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lecture34_umn - Lecture 34 Vibrational Spectroscopy Matrix...

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Lecture 34 December 7, 2009 Vibrational Spectroscopy Matrix Isolation Spectroscopy
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Molecular Vibrations With a defined PES, it is possible to formulate and solve Schrödinger equations for nuclear motion (as opposed to electronic motion) (33-6) N is the number of atoms , m is the atomic mass , V is the potential energy from the PES as a functions of the 3 N nuclear coordinates q , Ξ is the nuclear wave function that is expressed in those coordinates. Solution of eq. 33-6 provides entry into the realms of rotational and vibrational spectroscopy. " 1 2 m i i N # $ i 2 + V q ( ) % & ( ) * + q ( ) = E + q ( )
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Shrödinger Vibrational Equation Diatomic case: eq. 33-6 is a function of only a single variable , the interatomic distance r . Challenge: we do not know the potential energy function V ( r) . At a level of theory compute V point by point . Those points may then be fit to a polynomial, Morse , etc. One-dimensional Schrödinger equation solved using standard numerical recipes. Harmonic oscillator equation would be recovered if we fit V to a parabolic form. " 1 2 m i i N # $ i 2 + V q ( ) % & ( ) * + q ( ) = E + q ( ) " 1 2 μ # 2 r 2 + 1 2 k ( r " r eq ) 2 $ % & ( ) * nuc ( r ) = E * nuc ( r ) 33-6
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One-Dimensional Case Simplification when the potential is harmonic . The quantum mechanical harmonic oscillator : analytical solutions to the Schrödinger equation Products of Hermite polynomials and gaussian functions with eigenvalues (34-1) e n is the vibrational quantum number beginning at 0 and (34-2) μ reduced mass; k is the bond force constant, second derivative of the energy with respect to bond stretching at the equilibrium bond length (see eq. 9-3). E = n + 1 2 " # $ % & h ( " = 1 2 # k μ
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Polyatomic Case Multi-dimensional Taylor expansion: (34-3) q : vector of atomic coordinates. q eq vector at equilibrium structure. H : hessian matrix. Difficult to solve (34-3) in the indicated coordinates.
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

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lecture34_umn - Lecture 34 Vibrational Spectroscopy Matrix...

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