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Unformatted text preview: Chem 120A Molecular states continued 04/19/06 Spring 2006 Lecture 34 READING: Engel (QCS): Ch. 12-13, 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 LCAO-MO. Recall that the H + 2 Hamiltonian under the Born-Oppenheimer 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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