Normal Mode Analysis ©David Ronis Mcgill University
Chemistry 365: Normal Mode Analysis ©David Ronis McGill University 1. Quantum Mechanical Treatment Our starting point is the Schrodinger wav e equation: N −2 ∂2 − h + → → Ψ → → = Ψ → → Σ → U(r1,...,r N ) (r1,...,r N ) E (r1,...,r N ), (1.1) = ∂ 2 i 1 2mi ri where N is the number of atoms in the molecule, mi is the mass of the i’th atom, and → → U(r1,...,r N )isthe effective potential for the nuclear motion, e.g., as is obtained in the Born- Oppenheimer approximation. If the amplitude of the vibrational motion is small, then the vibrational part of the Hamil- tonian associated with Eq. (1.1) can be written as: N −2 ∂2 N ↔ → → ≈− h + + 1 ∆ ∆ Hvib Σ →2 U0 Σ Ki, j: i j,(1.2) i=1 2mi ∂∆ 2 i, j=1 i → ∆ ≡ → − → → where U0 is the minimum value of the potential energy, i ri Ri, Ri is the equilibrium posi- tion of the i’th atom, and 2 ↔ ∂ ≡ U Ki, j → → (1.3) ∂ ∂ → ri r j → r k = Rk is the matrix of (harmonic) force constants. Henceforth, we will shift the zero of energy so as to = make U0 0. Note that in obtaining Eq. (1.2), we have neglected anharmonic (i.e., cubic and higher order) corrections to the vibrational motion. The next and most confusing step is to change to matrix notation. We introduce a column vector containing the displacements as: ∆≡ ∆x ∆y ∆z ∆x ∆y ∆z T [ 1 , 1, 1,..., N , N , N ] ,(1.4) where "T"denotes a matrix transpose.
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