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lecture17_umn

# lecture17_umn - Lecture 17 Recapitulation of the Schrdinger...

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Lecture 17 October 21, 2009 Recapitulation of the Schrödinger Equation and its Eigenfunctions and Eigenvalues The Variational Principle The Born-Oppenheimer Approximation Construction of Trial Wave Functions: LCAO Basis Set Approach

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The Schrödinger Equation The operator that returns the system energy, E , as an eigenvalue is called the Hamiltonian operator, H . H Ψ = E Ψ (17-1) Typical form of H : five contributions to the total energy: H = " h 2 2 m e i # \$ i 2 " h 2 2 m k k # \$ k 2 " e 2 Z k 4 %& 0 r ik k # i # + e 2 4 %& 0 r ij i < j # + e 2 Z k Z l 4 %& 0 r kl k < l # (17-2)
Solutions of the Schrödinger Equation Eq. 17-1 has many acceptable eigenfunctions Ψ for a given molecule, each characterized by a different associated eigenvalue E . There is a complete set of Ψ i with eigenvalues E i . Assume that these wave functions are orthonormal: (17-3) ! """ i * ! j dx dy dz = # ij

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Determining Molecular Energy Easier notation: (17-4) Take eq. 17-1 for a specific Ψ i , multiply on the left by Ψ j * and integrate. (17-5) Move E outside the integral on the r.h.s. and use eq. 17-4 (17-6) ! " i * ! j d r = # ij ! " j * H ! i d r = ! " j * E i ! i d r ! " j * H ! i d r = E i # ij
How do we obtain the wave function? We do not have a prescription for obtaining the orthonormal set of molecular wave functions. Assume we can pick an arbitrary function, Φ . Since we defined the set of orthonormal wave functions Ψ i to be complete (and perhaps infinite), the function Φ must be some linear combination of the Ψ i (17-7) " = c i # i i \$

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Φ as linear combination of Ψ i (17-7) we do not yet know the individual Ψ i , neither the coefficients c i Normality of Φ imposes a constraint on the coefficients (17-8) Let us evaluate the energy associated with Φ .
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lecture17_umn - Lecture 17 Recapitulation of the Schrdinger...

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