lecture25_umn

lecture25_umn - Lecture 25 Recapitulation of the...

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Lecture 25 November 11, 2009 Recapitulation of the Variational Principle Recapitulation of the Secular Equation Hückel Theory
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Variational Method: Recap We cannot determine exact wave functions by analytical solution of the Schrödinger equation Guess at the wave function : Φ Variational principle : (25-1) E 0 is the correct ground-state energy Lower-limit condition : convenient way of evaluating the quality of different guesses (lower is better) It permits us to use the tools of variational calculus ! " * H ! d r ! " * ! d r # E 0
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LCAO The parameters are coefficients that describe how molecular orbitals are built up as linear combinations of atomic orbitals . Many-electron wave functions Φ can be written as antisymmetrized Hartree products—Slater determinants—of one-electron orbitals φ , defined as (25-2) set of N atomic-orbital basis functions ϕ i is called the basis set and each has associated with it some coefficient a i Use the variational principle to find the optimal coefficients . ! = a i " i i = 1 N #
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LCAO: Energy and Minimization For a given one-electron orbital we evaluate (25-3) H ij and S ij resonance and overlap integrals Minimization condition (25-4) N equations which must be satisfied in order for equation 25-4 to hold true . (25-5) E = a i * " i * i # $ % & ( ) * H a j j j # $ % & & ( ) ) d r a i * i * i # $ % & ( ) * a j j j # $ % & & ( ) ) d r = a i * a j ij # i * * H j d r a i * a j ij # i * * j d r = a i * a j ij # ij H a i * a j ij # ij S ! E ! a k = 0 " k a i i = 1 N ! H ki ES ki ( ) = 0 " k
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The Secular Equation . (25-5) these equations can be solved for the variables a i if and only if (25-6) Secular Equation N roots ( N different values of E ).
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lecture25_umn - Lecture 25 Recapitulation of the...

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