3502_25_nove11_LG

3502_25_nove11_LG - Chem 3502/5502 Physical Chemistry II...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 25, November 11, 2009 (Some material in this lecture has been adapted from Cramer, C. J. Essentials of Computational Chemistry , Wiley, Chichester: 2002; pp. 96-109.) Recapitulation of the Variational Principle and the Secular Equation Recall that for any system where we cannot determine exact wave functions by analytical solution of the Schrödinger equation (because the differential equation is simply too difficult to solve), we can make a guess at the wave function, which we will designate Φ , and the variational principle tells us that the expectation value of the Hamiltonian for Φ is governed by the equation ! " * H ! d r ! " * ! d r # E 0 (25-1) where E 0 is the correct ground-state energy. Not only does this lower-limit condition provide us with a convenient way of evaluating the quality of different guesses (lower is better), but it also permits us to use the tools of variational calculus to identify minimizing values for any parameters that appear in the definition of Φ . In the LCAO (linear combination of atomic orbitals) approach, the parameters are coefficients that describe how molecular orbitals (remember, orbital is another word for a one -electron wave function contributing to a many -electron wave function) are built up as linear combinations of atomic orbitals. In particular, many-electron wave functions Φ can be written as antisymmetrized Hartree products—i.e., Slater determinants—of such one- electron orbitals φ , where the one-electron orbitals are defined as ! = a i " i i = 1 N # (25-2) where the set of N atomic-orbital basis functions ϕ i is called the “basis set” and each has associated with it some coefficient a i , where we will use the variational principle to find the optimal coefficients. To be specific, for a given one-electron orbital we evaluate
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25-2 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 . (25-3) where the shorthand notation H ij and S ij is used for the resonance and overlap integrals in the numerator and denominator, respectively. If we impose the minimization condition ! E ! a k = 0 " k (25-4) we get N equations which must be satisfied in order for equation 25-4 to hold true, namely a i i = 1 N ! H ki ES ki ( ) = 0 " k . (25-5) these equations can be solved for the variables a i if and only if H 11 ES 11 H 12 ES 12 ! H 1 N ES 1 N H 21 ES 21 H 22 ES 22 ! H 2 N ES 2 N " " # " H N 1 ES N 1 H N 2 ES N 2 ! H NN ES NN = 0 (25-6) where equation 25-6 is called the secular equation. There are N roots (i.e., N different values of E ) which permit the secular equation to be true. For each such value E j there will be a different set of coefficients, a ij , which can be found by solving the set of linear equations 25-5 using that specific E j , and those coefficients will define an optimal associated wave function φ j within the given basis set.
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Spring '08 term at Minnesota.

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3502_25_nove11_LG - Chem 3502/5502 Physical Chemistry II...

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