Ψ g = N g (1s a + 1s b ) Themes. Take Home Message. Energy Decomposition. Multiple model wave fns possible. Begin describing brain-friendly MO

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1 Themes Multiple model wave fns possible Different explanations of bonding Some you have seen before Errors vary (not consistently better or worse) Convenience varies Brain v. algebra v. computer Where is insight located? Begin describing brain-friendly MO Ψ g = N g (1s a + 1s b ) E g = H aa + H ab = E 1 s + V nn' J i j K J S ab y z 1 + S ab k 1 + S ab { K = potential energy integral Delocalization creates overlap density, φ a φ b K = interaction with incoming proton only significant if orbitals overlap (short R) only attractive if electron density enhanced in bond region (no node) Explanation ignores orbital contraction Contraction (ζ) changes E 1s & S ab & K, but not V nn or J (much) Energy Decomposition (1 au = 7. ev = 65 kj/mol) Take Home Message E can be sliced in different ways Changes in KE are important for chemistry Leftover E changes KE changes correlate with no node v. node between nuclei still ok w/ bonding overlap of orbitals containing 1 or electrons gives bond Change in KE, PE, E total due to contraction of two noninteracting H +1/ 1

2 H electrons Heitler-London model ( valence bond ) ψ H1, L = N Hab + bal Hαβ βαl Left-Right correlated electrons Not an independent particle model But not a flexible model Equivalent to aα bβ + bα aβ H electrons Hartree-Fock model ( molecular orbital ) ψ H1, L = N Hσ σl Hαβ βαl σ = N σ Ha + bl independent particle model electrons occupy same orbital equivalent to σα σβ H Comparison R R e E VB & E MO similar even though assumptions differ R long Break bond E VB E H (correct dissociation) E MO >> E H (incorrect) Convenience? Convenient for What? VB models Brain-friendly? Electrons paired in bonds (like Lewis structure) Delocalization problems: polar bonds, pi systems Algebra-unfriendly, computer-unfriendly MO models Brain-unfriendly? Every electron delocalized over full molecule Algebra-friendly, computer-friendly

3 HF Bond Orbital Generating MO Shapes & Energies in a brain-friendly way Problem 10.8 Chapters 13. & 14.7 ψ H1, L = N Hσ σl Hαβ βαl σ = N σ Hc H H + c F FL Atoms not identical Polar bond Expect c F > c H Use variation method to determine c i Adjust c i to minimize E model E model Simplified Hamiltonian ˆ H eff = Tˆ + VˆH + VˆF Can t assume normalized MO E model = Ÿ σ Ĥ σ Ÿ σσ But assume normalized AO = Ÿ Hc H H + c F FL Ĥ Hc H H + c F FL Ÿ Hc H H + c F FL Hc H H + c F FL E model = c H H HH + c F H FF + c H c F H HF c H + c F + c H c F S HF Vary c H & c F To Optimize E model E model = c H H HH + c F H FF + c H c F H HF c H + c F + c H c F S HF de/dci = 0 (must repeat for c H & c F ) E H c H + c F SL = c H H HH E H c F + c H SL = c F H FF + c F H HF + c H H HF Rewrite secular eqns as matrix product J 0 0 N = J H HH E H FF E N J c H N c F 3

4 Secular Determinant J 0 0 N = J H HH E H HF ES Either c H = c F = 0 Or secular determinant = 0 H FF E N J c H N c F 0 = À H HH E H HF ES H FF E À Strategy choose AO to mix (basis set) calculate H & S integrals Find E that makes determinant vanish Plug E into determinant & solve secular eqns for c i Demonstrate for HH 0 = À H aa E H ab E S H ab E S H bb E À = À H aa E H ab E S H ab E S H aa E À H aa = H bb Characteristic polynomial = 0 Find roots Short-cut to E H aa E = H+ ê L HH ab ESL E + = H aa H ab 1 S E = H aa + H ab 1 + S Short-cut to c i Demonstrate for HH 0 = À H aa E H ab ES H ab ES H bb E À = À H aa E H ab E S H ab E S H aa E À H aa E = H+ ê L HH ab ESL Substitute E + or E - into secular eqs Either eq gives relative signs of c i Normalization sets magnitude of each c i E + J 0 0 N = J x x x x N J c a N c c a = c b b E J 0 0 N = J x x x x N J c a N c c a = c b b E + = H aa H ab 1 S E < E a, E b < E + Taking Stock E bonding MO (no node) E + antibonding MO (node) E = H aa + H ab 1 + S 4

5 Return to HF Engel, 13. H HH = -13.6, H FF (p) = -18.6, H HF = -8.3 ev S = = À H HH E H HF ES H HF ES H FF E À = À 13.6 E E À E 18.6 E Solve quadratic for E -19.6, Compare to H HH & H FF Solve for c H /c F c H = 0.4 c F c H = -1.6 c F Observations N AO N MO (E + c i ) All MO constructed from same group of AO (basis set) No node is good node Polar MO possible Atom w/ more negative (lower) H aa dominates bonding MO Atom w/ higher H aa dominates antibonding MO Questions for Next Time F s H 1s s even lower than p F (s, p) H 1s 3 AO 3 MO What if more than atoms? Is the algebra really brain-friendly yet? Make integrals more transparent Make integrals model more transparent What do different parts of secular determinant do? 5