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Unformatted text preview: 04Feb2011 Chemistry 21b – Spectroscopy Lecture # 14 – The Quantum & Group Theoretical Treatment of Molecular Vibrations The classical normal mode solution derived in Lecture #13 provides a very straightforward quantum mechanical generalization of the one-dimensional harmonic oscillator. In normal coordinates, the vibrational Schr¨ odinger equation becomes: parenleftBigg − ¯ h 2 2 3 N summationdisplay i =1 ∂ 2 ∂Q 2 i + 1 2 3 N summationdisplay i =1 λ i Q 2 i parenrightBigg Ψ = E Ψ . Since the Hamiltonian is now a linear sum over 3 N (or really 3 N − 6 / 3 N − 5 since the translational and rotational λ i are zero) harmonic oscillator equations, this means that the overall vibrational wavefunction Ψ may be written as a product of one-dimensional wavefunctions, or Ψ = ψ n 1 ( Q 1 ) ψ n 2 ( Q 2 ) · · · ψ n 3 N- 6 ( Q 3 N − 6 ) , where n i = the vibrational quantum number of the i th normal mode. The energy of each normal mode is simply E n k = ( n k + 1 2 ) hν k , and the total energy is E n k = 3 N − 6 summationdisplay k =1 ( n k + 1 2 ) hν k . For linear molecules, the sums above run over 3 N − 5, of course. The total zero-point energy for the molecule is now E = 3 N − 6 summationdisplay k =1 1 2 hν k , The individual ψ n i are simply those of a one-dimensional harmonic oscillator in the individual normal mode coordinates. That is, they are the product of a Hermite polynomial times a e − α k Q 2 k / 2 exponential. In explicit terms, the normalized quantum mechanical normal mode eigenfunctions are given by ψ n k ( Q k ) = 1 (2 n k n k !) 1 / 2 parenleftBig α k π parenrightBig 1 / 4 e − α k Q 2 k / 2 H n k ( α 1 / 2 k Q k ) , where H n k is the Hermite polynomial of order n k and α k = 2 πν k / ¯ h . Thus, in the harmonic limit the normal modes are uncoupled . States with one normal mode quantum number equal to one and all others equal to zero are called fundamental vibrations, or modes. States in which one normal mode quantum is equal to two or more, and all others zero, 105 are called overtones . Finally, states in which two or more normal mode quantum numbers are equal to or greater than one are called combination modes. The intensities of various fundamental, overtone, and combination bands are governed, as always, by the electric dipole selection rules. If we again expand the dipole moment function in a power series, as before, in the normal mode coordinates, we have μ g = μ ◦ g + 3 N − 6 summationdisplay i =1 parenleftbigg ∂μ g ∂Q i parenrightbigg 0= e Q i + ... g = x,y,z through first order. For the k th normal mode, the harmonic oscillator selections rules as thus generalized to n 1 = n ′ 1 ,n 2 = n ′ 2 ,...,n k = n ′ k ± 1 ,...,n 3 N − 6 = n ′ 3 N − 6 , with an absolute intensity that is proportional to | ( ∂μ g /∂Q k ) 0= e | 2 , the square of the dipole derivative evaluated at the equilibrium geometry. Again we see that the normal modes are uncoupled, and that, to first order, the overtones and combination bands are not...
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- Fall '10
- Mole, Normal mode, normal modes, Γ3N