilibrium internuclear distance. The Morse potential function more closely resembles the potential energy of vibrations in a molecule for all internuclear distances-anharmonic oscillator Fig. 12-1 model. Rotational-Vibrational Transitions The harmonic oscillator has already been treated and the model may be applied to molecules in their lower vibrational states. The allowed energies of the quantum harmonic oscillator are Where: and k is the force constant and μ is the reduced mass. Effect of Hydrogen Bonding • Hydrogen bonding alters the force constant of both groups: – the X–H stretching bands move to lower frequency – the stretching frequency of the acceptor group (B) is also reduced, but to a lesser degree – The X–H bending vibration usually shifts to a shorter wavelength Fingerprint region In the region from 1300 to 400 cm-1, vibrational frequencies are affected by the entire molecule, as the broader ranges for group absorptions in the figure below – fingerprint region. Absorption in this fingerprint region is characteristic of the molecule as a whole. This region finds widespread use for identification purpose by comparison with library spectra. Improvements to Harmonic Oscilla Harmonic oscillator: • Energy levels are equally spaced on the ladder Real molecule • Energy levels get closer together as energy increases • Near dissociation, energy levels become continuous Morse Potential • The simplest approximation to the real potential energy curve is the Morse Potential 1 2 x 2 k V( r ) De 1 e where 2De • The solutions are: 2 1 1 2 n n hn e n h n e xe 2 2 n e where xe 4De • The energy levels get closer and closer together with increasing v Quantum Harmonic Oscillator • Zero point energy: lowest energy corresponds to n = 0 1 1 0 0 hn hn 2 2 • Just as for the particle in a box, confining the oscillator leads to quantization of energy and non zero minimum energy • The molecule has vibrational energy even in the lowest possible energy level (and even at T = 0 K.
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