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Unformatted text preview: Chem 120A Molecular states continued 04/19/06 Spring 2006 Lecture 34 READING: Engel (QCS): Ch. 1213, section B.2 McQuarrie and Simon: Ch. 7.2, Ch. 9 H + 2 Continued In Lecture 34 we saw that using the variational principle for a linear combination of atomic wavefunctions for the ground and first excited states of H + 2 led to two secular equations. In order to determine the energies and values of c 1 and c 2 for the two states, we set up the secular determinant,  H ES  = . (1) The determinant is made up of matrix elements of the Hamiltonian, H i j = R f i H f j d , and the overlap matrix, S i j = R f 1 f 2 d . The solutions to Eq. 1 are E = E n , where E n are the eigenenergies of the secular equations. These energies are guaranteed to be an upper bound to the true ground state. Our variational wavefunc tions are linear combinations of atomic orbitals (LCAO) which form molecular orbitals (MO). This type of approximate solution to a molecular Schrodinger equation is called LCAOMO. Recall that the H + 2 Hamiltonian under the BornOppenheimer approximation is H = 2 2 1 r A 1 r B + 1 R AB , (2) where the coordinate system is shown in Fig. 1. The variational wavefunctions are of the form = c 1 1 s A + AB Figure 1: Coordinates for the H + 2 system. c 2 1 s B . Because there are two different atomic wavefunctions in our linear combination (1 s A and 1 s B ), the Chem 120A, Spring 2006, Lecture 34 1 Atomic orbitals r psi(r) 1sA(r) 1sB(r) A B Figure 2: The two atomic orbitals for nuclei A and B . secular determinant is 2 2. Expanding out the determinant, H 11 S 11 H 12 S 12 H 21 S 21 H 22 S 22 = . (3) We must evaluate each of the relevant Hamiltonian and overlap matrix elements in order to solve for . For the overlap matrix, we have S 11 = S 22 = Z 1 s...
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 Spring '07
 Whaley
 Physical chemistry, Atom, Mole, pH

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