# Oct 5 - Vibrational Spectra V(x = kx2 X = R-Re Harmonic...

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Vibrational Spectra

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V(x) = ½ kx 2 X = R-R e Born- Oppenheimer Approximation of MO energy Harmonic Oscillator Approximation
V(x) = ½ kx 2 X = R-R e But we can describe the potential V(x) by expanding V(x) as a Taylor Series about the point x = 0 A constant; set V(0) = 0 for convenience The first derivative dV/dx is zero when x = 0 This is the first finite term of the expansion! Additional terms in the expansion are also finite, but increasingly smaller. To a first approximation we may ignore them Therefore 0

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PHYSICAL CHEMISTRY: QUANTA, MATTER, AND CHANGE 2E| PETER ATKINS| JULIO DE PAULA | RONALD FRIEDMAN ©2014 W. H. FREEMAN D COMPANY
Review the Schrodinger Equation for the harmonic oscillator: The solution to the Schrodinger Equation for the harmonic oscillator : N v = normalization constant H v (y) = Hermite polynomial exp(-y 2 /2) = Gaussian function

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The Harmonic Oscillator The solution to the Schrodinger Equation for the harmonic oscillator : N v = normalization constant H v (y) = Hermite polynomial exp(-y 2 /2) = Gaussian function 8B.8
The Harmonic Oscillator The eigenvalues of the Schrodinger Equation for the Harmonic Oscillator represent the allowed quantum mechanical energies of the harmonic oscillator 8B.4  is the radial frequency of oscillation; this could also be converted to  easy to confuse with v ] These frequencies are in the infrared region of the electromagnetic spectrum so we need an IR spectrophotometer

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Selection Rules for IR spectroscopy Inducing a vibrational energy transition requires creating overlap between two vibrational energy levels using the electric dipole moment operator Now we want  to create overlap between two vibrational wavefunctions We need an function for  as the molecule vibrates so we utilize a Taylor Series expansion to describe m near x = 0 [the equilibrium position of the harmonic oscillator] Text p 505
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