( ) ( ) SALCs as basis functions. LCAO-MO Theory. Rewriting the Schrödinger Eqn. Matrix form of Schrödinger Eqn. Hφ j. φ 1. φ 3. φ 2. φ j. φ i.

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1 LCAO-MO Theory In molecular orbital theory the MOs, { }, are usually expanded as combinations of atomic orbitals, {φ i }. For the µ th MO, we write this expansion as = c iµ φ i = c 1µ φ 1 + c µ φ + c 3µ φ 3 +! i The MOs are solutions to an effective Schrödinger equation: H = E µ Read: Harris & Bertolucci, Symmetry and Spectroscopy..., Chapter 4, pp SALCs as basis functions Instead of AO basis functions, we can use precombined SALCs as basis functions. So, when we write the expansion = c iµ φ i = c 1µ φ 1 + c µ φ + c 3µ φ 3 +! i the basis, {φ 1, φ, }, may be AOs or they may be SALCs. The ultimate form of the MOs won t change, but the coefficients in this expansion will depend on the basis. Example: Express π MOs for allyl in AO and SALC bases. Rewriting the Schrödinger Eqn. H = E µ H = (H ) Now, we introduce the LCAO expansion, = c µ φ = c 1µ φ 1 + c µ φ + c 3µ φ 3 +! So we obtain a different form of the eqn.: = c µ (H )φ (H ) c µ φ Matrix form of Schrödinger Eqn. c µ (H )φ Mult. on the left by φ i and integrate: ( ) c µ φ i Hφ dτ φ i φ dτ H i Hφ dτ S i φ dτ ( ) c µ H i S i For a proper treatment using variational method, see for example Levine s Quantum Chemistry text.

2 Matrix Schrödinger Eqn., cont. (H) i = H i ( H i S i ) c µ in matrix form, this can be written as [H S]c µ or Hc µ = E µ Sc µ where the matrix elements are φ i Hφ dτ (S) i = S i φ i φ dτ A 3 basis function example ( )c 1µ + ( H 1 S 1 )c µ + ( H 13 S 13 )c 3µ ( H 1 S 1 )c 1µ + ( )c µ + ( H 3 S 3 )c 3µ ( H 31 S 31 )c 1µ + ( H 3 S 3 )c µ + ( H 33 )c 3µ H 1 S 1 H 13 S 13 H 1 S 1 H 3 S 3 H 31 S 31 H 3 S 3 H 33 This is called the Secular Equation. c 1µ c µ c 3µ = SALCs as the Expansion Basis Previously, we introduced the LCAO expansion, = c µ φ = c 1µφ 1 + c µ φ + c 3µ φ 3 +! If the functions {φ } are "precombined" SALCs, then the basis functions belong to irred. reps. We can use what we know about zero - value integrals H i Hφ dτ and S i φ dτ if Γ i Γ contains the totally symmetric representation. φ i and φ must belong to the same IR. The Matrix Equation when the basis is the set of MOs If we choose the MOs as basis functions to begin with, the equation simplifies drastically: H i = E i δ i for all i and S i = δ i E 1! c 1µ E! c µ = " " # " " "! E m E µ c mµ When E µ = one of the E i, the matrix is singular

3 What s the use of Group Theory? Using SALCs as a basis simplifies the Secular Eqn. For example, when SALCs are used as the basis functions, the secular equation for H O becomes: H 1 S 1 H 13 S 13 H 1 S 1 H 3 S 3 H 31 S 31 H 65 S 65 H 33 H 44 H 55 H 56 S 56 H 65 S 65 H 66 c 1µ c µ c 3µ c 4µ c 5µ c 6µ = Example, Usefulness of SALCs In the pictorial construction of the MOs of water given earlier, how many H i s would have been zero if we had used AOs (on the O atom and both H atoms) for the basis functions? How many are nonzero with the SALCs that were used? (Note: H i s involving two AOs centered on the same atom are automatically zero) Another Example Naphthalene π-orbital SALCs In problem 6.1, we constructed SALCs for the π orbitals of naphthalene. The IRs spanned by the pπ orbitals were ( 3), B 3g ( 3), B g ( ), A u ( ) If we used the pπ orbitals as our AO basis in the secular eqn., we would have had one 1 1 matrix eqn. Using SALCs, we have two 3 3 s (for and B 3g ) and two s (for B g and A u ). B 3g B g A u B 3g B3g B g A u

4 H 1 B1u H 13 H 1 H 3 H 31 H 3 H 33 Form of the Naphthalene π Orbital Secular Equation H 44 B3g H 45 H 46 H 54 H 55 H 56 H 64 H 65 H 66 Au H 77 H 78 H 87 H 88 Bg H 99 H 9,1 H 1,9 H 1,1 The Content and Solutions of the Secular Equation We need to calculate or estimate the matrix elements (the H i s and S i s) Once the matrix elements are known, we need to find solutions of the secular equation energies are found by finding the roots of the secular determinant eigenfunctions (the MO coefficients) are found by plugging the energies back into the secular eqn. The Crudest Approximation: In the Hückel approximation (applied to π systems where all the atoms are of the same type) the following substitutions are made (in this case the basis functions are understood to be AOs, not SALCs): a. H ii = α (can choose, if all atoms carbon) b. H i = β if i and are neighbors c. H i otherwise d. S ii = 1 (normalization) and S i Neglect of Overlap: Consequences A general secular determinant is E H 1 ES 1 H 1 ES 1 E expanding the determinant gives E H + H H S S 1 E + H H H S 1 solutions are of the form: E ± = b ± D (1 S 1 ) ; b = + H 1 S 1 ; D = b 4(1 S 1 )( H 1 )

5 Neglect of Overlap: Consequences A secular determinant w/o overlap is E H 1 H 1 E expanding the determinant gives Neglect of Overlap: Consequences E+ destabilization E+ E + E + H 1 solutions are of the form: E ± = b ± D ; b = + ; D = ( ) + 4H 1 E- Neglect of Overlap Overlap Included stabilization destabilization = E- stabilization destabilization > stabilization Examples Trivial example: ethylene π orbitals Cyclopropenium ion benzene Hückel π bond-order resonance other C N symmetry systems T N symmetry systems D 6h E C 6 C 3 C 3C 3C i S 3 S 6 σ h 3σ d 3σ v A 1g x + y, z A g R z B 1g B g E 1g (R x, R y ) (xz, yz) E g (x y,xy) A 1u A u z B u E 1u (x, y) E u

6 Fixing Accidental Degeneracies Spin Density for Benzyl Radical Use Change button, then click the atoms shown, then type in.1 for the alpha value, then click Ok! Turn on Verbose option, then read coefficients!